Daily Papers

Self-supervised monocular scene flow estimation, aiming to understand both 3D structures and 3D motions from two temporally consecutive monocular images, has received increasing attention for its simple and economical sensor setup. However, the accuracy of current methods suffers from the bottleneck of less-efficient network architecture and lack of motion rigidity for regularization. In this paper, we propose a superior model named EMR-MSF by borrowing the advantages of network architecture design under the scope of supervised learning. We further impose explicit and robust geometric constraints with an elaborately constructed ego-motion aggregation module where a rigidity soft mask is proposed to filter out dynamic regions for stable ego-motion estimation using static regions. Moreover, we propose a motion consistency loss along with a mask regularization loss to fully exploit static regions. Several efficient training strategies are integrated including a gradient detachment technique and an enhanced view synthesis process for better performance. Our proposed method outperforms the previous self-supervised works by a large margin and catches up to the performance of supervised methods. On the KITTI scene flow benchmark, our approach improves the SF-all metric of the state-of-the-art self-supervised monocular method by 44% and demonstrates superior performance across sub-tasks including depth and visual odometry, amongst other self-supervised single-task or multi-task methods.

Recent research shows that when Gradient Descent (GD) is applied to neural networks, the loss almost never decreases monotonically. Instead, the loss oscillates as gradient descent converges to its ''Edge of Stability'' (EoS). Here, we find a quantity that does decrease monotonically throughout GD training: the sharpness attained by the gradient flow solution (GFS)-the solution that would be obtained if, from now until convergence, we train with an infinitesimal step size. Theoretically, we analyze scalar neural networks with the squared loss, perhaps the simplest setting where the EoS phenomena still occur. In this model, we prove that the GFS sharpness decreases monotonically. Using this result, we characterize settings where GD provably converges to the EoS in scalar networks. Empirically, we show that GD monotonically decreases the GFS sharpness in a squared regression model as well as practical neural network architectures.

We identify and formalize a fundamental gradient descent phenomenon resulting in a learning proclivity in over-parameterized neural networks. Gradient Starvation arises when cross-entropy loss is minimized by capturing only a subset of features relevant for the task, despite the presence of other predictive features that fail to be discovered. This work provides a theoretical explanation for the emergence of such feature imbalance in neural networks. Using tools from Dynamical Systems theory, we identify simple properties of learning dynamics during gradient descent that lead to this imbalance, and prove that such a situation can be expected given certain statistical structure in training data. Based on our proposed formalism, we develop guarantees for a novel regularization method aimed at decoupling feature learning dynamics, improving accuracy and robustness in cases hindered by gradient starvation. We illustrate our findings with simple and real-world out-of-distribution (OOD) generalization experiments.

Group-based policy optimization methods like GRPO and GSPO have become standard for training multimodal models, leveraging group-wise rollouts and relative advantage estimation. However, they suffer from a critical gradient vanishing problem when all responses within a group receive identical rewards, causing advantage estimates to collapse and training signals to diminish. Existing attempts to mitigate this issue fall into two paradigms: filtering-based and sampling-based methods. Filtering-based methods first generate rollouts broadly and then retroactively filter out uninformative groups, leading to substantial computational overhead. Sampling-based methods proactively select effective samples before rollout but rely on static criteria or prior dataset knowledge, lacking real-time adaptability. To address these issues, we propose VADE, a Variance-Aware Dynamic sampling framework via online sample-level difficulty Estimation. Our framework integrates three key components: online sample-level difficulty estimation using Beta distributions, a Thompson sampler that maximizes information gain through the estimated correctness probability, and a two-scale prior decay mechanism that maintains robust estimation under policy evolution. This three components design enables VADE to dynamically select the most informative samples, thereby amplifying training signals while eliminating extra rollout costs. Extensive experiments on multimodal reasoning benchmarks show that VADE consistently outperforms strong baselines in both performance and sample efficiency, while achieving a dramatic reduction in computational overhead. More importantly, our framework can serves as a plug-and-play component to be seamlessly integrated into existing group-based RL algorithms. Code and models are available at https://VADE-RL.github.io.

In Federated Learning (FL) and many other distributed training frameworks, collaborators can hold their private data locally and only share the network weights trained with the local data after multiple iterations. Gradient inversion is a family of privacy attacks that recovers data from its generated gradients. Seemingly, FL can provide a degree of protection against gradient inversion attacks on weight updates, since the gradient of a single step is concealed by the accumulation of gradients over multiple local iterations. In this work, we propose a principled way to extend gradient inversion attacks to weight updates in FL, thereby better exposing weaknesses in the presumed privacy protection inherent in FL. In particular, we propose a surrogate model method based on the characteristic of two-dimensional gradient flow and low-rank property of local updates. Our method largely boosts the ability of gradient inversion attacks on weight updates containing many iterations and achieves state-of-the-art (SOTA) performance. Additionally, our method runs up to 100times faster than the SOTA baseline in the common FL scenario. Our work re-evaluates and highlights the privacy risk of sharing network weights. Our code is available at https://github.com/JunyiZhu-AI/surrogate_model_extension.

Plasticity, the ability of a neural network to quickly change its predictions in response to new information, is essential for the adaptability and robustness of deep reinforcement learning systems. Deep neural networks are known to lose plasticity over the course of training even in relatively simple learning problems, but the mechanisms driving this phenomenon are still poorly understood. This paper conducts a systematic empirical analysis into plasticity loss, with the goal of understanding the phenomenon mechanistically in order to guide the future development of targeted solutions. We find that loss of plasticity is deeply connected to changes in the curvature of the loss landscape, but that it typically occurs in the absence of saturated units or divergent gradient norms. Based on this insight, we identify a number of parameterization and optimization design choices which enable networks to better preserve plasticity over the course of training. We validate the utility of these findings in larger-scale learning problems by applying the best-performing intervention, layer normalization, to a deep RL agent trained on the Arcade Learning Environment.

Advancing towards generalist agents necessitates the concurrent processing of multiple tasks using a unified model, thereby underscoring the growing significance of simultaneous model training on multiple downstream tasks. A common issue in multi-task learning is the occurrence of gradient conflict, which leads to potential competition among different tasks during joint training. This competition often results in improvements in one task at the expense of deterioration in another. Although several optimization methods have been developed to address this issue by manipulating task gradients for better task balancing, they cannot decrease the incidence of gradient conflict. In this paper, we systematically investigate the occurrence of gradient conflict across different methods and propose a strategy to reduce such conflicts through sparse training (ST), wherein only a portion of the model's parameters are updated during training while keeping the rest unchanged. Our extensive experiments demonstrate that ST effectively mitigates conflicting gradients and leads to superior performance. Furthermore, ST can be easily integrated with gradient manipulation techniques, thus enhancing their effectiveness.

There are two widely known issues with properly training Recurrent Neural Networks, the vanishing and the exploding gradient problems detailed in Bengio et al. (1994). In this paper we attempt to improve the understanding of the underlying issues by exploring these problems from an analytical, a geometric and a dynamical systems perspective. Our analysis is used to justify a simple yet effective solution. We propose a gradient norm clipping strategy to deal with exploding gradients and a soft constraint for the vanishing gradients problem. We validate empirically our hypothesis and proposed solutions in the experimental section.

Deep neural networks are vulnerable to universal adversarial perturbation (UAP), an instance-agnostic perturbation capable of fooling the target model for most samples. Compared to instance-specific adversarial examples, UAP is more challenging as it needs to generalize across various samples and models. In this paper, we examine the serious dilemma of UAP generation methods from a generalization perspective -- the gradient vanishing problem using small-batch stochastic gradient optimization and the local optima problem using large-batch optimization. To address these problems, we propose a simple and effective method called Stochastic Gradient Aggregation (SGA), which alleviates the gradient vanishing and escapes from poor local optima at the same time. Specifically, SGA employs the small-batch training to perform multiple iterations of inner pre-search. Then, all the inner gradients are aggregated as a one-step gradient estimation to enhance the gradient stability and reduce quantization errors. Extensive experiments on the standard ImageNet dataset demonstrate that our method significantly enhances the generalization ability of UAP and outperforms other state-of-the-art methods. The code is available at https://github.com/liuxuannan/Stochastic-Gradient-Aggregation.

Deep Gradient Leakage (DGL) is a highly effective attack that recovers private training images from gradient vectors. This attack casts significant privacy challenges on distributed learning from clients with sensitive data, where clients are required to share gradients. Defending against such attacks requires but lacks an understanding of when and how privacy leakage happens, mostly because of the black-box nature of deep networks. In this paper, we propose a novel Inversion Influence Function (I^2F) that establishes a closed-form connection between the recovered images and the private gradients by implicitly solving the DGL problem. Compared to directly solving DGL, I^2F is scalable for analyzing deep networks, requiring only oracle access to gradients and Jacobian-vector products. We empirically demonstrate that I^2F effectively approximated the DGL generally on different model architectures, datasets, attack implementations, and noise-based defenses. With this novel tool, we provide insights into effective gradient perturbation directions, the unfairness of privacy protection, and privacy-preferred model initialization. Our codes are provided in https://github.com/illidanlab/inversion-influence-function.

Early stopping monitors global validation loss and halts all parameter updates simultaneously, which is computationally costly for large transformers due to the extended time required for validation inference. We propose GradES, a novel gradient-based early stopping approach that operates within transformer components (attention projections and Feed-Forward layer matrices). We found that different components converge at varying rates during fine-tuning. GradES tracks the magnitude of gradients in backpropagation for these matrices during training. When a projection matrix's gradients fall below a convergence threshold tau, we exclude that projection matrix from further updates individually, eliminating costly validation passes while allowing slow converging matrices to continue learning. By strategically freezing parameters when their gradients converge, GradES speeds up training time by 1.57--7.22times while simultaneously enhancing generalization through early prevention of overfitting, resulting in 1.2% higher average accuracy.

Exchanging gradients is a widely used method in modern multi-node machine learning system (e.g., distributed training, collaborative learning). For a long time, people believed that gradients are safe to share: i.e., the training data will not be leaked by gradient exchange. However, we show that it is possible to obtain the private training data from the publicly shared gradients. We name this leakage as Deep Leakage from Gradient and empirically validate the effectiveness on both computer vision and natural language processing tasks. Experimental results show that our attack is much stronger than previous approaches: the recovery is pixel-wise accurate for images and token-wise matching for texts. We want to raise people's awareness to rethink the gradient's safety. Finally, we discuss several possible strategies to prevent such deep leakage. The most effective defense method is gradient pruning.

Clipping the gradient is a known approach to improving gradient descent, but requires hand selection of a clipping threshold hyperparameter. We present AutoClip, a simple method for automatically and adaptively choosing a gradient clipping threshold, based on the history of gradient norms observed during training. Experimental results show that applying AutoClip results in improved generalization performance for audio source separation networks. Observation of the training dynamics of a separation network trained with and without AutoClip show that AutoClip guides optimization into smoother parts of the loss landscape. AutoClip is very simple to implement and can be integrated readily into a variety of applications across multiple domains.

It has been observed that the input space of deep neural network classifiers can exhibit `fragmentation', where the model function rapidly changes class as the input space is traversed. The severity of this fragmentation tends to follow the double descent curve, achieving a maximum at the interpolation regime. We study this phenomenon in the context of image classification and ask whether fragmentation could be predictive of generalization performance. Using a fragmentation-based complexity measure, we show this to be possible by achieving good performance on the PGDL (Predicting Generalization in Deep Learning) benchmark. In addition, we report on new observations related to fragmentation, namely (i) fragmentation is not limited to the input space but occurs in the hidden representations as well, (ii) fragmentation follows the trends in the validation error throughout training, and (iii) fragmentation is not a direct result of increased weight norms. Together, this indicates that fragmentation is a phenomenon worth investigating further when studying the generalization ability of deep neural networks.

Evaluating the robustness of a defense model is a challenging task in adversarial robustness research. Obfuscated gradients have previously been found to exist in many defense methods and cause a false signal of robustness. In this paper, we identify a more subtle situation called Imbalanced Gradients that can also cause overestimated adversarial robustness. The phenomenon of imbalanced gradients occurs when the gradient of one term of the margin loss dominates and pushes the attack towards to a suboptimal direction. To exploit imbalanced gradients, we formulate a Margin Decomposition (MD) attack that decomposes a margin loss into individual terms and then explores the attackability of these terms separately via a two-stage process. We also propose a multi-targeted and ensemble version of our MD attack. By investigating 24 defense models proposed since 2018, we find that 11 models are susceptible to a certain degree of imbalanced gradients and our MD attack can decrease their robustness evaluated by the best standalone baseline attack by more than 1%. We also provide an in-depth investigation on the likely causes of imbalanced gradients and effective countermeasures. Our code is available at https://github.com/HanxunH/MDAttack.

Federated Learning (FL) is a rising approach towards collaborative and privacy-preserving machine learning where large-scale medical datasets remain localized to each client. However, the issue of data heterogeneity among clients often compels local models to diverge, leading to suboptimal global models. To mitigate the impact of data heterogeneity on FL performance, we start with analyzing how FL training influence FL performance by decomposing the global loss into three terms: local loss, distribution shift loss and aggregation loss. Remarkably, our loss decomposition reveals that existing local training-based FL methods attempt to reduce the distribution shift loss, while the global aggregation-based FL methods propose better aggregation strategies to reduce the aggregation loss. Nevertheless, a comprehensive joint effort to minimize all three terms is currently limited in the literature, leading to subpar performance when dealing with data heterogeneity challenges. To fill this gap, we propose a novel FL method based on global loss decomposition, called FedLD, to jointly reduce these three loss terms. Our FedLD involves a margin control regularization in local training to reduce the distribution shift loss, and a principal gradient-based server aggregation strategy to reduce the aggregation loss. Notably, under different levels of data heterogeneity, our strategies achieve better and more robust performance on retinal and chest X-ray classification compared to other FL algorithms. Our code is available at https://github.com/Zeng-Shuang/FedLD.

