Daily Papers

This work proposes a Momentum-Enabled Kronecker-Factor-Based Optimizer Using Rank-1 updates, called MKOR, that improves the training time and convergence properties of deep neural networks (DNNs). Second-order techniques, while enjoying higher convergence rates vs first-order counterparts, have cubic complexity with respect to either the model size and/or the training batch size. Hence they exhibit poor scalability and performance in transformer models, e.g. large language models (LLMs), because the batch sizes in these models scale by the attention mechanism sequence length, leading to large model size and batch sizes. MKOR's complexity is quadratic with respect to the model size, alleviating the computation bottlenecks in second-order methods. Because of their high computation complexity, state-of-the-art implementations of second-order methods can only afford to update the second order information infrequently, and thus do not fully exploit the promise of better convergence from these updates. By reducing the communication complexity of the second-order updates as well as achieving a linear communication complexity, MKOR increases the frequency of second order updates. We also propose a hybrid version of MKOR (called MKOR-H) that mid-training falls backs to a first order optimizer if the second order updates no longer accelerate convergence. Our experiments show that MKOR outperforms state -of-the-art first order methods, e.g. the LAMB optimizer, and best implementations of second-order methods, i.e. KAISA/KFAC, up to 2.57x and 1.85x respectively on BERT-Large-Uncased on 64 GPUs.

In the deep learning (DL) era, parsing models are extremely simplified with little hurt on performance, thanks to the remarkable capability of multi-layer BiLSTMs in context representation. As the most popular graph-based dependency parser due to its high efficiency and performance, the biaffine parser directly scores single dependencies under the arc-factorization assumption, and adopts a very simple local token-wise cross-entropy training loss. This paper for the first time presents a second-order TreeCRF extension to the biaffine parser. For a long time, the complexity and inefficiency of the inside-outside algorithm hinder the popularity of TreeCRF. To address this issue, we propose an effective way to batchify the inside and Viterbi algorithms for direct large matrix operation on GPUs, and to avoid the complex outside algorithm via efficient back-propagation. Experiments and analysis on 27 datasets from 13 languages clearly show that techniques developed before the DL era, such as structural learning (global TreeCRF loss) and high-order modeling are still useful, and can further boost parsing performance over the state-of-the-art biaffine parser, especially for partially annotated training data. We release our code at https://github.com/yzhangcs/crfpar.

The diffusion models (DMs) have demonstrated the remarkable capability of generating images via learning the noised score function of data distribution. Current DM sampling techniques typically rely on first-order Langevin dynamics at each noise level, with efforts concentrated on refining inter-level denoising strategies. While leveraging additional second-order Hessian geometry to enhance the sampling quality of Langevin is a common practice in Markov chain Monte Carlo (MCMC), the naive attempts to utilize Hessian geometry in high-dimensional DMs lead to quadratic-complexity computational costs, rendering them non-scalable. In this work, we introduce a novel Levenberg-Marquardt-Langevin (LML) method that approximates the diffusion Hessian geometry in a training-free manner, drawing inspiration from the celebrated Levenberg-Marquardt optimization algorithm. Our approach introduces two key innovations: (1) A low-rank approximation of the diffusion Hessian, leveraging the DMs' inherent structure and circumventing explicit quadratic-complexity computations; (2) A damping mechanism to stabilize the approximated Hessian. This LML approximated Hessian geometry enables the diffusion sampling to execute more accurate steps and improve the image generation quality. We further conduct a theoretical analysis to substantiate the approximation error bound of low-rank approximation and the convergence property of the damping mechanism. Extensive experiments across multiple pretrained DMs validate that the LML method significantly improves image generation quality, with negligible computational overhead.

Obtaining large pre-trained models that can be fine-tuned to new tasks with limited annotated samples has remained an open challenge for medical imaging data. While pre-trained deep networks on ImageNet and vision-language foundation models trained on web-scale data are prevailing approaches, their effectiveness on medical tasks is limited due to the significant domain shift between natural and medical images. To bridge this gap, we introduce LVM-Med, the first family of deep networks trained on large-scale medical datasets. We have collected approximately 1.3 million medical images from 55 publicly available datasets, covering a large number of organs and modalities such as CT, MRI, X-ray, and Ultrasound. We benchmark several state-of-the-art self-supervised algorithms on this dataset and propose a novel self-supervised contrastive learning algorithm using a graph-matching formulation. The proposed approach makes three contributions: (i) it integrates prior pair-wise image similarity metrics based on local and global information; (ii) it captures the structural constraints of feature embeddings through a loss function constructed via a combinatorial graph-matching objective; and (iii) it can be trained efficiently end-to-end using modern gradient-estimation techniques for black-box solvers. We thoroughly evaluate the proposed LVM-Med on 15 downstream medical tasks ranging from segmentation and classification to object detection, and both for the in and out-of-distribution settings. LVM-Med empirically outperforms a number of state-of-the-art supervised, self-supervised, and foundation models. For challenging tasks such as Brain Tumor Classification or Diabetic Retinopathy Grading, LVM-Med improves previous vision-language models trained on 1 billion masks by 6-7% while using only a ResNet-50.

We present new algorithms for optimizing non-smooth, non-convex stochastic objectives based on a novel analysis technique. This improves the current best-known complexity for finding a (delta,epsilon)-stationary point from O(epsilon^{-4}delta^{-1}) stochastic gradient queries to O(epsilon^{-3}delta^{-1}), which we also show to be optimal. Our primary technique is a reduction from non-smooth non-convex optimization to online learning, after which our results follow from standard regret bounds in online learning. For deterministic and second-order smooth objectives, applying more advanced optimistic online learning techniques enables a new complexity of O(epsilon^{-1.5}delta^{-0.5}). Our techniques also recover all optimal or best-known results for finding epsilon stationary points of smooth or second-order smooth objectives in both stochastic and deterministic settings.

This paper provides a review and commentary on the past, present, and future of numerical optimization algorithms in the context of machine learning applications. Through case studies on text classification and the training of deep neural networks, we discuss how optimization problems arise in machine learning and what makes them challenging. A major theme of our study is that large-scale machine learning represents a distinctive setting in which the stochastic gradient (SG) method has traditionally played a central role while conventional gradient-based nonlinear optimization techniques typically falter. Based on this viewpoint, we present a comprehensive theory of a straightforward, yet versatile SG algorithm, discuss its practical behavior, and highlight opportunities for designing algorithms with improved performance. This leads to a discussion about the next generation of optimization methods for large-scale machine learning, including an investigation of two main streams of research on techniques that diminish noise in the stochastic directions and methods that make use of second-order derivative approximations.

This work studies the estimation of many statistical quantiles under differential privacy. More precisely, given a distribution and access to i.i.d. samples from it, we study the estimation of the inverse of its cumulative distribution function (the quantile function) at specific points. For instance, this task is of key importance in private data generation. We present two different approaches. The first one consists in privately estimating the empirical quantiles of the samples and using this result as an estimator of the quantiles of the distribution. In particular, we study the statistical properties of the recently published algorithm introduced by Kaplan et al. 2022 that privately estimates the quantiles recursively. The second approach is to use techniques of density estimation in order to uniformly estimate the quantile function on an interval. In particular, we show that there is a tradeoff between the two methods. When we want to estimate many quantiles, it is better to estimate the density rather than estimating the quantile function at specific points.

Subspace representation is a fundamental technique in various fields of machine learning. Analyzing a geometrical relationship among multiple subspaces is essential for understanding subspace series' temporal and/or spatial dynamics. This paper proposes the second-order difference subspace, a higher-order extension of the first-order difference subspace between two subspaces that can analyze the geometrical difference between them. As a preliminary for that, we extend the definition of the first-order difference subspace to the more general setting that two subspaces with different dimensions have an intersection. We then define the second-order difference subspace by combining the concept of first-order difference subspace and principal component subspace (Karcher mean) between two subspaces, motivated by the second-order central difference method. We can understand that the first/second-order difference subspaces correspond to the velocity and acceleration of subspace dynamics from the viewpoint of a geodesic on a Grassmann manifold. We demonstrate the validity and naturalness of our second-order difference subspace by showing numerical results on two applications: temporal shape analysis of a 3D object and time series analysis of a biometric signal.

We analyze Newton's method with lazy Hessian updates for solving general possibly non-convex optimization problems. We propose to reuse a previously seen Hessian for several iterations while computing new gradients at each step of the method. This significantly reduces the overall arithmetical complexity of second-order optimization schemes. By using the cubic regularization technique, we establish fast global convergence of our method to a second-order stationary point, while the Hessian does not need to be updated each iteration. For convex problems, we justify global and local superlinear rates for lazy Newton steps with quadratic regularization, which is easier to compute. The optimal frequency for updating the Hessian is once every d iterations, where d is the dimension of the problem. This provably improves the total arithmetical complexity of second-order algorithms by a factor d.

Anchor-based 3D Gaussian splatting (3D-GS) exploits anchor features in 3D Gaussian prediction, which has achieved impressive 3D rendering quality with reduced Gaussian redundancy. On the other hand, it often encounters the dilemma among anchor features, model size, and rendering quality - large anchor features lead to large 3D models and high-quality rendering whereas reducing anchor features degrades Gaussian attribute prediction which leads to clear artifacts in the rendered textures and geometries. We design SOGS, an anchor-based 3D-GS technique that introduces second-order anchors to achieve superior rendering quality and reduced anchor features and model size simultaneously. Specifically, SOGS incorporates covariance-based second-order statistics and correlation across feature dimensions to augment features within each anchor, compensating for the reduced feature size and improving rendering quality effectively. In addition, it introduces a selective gradient loss to enhance the optimization of scene textures and scene geometries, leading to high-quality rendering with small anchor features. Extensive experiments over multiple widely adopted benchmarks show that SOGS achieves superior rendering quality in novel view synthesis with clearly reduced model size.

In the training of large language models, momentum is widely used and often demonstrated to achieve significant acceleration. However, storing momentum typically presents memory challenges. In this paper, we propose AdaPM, an adaptive training strategy that leverages partial momentum to implement a memory-efficient optimizer. To this end, AdaPM utilizes a non-uniform momentum design: for most blocks, full momentum is not necessary to preserve the performance of the optimization. In the momentum design of AdaPM, to mitigate the bias and performance loss caused by partial momentum, we enhance the partial momentum by a bias correction technique. Empirically, we verify that our approach reduces memory by over 90% in momentum while maintaining both efficiency and performance for pretraining various language models ranging from 60M to 1.5B, as well as for supervised fine-tuning and RLHF. AdaPM can further reduce memory by up to 95% in optimizer states by combining the memory-efficient technique on the second-order statistic, saving over 30% GPU hours for pretraining GPT-2 1.5B.

In this work, we develop new optimization algorithms that use approximate second-order information combined with the gradient regularization technique to achieve fast global convergence rates for both convex and non-convex objectives. The key innovation of our analysis is a novel notion called Gradient-Normalized Smoothness, which characterizes the maximum radius of a ball around the current point that yields a good relative approximation of the gradient field. Our theory establishes a natural intrinsic connection between Hessian approximation and the linearization of the gradient. Importantly, Gradient-Normalized Smoothness does not depend on the specific problem class of the objective functions, while effectively translating local information about the gradient field and Hessian approximation into the global behavior of the method. This new concept equips approximate second-order algorithms with universal global convergence guarantees, recovering state-of-the-art rates for functions with H\"older-continuous Hessians and third derivatives, quasi-self-concordant functions, as well as smooth classes in first-order optimization. These rates are achieved automatically and extend to broader classes, such as generalized self-concordant functions. We demonstrate direct applications of our results for global linear rates in logistic regression and softmax problems with approximate Hessians, as well as in non-convex optimization using Fisher and Gauss-Newton approximations.