While deep learning and deep reinforcement learning (RL) systems have demonstrated impressive results in domains such as image classification, game playing, and robotic control, data efficiency remains a major challenge. Multi-task learning has emerged as a promising approach for sharing structure across multiple tasks to enable more efficient learning. However, the multi-task setting presents a number of optimization challenges, making it difficult to realize large efficiency gains compared to learning tasks independently. The reasons why multi-task learning is so challenging compared to single-task learning are not fully understood. In this work, we identify a set of three conditions of the multi-task optimization landscape that cause detrimental gradient interference, and develop a simple yet general approach for avoiding such interference between task gradients. We propose a form of gradient surgery that projects a task's gradient onto the normal plane of the gradient of any other task that has a conflicting gradient. On a series of challenging multi-task supervised and multi-task RL problems, this approach leads to substantial gains in efficiency and performance. Further, it is model-agnostic and can be combined with previously-proposed multi-task architectures for enhanced performance.

Machine unlearning, the removal of a training subset's influence from a deployed model, is critical for privacy preservation and model reliability, yet gradient ascent on forget samples often harms retained knowledge. Existing approaches face a persistent tradeoff between effective forgetting and preservation on the retain set. While previous methods provide useful heuristics, they often lack a formal analysis on how exactly forgetting updates harm retained knowledge, and whether the side effects can be removed with theoretical guarantees. To explore a theoretically sound and simple solution, we start from the first principle on how performance on the retain set is actually affected: a first-order analysis of the local change of the retain loss under small parameter updates during model training. We start from a crisp equivalence: the retain loss is unchanged to first order iff the update direction is orthogonal to the subspace spanned by retain gradients ("retain-invariant"). This identifies the entangled component as the tangential part of forget update within the retain-gradient subspace, and characterizes disentanglement as orthogonality. Guided by this, we propose the Geometric-disentanglement Unlearning (GU) that decomposes any candidate forget gradient update into tangential and normal components to retain space and executes only the normal component. Under a standard trust-region budget, the projected direction aligned with the raw forget gradient is optimal among all first-order retain-invariant moves, and we also derive the optimal projected direction for joint forget-retain updating objectives. Our method is plug-and-play and can be attached to existing gradient-based unlearning procedures to mitigate side effects. GU achieves consistent improvement on various methods across three benchmarks TOFU, MUSE, and WMDP.

We study the properties of common loss surfaces through their Hessian matrix. In particular, in the context of deep learning, we empirically show that the spectrum of the Hessian is composed of two parts: (1) the bulk centered near zero, (2) and outliers away from the bulk. We present numerical evidence and mathematical justifications to the following conjectures laid out by Sagun et al. (2016): Fixing data, increasing the number of parameters merely scales the bulk of the spectrum; fixing the dimension and changing the data (for instance adding more clusters or making the data less separable) only affects the outliers. We believe that our observations have striking implications for non-convex optimization in high dimensions. First, the flatness of such landscapes (which can be measured by the singularity of the Hessian) implies that classical notions of basins of attraction may be quite misleading. And that the discussion of wide/narrow basins may be in need of a new perspective around over-parametrization and redundancy that are able to create large connected components at the bottom of the landscape. Second, the dependence of small number of large eigenvalues to the data distribution can be linked to the spectrum of the covariance matrix of gradients of model outputs. With this in mind, we may reevaluate the connections within the data-architecture-algorithm framework of a model, hoping that it would shed light into the geometry of high-dimensional and non-convex spaces in modern applications. In particular, we present a case that links the two observations: small and large batch gradient descent appear to converge to different basins of attraction but we show that they are in fact connected through their flat region and so belong to the same basin.

We show that in a variety of large-scale deep learning scenarios the gradient dynamically converges to a very small subspace after a short period of training. The subspace is spanned by a few top eigenvectors of the Hessian (equal to the number of classes in the dataset), and is mostly preserved over long periods of training. A simple argument then suggests that gradient descent may happen mostly in this subspace. We give an example of this effect in a solvable model of classification, and we comment on possible implications for optimization and learning.

Gradient clipping is a popular modification to standard (stochastic) gradient descent, at every iteration limiting the gradient norm to a certain value c >0. It is widely used for example for stabilizing the training of deep learning models (Goodfellow et al., 2016), or for enforcing differential privacy (Abadi et al., 2016). Despite popularity and simplicity of the clipping mechanism, its convergence guarantees often require specific values of c and strong noise assumptions. In this paper, we give convergence guarantees that show precise dependence on arbitrary clipping thresholds c and show that our guarantees are tight with both deterministic and stochastic gradients. In particular, we show that (i) for deterministic gradient descent, the clipping threshold only affects the higher-order terms of convergence, (ii) in the stochastic setting convergence to the true optimum cannot be guaranteed under the standard noise assumption, even under arbitrary small step-sizes. We give matching upper and lower bounds for convergence of the gradient norm when running clipped SGD, and illustrate these results with experiments.

As deep learning continues to advance, the transparency of neural network decision-making remains a critical challenge, limiting trust and applicability in high-stakes domains. Class Activation Mapping (CAM) techniques have emerged as a key approach toward visualizing model decisions, yet existing methods face inherent trade-offs. Gradient-based CAM variants suffer from sensitivity to gradient perturbations due to gradient noise, leading to unstable and unreliable explanations. Conversely, gradient-free approaches mitigate gradient instability but incur significant computational overhead and inference latency. To address these limitations, we propose a Cluster Filter Class Activation Map (CF-CAM) technique, a novel framework that reintroduces gradient-based weighting while enhancing robustness against gradient noise. CF-CAM utilizes hierarchical importance weighting strategy to balance discriminative feature preservation and noise elimination. A density-aware channel clustering method via Density-Based Spatial Clustering of Applications with Noise (DBSCAN) groups semantically relevant feature channels and discard noise-prone activations. Additionally, cluster-conditioned gradient filtering leverages Gaussian filters to refine gradient signals, preserving edge-aware localization while suppressing noise impact. Experiment results demonstrate that CF-CAM achieves superior interpretability performance while enhancing computational efficiency, outperforming state-of-the-art CAM methods in faithfulness and robustness. By effectively mitigating gradient instability without excessive computational cost, CF-CAM provides a competitive solution for enhancing the interpretability of deep neural networks in critical applications such as autonomous driving and medical diagnosis.

In this paper, we first investigate a visual quality degradation problem observed in recent high-resolution virtual try-on approach. The tendency is empirically found that the textures of clothes are squeezed at the sleeve, as visualized in the upper row of Fig.1(a). A main reason for the issue arises from a gradient conflict between two popular losses, the Total Variation (TV) and adversarial losses. Specifically, the TV loss aims to disconnect boundaries between the sleeve and torso in a warped clothing mask, whereas the adversarial loss aims to combine between them. Such contrary objectives feedback the misaligned gradients to a cascaded appearance flow estimation, resulting in undesirable squeezing artifacts. To reduce this, we propose a Sequential Deformation (SD-VITON) that disentangles the appearance flow prediction layers into TV objective-dominant (TVOB) layers and a task-coexistence (TACO) layer. Specifically, we coarsely fit the clothes onto a human body via the TVOB layers, and then keep on refining via the TACO layer. In addition, the bottom row of Fig.1(a) shows a different type of squeezing artifacts around the waist. To address it, we further propose that we first warp the clothes into a tucked-out shirts style, and then partially erase the texture from the warped clothes without hurting the smoothness of the appearance flows. Experimental results show that our SD-VITON successfully resolves both types of artifacts and outperforms the baseline methods. Source code will be available at https://github.com/SHShim0513/SD-VITON.

Training large language models (LLMs) presents numerous challenges, including gradient instability and loss spikes. These phenomena can lead to catastrophic divergence, requiring costly checkpoint restoration and data batch skipping. Traditional gradient clipping techniques, such as constant or norm-based methods, fail to address these issues effectively due to their reliance on fixed thresholds or heuristics, leading to inefficient learning and requiring frequent manual intervention. In this work, we propose ZClip, an adaptive gradient clipping algorithm that dynamically adjusts the clipping threshold based on statistical properties of gradient norms over time. Unlike prior reactive strategies, ZClip proactively adapts to training dynamics without making any prior assumptions on the scale and the temporal evolution of gradient norms. At its core, it leverages z-score-based anomaly detection to identify and mitigate large gradient spikes, preventing malignant loss spikes while not interfering with convergence otherwise. Our code is available at: https://github.com/bluorion-com/ZClip.

How to ensure fairness is an important topic in federated learning (FL). Recent studies have investigated how to reward clients based on their contribution (collaboration fairness), and how to achieve uniformity of performance across clients (performance fairness). Despite achieving progress on either one, we argue that it is critical to consider them together, in order to engage and motivate more diverse clients joining FL to derive a high-quality global model. In this work, we propose a novel method to optimize both types of fairness simultaneously. Specifically, we propose to estimate client contribution in gradient and data space. In gradient space, we monitor the gradient direction differences of each client with respect to others. And in data space, we measure the prediction error on client data using an auxiliary model. Based on this contribution estimation, we propose a FL method, federated training via contribution estimation (FedCE), i.e., using estimation as global model aggregation weights. We have theoretically analyzed our method and empirically evaluated it on two real-world medical datasets. The effectiveness of our approach has been validated with significant performance improvements, better collaboration fairness, better performance fairness, and comprehensive analytical studies.

Gradient inversion attacks on federated learning systems reconstruct client training data from exchanged gradient information. To defend against such attacks, a variety of defense mechanisms were proposed. However, they usually lead to an unacceptable trade-off between privacy and model utility. Recent observations suggest that dropout could mitigate gradient leakage and improve model utility if added to neural networks. Unfortunately, this phenomenon has not been systematically researched yet. In this work, we thoroughly analyze the effect of dropout on iterative gradient inversion attacks. We find that state of the art attacks are not able to reconstruct the client data due to the stochasticity induced by dropout during model training. Nonetheless, we argue that dropout does not offer reliable protection if the dropout induced stochasticity is adequately modeled during attack optimization. Consequently, we propose a novel Dropout Inversion Attack (DIA) that jointly optimizes for client data and dropout masks to approximate the stochastic client model. We conduct an extensive systematic evaluation of our attack on four seminal model architectures and three image classification datasets of increasing complexity. We find that our proposed attack bypasses the protection seemingly induced by dropout and reconstructs client data with high fidelity. Our work demonstrates that privacy inducing changes to model architectures alone cannot be assumed to reliably protect from gradient leakage and therefore should be combined with complementary defense mechanisms.

Neural network training is commonly accelerated by using multiple synchronized workers to compute gradient updates in parallel. Asynchronous methods remove synchronization overheads and improve hardware utilization at the cost of introducing gradient delay, which impedes optimization and can lead to lower final model performance. We introduce Adaptive Braking (AB), a modification for momentum-based optimizers that mitigates the effects of gradient delay. AB dynamically scales the gradient based on the alignment of the gradient and the velocity. This can dampen oscillations along high curvature directions of the loss surface, stabilizing and accelerating asynchronous training. We show that applying AB on top of SGD with momentum enables training ResNets on CIFAR-10 and ImageNet-1k with delays D geq 32 update steps with minimal drop in final test accuracy.

While previous distribution shift detection approaches can identify if a shift has occurred, these approaches cannot localize which specific features have caused a distribution shift -- a critical step in diagnosing or fixing any underlying issue. For example, in military sensor networks, users will want to detect when one or more of the sensors has been compromised, and critically, they will want to know which specific sensors might be compromised. Thus, we first define a formalization of this problem as multiple conditional distribution hypothesis tests and propose both non-parametric and parametric statistical tests. For both efficiency and flexibility, we then propose to use a test statistic based on the density model score function (i.e. gradient with respect to the input) -- which can easily compute test statistics for all dimensions in a single forward and backward pass. Any density model could be used for computing the necessary statistics including deep density models such as normalizing flows or autoregressive models. We additionally develop methods for identifying when and where a shift occurs in multivariate time-series data and show results for multiple scenarios using realistic attack models on both simulated and real world data.

Reinforcement learning has significantly enhanced the reasoning capabilities of Large Language Models (LLMs) in complex problem-solving tasks. Recently, the introduction of DeepSeek R1 has inspired a surge of interest in leveraging rule-based rewards as a low-cost alternative for computing advantage functions and guiding policy optimization. However, a common challenge observed across many replication and extension efforts is that when multiple sampled responses under a single prompt converge to identical outcomes, whether correct or incorrect, the group-based advantage degenerates to zero. This leads to vanishing gradients and renders the corresponding samples ineffective for learning, ultimately limiting training efficiency and downstream performance. To address this issue, we propose a consistency-aware policy optimization framework that introduces a structured global reward based on outcome consistency, the global loss based on it ensures that, even when model outputs show high intra-group consistency, the training process still receives meaningful learning signals, which encourages the generation of correct and self-consistent reasoning paths from a global perspective. Furthermore, we incorporate an entropy-based soft blending mechanism that adaptively balances local advantage estimation with global optimization, enabling dynamic transitions between exploration and convergence throughout training. Our method introduces several key innovations in both reward design and optimization strategy. We validate its effectiveness through substantial performance gains on multiple mathematical reasoning benchmarks, highlighting the proposed framework's robustness and general applicability. Code of this work has been released at https://github.com/hijih/copo-code.git.

We present a class of novel optimisers for training neural networks that makes use of the Riemannian metric naturally induced when the loss landscape is embedded in higher-dimensional space. This is the same metric that underlies common visualisations of loss landscapes. By taking this geometric perspective literally and using the induced metric, we develop a new optimiser and compare it to existing methods, namely: SGD, Adam, AdamW, and Muon, across a range of tasks and architectures. Empirically, we conclude that this new class of optimisers is highly effective in low dimensional examples, and provides slight improvement over state-of-the-art methods for training neural networks. These new optimisers have theoretically desirable properties. In particular, the effective learning rate is automatically decreased in regions of high curvature acting as a smoothed out form of gradient clipping. Similarly, one variant of these optimisers can also be viewed as inducing an effective scheduled learning rate and decoupled weight decay is the natural choice from our geometric perspective. The basic method can be used to modify any existing preconditioning method. The new optimiser has a computational complexity comparable to that of Adam.

Gradient regularization (GR) is a method that penalizes the gradient norm of the training loss during training. While some studies have reported that GR can improve generalization performance, little attention has been paid to it from the algorithmic perspective, that is, the algorithms of GR that efficiently improve the performance. In this study, we first reveal that a specific finite-difference computation, composed of both gradient ascent and descent steps, reduces the computational cost of GR. Next, we show that the finite-difference computation also works better in the sense of generalization performance. We theoretically analyze a solvable model, a diagonal linear network, and clarify that GR has a desirable implicit bias to so-called rich regime and finite-difference computation strengthens this bias. Furthermore, finite-difference GR is closely related to some other algorithms based on iterative ascent and descent steps for exploring flat minima. In particular, we reveal that the flooding method can perform finite-difference GR in an implicit way. Thus, this work broadens our understanding of GR for both practice and theory.