The increasing complexity and parameter count of Convolutional Neural Networks (CNNs) and Transformers pose challenges in terms of computational efficiency and resource demands. Pruning has been identified as an effective strategy to address these challenges by removing redundant elements such as neurons, channels, or connections, thereby enhancing computational efficiency without heavily compromising performance. This paper builds on the foundational work of Optimal Brain Damage (OBD) by advancing the methodology of parameter importance estimation using the Hessian matrix. Unlike previous approaches that rely on approximations, we introduce Optimal Brain Apoptosis (OBA), a novel pruning method that calculates the Hessian-vector product value directly for each parameter. By decomposing the Hessian matrix across network layers and identifying conditions under which inter-layer Hessian submatrices are non-zero, we propose a highly efficient technique for computing the second-order Taylor expansion of parameters. This approach allows for a more precise pruning process, particularly in the context of CNNs and Transformers, as validated in our experiments including VGG19, ResNet32, ResNet50, and ViT-B/16 on CIFAR10, CIFAR100 and Imagenet datasets. Our code is available at https://github.com/NEU-REAL/OBA.

Despite the remarkable success of diffusion models in image generation, slow sampling remains a persistent issue. To accelerate the sampling process, prior studies have reformulated diffusion sampling as an ODE/SDE and introduced higher-order numerical methods. However, these methods often produce divergence artifacts, especially with a low number of sampling steps, which limits the achievable acceleration. In this paper, we investigate the potential causes of these artifacts and suggest that the small stability regions of these methods could be the principal cause. To address this issue, we propose two novel techniques. The first technique involves the incorporation of Heavy Ball (HB) momentum, a well-known technique for improving optimization, into existing diffusion numerical methods to expand their stability regions. We also prove that the resulting methods have first-order convergence. The second technique, called Generalized Heavy Ball (GHVB), constructs a new high-order method that offers a variable trade-off between accuracy and artifact suppression. Experimental results show that our techniques are highly effective in reducing artifacts and improving image quality, surpassing state-of-the-art diffusion solvers on both pixel-based and latent-based diffusion models for low-step sampling. Our research provides novel insights into the design of numerical methods for future diffusion work.

Scene flow estimation is the task of describing the 3D motion field between temporally successive point clouds. State-of-the-art methods use strong priors and test-time optimization techniques, but require on the order of tens of seconds to process full-size point clouds, making them unusable as computer vision primitives for real-time applications such as open world object detection. Feedforward methods are considerably faster, running on the order of tens to hundreds of milliseconds for full-size point clouds, but require expensive human supervision. To address both limitations, we propose Scene Flow via Distillation, a simple, scalable distillation framework that uses a label-free optimization method to produce pseudo-labels to supervise a feedforward model. Our instantiation of this framework, ZeroFlow, achieves state-of-the-art performance on the Argoverse 2 Self-Supervised Scene Flow Challenge while using zero human labels by simply training on large-scale, diverse unlabeled data. At test-time, ZeroFlow is over 1000x faster than label-free state-of-the-art optimization-based methods on full-size point clouds (34 FPS vs 0.028 FPS) and over 1000x cheaper to train on unlabeled data compared to the cost of human annotation (\394 vs ~750,000). To facilitate further research, we will release our code, trained model weights, and high quality pseudo-labels for the Argoverse 2 and Waymo Open datasets.

Coordinate networks are widely used in computer vision due to their ability to represent signals as compressed, continuous entities. However, training these networks with first-order optimizers can be slow, hindering their use in real-time applications. Recent works have opted for shallow voxel-based representations to achieve faster training, but this sacrifices memory efficiency. This work proposes a solution that leverages second-order optimization methods to significantly reduce training times for coordinate networks while maintaining their compressibility. Experiments demonstrate the effectiveness of this approach on various signal modalities, such as audio, images, videos, shape reconstruction, and neural radiance fields.

Optimization in machine learning, both theoretical and applied, is presently dominated by first-order gradient methods such as stochastic gradient descent. Second-order optimization methods, that involve second derivatives and/or second order statistics of the data, are far less prevalent despite strong theoretical properties, due to their prohibitive computation, memory and communication costs. In an attempt to bridge this gap between theoretical and practical optimization, we present a scalable implementation of a second-order preconditioned method (concretely, a variant of full-matrix Adagrad), that along with several critical algorithmic and numerical improvements, provides significant convergence and wall-clock time improvements compared to conventional first-order methods on state-of-the-art deep models. Our novel design effectively utilizes the prevalent heterogeneous hardware architecture for training deep models, consisting of a multicore CPU coupled with multiple accelerator units. We demonstrate superior performance compared to state-of-the-art on very large learning tasks such as machine translation with Transformers, language modeling with BERT, click-through rate prediction on Criteo, and image classification on ImageNet with ResNet-50.

Second-order optimization has been developed to accelerate the training of deep neural networks and it is being applied to increasingly larger-scale models. In this study, towards training on further larger scales, we identify a specific parameterization for second-order optimization that promotes feature learning in a stable manner even if the network width increases significantly. Inspired by a maximal update parameterization, we consider a one-step update of the gradient and reveal the appropriate scales of hyperparameters including random initialization, learning rates, and damping terms. Our approach covers two major second-order optimization algorithms, K-FAC and Shampoo, and we demonstrate that our parameterization achieves higher generalization performance in feature learning. In particular, it enables us to transfer the hyperparameters across models with different widths.

Linear transformation of the inputs alters the training performance of feed-forward networks that are otherwise equivalent. However, most linear transforms are viewed as a pre-processing operation separate from the actual training. Starting from equivalent networks, it is shown that pre-processing inputs using linear transformation are equivalent to multiplying the negative gradient matrix with an autocorrelation matrix per training iteration. Second order method is proposed to find the autocorrelation matrix that maximizes learning in a given iteration. When the autocorrelation matrix is diagonal, the method optimizes input gains. This optimal input gain (OIG) approach is used to improve two first-order two-stage training algorithms, namely back-propagation (BP) and hidden weight optimization (HWO), which alternately update the input weights and solve linear equations for output weights. Results show that the proposed OIG approach greatly enhances the performance of the first-order algorithms, often allowing them to rival the popular Levenberg-Marquardt approach with far less computation. It is shown that HWO is equivalent to BP with Whitening transformation applied to the inputs. HWO effectively combines Whitening transformation with learning. Thus, OIG improved HWO could be a significant building block to more complex deep learning architectures.