Quantization scale and bit-width are the most important parameters when considering how to quantize a neural network. Prior work focuses on optimizing quantization scales in a global manner through gradient methods (gradient descent \& Hessian analysis). Yet, when applying perturbations to quantization scales, we observe a very jagged, highly non-smooth test loss landscape. In fact, small perturbations in quantization scale can greatly affect accuracy, yielding a 0.5-0.8% accuracy boost in 4-bit quantized vision transformers (ViTs). In this regime, gradient methods break down, since they cannot reliably reach local minima. In our work, dubbed Evol-Q, we use evolutionary search to effectively traverse the non-smooth landscape. Additionally, we propose using an infoNCE loss, which not only helps combat overfitting on the small calibration dataset (1,000 images) but also makes traversing such a highly non-smooth surface easier. Evol-Q improves the top-1 accuracy of a fully quantized ViT-Base by 10.30%, 0.78%, and 0.15% for 3-bit, 4-bit, and 8-bit weight quantization levels. Extensive experiments on a variety of CNN and ViT architectures further demonstrate its robustness in extreme quantization scenarios. Our code is available at https://github.com/enyac-group/evol-q

Modern deep neural networks have achieved impressive performance on tasks from image classification to natural language processing. Surprisingly, these complex systems with massive amounts of parameters exhibit the same structural properties in their last-layer features and classifiers across canonical datasets when training until convergence. In particular, it has been observed that the last-layer features collapse to their class-means, and those class-means are the vertices of a simplex Equiangular Tight Frame (ETF). This phenomenon is known as Neural Collapse (NC). Recent papers have theoretically shown that NC emerges in the global minimizers of training problems with the simplified "unconstrained feature model". In this context, we take a step further and prove the NC occurrences in deep linear networks for the popular mean squared error (MSE) and cross entropy (CE) losses, showing that global solutions exhibit NC properties across the linear layers. Furthermore, we extend our study to imbalanced data for MSE loss and present the first geometric analysis of NC under bias-free setting. Our results demonstrate the convergence of the last-layer features and classifiers to a geometry consisting of orthogonal vectors, whose lengths depend on the amount of data in their corresponding classes. Finally, we empirically validate our theoretical analyses on synthetic and practical network architectures with both balanced and imbalanced scenarios.

Understanding the properties of neural networks trained via stochastic gradient descent (SGD) is at the heart of the theory of deep learning. In this work, we take a mean-field view, and consider a two-layer ReLU network trained via SGD for a univariate regularized regression problem. Our main result is that SGD is biased towards a simple solution: at convergence, the ReLU network implements a piecewise linear map of the inputs, and the number of "knot" points - i.e., points where the tangent of the ReLU network estimator changes - between two consecutive training inputs is at most three. In particular, as the number of neurons of the network grows, the SGD dynamics is captured by the solution of a gradient flow and, at convergence, the distribution of the weights approaches the unique minimizer of a related free energy, which has a Gibbs form. Our key technical contribution consists in the analysis of the estimator resulting from this minimizer: we show that its second derivative vanishes everywhere, except at some specific locations which represent the "knot" points. We also provide empirical evidence that knots at locations distinct from the data points might occur, as predicted by our theory.

Recent work by Baratin et al. (2021) sheds light on an intriguing pattern that occurs during the training of deep neural networks: some layers align much more with data compared to other layers (where the alignment is defined as the euclidean product of the tangent features matrix and the data labels matrix). The curve of the alignment as a function of layer index (generally) exhibits an ascent-descent pattern where the maximum is reached for some hidden layer. In this work, we provide the first explanation for this phenomenon. We introduce the Equilibrium Hypothesis which connects this alignment pattern to signal propagation in deep neural networks. Our experiments demonstrate an excellent match with the theoretical predictions.

Momentum based optimizers are central to a wide range of machine learning applications. These typically rely on an Exponential Moving Average (EMA) of gradients, which decays exponentially the present contribution of older gradients. This accounts for gradients being local linear approximations which lose their relevance as the iterate moves along the loss landscape. This work questions the use of a single EMA to accumulate past gradients and empirically demonstrates how this choice can be sub-optimal: a single EMA cannot simultaneously give a high weight to the immediate past, and a non-negligible weight to older gradients. Building on this observation, we propose AdEMAMix, a simple modification of the Adam optimizer with a mixture of two EMAs to better take advantage of past gradients. Our experiments on language modeling and image classification show -- quite surprisingly -- that gradients can stay relevant for tens of thousands of steps. They help to converge faster, and often to lower minima: e.g., a 1.3B parameter AdEMAMix LLM trained on 101B tokens performs comparably to an AdamW model trained on 197B tokens (+95%). Moreover, our method significantly slows-down model forgetting during training. Our work motivates further exploration of different types of functions to leverage past gradients, beyond EMAs.

Contrastive learning has been applied successfully to learn vector representations of text. Previous research demonstrated that learning high-quality representations benefits from batch-wise contrastive loss with a large number of negatives. In practice, the technique of in-batch negative is used, where for each example in a batch, other batch examples' positives will be taken as its negatives, avoiding encoding extra negatives. This, however, still conditions each example's loss on all batch examples and requires fitting the entire large batch into GPU memory. This paper introduces a gradient caching technique that decouples backpropagation between contrastive loss and the encoder, removing encoder backward pass data dependency along the batch dimension. As a result, gradients can be computed for one subset of the batch at a time, leading to almost constant memory usage.

A detailed measurement is made of the metallicity distributions, kinematics and dynamics of the thin and thick disks, across a large disk volume (5.0 leq R leq 15.0 kpc and |Z| leq3.0 kpc), by using the LAMOST-APOGEE red clump stars. The metallicity distributions results show that the radial metallicity gradient Delta[Fe/H]/DeltaR of the thin disk weakens with |Z| from -0.06 dex kpc^{-1} at around |Z| < 0.25 kpc to -0.02 dex kpc^{-1} at around |Z| > 2.75 kpc, while the thick disk displays a global weak positive Delta[Fe/H]/DeltaR, generally weaker than 0.01 dex kpc^{-1}. The vertical metallicity gradient Delta[Fe/H]/Delta|Z| weakened steadily from -0.36 dex kpc^{-1} at R sim 5.5 kpc to -0.05 dex kpc^{-1} at around R > 11.5 kpc for the thin disk, while the thick disk presents an almost constant value (nearly -0.06 sim -0.08 dex kpc^{-1}) for all the R bins. These results indicate the contribution of the radial migration to the disk evolution, and the obvious north-south asymmetry in [Fe/H] may be linked to the disk warp and/or the disk perturbation events. The oscillations of the corrected Delta[Fe/H]/Delta|Z| with R are likely because of the resonances with the Galactic Bar. Our detailed measurements of DeltaV_{phi}/Delta[Fe/H] indicate an "inside-out" and "upside-down" star formation scenario for the thick disk. The results of eccentricity distributions and [alpha/Fe]--velocity dispersion relations are likely to suggest that the thick disk stars require an obvious contribution from other heating mechanisms such as merger and accretion, or born in the chaotic mergers of gas-rich systems and/or turbulent interstellar medium.

Binarization of neural networks is a dominant paradigm in neural networks compression. The pioneering work BinaryConnect uses Straight Through Estimator (STE) to mimic the gradients of the sign function, but it also causes the crucial inconsistency problem. Most of the previous methods design different estimators instead of STE to mitigate it. However, they ignore the fact that when reducing the estimating error, the gradient stability will decrease concomitantly. These highly divergent gradients will harm the model training and increase the risk of gradient vanishing and gradient exploding. To fully take the gradient stability into consideration, we present a new perspective to the BNNs training, regarding it as the equilibrium between the estimating error and the gradient stability. In this view, we firstly design two indicators to quantitatively demonstrate the equilibrium phenomenon. In addition, in order to balance the estimating error and the gradient stability well, we revise the original straight through estimator and propose a power function based estimator, Rectified Straight Through Estimator (ReSTE for short). Comparing to other estimators, ReSTE is rational and capable of flexibly balancing the estimating error with the gradient stability. Extensive experiments on CIFAR-10 and ImageNet datasets show that ReSTE has excellent performance and surpasses the state-of-the-art methods without any auxiliary modules or losses.

A central challenge to many fields of science and engineering involves minimizing non-convex error functions over continuous, high dimensional spaces. Gradient descent or quasi-Newton methods are almost ubiquitously used to perform such minimizations, and it is often thought that a main source of difficulty for the ability of these local methods to find the global minimum is the proliferation of local minima with much higher error than the global minimum. Here we argue, based on results from statistical physics, random matrix theory, and neural network theory, that a deeper and more profound difficulty originates from the proliferation of saddle points, not local minima, especially in high dimensional problems of practical interest. Such saddle points are surrounded by high error plateaus that can dramatically slow down learning, and give the illusory impression of the existence of a local minimum. Motivated by these arguments, we propose a new algorithm, the saddle-free Newton method, that can rapidly escape high dimensional saddle points, unlike gradient descent and quasi-Newton methods. We apply this algorithm to deep neural network training, and provide preliminary numerical evidence for its superior performance.

The optimization of multilayer neural networks typically leads to a solution with zero training error, yet the landscape can exhibit spurious local minima and the minima can be disconnected. In this paper, we shed light on this phenomenon: we show that the combination of stochastic gradient descent (SGD) and over-parameterization makes the landscape of multilayer neural networks approximately connected and thus more favorable to optimization. More specifically, we prove that SGD solutions are connected via a piecewise linear path, and the increase in loss along this path vanishes as the number of neurons grows large. This result is a consequence of the fact that the parameters found by SGD are increasingly dropout stable as the network becomes wider. We show that, if we remove part of the neurons (and suitably rescale the remaining ones), the change in loss is independent of the total number of neurons, and it depends only on how many neurons are left. Our results exhibit a mild dependence on the input dimension: they are dimension-free for two-layer networks and depend linearly on the dimension for multilayer networks. We validate our theoretical findings with numerical experiments for different architectures and classification tasks.

Contrastive learning (CL) continuously achieves significant breakthroughs across multiple domains. However, the most common InfoNCE-based methods suffer from some dilemmas, such as uniformity-tolerance dilemma (UTD) and gradient reduction, both of which are related to a P_{ij} term. It has been identified that UTD can lead to unexpected performance degradation. We argue that the fixity of temperature is to blame for UTD. To tackle this challenge, we enrich the CL loss family by presenting a Model-Aware Contrastive Learning (MACL) strategy, whose temperature is adaptive to the magnitude of alignment that reflects the basic confidence of the instance discrimination task, then enables CL loss to adjust the penalty strength for hard negatives adaptively. Regarding another dilemma, the gradient reduction issue, we derive the limits of an involved gradient scaling factor, which allows us to explain from a unified perspective why some recent approaches are effective with fewer negative samples, and summarily present a gradient reweighting to escape this dilemma. Extensive remarkable empirical results in vision, sentence, and graph modality validate our approach's general improvement for representation learning and downstream tasks.

Many machine learning methods operate by inverting a neural network at inference time, which has become a popular technique for solving inverse problems in computer vision, robotics, and graphics. However, these methods often involve gradient descent through a highly non-convex loss landscape, causing the optimization process to be unstable and slow. We introduce a method that learns a loss landscape where gradient descent is efficient, bringing massive improvement and acceleration to the inversion process. We demonstrate this advantage on a number of methods for both generative and discriminative tasks, including GAN inversion, adversarial defense, and 3D human pose reconstruction.

Normalization techniques are a boon for modern deep learning. They let weights converge more quickly with often better generalization performances. It has been argued that the normalization-induced scale invariance among the weights provides an advantageous ground for gradient descent (GD) optimizers: the effective step sizes are automatically reduced over time, stabilizing the overall training procedure. It is often overlooked, however, that the additional introduction of momentum in GD optimizers results in a far more rapid reduction in effective step sizes for scale-invariant weights, a phenomenon that has not yet been studied and may have caused unwanted side effects in the current practice. This is a crucial issue because arguably the vast majority of modern deep neural networks consist of (1) momentum-based GD (e.g. SGD or Adam) and (2) scale-invariant parameters. In this paper, we verify that the widely-adopted combination of the two ingredients lead to the premature decay of effective step sizes and sub-optimal model performances. We propose a simple and effective remedy, SGDP and AdamP: get rid of the radial component, or the norm-increasing direction, at each optimizer step. Because of the scale invariance, this modification only alters the effective step sizes without changing the effective update directions, thus enjoying the original convergence properties of GD optimizers. Given the ubiquity of momentum GD and scale invariance in machine learning, we have evaluated our methods against the baselines on 13 benchmarks. They range from vision tasks like classification (e.g. ImageNet), retrieval (e.g. CUB and SOP), and detection (e.g. COCO) to language modelling (e.g. WikiText) and audio classification (e.g. DCASE) tasks. We verify that our solution brings about uniform gains in those benchmarks. Source code is available at https://github.com/clovaai/AdamP.

Loss landscape is a useful tool to characterize and compare neural network models. The main challenge for analysis of loss landscape for the deep neural networks is that they are generally highly non-convex in very high dimensional space. In this paper, we develop "the roughness"concept for understanding such landscapes in high dimensions and apply this technique to study two neural network models arising from solving differential equations. Our main innovation is the proposal of a well-defined and easy-to-compute roughness index (RI) which is based on the mean and variance of the (normalized) total variation for one-dimensional functions projected on randomly sampled directions. A large RI at the local minimizer hints an oscillatory landscape profile and indicates a severe challenge for the first-order optimization method. Particularly, we observe the increasing-then-decreasing pattern for RI along the gradient descent path in most models. We apply our method to two types of loss functions used to solve partial differential equations (PDEs) when the solution of PDE is parametrized by neural networks. Our empirical results on these PDE problems reveal important and consistent observations that the landscapes from the deep Galerkin method around its local minimizers are less rough than the deep Ritz method.