Two-point zeroth order methods are important in many applications of zeroth-order optimization, such as robotics, wind farms, power systems, online optimization, and adversarial robustness to black-box attacks in deep neural networks, where the problem may be high-dimensional and/or time-varying. Most problems in these applications are nonconvex and contain saddle points. While existing works have shown that zeroth-order methods utilizing Omega(d) function valuations per iteration (with d denoting the problem dimension) can escape saddle points efficiently, it remains an open question if zeroth-order methods based on two-point estimators can escape saddle points. In this paper, we show that by adding an appropriate isotropic perturbation at each iteration, a zeroth-order algorithm based on 2m (for any 1 leq m leq d) function evaluations per iteration can not only find epsilon-second order stationary points polynomially fast, but do so using only Oleft(d{mepsilon^{2}psi}right) function evaluations, where psi geq Omegaleft(epsilonright) is a parameter capturing the extent to which the function of interest exhibits the strict saddle property.

Gradient-based meta-learning has proven to be highly effective at learning model initializations, representations, and update rules that allow fast adaptation from a few samples. The core idea behind these approaches is to use fast adaptation and generalization -- two second-order metrics -- as training signals on a meta-training dataset. However, little attention has been given to other possible second-order metrics. In this paper, we investigate a different training signal -- robustness to catastrophic interference -- and demonstrate that representations learned by directing minimizing interference are more conducive to incremental learning than those learned by just maximizing fast adaptation.

Popular machine learning approaches forgo second-order information due to the difficulty of computing curvature in high dimensions. We present FOSI, a novel meta-algorithm that improves the performance of any base first-order optimizer by efficiently incorporating second-order information during the optimization process. In each iteration, FOSI implicitly splits the function into two quadratic functions defined on orthogonal subspaces, then uses a second-order method to minimize the first, and the base optimizer to minimize the other. We formally analyze FOSI's convergence and the conditions under which it improves a base optimizer. Our empirical evaluation demonstrates that FOSI improves the convergence rate and optimization time of first-order methods such as Heavy-Ball and Adam, and outperforms second-order methods (K-FAC and L-BFGS).

Backpropagation, the cornerstone of deep learning, is limited to computing gradients for continuous variables. This limitation poses challenges for problems involving discrete latent variables. To address this issue, we propose a novel approach to approximate the gradient of parameters involved in generating discrete latent variables. First, we examine the widely used Straight-Through (ST) heuristic and demonstrate that it works as a first-order approximation of the gradient. Guided by our findings, we propose ReinMax, which achieves second-order accuracy by integrating Heun's method, a second-order numerical method for solving ODEs. ReinMax does not require Hessian or other second-order derivatives, thus having negligible computation overheads. Extensive experimental results on various tasks demonstrate the superiority of ReinMax over the state of the art. Implementations are released at https://github.com/microsoft/ReinMax.

Second-order training methods have better convergence properties than gradient descent but are rarely used in practice for large-scale training due to their computational overhead. This can be viewed as a hardware limitation (imposed by digital computers). Here we show that natural gradient descent (NGD), a second-order method, can have a similar computational complexity per iteration to a first-order method, when employing appropriate hardware. We present a new hybrid digital-analog algorithm for training neural networks that is equivalent to NGD in a certain parameter regime but avoids prohibitively costly linear system solves. Our algorithm exploits the thermodynamic properties of an analog system at equilibrium, and hence requires an analog thermodynamic computer. The training occurs in a hybrid digital-analog loop, where the gradient and Fisher information matrix (or any other positive semi-definite curvature matrix) are calculated at given time intervals while the analog dynamics take place. We numerically demonstrate the superiority of this approach over state-of-the-art digital first- and second-order training methods on classification tasks and language model fine-tuning tasks.

Stochastic gradient descent method and its variants constitute the core optimization algorithms that achieve good convergence rates for solving machine learning problems. These rates are obtained especially when these algorithms are fine-tuned for the application at hand. Although this tuning process can require large computational costs, recent work has shown that these costs can be reduced by line search methods that iteratively adjust the stepsize. We propose an alternative approach to stochastic line search by using a new algorithm based on forward step model building. This model building step incorporates second-order information that allows adjusting not only the stepsize but also the search direction. Noting that deep learning model parameters come in groups (layers of tensors), our method builds its model and calculates a new step for each parameter group. This novel diagonalization approach makes the selected step lengths adaptive. We provide convergence rate analysis, and experimentally show that the proposed algorithm achieves faster convergence and better generalization in well-known test problems. More precisely, SMB requires less tuning, and shows comparable performance to other adaptive methods.

Sampling from the posterior distribution poses a major computational challenge in solving inverse problems using latent diffusion models. Common methods rely on Tweedie's first-order moments, which are known to induce a quality-limiting bias. Existing second-order approximations are impractical due to prohibitive computational costs, making standard reverse diffusion processes intractable for posterior sampling. This paper introduces Second-order Tweedie sampler from Surrogate Loss (STSL), a novel sampler that offers efficiency comparable to first-order Tweedie with a tractable reverse process using second-order approximation. Our theoretical results reveal that the second-order approximation is lower bounded by our surrogate loss that only requires O(1) compute using the trace of the Hessian, and by the lower bound we derive a new drift term to make the reverse process tractable. Our method surpasses SoTA solvers PSLD and P2L, achieving 4X and 8X reduction in neural function evaluations, respectively, while notably enhancing sampling quality on FFHQ, ImageNet, and COCO benchmarks. In addition, we show STSL extends to text-guided image editing and addresses residual distortions present from corrupted images in leading text-guided image editing methods. To our best knowledge, this is the first work to offer an efficient second-order approximation in solving inverse problems using latent diffusion and editing real-world images with corruptions.

We study the sparse phase retrieval problem, which seeks to recover a sparse signal from a limited set of magnitude-only measurements. In contrast to prevalent sparse phase retrieval algorithms that primarily use first-order methods, we propose an innovative second-order algorithm that employs a Newton-type method with hard thresholding. This algorithm overcomes the linear convergence limitations of first-order methods while preserving their hallmark per-iteration computational efficiency. We provide theoretical guarantees that our algorithm converges to the s-sparse ground truth signal x^{natural} in R^n (up to a global sign) at a quadratic convergence rate after at most O(log (Vertx^{natural} Vert /x_{min}^{natural})) iterations, using Omega(s^2log n) Gaussian random samples. Numerical experiments show that our algorithm achieves a significantly faster convergence rate than state-of-the-art methods.