In distributed computing, slower nodes (stragglers) usually become a bottleneck. Gradient Coding (GC), introduced by Tandon et al., is an efficient technique that uses principles of error-correcting codes to distribute gradient computation in the presence of stragglers. In this paper, we consider the distributed computation of a sequence of gradients {g(1),g(2),ldots,g(J)}, where processing of each gradient g(t) starts in round-t and finishes by round-(t+T). Here Tgeq 0 denotes a delay parameter. For the GC scheme, coding is only across computing nodes and this results in a solution where T=0. On the other hand, having T>0 allows for designing schemes which exploit the temporal dimension as well. In this work, we propose two schemes that demonstrate improved performance compared to GC. Our first scheme combines GC with selective repetition of previously unfinished tasks and achieves improved straggler mitigation. In our second scheme, which constitutes our main contribution, we apply GC to a subset of the tasks and repetition for the remainder of the tasks. We then multiplex these two classes of tasks across workers and rounds in an adaptive manner, based on past straggler patterns. Using theoretical analysis, we demonstrate that our second scheme achieves significant reduction in the computational load. In our experiments, we study a practical setting of concurrently training multiple neural networks over an AWS Lambda cluster involving 256 worker nodes, where our framework naturally applies. We demonstrate that the latter scheme can yield a 16\% improvement in runtime over the baseline GC scheme, in the presence of naturally occurring, non-simulated stragglers.

A major challenge in training large-scale machine learning models is configuring the training process to maximize model performance, i.e., finding the best training setup from a vast design space. In this work, we unlock a gradient-based approach to this problem. We first introduce an algorithm for efficiently calculating metagradients -- gradients through model training -- at scale. We then introduce a "smooth model training" framework that enables effective optimization using metagradients. With metagradient descent (MGD), we greatly improve on existing dataset selection methods, outperform accuracy-degrading data poisoning attacks by an order of magnitude, and automatically find competitive learning rate schedules.

We introduce Gradient Descent with Adaptive Momentum Scaling (Grams), a novel optimization algorithm that decouples the direction and magnitude of parameter updates in deep learning. Unlike traditional optimizers that directly integrate momentum into updates, Grams separates the update direction, derived from current gradients, from momentum, which is used solely for adaptive magnitude scaling. This approach enables Grams to achieve improved loss descent compared to state-of-the-art cautious and momentum-based optimizers. We establish a global convergence guarantee for Grams and validate its effectiveness through extensive empirical evaluations. The results demonstrate Grams' superior performance, including faster convergence and better generalization, compared to widely-used optimizers such as Adam, Lion, and their cautious variants. Our results highlight Grams' potential as a transformative approach for efficient optimization in large-scale machine learning.

In this work, we study the evolution of the loss Hessian across many classification tasks in order to understand the effect the curvature of the loss has on the training dynamics. Whereas prior work has focused on how different learning rates affect the loss Hessian observed during training, we also analyze the effects of model initialization, architectural choices, and common training heuristics such as gradient clipping and learning rate warmup. Our results demonstrate that successful model and hyperparameter choices allow the early optimization trajectory to either avoid -- or navigate out of -- regions of high curvature and into flatter regions that tolerate a higher learning rate. Our results suggest a unifying perspective on how disparate mitigation strategies for training instability ultimately address the same underlying failure mode of neural network optimization, namely poor conditioning. Inspired by the conditioning perspective, we show that learning rate warmup can improve training stability just as much as batch normalization, layer normalization, MetaInit, GradInit, and Fixup initialization.

We identify a new phenomenon in neural network optimization which arises from the interaction of depth and a particular heavy-tailed structure in natural data. Our result offers intuitive explanations for several previously reported observations about network training dynamics. In particular, it implies a conceptually new cause for progressive sharpening and the edge of stability; we also highlight connections to other concepts in optimization and generalization including grokking, simplicity bias, and Sharpness-Aware Minimization. Experimentally, we demonstrate the significant influence of paired groups of outliers in the training data with strong opposing signals: consistent, large magnitude features which dominate the network output throughout training and provide gradients which point in opposite directions. Due to these outliers, early optimization enters a narrow valley which carefully balances the opposing groups; subsequent sharpening causes their loss to rise rapidly, oscillating between high on one group and then the other, until the overall loss spikes. We describe how to identify these groups, explore what sets them apart, and carefully study their effect on the network's optimization and behavior. We complement these experiments with a mechanistic explanation on a toy example of opposing signals and a theoretical analysis of a two-layer linear network on a simple model. Our finding enables new qualitative predictions of training behavior which we confirm experimentally. It also provides a new lens through which to study and improve modern training practices for stochastic optimization, which we highlight via a case study of Adam versus SGD.

Training stability is a persistent challenge in the pre-training of large language models (LLMs), particularly for architectures such as Post-Norm Transformers, which are prone to gradient explosion and dissipation. In this paper, we propose Scale-Distribution Decoupling (SDD), a novel approach that stabilizes training by explicitly decoupling the scale and distribution of the weight matrix in fully-connected layers. SDD applies a normalization mechanism to regulate activations and a learnable scaling vector to maintain well-conditioned gradients, effectively preventing gradient explosion and dissipation. This separation improves optimization efficiency, particularly in deep networks, by ensuring stable gradient propagation. Experimental results demonstrate that our method stabilizes training across various LLM architectures and outperforms existing techniques in different normalization configurations. Furthermore, the proposed method is lightweight and compatible with existing frameworks, making it a practical solution for stabilizing LLM training. Code is available at https://github.com/kaihemo/SDD.

The backpropagation algorithm has long been the canonical training method for neural networks. Modern paradigms are implicitly optimized for it, and numerous guidelines exist to ensure its proper use. Recently, synthetic gradients methods -where the error gradient is only roughly approximated - have garnered interest. These methods not only better portray how biological brains are learning, but also open new computational possibilities, such as updating layers asynchronously. Even so, they have failed to scale past simple tasks like MNIST or CIFAR-10. This is in part due to a lack of standards, leading to ill-suited models and practices forbidding such methods from performing to the best of their abilities. In this work, we focus on direct feedback alignment and present a set of best practices justified by observations of the alignment angles. We characterize a bottleneck effect that prevents alignment in narrow layers, and hypothesize it may explain why feedback alignment methods have yet to scale to large convolutional networks.

Ensuring faithful interpretability in large language models is imperative for trustworthy and reliable AI. A key obstacle is self-repair, a phenomenon where networks compensate for reduced signal in one component by amplifying others, masking the true importance of the ablated component. While prior work attributes self-repair to layer normalization and back-up components that compensate for ablated components, we identify a novel form occurring within the attention mechanism, where softmax redistribution conceals the influence of important attention scores. This leads traditional ablation and gradient-based methods to underestimate the significance of all components contributing to these attention scores. We introduce Gradient Interaction Modifications (GIM), a technique that accounts for self-repair during backpropagation. Extensive experiments across multiple large language models (Gemma 2B/9B, LLAMA 1B/3B/8B, Qwen 1.5B/3B) and diverse tasks demonstrate that GIM significantly improves faithfulness over existing circuit identification and feature attribution methods. Our work is a significant step toward better understanding the inner mechanisms of LLMs, which is crucial for improving them and ensuring their safety. Our code is available at https://github.com/JoakimEdin/gim.

Modern Large Language Models, such as the LLaMA, Qwen and DeepSeek series, predominantly adopt the Pre-LayerNorm (Pre-LN) Transformer architecture. While being stable during pretraining and scalable to large model sizes, Pre-LN suffers from an exponential growth in activation variance across layers, causing the residual path to dominate over sub-layer outputs and limiting the learning capacity of deeper layers. To mitigate this issue, we propose Gradient-Preserving Activation Scaling (GPAS), a simple technique that can be used in combination with existing approaches. GPAS works by scaling down the intermediate activations while keeping their gradients unchanged. This leaves information in the activations intact, and avoids the gradient vanishing problem associated with gradient downscaling. Extensive experiments across various model sizes from 71M to 1B show that GPAS achieves consistent performance gains. Beyond enhancing Pre-LN Transformers, GPAS also shows promise in improving alternative architectures such as Sandwich-LN and DeepNorm, demonstrating its versatility and potential for improving training dynamics in a wide range of settings.

A growing literature in computational neuroscience leverages gradient descent and learning algorithms that approximate it to study synaptic plasticity in the brain. However, the vast majority of this work ignores a critical underlying assumption: the choice of distance for synaptic changes - i.e. the geometry of synaptic plasticity. Gradient descent assumes that the distance is Euclidean, but many other distances are possible, and there is no reason that biology necessarily uses Euclidean geometry. Here, using the theoretical tools provided by mirror descent, we show that the distribution of synaptic weights will depend on the geometry of synaptic plasticity. We use these results to show that experimentally-observed log-normal weight distributions found in several brain areas are not consistent with standard gradient descent (i.e. a Euclidean geometry), but rather with non-Euclidean distances. Finally, we show that it should be possible to experimentally test for different synaptic geometries by comparing synaptic weight distributions before and after learning. Overall, our work shows that the current paradigm in theoretical work on synaptic plasticity that assumes Euclidean synaptic geometry may be misguided and that it should be possible to experimentally determine the true geometry of synaptic plasticity in the brain.

Stochastic Gradient Decent (SGD) is one of the core techniques behind the success of deep neural networks. The gradient provides information on the direction in which a function has the steepest rate of change. The main problem with basic SGD is to change by equal sized steps for all parameters, irrespective of gradient behavior. Hence, an efficient way of deep network optimization is to make adaptive step sizes for each parameter. Recently, several attempts have been made to improve gradient descent methods such as AdaGrad, AdaDelta, RMSProp and Adam. These methods rely on the square roots of exponential moving averages of squared past gradients. Thus, these methods do not take advantage of local change in gradients. In this paper, a novel optimizer is proposed based on the difference between the present and the immediate past gradient (i.e., diffGrad). In the proposed diffGrad optimization technique, the step size is adjusted for each parameter in such a way that it should have a larger step size for faster gradient changing parameters and a lower step size for lower gradient changing parameters. The convergence analysis is done using the regret bound approach of online learning framework. Rigorous analysis is made in this paper over three synthetic complex non-convex functions. The image categorization experiments are also conducted over the CIFAR10 and CIFAR100 datasets to observe the performance of diffGrad with respect to the state-of-the-art optimizers such as SGDM, AdaGrad, AdaDelta, RMSProp, AMSGrad, and Adam. The residual unit (ResNet) based Convolutional Neural Networks (CNN) architecture is used in the experiments. The experiments show that diffGrad outperforms other optimizers. Also, we show that diffGrad performs uniformly well for training CNN using different activation functions. The source code is made publicly available at https://github.com/shivram1987/diffGrad.

Modern deep-learning systems are specialized to problem settings in which training occurs once and then never again, as opposed to continual-learning settings in which training occurs continually. If deep-learning systems are applied in a continual learning setting, then it is well known that they may fail to remember earlier examples. More fundamental, but less well known, is that they may also lose their ability to learn on new examples, a phenomenon called loss of plasticity. We provide direct demonstrations of loss of plasticity using the MNIST and ImageNet datasets repurposed for continual learning as sequences of tasks. In ImageNet, binary classification performance dropped from 89\% accuracy on an early task down to 77\%, about the level of a linear network, on the 2000th task. Loss of plasticity occurred with a wide range of deep network architectures, optimizers, activation functions, batch normalization, dropout, but was substantially eased by L^2-regularization, particularly when combined with weight perturbation. Further, we introduce a new algorithm -- continual backpropagation -- which slightly modifies conventional backpropagation to reinitialize a small fraction of less-used units after each example and appears to maintain plasticity indefinitely.

Beyond minimizing a single training loss, many deep learning estimation pipelines rely on an auxiliary objective to quantify and encourage desirable properties of the model (e.g. performance on another dataset, robustness, agreement with a prior). Although the simplest approach to incorporating an auxiliary loss is to sum it with the training loss as a regularizer, recent works have shown that one can improve performance by blending the gradients beyond a simple sum; this is known as gradient surgery. We cast the problem as a constrained minimization problem where the auxiliary objective is minimized among the set of minimizers of the training loss. To solve this bilevel problem, we follow a parameter update direction that combines the training loss gradient and the orthogonal projection of the auxiliary gradient to the training gradient. In a setting where gradients come from mini-batches, we explain how, using a moving average of the training loss gradients, we can carefully maintain this critical orthogonality property. We demonstrate that our method, Bloop, can lead to much better performances on NLP and vision experiments than other gradient surgery methods without EMA.

Learning a policy with great generalization to unseen environments remains challenging but critical in visual reinforcement learning. Despite the success of augmentation combination in the supervised learning generalization, naively applying it to visual RL algorithms may damage the training efficiency, suffering from serve performance degradation. In this paper, we first conduct qualitative analysis and illuminate the main causes: (i) high-variance gradient magnitudes and (ii) gradient conflicts existed in various augmentation methods. To alleviate these issues, we propose a general policy gradient optimization framework, named Conflict-aware Gradient Agreement Augmentation (CG2A), and better integrate augmentation combination into visual RL algorithms to address the generalization bias. In particular, CG2A develops a Gradient Agreement Solver to adaptively balance the varying gradient magnitudes, and introduces a Soft Gradient Surgery strategy to alleviate the gradient conflicts. Extensive experiments demonstrate that CG2A significantly improves the generalization performance and sample efficiency of visual RL algorithms.

Time-dependent data-generating distributions have proven to be difficult for gradient-based training of neural networks, as the greedy updates result in catastrophic forgetting of previously learned knowledge. Despite the progress in the field of continual learning to overcome this forgetting, we show that a set of common state-of-the-art methods still suffers from substantial forgetting upon starting to learn new tasks, except that this forgetting is temporary and followed by a phase of performance recovery. We refer to this intriguing but potentially problematic phenomenon as the stability gap. The stability gap had likely remained under the radar due to standard practice in the field of evaluating continual learning models only after each task. Instead, we establish a framework for continual evaluation that uses per-iteration evaluation and we define a new set of metrics to quantify worst-case performance. Empirically we show that experience replay, constraint-based replay, knowledge-distillation, and parameter regularization methods are all prone to the stability gap; and that the stability gap can be observed in class-, task-, and domain-incremental learning benchmarks. Additionally, a controlled experiment shows that the stability gap increases when tasks are more dissimilar. Finally, by disentangling gradients into plasticity and stability components, we propose a conceptual explanation for the stability gap.

Deep ResNet architectures have achieved state of the art performance on many tasks. While they solve the problem of gradient vanishing, they might suffer from gradient exploding as the depth becomes large (Yang et al. 2017). Moreover, recent results have shown that ResNet might lose expressivity as the depth goes to infinity (Yang et al. 2017, Hayou et al. 2019). To resolve these issues, we introduce a new class of ResNet architectures, called Stable ResNet, that have the property of stabilizing the gradient while ensuring expressivity in the infinite depth limit.

This paper studies the problem of training a two-layer ReLU network for binary classification using gradient flow with small initialization. We consider a training dataset with well-separated input vectors: Any pair of input data with the same label are positively correlated, and any pair with different labels are negatively correlated. Our analysis shows that, during the early phase of training, neurons in the first layer try to align with either the positive data or the negative data, depending on its corresponding weight on the second layer. A careful analysis of the neurons' directional dynamics allows us to provide an O(log n{mu}) upper bound on the time it takes for all neurons to achieve good alignment with the input data, where n is the number of data points and mu measures how well the data are separated. After the early alignment phase, the loss converges to zero at a O(1{t}) rate, and the weight matrix on the first layer is approximately low-rank. Numerical experiments on the MNIST dataset illustrate our theoretical findings.

A generative adversarial network (GAN) has been a representative backbone model in generative artificial intelligence (AI) because of its powerful performance in capturing intricate data-generating processes. However, the GAN training is well-known for its notorious training instability, usually characterized by the occurrence of mode collapse. Through the lens of gradients' variance, this work particularly analyzes the training instability and inefficiency in the presence of mode collapse by linking it to multimodality in the target distribution. To ease the raised training issues from severe multimodality, we introduce a novel GAN training framework that leverages a series of tempered distributions produced via convex interpolation. With our newly developed GAN objective function, the generator can learn all the tempered distributions simultaneously, conceptually resonating with the parallel tempering in Statistics. Our simulation studies demonstrate the superiority of our approach over existing popular training strategies in both image and tabular data synthesis. We theoretically analyze that such significant improvement can arise from reducing the variance of gradient estimates by using the tempered distributions. Finally, we further develop a variant of the proposed framework aimed at generating fair synthetic data which is one of the growing interests in the field of trustworthy AI.

Fast adversarial training (FAT) is beneficial for improving the adversarial robustness of neural networks. However, previous FAT work has encountered a significant issue known as catastrophic overfitting when dealing with large perturbation budgets, \ie the adversarial robustness of models declines to near zero during training. To address this, we analyze the training process of prior FAT work and observe that catastrophic overfitting is accompanied by the appearance of loss convergence outliers. Therefore, we argue a moderately smooth loss convergence process will be a stable FAT process that solves catastrophic overfitting. To obtain a smooth loss convergence process, we propose a novel oscillatory constraint (dubbed ConvergeSmooth) to limit the loss difference between adjacent epochs. The convergence stride of ConvergeSmooth is introduced to balance convergence and smoothing. Likewise, we design weight centralization without introducing additional hyperparameters other than the loss balance coefficient. Our proposed methods are attack-agnostic and thus can improve the training stability of various FAT techniques. Extensive experiments on popular datasets show that the proposed methods efficiently avoid catastrophic overfitting and outperform all previous FAT methods. Code is available at https://github.com/FAT-CS/ConvergeSmooth.

Gradient inversion attacks aim to reconstruct local training data from intermediate gradients exposed in the federated learning framework. Despite successful attacks, all previous methods, starting from reconstructing a single data point and then relaxing the single-image limit to batch level, are only tested under hard label constraints. Even for single-image reconstruction, we still lack an analysis-based algorithm to recover augmented soft labels. In this work, we change the focus from enlarging batchsize to investigating the hard label constraints, considering a more realistic circumstance where label smoothing and mixup techniques are used in the training process. In particular, we are the first to initiate a novel algorithm to simultaneously recover the ground-truth augmented label and the input feature of the last fully-connected layer from single-input gradients, and provide a necessary condition for any analytical-based label recovery methods. Extensive experiments testify to the label recovery accuracy, as well as the benefits to the following image reconstruction. We believe soft labels in classification tasks are worth further attention in gradient inversion attacks.

Teams that have trained large Transformer-based models have reported training instabilities at large scale that did not appear when training with the same hyperparameters at smaller scales. Although the causes of such instabilities are of scientific interest, the amount of resources required to reproduce them has made investigation difficult. In this work, we seek ways to reproduce and study training stability and instability at smaller scales. First, we focus on two sources of training instability described in previous work: the growth of logits in attention layers (Dehghani et al., 2023) and divergence of the output logits from the log probabilities (Chowdhery et al., 2022). By measuring the relationship between learning rate and loss across scales, we show that these instabilities also appear in small models when training at high learning rates, and that mitigations previously employed at large scales are equally effective in this regime. This prompts us to investigate the extent to which other known optimizer and model interventions influence the sensitivity of the final loss to changes in the learning rate. To this end, we study methods such as warm-up, weight decay, and the muParam (Yang et al., 2022), and combine techniques to train small models that achieve similar losses across orders of magnitude of learning rate variation. Finally, to conclude our exploration we study two cases where instabilities can be predicted before they emerge by examining the scaling behavior of model activation and gradient norms.

Using backpropagation to compute gradients of objective functions for optimization has remained a mainstay of machine learning. Backpropagation, or reverse-mode differentiation, is a special case within the general family of automatic differentiation algorithms that also includes the forward mode. We present a method to compute gradients based solely on the directional derivative that one can compute exactly and efficiently via the forward mode. We call this formulation the forward gradient, an unbiased estimate of the gradient that can be evaluated in a single forward run of the function, entirely eliminating the need for backpropagation in gradient descent. We demonstrate forward gradient descent in a range of problems, showing substantial savings in computation and enabling training up to twice as fast in some cases.

Unrestricted adversarial attacks present a serious threat to deep learning models and adversarial defense techniques. They pose severe security problems for deep learning applications because they can effectively bypass defense mechanisms. However, previous attack methods often directly inject Projected Gradient Descent (PGD) gradients into the sampling of generative models, which are not theoretically provable and thus generate unrealistic examples by incorporating adversarial objectives, especially for GAN-based methods on large-scale datasets like ImageNet. In this paper, we propose a new method, called AdvDiff, to generate unrestricted adversarial examples with diffusion models. We design two novel adversarial guidance techniques to conduct adversarial sampling in the reverse generation process of diffusion models. These two techniques are effective and stable in generating high-quality, realistic adversarial examples by integrating gradients of the target classifier interpretably. Experimental results on MNIST and ImageNet datasets demonstrate that AdvDiff is effective in generating unrestricted adversarial examples, which outperforms state-of-the-art unrestricted adversarial attack methods in terms of attack performance and generation quality.

The presence of thermal gradients in alloys often leads to non-uniformity in concentration profiles, which can induce the thermomigration of microstructural features such as precipitates. To investigate such microstructural changes, we present a phase-field model that incorporates coupling between concentration and thermal gradients. First, we simulated the evolution of non-uniform concentration profiles in the single-phase regions of Fe-C and Fe-N alloy systems due to imposed thermal gradients. To validate our model with the classical experiments performed by Darken and Oriani, we studied the evolution of spatially varying concentration profiles where thermal gradients encompass single-phase and two-phase regions. We developed a parameterized thermodynamic description of the two-phase region of a binary alloy to systematically study the effect of interactions between chemically-driven and thermal gradient-driven diffusion of solute on the evolution of precipitates. Our simulations show how thermal gradient, precipitate size, and interparticle distance influence the migration and associated morphological changes of precipitates. The composition profiles and migration rates obtained from single-particle simulations show an exact match with our analytical model. We use twoparticle simulations to show conditions under which thermomigration induces the growth of the smaller particle and shrinkage of the larger one in contrast to the isothermal Ostwald ripening behavior. Our multiparticle simulations show similar behavior during coarsening. Moreover, in the presence of a thermal gradient, there is a shift in the center of mass of the precipitates towards the high-temperature region. Thus, our study offers new insights into the phenomena of microstructure evolution in the presence of thermal gradient.

Adversarial robustness is essential for security and reliability of machine learning systems. However, adversarial robustness enhanced by defense algorithms is easily erased as the neural network's weights update to learn new tasks. To address this vulnerability, it is essential to improve the capability of neural networks in terms of robust continual learning. Specially, we propose a novel gradient projection technique that effectively stabilizes sample gradients from previous data by orthogonally projecting back-propagation gradients onto a crucial subspace before using them for weight updates. This technique can maintaining robustness by collaborating with a class of defense algorithms through sample gradient smoothing. The experimental results on four benchmarks including Split-CIFAR100 and Split-miniImageNet, demonstrate that the superiority of the proposed approach in mitigating rapidly degradation of robustness during continual learning even when facing strong adversarial attacks.

Diffusion models have recently gained recognition for generating diverse and high-quality content, especially in the domain of image synthesis. These models excel not only in creating fixed-size images but also in producing panoramic images. However, existing methods often struggle with spatial layout consistency when producing high-resolution panoramas, due to the lack of guidance of the global image layout. In this paper, we introduce the Multi-Scale Diffusion (MSD) framework, a plug-and-play module that extends the existing panoramic image generation framework to multiple resolution levels. By utilizing gradient descent techniques, our method effectively incorporates structural information from low-resolution images into high-resolution outputs. A comprehensive evaluation of the proposed method was conducted, comparing it with the prior works in qualitative and quantitative dimensions. The evaluation results demonstrate that our method significantly outperforms others in generating coherent high-resolution panoramas.

Image generation using diffusion models have demonstrated outstanding learning capabilities, effectively capturing the full distribution of the training dataset. They are known to generate wide variations in sampled images, albeit with a trade-off in image fidelity. Guided sampling methods, such as classifier guidance (CG) and classifier-free guidance (CFG), focus sampling in well-learned high-probability regions to generate images of high fidelity, but each has its limitations. CG is computationally expensive due to the use of back-propagation for classifier gradient descent, while CFG, being gradient-free, is more efficient but compromises class label alignment compared to CG. In this work, we propose an efficient guidance method that fully utilizes a pre-trained classifier without using gradient descent. By using the classifier solely in inference mode, a time-adaptive reference class label and corresponding guidance scale are determined at each time step for guided sampling. Experiments on both class-conditioned and text-to-image generation diffusion models demonstrate that the proposed Gradient-free Classifier Guidance (GFCG) method consistently improves class prediction accuracy. We also show GFCG to be complementary to other guided sampling methods like CFG. When combined with the state-of-the-art Autoguidance (ATG), without additional computational overhead, it enhances image fidelity while preserving diversity. For ImageNet 512times512, we achieve a record FD_{DINOv2} of 23.09, while simultaneously attaining a higher classification Precision (94.3%) compared to ATG (90.2%)

Gradients can be employed for sensitivity analysis. Here, we leverage the advantages of the Loss Landscape to comprehend which independent variables impact the dependent variable. We seek to grasp the loss landscape by utilizing first, second, and third derivatives through automatic differentiation. we know that Spearman's rank correlation coefficient can detect the monotonic relationship between two variables. However, I have found that second-order gradients, with certain configurations and parameters, provide information that can be visualized similarly to Spearman results, In this approach, we incorporate a loss function with an activation function, resulting in a non-linear pattern. Each exploration of the loss landscape through retraining yields new valuable information. Furthermore, the first and third derivatives are also beneficial, as they indicate the extent to which independent variables influence the dependent variable.

Recent approaches have yielded promising results in distilling multi-step text-to-image diffusion models into one-step ones. The state-of-the-art efficient distillation technique, i.e., SwiftBrushv2 (SBv2), even surpasses the teacher model's performance with limited resources. However, our study reveals its instability when handling different diffusion model backbones due to using a fixed guidance scale within the Variational Score Distillation (VSD) loss. Another weakness of the existing one-step diffusion models is the missing support for negative prompt guidance, which is crucial in practical image generation. This paper presents SNOOPI, a novel framework designed to address these limitations by enhancing the guidance in one-step diffusion models during both training and inference. First, we effectively enhance training stability through Proper Guidance-SwiftBrush (PG-SB), which employs a random-scale classifier-free guidance approach. By varying the guidance scale of both teacher models, we broaden their output distributions, resulting in a more robust VSD loss that enables SB to perform effectively across diverse backbones while maintaining competitive performance. Second, we propose a training-free method called Negative-Away Steer Attention (NASA), which integrates negative prompts into one-step diffusion models via cross-attention to suppress undesired elements in generated images. Our experimental results show that our proposed methods significantly improve baseline models across various metrics. Remarkably, we achieve an HPSv2 score of 31.08, setting a new state-of-the-art benchmark for one-step diffusion models.

Loss of plasticity is a phenomenon where neural networks become more difficult to train during the course of learning. Continual learning algorithms seek to mitigate this effect by sustaining good predictive performance while maintaining network trainability. We develop new techniques for improving continual learning by first reconsidering how initialization can ensure trainability during early phases of learning. From this perspective, we derive new regularization strategies for continual learning that ensure beneficial initialization properties are better maintained throughout training. In particular, we investigate two new regularization techniques for continual learning: (i) Wasserstein regularization toward the initial weight distribution, which is less restrictive than regularizing toward initial weights; and (ii) regularizing weight matrix singular values, which directly ensures gradient diversity is maintained throughout training. We present an experimental analysis that shows these alternative regularizers can improve continual learning performance across a range of supervised learning tasks and model architectures. The alternative regularizers prove to be less sensitive to hyperparameters while demonstrating better training in individual tasks, sustaining trainability as new tasks arrive, and achieving better generalization performance.

To accelerate diffusion model inference, numerical solvers perform poorly at extremely small steps, while distillation techniques often introduce complexity and instability. This work presents an intermediate strategy, balancing performance and cost, by learning ODE integration using loss functions derived from the derivative-integral relationship, inspired by Monte Carlo integration and Picard iteration. From a geometric perspective, the losses operate by gradually extending the tangent to the secant, thus are named as secant losses. The target of secant losses is the same as that of diffusion models, or the diffusion model itself, leading to great training stability. By fine-tuning or distillation, the secant version of EDM achieves a 10-step FID of 2.14 on CIFAR-10, while the secant version of SiT-XL/2 attains a 4-step FID of 2.27 and an 8-step FID of 1.96 on ImageNet-256times256. Code is available at https://github.com/poppuppy/secant-expectation.