We introduce ADAHESSIAN, a second order stochastic optimization algorithm which dynamically incorporates the curvature of the loss function via ADAptive estimates of the HESSIAN. Second order algorithms are among the most powerful optimization algorithms with superior convergence properties as compared to first order methods such as SGD and Adam. The main disadvantage of traditional second order methods is their heavier per iteration computation and poor accuracy as compared to first order methods. To address these, we incorporate several novel approaches in ADAHESSIAN, including: (i) a fast Hutchinson based method to approximate the curvature matrix with low computational overhead; (ii) a root-mean-square exponential moving average to smooth out variations of the Hessian diagonal across different iterations; and (iii) a block diagonal averaging to reduce the variance of Hessian diagonal elements. We show that ADAHESSIAN achieves new state-of-the-art results by a large margin as compared to other adaptive optimization methods, including variants of Adam. In particular, we perform extensive tests on CV, NLP, and recommendation system tasks and find that ADAHESSIAN: (i) achieves 1.80%/1.45% higher accuracy on ResNets20/32 on Cifar10, and 5.55% higher accuracy on ImageNet as compared to Adam; (ii) outperforms AdamW for transformers by 0.13/0.33 BLEU score on IWSLT14/WMT14 and 2.7/1.0 PPL on PTB/Wikitext-103; (iii) outperforms AdamW for SqueezeBert by 0.41 points on GLUE; and (iv) achieves 0.032% better score than Adagrad for DLRM on the Criteo Ad Kaggle dataset. Importantly, we show that the cost per iteration of ADAHESSIAN is comparable to first order methods, and that it exhibits robustness towards its hyperparameters.

It is well known that accurate probabilistic predictors can be trained through empirical risk minimisation with proper scoring rules as loss functions. While such learners capture so-called aleatoric uncertainty of predictions, various machine learning methods have recently been developed with the goal to let the learner also represent its epistemic uncertainty, i.e., the uncertainty caused by a lack of knowledge and data. An emerging branch of the literature proposes the use of a second-order learner that provides predictions in terms of distributions on probability distributions. However, recent work has revealed serious theoretical shortcomings for second-order predictors based on loss minimisation. In this paper, we generalise these findings and prove a more fundamental result: There seems to be no loss function that provides an incentive for a second-order learner to faithfully represent its epistemic uncertainty in the same manner as proper scoring rules do for standard (first-order) learners. As a main mathematical tool to prove this result, we introduce the generalised notion of second-order scoring rules.

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.

We present a new accelerated stochastic second-order method that is robust to both gradient and Hessian inexactness, which occurs typically in machine learning. We establish theoretical lower bounds and prove that our algorithm achieves optimal convergence in both gradient and Hessian inexactness in this key setting. We further introduce a tensor generalization for stochastic higher-order derivatives. When the oracles are non-stochastic, the proposed tensor algorithm matches the global convergence of Nesterov Accelerated Tensor method. Both algorithms allow for approximate solutions of their auxiliary subproblems with verifiable conditions on the accuracy of the solution.

Second-order optimizers, maintaining a matrix termed a preconditioner, are superior to first-order optimizers in both theory and practice. The states forming the preconditioner and its inverse root restrict the maximum size of models trained by second-order optimizers. To address this, compressing 32-bit optimizer states to lower bitwidths has shown promise in reducing memory usage. However, current approaches only pertain to first-order optimizers. In this paper, we propose the first 4-bit second-order optimizers, exemplified by 4-bit Shampoo, maintaining performance similar to that of 32-bit ones. We show that quantizing the eigenvector matrix of the preconditioner in 4-bit Shampoo is remarkably better than quantizing the preconditioner itself both theoretically and experimentally. By rectifying the orthogonality of the quantized eigenvector matrix, we enhance the approximation of the preconditioner's eigenvector matrix, which also benefits the computation of its inverse 4-th root. Besides, we find that linear square quantization slightly outperforms dynamic tree quantization when quantizing second-order optimizer states. Evaluation on various networks for image classification demonstrates that our 4-bit Shampoo achieves comparable test accuracy to its 32-bit counterpart while being more memory-efficient. The source code will be made available.

We interpret the function of individual neurons in CLIP by automatically describing them using text. Analyzing the direct effects (i.e. the flow from a neuron through the residual stream to the output) or the indirect effects (overall contribution) fails to capture the neurons' function in CLIP. Therefore, we present the "second-order lens", analyzing the effect flowing from a neuron through the later attention heads, directly to the output. We find that these effects are highly selective: for each neuron, the effect is significant for <2% of the images. Moreover, each effect can be approximated by a single direction in the text-image space of CLIP. We describe neurons by decomposing these directions into sparse sets of text representations. The sets reveal polysemantic behavior - each neuron corresponds to multiple, often unrelated, concepts (e.g. ships and cars). Exploiting this neuron polysemy, we mass-produce "semantic" adversarial examples by generating images with concepts spuriously correlated to the incorrect class. Additionally, we use the second-order effects for zero-shot segmentation and attribute discovery in images. Our results indicate that a scalable understanding of neurons can be used for model deception and for introducing new model capabilities.

Deep learning optimizers are often motivated through a mix of convex and approximate second-order theory. We select three such methods -- Adam, Shampoo and Prodigy -- and argue that each method can instead be understood as a squarely first-order method without convexity assumptions. In fact, after switching off exponential moving averages, each method is equivalent to steepest descent under a particular norm. By generalizing this observation, we chart a new design space for training algorithms. Different operator norms should be assigned to different tensors based on the role that the tensor plays within the network. For example, while linear and embedding layers may have the same weight space of R^{mtimes n}, these layers play different roles and should be assigned different norms. We hope that this idea of carefully metrizing the neural architecture might lead to more stable, scalable and indeed faster training.

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.