Data imbalance is a common problem in machine learning that can have a critical effect on the performance of a model. Various solutions exist but their impact on the convergence of the learning dynamics is not understood. Here, we elucidate the significant negative impact of data imbalance on learning, showing that the learning curves for minority and majority classes follow sub-optimal trajectories when training with a gradient-based optimizer. This slowdown is related to the imbalance ratio and can be traced back to a competition between the optimization of different classes. Our main contribution is the analysis of the convergence of full-batch (GD) and stochastic gradient descent (SGD), and of variants that renormalize the contribution of each per-class gradient. We find that GD is not guaranteed to decrease the loss for each class but that this problem can be addressed by performing a per-class normalization of the gradient. With SGD, class imbalance has an additional effect on the direction of the gradients: the minority class suffers from a higher directional noise, which reduces the effectiveness of the per-class gradient normalization. Our findings not only allow us to understand the potential and limitations of strategies involving the per-class gradients, but also the reason for the effectiveness of previously used solutions for class imbalance such as oversampling.

Single image reflection removal is inherently ambiguous, as both the reflection and transmission components requiring separation may follow natural image statistics. Existing methods attempt to address the issue by using various types of low-level and physics-based cues as sources of reflection signals. However, these cues are not universally applicable, since they are only observable in specific capture scenarios. This leads to a significant performance drop when test images do not align with their assumptions. In this paper, we aim to explore a novel flexible interactive reflection removal approach that leverages various forms of sparse human guidance, such as points and bounding boxes, as auxiliary high-level prior to achieve robust reflection removal. However, incorporating the raw user guidance naively into the existing reflection removal network does not result in performance gains. To this end, we innovatively transform raw user input into a unified form -- reflection masks using an Interactive Segmentation Foundation Model. Such a design absorbs the quintessence of the foundational segmentation model and flexible human guidance, thereby mitigating the challenges of reflection separations. Furthermore, to fully utilize user guidance and reduce user annotation costs, we design a mask-guided reflection removal network, comprising our proposed self-adaptive prompt block. This block adaptively incorporates user guidance as anchors and refines transmission features via cross-attention mechanisms. Extensive results on real-world images validate that our method demonstrates state-of-the-art performance on various datasets with the help of flexible and sparse user guidance. Our code and dataset will be publicly available here https://github.com/ShawnChenn/FlexibleReflectionRemoval.

Stochastic Gradient Descent (SGD) stands as a cornerstone optimization algorithm with proven real-world empirical successes but relatively limited theoretical understanding. Recent research has illuminated a key factor contributing to its practical efficacy: the implicit regularization it instigates. Several studies have investigated the linear stability property of SGD in the vicinity of a stationary point as a predictive proxy for sharpness and generalization error in overparameterized neural networks (Wu et al., 2022; Jastrzebski et al., 2019; Cohen et al., 2021). In this paper, we delve deeper into the relationship between linear stability and sharpness. More specifically, we meticulously delineate the necessary and sufficient conditions for linear stability, contingent on hyperparameters of SGD and the sharpness at the optimum. Towards this end, we introduce a novel coherence measure of the loss Hessian that encapsulates pertinent geometric properties of the loss function that are relevant to the linear stability of SGD. It enables us to provide a simplified sufficient condition for identifying linear instability at an optimum. Notably, compared to previous works, our analysis relies on significantly milder assumptions and is applicable for a broader class of loss functions than known before, encompassing not only mean-squared error but also cross-entropy loss.

We consider a deep matrix factorization model of covariance matrices trained with the Bures-Wasserstein distance. While recent works have made important advances in the study of the optimization problem for overparametrized low-rank matrix approximation, much emphasis has been placed on discriminative settings and the square loss. In contrast, our model considers another interesting type of loss and connects with the generative setting. We characterize the critical points and minimizers of the Bures-Wasserstein distance over the space of rank-bounded matrices. For low-rank matrices the Hessian of this loss can theoretically blow up, which creates challenges to analyze convergence of optimizaton methods. We establish convergence results for gradient flow using a smooth perturbative version of the loss and convergence results for finite step size gradient descent under certain assumptions on the initial weights.

The loss landscape of neural networks is a critical aspect of their training, and understanding its properties is essential for improving their performance. In this paper, we investigate how the loss surface changes when the sample size increases, a previously unexplored issue. We theoretically analyze the convergence of the loss landscape in a fully connected neural network and derive upper bounds for the difference in loss function values when adding a new object to the sample. Our empirical study confirms these results on various datasets, demonstrating the convergence of the loss function surface for image classification tasks. Our findings provide insights into the local geometry of neural loss landscapes and have implications for the development of sample size determination techniques.

The radial gradient of gas-phase metallicity is a powerful probe of the chemical and structural evolution of star-forming galaxies, closely tied to disk formation and gas kinematics in the early universe. We present spatially resolved chemical and dynamical properties for a sample of 25 galaxies at 0.5 lesssim z lesssim 1.7 from the \msasd survey. These innovative observations provide 3D spectroscopy of galaxies at a spatial resolution approaching JWST's diffraction limit and a high spectral resolution of Rsimeq2700. The metallicity gradients measured in our galaxy sample range from -0.03 to 0.02 dex~kpc^{-1}. Most galaxies exhibit negative or flat radial gradients, indicating lower metallicity in the outskirts or uniform metallicity throughout the entire galaxy. We confirm a tight relationship between stellar mass and metallicity gradient at zsim1 with small intrinsic scatter of 0.02 dex~kpc^{-1}. Our results indicate that metallicity gradients become increasingly negative as stellar mass increases, likely because the more massive galaxies tend to be more ``disky". This relationship is consistent with the predictions from cosmological hydrodynamic zoom-in simulations with strong stellar feedback. This work presents the effort to harness the multiplexing capability of JWST NIRSpec/MSA in slit-stepping mode to map the chemical and kinematic profiles of high-redshift galaxies in large samples and at high spatial and spectral resolution.

Global stability of differentially rotating plasma is investigated using a generalized effective potential. We first, for a current-free system, obtain a general form of an effective potential in terms of the free energies of global curvature and gradients of rotation for non-axisymmetric disturbances. We then examine the stability of differentially rotating disks for several rotation profiles and present the associated effective potential for the onset of these instabilities in the MHD regime. In particular, results for global axisymmetric magnetorotational instability (MRI) as well as local and global non-axisymmetric modes are presented. The latter constitute two distinct non-axisymmetric modes, a high frequency local MRI and a global low-frequency non-axisymmetric mode (the magneto-curvature mode, introduced in Ebrahimi&Pharr, ApJ 2022), confined either between two Alfv\'enic resonances or an Alfv\'enic resonance and a boundary.

Flow-based generative models have achieved remarkable progress, with classifier-free guidance (CFG) becoming the standard for high-fidelity generation. However, the conventional practice of applying a strong, fixed guidance scale throughout inference is poorly suited for the rapid, few-step sampling required by modern applications. In this work, we uncover the root cause of this conflict: a fundamental sampling instability where the earliest steps are acutely sensitive to guidance. We trace this to a significant spike in the ratio of conditional to unconditional predictions--a spike that we prove to be an inherent property of the training data distribution itself, making it a almost inevitable challenge. Applying a high, static guidance value during this volatile initial phase leads to an exponential amplification of error, degrading image quality. To resolve this, we propose a simple, theoretically grounded, adaptive guidance schedule that automatically dampens the guidance scale at early steps based on the evolving ratio. Our method is lightweight, incurs no inference overhead, and is compatible with standard frameworks. Experiments across state-of-the-art image (SD3.5, Qwen-Image) and video (WAN2.1) models show our approach enables up to 3x faster sampling while maintaining or improving quality, robustness, and semantic alignment. Our findings highlight that adapting guidance to the sampling process, rather than fixing it, is critical for unlocking the full potential of fast, flow-based models.

Machine unlearning methods seek to revise pretrained models such that effects of certain training samples can be removed. In addition to effective erasure, low computational cost and general utility retention are also highly desirable. Existing unlearning methods usually involve iterative updates over the model parameters, which incurs a high computational cost. In this work, we propose an efficient method that only requires a one-time gradient computation, with which we modify only a single layer of model parameters. Specifically, we first identify a small number of model layers that lie on the Pareto front of high forget importance and low retain influence as critical layers. Then we search for a suitable step size and take a step along the gradient direction of a single critical layer while keeping other layers frozen. This method is highly modular and can be used to unlearn multiple concepts simultaneously in a controllable manner. We demonstrate the effectiveness and efficiency of this method on various models including CLIP, stable diffusion, and VLMs, surpassing other state-of-the-art methods.

Reinforcement Learning from Human Feedback (RLHF) has become the predominant approach for language model (LM) alignment. At its core, RLHF uses a margin-based loss for preference optimization, specifying ideal LM behavior only by the difference between preferred and dispreferred responses. In this paper, we identify a common pitfall of margin-based methods -- the under-specification of ideal LM behavior on preferred and dispreferred responses individually, which leads to two unintended consequences as the margin increases: (1) The probability of dispreferred (e.g., unsafe) responses may increase, resulting in potential safety alignment failures. (2) The probability of preferred responses may decrease, even when those responses are ideal. We demystify the reasons behind these problematic behaviors: margin-based losses couple the change in the preferred probability to the gradient of the dispreferred one, and vice versa, often preventing the preferred probability from increasing while the dispreferred one decreases, and thus causing a synchronized increase or decrease in both probabilities. We term this effect, inherent in margin-based objectives, gradient entanglement. Formally, we derive conditions for general margin-based alignment objectives under which gradient entanglement becomes concerning: the inner product of the gradients of preferred and dispreferred log-probabilities is large relative to the individual gradient norms. We theoretically investigate why such inner products can be large when aligning language models and empirically validate our findings. Empirical implications of our framework extend to explaining important differences in the training dynamics of various preference optimization algorithms, and suggesting potential algorithm designs to mitigate the under-specification issue of margin-based methods and thereby improving language model alignment.

Imposing known physical constraints, such as conservation laws, during neural network training introduces an inductive bias that can improve accuracy, reliability, convergence, and data efficiency for modeling physical dynamics. While such constraints can be softly imposed via loss function penalties, recent advancements in differentiable physics and optimization improve performance by incorporating PDE-constrained optimization as individual layers in neural networks. This enables a stricter adherence to physical constraints. However, imposing hard constraints significantly increases computational and memory costs, especially for complex dynamical systems. This is because it requires solving an optimization problem over a large number of points in a mesh, representing spatial and temporal discretizations, which greatly increases the complexity of the constraint. To address this challenge, we develop a scalable approach to enforce hard physical constraints using Mixture-of-Experts (MoE), which can be used with any neural network architecture. Our approach imposes the constraint over smaller decomposed domains, each of which is solved by an "expert" through differentiable optimization. During training, each expert independently performs a localized backpropagation step by leveraging the implicit function theorem; the independence of each expert allows for parallelization across multiple GPUs. Compared to standard differentiable optimization, our scalable approach achieves greater accuracy in the neural PDE solver setting for predicting the dynamics of challenging non-linear systems. We also improve training stability and require significantly less computation time during both training and inference stages.

Recently, GRPO-based reinforcement learning has shown remarkable progress in optimizing flow-matching models, effectively improving their alignment with task-specific rewards. Within these frameworks, the policy update relies on importance-ratio clipping to constrain overconfident positive and negative gradients. However, in practice, we observe a systematic shift in the importance-ratio distribution-its mean falls below 1 and its variance differs substantially across timesteps. This left-shifted and inconsistent distribution prevents positive-advantage samples from entering the clipped region, causing the mechanism to fail in constraining overconfident positive updates. As a result, the policy model inevitably enters an implicit over-optimization stage-while the proxy reward continues to increase, essential metrics such as image quality and text-prompt alignment deteriorate sharply, ultimately making the learned policy impractical for real-world use. To address this issue, we introduce GRPO-Guard, a simple yet effective enhancement to existing GRPO frameworks. Our method incorporates ratio normalization, which restores a balanced and step-consistent importance ratio, ensuring that PPO clipping properly constrains harmful updates across denoising timesteps. In addition, a gradient reweighting strategy equalizes policy gradients over noise conditions, preventing excessive updates from particular timestep regions. Together, these designs act as a regulated clipping mechanism, stabilizing optimization and substantially mitigating implicit over-optimization without relying on heavy KL regularization. Extensive experiments on multiple diffusion backbones (e.g., SD3.5M, Flux.1-dev) and diverse proxy tasks demonstrate that GRPO-Guard significantly reduces over-optimization while maintaining or even improving generation quality.

Forward Gradients - the idea of using directional derivatives in forward differentiation mode - have recently been shown to be utilizable for neural network training while avoiding problems generally associated with backpropagation gradient computation, such as locking and memorization requirements. The cost is the requirement to guess the step direction, which is hard in high dimensions. While current solutions rely on weighted averages over isotropic guess vector distributions, we propose to strongly bias our gradient guesses in directions that are much more promising, such as feedback obtained from small, local auxiliary networks. For a standard computer vision neural network, we conduct a rigorous study systematically covering a variety of combinations of gradient targets and gradient guesses, including those previously presented in the literature. We find that using gradients obtained from a local loss as a candidate direction drastically improves on random noise in Forward Gradient methods.

Minimizing inconsistencies across successive versions of an AI system is as crucial as reducing the overall error. In image classification, such inconsistencies manifest as negative flips, where an updated model misclassifies test samples that were previously classified correctly. This issue becomes increasingly pronounced as the number of training classes grows over time, since adding new categories reduces the margin of each class and may introduce conflicting patterns that undermine their learning process, thereby degrading performance on the original subset. To mitigate negative flips, we propose a novel approach that preserves the margins of the original model while learning an improved one. Our method encourages a larger relative margin between the previously learned and newly introduced classes by introducing an explicit margin-calibration term on the logits. However, overly constraining the logit margin for the new classes can significantly degrade their accuracy compared to a new independently trained model. To address this, we integrate a double-source focal distillation loss with the previous model and a new independently trained model, learning an appropriate decision margin from both old and new data, even under a logit margin calibration. Extensive experiments on image classification benchmarks demonstrate that our approach consistently reduces the negative flip rate with high overall accuracy.

This paper investigates the distinctions between gradient methods applied to non-differentiable functions (NGDMs) and classical gradient descents (GDs) designed for differentiable functions. First, we demonstrate significant differences in the convergence properties of NGDMs compared to GDs, challenging the applicability of the extensive neural network convergence literature based on L-smoothness to non-smooth neural networks. Next, we demonstrate the paradoxical nature of NGDM solutions for L_{1}-regularized problems, showing that increasing the regularization penalty leads to an increase in the L_{1} norm of optimal solutions in NGDMs. Consequently, we show that widely adopted L_{1} penalization-based techniques for network pruning do not yield expected results. Finally, we explore the Edge of Stability phenomenon, indicating its inapplicability even to Lipschitz continuous convex differentiable functions, leaving its relevance to non-convex non-differentiable neural networks inconclusive. Our analysis exposes misguided interpretations of NGDMs in widely referenced papers and texts due to an overreliance on strong smoothness assumptions, emphasizing the necessity for a nuanced understanding of foundational assumptions in the analysis of these systems.

We consider the training of the first layer of vision models and notice the clear relationship between pixel values and gradient update magnitudes: the gradients arriving at the weights of a first layer are by definition directly proportional to (normalized) input pixel values. Thus, an image with low contrast has a smaller impact on learning than an image with higher contrast, and a very bright or very dark image has a stronger impact on the weights than an image with moderate brightness. In this work, we propose performing gradient descent on the embeddings produced by the first layer of the model. However, switching to discrete inputs with an embedding layer is not a reasonable option for vision models. Thus, we propose the conceptual procedure of (i) a gradient descent step on first layer activations to construct an activation proposal, and (ii) finding the optimal weights of the first layer, i.e., those weights which minimize the squared distance to the activation proposal. We provide a closed form solution of the procedure and adjust it for robust stochastic training while computing everything efficiently. Empirically, we find that TrAct (Training Activations) speeds up training by factors between 1.25x and 4x while requiring only a small computational overhead. We demonstrate the utility of TrAct with different optimizers for a range of different vision models including convolutional and transformer architectures.

Based on the weight-sharing mechanism, one-shot NAS methods train a supernet and then inherit the pre-trained weights to evaluate sub-models, largely reducing the search cost. However, several works have pointed out that the shared weights suffer from different gradient descent directions during training. And we further find that large gradient variance occurs during supernet training, which degrades the supernet ranking consistency. To mitigate this issue, we propose to explicitly minimize the gradient variance of the supernet training by jointly optimizing the sampling distributions of PAth and DAta (PA&DA). We theoretically derive the relationship between the gradient variance and the sampling distributions, and reveal that the optimal sampling probability is proportional to the normalized gradient norm of path and training data. Hence, we use the normalized gradient norm as the importance indicator for path and training data, and adopt an importance sampling strategy for the supernet training. Our method only requires negligible computation cost for optimizing the sampling distributions of path and data, but achieves lower gradient variance during supernet training and better generalization performance for the supernet, resulting in a more consistent NAS. We conduct comprehensive comparisons with other improved approaches in various search spaces. Results show that our method surpasses others with more reliable ranking performance and higher accuracy of searched architectures, showing the effectiveness of our method. Code is available at https://github.com/ShunLu91/PA-DA.

Recent studies of gradient descent with large step sizes have shown that there is often a regime with an initial increase in the largest eigenvalue of the loss Hessian (progressive sharpening), followed by a stabilization of the eigenvalue near the maximum value which allows convergence (edge of stability). These phenomena are intrinsically non-linear and do not happen for models in the constant Neural Tangent Kernel (NTK) regime, for which the predictive function is approximately linear in the parameters. As such, we consider the next simplest class of predictive models, namely those that are quadratic in the parameters, which we call second-order regression models. For quadratic objectives in two dimensions, we prove that this second-order regression model exhibits progressive sharpening of the NTK eigenvalue towards a value that differs slightly from the edge of stability, which we explicitly compute. In higher dimensions, the model generically shows similar behavior, even without the specific structure of a neural network, suggesting that progressive sharpening and edge-of-stability behavior aren't unique features of neural networks, and could be a more general property of discrete learning algorithms in high-dimensional non-linear models.

Training deep neural networks for classification often includes minimizing the training loss beyond the zero training error point. In this phase of training, a "neural collapse" behavior has been observed: the variability of features (outputs of the penultimate layer) of within-class samples decreases and the mean features of different classes approach a certain tight frame structure. Recent works analyze this behavior via idealized unconstrained features models where all the minimizers exhibit exact collapse. However, with practical networks and datasets, the features typically do not reach exact collapse, e.g., because deep layers cannot arbitrarily modify intermediate features that are far from being collapsed. In this paper, we propose a richer model that can capture this phenomenon by forcing the features to stay in the vicinity of a predefined features matrix (e.g., intermediate features). We explore the model in the small vicinity case via perturbation analysis and establish results that cannot be obtained by the previously studied models. For example, we prove reduction in the within-class variability of the optimized features compared to the predefined input features (via analyzing gradient flow on the "central-path" with minimal assumptions), analyze the minimizers in the near-collapse regime, and provide insights on the effect of regularization hyperparameters on the closeness to collapse. We support our theory with experiments in practical deep learning settings.

We introduce Reverse Derivative Ascent: a categorical analogue of gradient based methods for machine learning. Our algorithm is defined at the level of so-called reverse differential categories. It can be used to learn the parameters of models which are expressed as morphisms of such categories. Our motivating example is boolean circuits: we show how our algorithm can be applied to such circuits by using the theory of reverse differential categories. Note our methodology allows us to learn the parameters of boolean circuits directly, in contrast to existing binarised neural network approaches. Moreover, we demonstrate its empirical value by giving experimental results on benchmark machine learning datasets.

Real-world datasets exhibit imbalances of varying types and degrees. Several techniques based on re-weighting and margin adjustment of loss are often used to enhance the performance of neural networks, particularly on minority classes. In this work, we analyze the class-imbalanced learning problem by examining the loss landscape of neural networks trained with re-weighting and margin-based techniques. Specifically, we examine the spectral density of Hessian of class-wise loss, through which we observe that the network weights converge to a saddle point in the loss landscapes of minority classes. Following this observation, we also find that optimization methods designed to escape from saddle points can be effectively used to improve generalization on minority classes. We further theoretically and empirically demonstrate that Sharpness-Aware Minimization (SAM), a recent technique that encourages convergence to a flat minima, can be effectively used to escape saddle points for minority classes. Using SAM results in a 6.2\% increase in accuracy on the minority classes over the state-of-the-art Vector Scaling Loss, leading to an overall average increase of 4\% across imbalanced datasets. The code is available at: https://github.com/val-iisc/Saddle-LongTail.

Training large language models (LLMs) is often bottlenecked by extreme memory demands, with optimizer states dominating the footprint. Recent works mitigates this cost by projecting gradients into low-dimensional subspaces using sophisticated update strategies. In this paper, we analyze the dynamics of gradient space and its underlying subspaces. We find that while a small subspace captures most gradient energy, a significant portion still resides in the residual bulk; moreover, the influence of the core subspace diminishes over time and in deeper layers. We also observe that the gradient space exhibits near-flat curvature, calling for algorithms that explicitly account for this geometry. Motivated by these insights, we introduce a suite of randomized algorithms, GrassWalk and GrassJump, which exploit subspace and achieve state-of-the-art memory savings while improving performance on LLaMA-1B and LLaMA-7B pretraining.

Krylov subspace, which is generated by multiplying a given vector by the matrix of a linear transformation and its successive powers, has been extensively studied in classical optimization literature to design algorithms that converge quickly for large linear inverse problems. For example, the conjugate gradient method (CG), one of the most popular Krylov subspace methods, is based on the idea of minimizing the residual error in the Krylov subspace. However, with the recent advancement of high-performance diffusion solvers for inverse problems, it is not clear how classical wisdom can be synergistically combined with modern diffusion models. In this study, we propose a novel and efficient diffusion sampling strategy that synergistically combines the diffusion sampling and Krylov subspace methods. Specifically, we prove that if the tangent space at a denoised sample by Tweedie's formula forms a Krylov subspace, then the CG initialized with the denoised data ensures the data consistency update to remain in the tangent space. This negates the need to compute the manifold-constrained gradient (MCG), leading to a more efficient diffusion sampling method. Our method is applicable regardless of the parametrization and setting (i.e., VE, VP). Notably, we achieve state-of-the-art reconstruction quality on challenging real-world medical inverse imaging problems, including multi-coil MRI reconstruction and 3D CT reconstruction. Moreover, our proposed method achieves more than 80 times faster inference time than the previous state-of-the-art method. Code is available at https://github.com/HJ-harry/DDS

We introduce ORIGEN, the first zero-shot method for 3D orientation grounding in text-to-image generation across multiple objects and diverse categories. While previous work on spatial grounding in image generation has mainly focused on 2D positioning, it lacks control over 3D orientation. To address this, we propose a reward-guided sampling approach using a pretrained discriminative model for 3D orientation estimation and a one-step text-to-image generative flow model. While gradient-ascent-based optimization is a natural choice for reward-based guidance, it struggles to maintain image realism. Instead, we adopt a sampling-based approach using Langevin dynamics, which extends gradient ascent by simply injecting random noise--requiring just a single additional line of code. Additionally, we introduce adaptive time rescaling based on the reward function to accelerate convergence. Our experiments show that ORIGEN outperforms both training-based and test-time guidance methods across quantitative metrics and user studies.

Denoising diffusion models excel at generating high-quality images conditioned on text prompts, yet their effectiveness heavily relies on careful guidance during the sampling process. Classifier-Free Guidance (CFG) provides a widely used mechanism for steering generation by setting the guidance scale, which balances image quality and prompt alignment. However, the choice of the guidance scale has a critical impact on the convergence toward a visually appealing and prompt-adherent image. In this work, we propose an annealing guidance scheduler which dynamically adjusts the guidance scale over time based on the conditional noisy signal. By learning a scheduling policy, our method addresses the temperamental behavior of CFG. Empirical results demonstrate that our guidance scheduler significantly enhances image quality and alignment with the text prompt, advancing the performance of text-to-image generation. Notably, our novel scheduler requires no additional activations or memory consumption, and can seamlessly replace the common classifier-free guidance, offering an improved trade-off between prompt alignment and quality.

Fully quantized training (FQT) accelerates the training of deep neural networks by quantizing the activations, weights, and gradients into lower precision. To explore the ultimate limit of FQT (the lowest achievable precision), we make a first attempt to 1-bit FQT. We provide a theoretical analysis of FQT based on Adam and SGD, revealing that the gradient variance influences the convergence of FQT. Building on these theoretical results, we introduce an Activation Gradient Pruning (AGP) strategy. The strategy leverages the heterogeneity of gradients by pruning less informative gradients and enhancing the numerical precision of remaining gradients to mitigate gradient variance. Additionally, we propose Sample Channel joint Quantization (SCQ), which utilizes different quantization strategies in the computation of weight gradients and activation gradients to ensure that the method is friendly to low-bitwidth hardware. Finally, we present a framework to deploy our algorithm. For fine-tuning VGGNet-16 and ResNet-18 on multiple datasets, our algorithm achieves an average accuracy improvement of approximately 6%, compared to per-sample quantization. Moreover, our training speedup can reach a maximum of 5.13x compared to full precision training.

Deep neural networks have a good success record and are thus viewed as the best architecture choice for complex applications. Their main shortcoming has been, for a long time, the vanishing gradient which prevented the numerical optimization algorithms from acceptable convergence. A breakthrough has been achieved by the concept of residual connections -- an identity mapping parallel to a conventional layer. This concept is applicable to stacks of layers of the same dimension and substantially alleviates the vanishing gradient problem. A stack of residual connection layers can be expressed as an expansion of terms similar to the Taylor expansion. This expansion suggests the possibility of truncating the higher-order terms and receiving an architecture consisting of a single broad layer composed of all initially stacked layers in parallel. In other words, a sequential deep architecture is substituted by a parallel shallow one. Prompted by this theory, we investigated the performance capabilities of the parallel architecture in comparison to the sequential one. The computer vision datasets MNIST and CIFAR10 were used to train both architectures for a total of 6912 combinations of varying numbers of convolutional layers, numbers of filters, kernel sizes, and other meta parameters. Our findings demonstrate a surprising equivalence between the deep (sequential) and shallow (parallel) architectures. Both layouts produced similar results in terms of training and validation set loss. This discovery implies that a wide, shallow architecture can potentially replace a deep network without sacrificing performance. Such substitution has the potential to simplify network architectures, improve optimization efficiency, and accelerate the training process.

This paper introduces a new biologically-inspired training method named Continual Learning through Adjustment Suppression and Sparsity Promotion (CLASSP). CLASSP is based on two main principles observed in neuroscience, particularly in the context of synaptic transmission and Long-Term Potentiation (LTP). The first principle is a decay rate over the weight adjustment, which is implemented as a generalization of the AdaGrad optimization algorithm. This means that weights that have received many updates should have lower learning rates as they likely encode important information about previously seen data. However, this principle results in a diffuse distribution of updates throughout the model, as it promotes updates for weights that haven't been previously updated, while a sparse update distribution is preferred to leave weights unassigned for future tasks. Therefore, the second principle introduces a threshold on the loss gradient. This promotes sparse learning by updating a weight only if the loss gradient with respect to that weight is above a certain threshold, i.e. only updating weights with a significant impact on the current loss. Both principles reflect phenomena observed in LTP, where a threshold effect and a gradual saturation of potentiation have been observed. CLASSP is implemented in a Python/PyTorch class, making it applicable to any model. When compared with Elastic Weight Consolidation (EWC) using Computer Vision and sentiment analysis datasets, CLASSP demonstrates superior performance in terms of accuracy and memory footprint.

In this paper we argue that ReLU networks learn an implicit linear model we can actually tap into. We describe that alleged model formally and show that we can approximately pull its decision boundary back to the input space with certain simple modification to the backward pass. The resulting gradients (called excitation pullbacks) reveal high-resolution input- and target-specific features of remarkable perceptual alignment on a number of popular ImageNet-pretrained deep architectures. This strongly suggests that neural networks do, in fact, rely on learned interpretable patterns that can be recovered after training. Thus, our findings may have profound implications for knowledge discovery and the development of dependable artificial systems.