In training neural networks, it is common practice to use partial gradients computed over batches, mostly very small subsets of the training set. This approach is motivated by the argument that such a partial gradient is close to the true one, with precision growing only with the square root of the batch size. A theoretical justification is with the help of stochastic approximation theory. However, the conditions for the validity of this theory are not satisfied in the usual learning rate schedules. Batch processing is also difficult to combine with efficient second-order optimization methods. This proposal is based on another hypothesis: the loss minimum of the training set can be expected to be well-approximated by the minima of its subsets. Such subset minima can be computed in a fraction of the time necessary for optimizing over the whole training set. This hypothesis has been tested with the help of the MNIST, CIFAR-10, and CIFAR-100 image classification benchmarks, optionally extended by training data augmentation. The experiments have confirmed that results equivalent to conventional training can be reached. In summary, even small subsets are representative if the overdetermination ratio for the given model parameter set sufficiently exceeds unity. The computing expense can be reduced to a tenth or less.

Our research aims to develop machines that learn to perceive visual motion as do humans. While recent advances in computer vision (CV) have enabled DNN-based models to accurately estimate optical flow in naturalistic images, a significant disparity remains between CV models and the biological visual system in both architecture and behavior. This disparity includes humans' ability to perceive the motion of higher-order image features (second-order motion), which many CV models fail to capture because of their reliance on the intensity conservation law. Our model architecture mimics the cortical V1-MT motion processing pathway, utilizing a trainable motion energy sensor bank and a recurrent graph network. Supervised learning employing diverse naturalistic videos allows the model to replicate psychophysical and physiological findings about first-order (luminance-based) motion perception. For second-order motion, inspired by neuroscientific findings, the model includes an additional sensing pathway with nonlinear preprocessing before motion energy sensing, implemented using a simple multilayer 3D CNN block. When exploring how the brain acquired the ability to perceive second-order motion in natural environments, in which pure second-order signals are rare, we hypothesized that second-order mechanisms were critical when estimating robust object motion amidst optical fluctuations, such as highlights on glossy surfaces. We trained our dual-pathway model on novel motion datasets with varying material properties of moving objects. We found that training to estimate object motion from non-Lambertian materials naturally endowed the model with the capacity to perceive second-order motion, as can humans. The resulting model effectively aligns with biological systems while generalizing to both first- and second-order motion phenomena in natural scenes.

First-order optimization methods are currently the mainstream in training deep neural networks (DNNs). Optimizers like Adam incorporate limited curvature information by employing the diagonal matrix preconditioning of the stochastic gradient during the training. Despite their widespread, second-order optimization algorithms exhibit superior convergence properties compared to their first-order counterparts e.g. Adam and SGD. However, their practicality in training DNNs are still limited due to increased per-iteration computations and suboptimal accuracy compared to the first order methods. We present AdaFisher--an adaptive second-order optimizer that leverages a block-diagonal approximation to the Fisher information matrix for adaptive gradient preconditioning. AdaFisher aims to bridge the gap between enhanced convergence capabilities and computational efficiency in second-order optimization framework for training DNNs. Despite the slow pace of second-order optimizers, we showcase that AdaFisher can be reliably adopted for image classification, language modelling and stand out for its stability and robustness in hyperparameter tuning. We demonstrate that AdaFisher outperforms the SOTA optimizers in terms of both accuracy and convergence speed. Code available from https://github.com/AtlasAnalyticsLab/AdaFisher{https://github.com/AtlasAnalyticsLab/AdaFisher}

Shampoo, a second-order optimization algorithm which uses a Kronecker product preconditioner, has recently garnered increasing attention from the machine learning community. The preconditioner used by Shampoo can be viewed either as an approximation of the Gauss--Newton component of the Hessian or the covariance matrix of the gradients maintained by Adagrad. We provide an explicit and novel connection between the optimal Kronecker product approximation of these matrices and the approximation made by Shampoo. Our connection highlights a subtle but common misconception about Shampoo's approximation. In particular, the square of the approximation used by the Shampoo optimizer is equivalent to a single step of the power iteration algorithm for computing the aforementioned optimal Kronecker product approximation. Across a variety of datasets and architectures we empirically demonstrate that this is close to the optimal Kronecker product approximation. Additionally, for the Hessian approximation viewpoint, we empirically study the impact of various practical tricks to make Shampoo more computationally efficient (such as using the batch gradient and the empirical Fisher) on the quality of Hessian approximation.

Model order reduction has been extensively studied over the last two decades. Projection-based methods such as the Proper Orthogonal Decomposition and the Reduced Basis Method enjoy the important advantages of Galerkin methods in the derivation of the reduced problem, but are limited to linear data compression for which the reduced solution is sought as a linear combination of spatial modes. Nonlinear data compression must be used when the solution manifold is not embedded in a low-dimensional subspace. Early methods involve piecewise linear data compression, by constructing a dictionary of reduced-order models tailored to a partition of the solution manifold. In this work, we introduce the concept of optimal partition of the solution manifold in terms of normalized Kolmogorov widths, and prove that the optimal partitions can be found by means of a representative-based clustering algorithm using the sine dissimilarity measure on the solution manifold.

This paper studies the fitting of Hessian or its inverse with stochastic Hessian-vector products. A Hessian fitting criterion, which can be used to derive most of the commonly used methods, e.g., BFGS, Gaussian-Newton, AdaGrad, etc., is used for the analysis. Our studies reveal different convergence rates for different Hessian fitting methods, e.g., sublinear rates for gradient descent in the Euclidean space and a commonly used closed-form solution, linear rates for gradient descent on the manifold of symmetric positive definite (SPL) matrices and certain Lie groups. The Hessian fitting problem is further shown to be strongly convex under mild conditions on a specific yet general enough Lie group. To confirm our analysis, these methods are tested under different settings like noisy Hessian-vector products, time varying Hessians, and low precision arithmetic. These findings are useful for stochastic second order optimizations that rely on fast, robust and accurate Hessian estimations.