Negative guidance -- explicitly suppressing unwanted attributes -- remains a fundamental challenge in diffusion models, particularly in few-step sampling regimes. While Classifier-Free Guidance (CFG) works well in standard settings, it fails under aggressive sampling step compression due to divergent predictions between positive and negative branches. We present Normalized Attention Guidance (NAG), an efficient, training-free mechanism that applies extrapolation in attention space with L1-based normalization and refinement. NAG restores effective negative guidance where CFG collapses while maintaining fidelity. Unlike existing approaches, NAG generalizes across architectures (UNet, DiT), sampling regimes (few-step, multi-step), and modalities (image, video), functioning as a universal plug-in with minimal computational overhead. Through extensive experimentation, we demonstrate consistent improvements in text alignment (CLIP Score), fidelity (FID, PFID), and human-perceived quality (ImageReward). Our ablation studies validate each design component, while user studies confirm significant preference for NAG-guided outputs. As a model-agnostic inference-time approach requiring no retraining, NAG provides effortless negative guidance for all modern diffusion frameworks -- pseudocode in the Appendix!

Machine learning models fail to perform when facing out-of-distribution (OOD) domains, a challenging task known as domain generalization (DG). In this work, we develop a novel DG training strategy, we call PGrad, to learn a robust gradient direction, improving models' generalization ability on unseen domains. The proposed gradient aggregates the principal directions of a sampled roll-out optimization trajectory that measures the training dynamics across all training domains. PGrad's gradient design forces the DG training to ignore domain-dependent noise signals and updates all training domains with a robust direction covering main components of parameter dynamics. We further improve PGrad via bijection-based computational refinement and directional plus length-based calibrations. Our theoretical proof connects PGrad to the spectral analysis of Hessian in training neural networks. Experiments on DomainBed and WILDS benchmarks demonstrate that our approach effectively enables robust DG optimization and leads to smoothly decreased loss curves. Empirically, PGrad achieves competitive results across seven datasets, demonstrating its efficacy across both synthetic and real-world distributional shifts. Code is available at https://github.com/QData/PGrad.

The remarkable capabilities of pretrained image diffusion models have been utilized not only for generating fixed-size images but also for creating panoramas. However, naive stitching of multiple images often results in visible seams. Recent techniques have attempted to address this issue by performing joint diffusions in multiple windows and averaging latent features in overlapping regions. However, these approaches, which focus on seamless montage generation, often yield incoherent outputs by blending different scenes within a single image. To overcome this limitation, we propose SyncDiffusion, a plug-and-play module that synchronizes multiple diffusions through gradient descent from a perceptual similarity loss. Specifically, we compute the gradient of the perceptual loss using the predicted denoised images at each denoising step, providing meaningful guidance for achieving coherent montages. Our experimental results demonstrate that our method produces significantly more coherent outputs compared to previous methods (66.35% vs. 33.65% in our user study) while still maintaining fidelity (as assessed by GIQA) and compatibility with the input prompt (as measured by CLIP score).

Pretrained language models are commonly aligned with human preferences and downstream tasks via reinforcement finetuning (RFT), which entails maximizing a (possibly learned) reward function using policy gradient algorithms. This work highlights a fundamental optimization obstacle in RFT: we prove that the expected gradient for an input vanishes when its reward standard deviation under the model is small, even if the expected reward is far from optimal. Through experiments on an RFT benchmark and controlled environments, as well as a theoretical analysis, we then demonstrate that vanishing gradients due to small reward standard deviation are prevalent and detrimental, leading to extremely slow reward maximization. Lastly, we explore ways to overcome vanishing gradients in RFT. We find the common practice of an initial supervised finetuning (SFT) phase to be the most promising candidate, which sheds light on its importance in an RFT pipeline. Moreover, we show that a relatively small number of SFT optimization steps on as few as 1% of the input samples can suffice, indicating that the initial SFT phase need not be expensive in terms of compute and data labeling efforts. Overall, our results emphasize that being mindful for inputs whose expected gradient vanishes, as measured by the reward standard deviation, is crucial for successful execution of RFT.

While backpropagation (BP) is the mainstream approach for gradient computation in neural network training, its heavy reliance on the chain rule of differentiation constrains the designing flexibility of network architecture and training pipelines. We avoid the recursive computation in BP and develop a unified likelihood ratio (ULR) method for gradient estimation with just one forward propagation. Not only can ULR be extended to train a wide variety of neural network architectures, but the computation flow in BP can also be rearranged by ULR for better device adaptation. Moreover, we propose several variance reduction techniques to further accelerate the training process. Our experiments offer numerical results across diverse aspects, including various neural network training scenarios, computation flow rearrangement, and fine-tuning of pre-trained models. All findings demonstrate that ULR effectively enhances the flexibility of neural network training by permitting localized module training without compromising the global objective and significantly boosts the network robustness.

It is believed that Gradient Descent (GD) induces an implicit bias towards good generalization in training machine learning models. This paper provides a fine-grained analysis of the dynamics of GD for the matrix sensing problem, whose goal is to recover a low-rank ground-truth matrix from near-isotropic linear measurements. It is shown that GD with small initialization behaves similarly to the greedy low-rank learning heuristics (Li et al., 2020) and follows an incremental learning procedure (Gissin et al., 2019): GD sequentially learns solutions with increasing ranks until it recovers the ground truth matrix. Compared to existing works which only analyze the first learning phase for rank-1 solutions, our result provides characterizations for the whole learning process. Moreover, besides the over-parameterized regime that many prior works focused on, our analysis of the incremental learning procedure also applies to the under-parameterized regime. Finally, we conduct numerical experiments to confirm our theoretical findings.

Research on loss surface geometry, such as Sharpness-Aware Minimization (SAM), shows that flatter minima improve generalization. Recent studies further reveal that flatter minima can also reduce the domain generalization (DG) gap. However, existing flatness-based DG techniques predominantly operate within a full-precision training process, which is impractical for deployment on resource-constrained edge devices that typically rely on lower bit-width representations (e.g., 4 bits, 3 bits). Consequently, low-precision quantization-aware training is critical for optimizing these techniques in real-world applications. In this paper, we observe a significant degradation in performance when applying state-of-the-art DG-SAM methods to quantized models, suggesting that current approaches fail to preserve generalizability during the low-precision training process. To address this limitation, we propose a novel Gradient-Adaptive Quantization-Aware Training (GAQAT) framework for DG. Our approach begins by identifying the scale-gradient conflict problem in low-precision quantization, where the task loss and smoothness loss induce conflicting gradients for the scaling factors of quantizers, with certain layers exhibiting opposing gradient directions. This conflict renders the optimization of quantized weights highly unstable. To mitigate this, we further introduce a mechanism to quantify gradient inconsistencies and selectively freeze the gradients of scaling factors, thereby stabilizing the training process and enhancing out-of-domain generalization. Extensive experiments validate the effectiveness of the proposed GAQAT framework. On PACS, our 3-bit and 4-bit models outperform direct DG-QAT integration by up to 4.5%. On DomainNet, the 4-bit model achieves near-lossless performance compared to full precision, with improvements of 1.39% (4-bit) and 1.06% (3-bit) over the SOTA QAT baseline.

Recent studies empirically demonstrate the positive relationship between the transferability of neural networks and the within-class variation of the last layer features. The recently discovered Neural Collapse (NC) phenomenon provides a new perspective of understanding such last layer geometry of neural networks. In this paper, we propose a novel metric, named Variability Collapse Index (VCI), to quantify the variability collapse phenomenon in the NC paradigm. The VCI metric is well-motivated and intrinsically related to the linear probing loss on the last layer features. Moreover, it enjoys desired theoretical and empirical properties, including invariance under invertible linear transformations and numerical stability, that distinguishes it from previous metrics. Our experiments verify that VCI is indicative of the variability collapse and the transferability of pretrained neural networks.

MeanFlow has recently emerged as a powerful framework for few-step generative modeling trained from scratch, but its success is not yet fully understood. In this work, we show that the MeanFlow objective naturally decomposes into two parts: trajectory flow matching and trajectory consistency. Through gradient analysis, we find that these terms are strongly negatively correlated, causing optimization conflict and slow convergence. Motivated by these insights, we introduce alpha-Flow, a broad family of objectives that unifies trajectory flow matching, Shortcut Model, and MeanFlow under one formulation. By adopting a curriculum strategy that smoothly anneals from trajectory flow matching to MeanFlow, alpha-Flow disentangles the conflicting objectives, and achieves better convergence. When trained from scratch on class-conditional ImageNet-1K 256x256 with vanilla DiT backbones, alpha-Flow consistently outperforms MeanFlow across scales and settings. Our largest alpha-Flow-XL/2+ model achieves new state-of-the-art results using vanilla DiT backbones, with FID scores of 2.58 (1-NFE) and 2.15 (2-NFE).

Real-world analog systems intrinsically suffer from noise that can impede model convergence and accuracy on a variety of deep learning models. We demonstrate that differentiable activations like GELU and SiLU enable robust propagation of gradients which help to mitigate analog quantization error that is ubiquitous to all analog systems. We perform analysis and training of convolutional, linear, and transformer networks in the presence of quantized noise. Here, we are able to demonstrate that continuously differentiable activation functions are significantly more noise resilient over conventional rectified activations. As in the case of ReLU, the error in gradients are 100x higher than those in GELU near zero. Our findings provide guidance for selecting appropriate activations to realize performant and reliable hardware implementations across several machine learning domains such as computer vision, signal processing, and beyond.

This paper considers the Pointer Value Retrieval (PVR) benchmark introduced in [ZRKB21], where a 'reasoning' function acts on a string of digits to produce the label. More generally, the paper considers the learning of logical functions with gradient descent (GD) on neural networks. It is first shown that in order to learn logical functions with gradient descent on symmetric neural networks, the generalization error can be lower-bounded in terms of the noise-stability of the target function, supporting a conjecture made in [ZRKB21]. It is then shown that in the distribution shift setting, when the data withholding corresponds to freezing a single feature (referred to as canonical holdout), the generalization error of gradient descent admits a tight characterization in terms of the Boolean influence for several relevant architectures. This is shown on linear models and supported experimentally on other models such as MLPs and Transformers. In particular, this puts forward the hypothesis that for such architectures and for learning logical functions such as PVR functions, GD tends to have an implicit bias towards low-degree representations, which in turn gives the Boolean influence for the generalization error under quadratic loss.

Task arithmetic has emerged as a simple yet powerful technique for model merging, enabling the combination of multiple finetuned models into one. Despite its empirical success, a clear theoretical explanation of why and when it works is lacking. This paper provides a rigorous theoretical foundation for task arithmetic by establishing a connection between task vectors and gradients of the task losses. We show that under standard gradient descent, a task vector generated from one epoch of finetuning is exactly equivalent to the negative gradient of the loss, scaled by the learning rate. For the practical multi-epoch setting, we prove that this equivalence holds approximately, with a second-order error term that we explicitly bound for feed-forward networks. Our empirical analysis across seven vision benchmarks corroborates our theory, demonstrating that the first-epoch gradient dominates the finetuning trajectory in both norm and direction. A key implication is that merging models finetuned for only a single epoch often yields performance comparable to merging fully converged models. These findings reframe task arithmetic as a form of approximate multitask learning, providing a clear rationale for its effectiveness and highlighting the critical role of early training dynamics in model merging.

Federated Learning (FL) has recently emerged as a promising distributed machine learning framework to preserve clients' privacy, by allowing multiple clients to upload the gradients calculated from their local data to a central server. Recent studies find that the exchanged gradients also take the risk of privacy leakage, e.g., an attacker can invert the shared gradients and recover sensitive data against an FL system by leveraging pre-trained generative adversarial networks (GAN) as prior knowledge. However, performing gradient inversion attacks in the latent space of the GAN model limits their expression ability and generalizability. To tackle these challenges, we propose Gradient Inversion over Feature Domains (GIFD), which disassembles the GAN model and searches the feature domains of the intermediate layers. Instead of optimizing only over the initial latent code, we progressively change the optimized layer, from the initial latent space to intermediate layers closer to the output images. In addition, we design a regularizer to avoid unreal image generation by adding a small {l_1} ball constraint to the searching range. We also extend GIFD to the out-of-distribution (OOD) setting, which weakens the assumption that the training sets of GANs and FL tasks obey the same data distribution. Extensive experiments demonstrate that our method can achieve pixel-level reconstruction and is superior to the existing methods. Notably, GIFD also shows great generalizability under different defense strategy settings and batch sizes.

How momentum, energy, and magnetic fields are transported in the presence of macroscopic gradients is a fundamental question in plasma physics. Answering this question is especially challenging for weakly collisional, magnetized plasmas, where macroscopic gradients influence the plasma's microphysical structure. In this paper, we introduce thermodynamic forcing, a new method for systematically modeling how macroscopic gradients in magnetized or unmagnetized plasmas shape the distribution functions of constituent particles. In this method, we propose to apply an anomalous force to those particles inducing the anisotropy that would naturally emerge due to macroscopic gradients in weakly collisional plasmas. We implement thermodynamic forcing in particle-in-cell (TF-PIC) simulations using a modified Vay particle pusher and validate it against analytic solutions of the equations of motion. We then carry out a series of simulations of electron-proton plasmas with periodic boundary conditions using TF-PIC. First, we confirm that the properties of two electron-scale kinetic instabilities -- one driven by a temperature gradient and the other by pressure anisotropy -- are consistent with previous results. Then, we demonstrate that in the presence of multiple macroscopic gradients, the saturated state can differ significantly from current expectations. This work enables, for the first time, systematic and self-consistent transport modeling in weakly collisional plasmas, with broad applications in astrophysics, laser-plasma physics, and inertial confinement fusion.

The recently proposed term "heterostructured (HS) materials" serves as an umbrella classification encompassing a wide range of materials that hold great promise for enhanced mechanical properties. Most HS materials exhibit back-stress strengthening, as is typical for all plastically non-homogeneous materials. To better embody the distinctiveness of materials crafted via innovative heterostructuring, here we introduce the concept of "structural gradient hardening" (SGH), which captures an essential feature of HS materials and complements traditional strengthening mechanisms. SGH refers to the extra strengthening that arises from a characteristic gradient structure introduced by heterostructuring, beyond what is predicted by the rule of mixtures. This distinction is useful, as the overall back stress can in fact be partitioned into Type I and Type II components, with the latter specifically quantifying the extra hardening originating from the structural and strain gradients established by heterostructuring, as articulated in this Viewpoint article.