We use automated theorem provers to significantly shorten a formal development in higher order set theory. The development includes many standard theorems such as the fundamental theorem of arithmetic and irrationality of square root of two. Higher order automated theorem provers are particularly useful here, since the underlying framework of higher order set theory coincides with the classical extensional higher order logic of (most) higher order automated theorem provers, so no significant translation or encoding is required. Additionally, many subgoals are first order and so first order automated provers often suffice. We compare the performance of different provers on the subgoals generated from the development. We also discuss possibilities for proof reconstruction, i.e., obtaining formal proof terms when an automated theorem prover claims to have proven the subgoal.

Editing signals using large pre-trained models, in a zero-shot manner, has recently seen rapid advancements in the image domain. However, this wave has yet to reach the audio domain. In this paper, we explore two zero-shot editing techniques for audio signals, which use DDPM inversion on pre-trained diffusion models. The first, adopted from the image domain, allows text-based editing. The second, is a novel approach for discovering semantically meaningful editing directions without supervision. When applied to music signals, this method exposes a range of musically interesting modifications, from controlling the participation of specific instruments to improvisations on the melody. Samples can be found on our examples page in https://hilamanor.github.io/AudioEditing/ and code can be found in https://github.com/hilamanor/AudioEditing/ .

We develop a spacetime neural network method with second order optimization for solving quantum dynamics from the high dimensional Schr\"{o}dinger equation. In contrast to the standard iterative first order optimization and the time-dependent variational principle, our approach utilizes the implicit mid-point method and generates the solution for all spatial and temporal values simultaneously after optimization. We demonstrate the method in the Schr\"{o}dinger equation with a self-normalized autoregressive spacetime neural network construction. Future explorations for solving different high dimensional differential equations are discussed.

Implicit layer deep learning techniques, like Neural Differential Equations, have become an important modeling framework due to their ability to adapt to new problems automatically. Training a neural differential equation is effectively a search over a space of plausible dynamical systems. However, controlling the computational cost for these models is difficult since it relies on the number of steps the adaptive solver takes. Most prior works have used higher-order methods to reduce prediction timings while greatly increasing training time or reducing both training and prediction timings by relying on specific training algorithms, which are harder to use as a drop-in replacement due to strict requirements on automatic differentiation. In this manuscript, we use internal cost heuristics of adaptive differential equation solvers at stochastic time points to guide the training toward learning a dynamical system that is easier to integrate. We "close the black-box" and allow the use of our method with any adjoint technique for gradient calculations of the differential equation solution. We perform experimental studies to compare our method to global regularization to show that we attain similar performance numbers without compromising the flexibility of implementation on ordinary differential equations (ODEs) and stochastic differential equations (SDEs). We develop two sampling strategies to trade off between performance and training time. Our method reduces the number of function evaluations to 0.556-0.733x and accelerates predictions by 1.3-2x.

We study the problem of training neural networks with quantized parameters. Learning low-precision quantized parameters by enabling computation of gradients via the Straight-Through Estimator (STE) can be challenging. While the STE enables back-propagation, which is a first-order method, recent works have explored the use of zeroth-order (ZO) gradient descent for fine-tuning. We note that the STE provides high-quality biased gradients, and ZO gradients are unbiased but can be expensive. We thus propose First-Order-Guided Zeroth-Order Gradient Descent (FOGZO) that reduces STE bias while reducing computations relative to ZO methods. Empirically, we show FOGZO improves the tradeoff between quality and training time in Quantization-Aware Pre-Training. Specifically, versus STE at the same number of iterations, we show a 1-8\% accuracy improvement for DeiT Tiny/Small, 1-2\% accuracy improvement on ResNet 18/50, and 1-22 perplexity point improvement for LLaMA models with up to 0.3 billion parameters. For the same loss, FOGZO yields a 796times reduction in computation versus n-SPSA for a 2-layer MLP on MNIST. Code is available at https://github.com/1733116199/fogzo.

Uncertainty quantification has received increasing attention in machine learning in the recent past. In particular, a distinction between aleatoric and epistemic uncertainty has been found useful in this regard. The latter refers to the learner's (lack of) knowledge and appears to be especially difficult to measure and quantify. In this paper, we analyse a recent proposal based on the idea of a second-order learner, which yields predictions in the form of distributions over probability distributions. While standard (first-order) learners can be trained to predict accurate probabilities, namely by minimising suitable loss functions on sample data, we show that loss minimisation does not work for second-order predictors: The loss functions proposed for inducing such predictors do not incentivise the learner to represent its epistemic uncertainty in a faithful way.

Kernel online convex optimization (KOCO) is a framework combining the expressiveness of non-parametric kernel models with the regret guarantees of online learning. First-order KOCO methods such as functional gradient descent require only O(t) time and space per iteration, and, when the only information on the losses is their convexity, achieve a minimax optimal O(T) regret. Nonetheless, many common losses in kernel problems, such as squared loss, logistic loss, and squared hinge loss posses stronger curvature that can be exploited. In this case, second-order KOCO methods achieve O(log(Det(K))) regret, which we show scales as O(d_{eff}log T), where d_{eff} is the effective dimension of the problem and is usually much smaller than O(T). The main drawback of second-order methods is their much higher O(t^2) space and time complexity. In this paper, we introduce kernel online Newton step (KONS), a new second-order KOCO method that also achieves O(d_{eff}log T) regret. To address the computational complexity of second-order methods, we introduce a new matrix sketching algorithm for the kernel matrix K_t, and show that for a chosen parameter γleq 1 our Sketched-KONS reduces the space and time complexity by a factor of γ^2 to O(t^2γ^2) space and time per iteration, while incurring only 1/γ times more regret.

We give an improved theoretical analysis of score-based generative modeling. Under a score estimate with small L^2 error (averaged across timesteps), we provide efficient convergence guarantees for any data distribution with second-order moment, by either employing early stopping or assuming smoothness condition on the score function of the data distribution. Our result does not rely on any log-concavity or functional inequality assumption and has a logarithmic dependence on the smoothness. In particular, we show that under only a finite second moment condition, approximating the following in reverse KL divergence in epsilon-accuracy can be done in tilde Oleft(d log (1/delta){epsilon}right) steps: 1) the variance-delta Gaussian perturbation of any data distribution; 2) data distributions with 1/delta-smooth score functions. Our analysis also provides a quantitative comparison between different discrete approximations and may guide the choice of discretization points in practice.

In an industrial group like Safran, numerical simulations of physical phenomena are integral to most design processes. At Safran's corporate research center, we enhance these processes by developing fast and reliable surrogate models for various physics. We focus here on two technologies developed in recent years. The first is a physical reduced-order modeling method for non-linear structural mechanics and thermal analysis, used for calculating the lifespan of high-pressure turbine blades and performing heat analysis of high-pressure compressors. The second technology involves learning physics simulations with non-parameterized geometrical variability using classical machine learning tools, such as Gaussian process regression. Finally, we present our contributions to the open-source and open-data community.

Nonlinear model order reduction has opened the door to parameter optimization and uncertainty quantification in complex physics problems governed by nonlinear equations. In particular, the computational cost of solving these equations can be reduced by means of local reduced-order bases. This article examines the benefits of a physics-informed cluster analysis for the construction of cluster-specific reduced-order bases. We illustrate that the choice of the dissimilarity measure for clustering is fundamental and highly affects the performances of the local reduced-order bases. It is shown that clustering with an angle-based dissimilarity on simulation data efficiently decreases the intra-cluster Kolmogorov N-width. Additionally, an a priori efficiency criterion is introduced to assess the relevance of a ROM-net, a methodology for the reduction of nonlinear physics problems introduced in our previous work in [T. Daniel, F. Casenave, N. Akkari, D. Ryckelynck, Model order reduction assisted by deep neural networks (ROM-net), Advanced Modeling and Simulation in Engineering Sciences 7 (16), 2020]. This criterion also provides engineers with a very practical method for ROM-nets' hyperparameters calibration under constrained computational costs for the training phase. On five different physics problems, our physics-informed clustering strategy significantly outperforms classic strategies for the construction of local reduced-order bases in terms of projection errors.

In the past couple of years, various approaches to representing and quantifying different types of predictive uncertainty in machine learning, notably in the setting of classification, have been proposed on the basis of second-order probability distributions, i.e., predictions in the form of distributions on probability distributions. A completely conclusive solution has not yet been found, however, as shown by recent criticisms of commonly used uncertainty measures associated with second-order distributions, identifying undesirable theoretical properties of these measures. In light of these criticisms, we propose a set of formal criteria that meaningful uncertainty measures for predictive uncertainty based on second-order distributions should obey. Moreover, we provide a general framework for developing uncertainty measures to account for these criteria, and offer an instantiation based on the Wasserstein distance, for which we prove that all criteria are satisfied.

This paper introduces a novel training methodology that enables a small Transformer model to generalize the addition of two-digit numbers to numbers with unseen lengths of digits. The proposed approach employs an autoregressive generation technique, processing from right to left, which mimics a common manual method for adding large numbers. To the best of my knowledge, this methodology has not been previously explored in the literature. All results are reproducible, and the corresponding R code is available at: https://github.com/AGPatriota/ALGA-R/.

We address the problem of synthesizing multi-view optical illusions: images that change appearance upon a transformation, such as a flip or rotation. We propose a simple, zero-shot method for obtaining these illusions from off-the-shelf text-to-image diffusion models. During the reverse diffusion process, we estimate the noise from different views of a noisy image, and then combine these noise estimates together and denoise the image. A theoretical analysis suggests that this method works precisely for views that can be written as orthogonal transformations, of which permutations are a subset. This leads to the idea of a visual anagram--an image that changes appearance under some rearrangement of pixels. This includes rotations and flips, but also more exotic pixel permutations such as a jigsaw rearrangement. Our approach also naturally extends to illusions with more than two views. We provide both qualitative and quantitative results demonstrating the effectiveness and flexibility of our method. Please see our project webpage for additional visualizations and results: https://dangeng.github.io/visual_anagrams/

Optimizing neural networks with loss that contain high-dimensional and high-order differential operators is expensive to evaluate with back-propagation due to O(d^{k}) scaling of the derivative tensor size and the O(2^{k-1}L) scaling in the computation graph, where d is the dimension of the domain, L is the number of ops in the forward computation graph, and k is the derivative order. In previous works, the polynomial scaling in d was addressed by amortizing the computation over the optimization process via randomization. Separately, the exponential scaling in k for univariate functions (d=1) was addressed with high-order auto-differentiation (AD). In this work, we show how to efficiently perform arbitrary contraction of the derivative tensor of arbitrary order for multivariate functions, by properly constructing the input tangents to univariate high-order AD, which can be used to efficiently randomize any differential operator. When applied to Physics-Informed Neural Networks (PINNs), our method provides >1000times speed-up and >30times memory reduction over randomization with first-order AD, and we can now solve 1-million-dimensional PDEs in 8 minutes on a single NVIDIA A100 GPU. This work opens the possibility of using high-order differential operators in large-scale problems.

Computing the matrix square root or its inverse in a differentiable manner is important in a variety of computer vision tasks. Previous methods either adopt the Singular Value Decomposition (SVD) to explicitly factorize the matrix or use the Newton-Schulz iteration (NS iteration) to derive the approximate solution. However, both methods are not computationally efficient enough in either the forward pass or in the backward pass. In this paper, we propose two more efficient variants to compute the differentiable matrix square root. For the forward propagation, one method is to use Matrix Taylor Polynomial (MTP), and the other method is to use Matrix Pad\'e Approximants (MPA). The backward gradient is computed by iteratively solving the continuous-time Lyapunov equation using the matrix sign function. Both methods yield considerable speed-up compared with the SVD or the Newton-Schulz iteration. Experimental results on the de-correlated batch normalization and second-order vision transformer demonstrate that our methods can also achieve competitive and even slightly better performances. The code is available at https://github.com/KingJamesSong/FastDifferentiableMatSqrt{https://github.com/KingJamesSong/FastDifferentiableMatSqrt}.

We consider stochastic unconstrained bilevel optimization problems when only the first-order gradient oracles are available. While numerous optimization methods have been proposed for tackling bilevel problems, existing methods either tend to require possibly expensive calculations regarding Hessians of lower-level objectives, or lack rigorous finite-time performance guarantees. In this work, we propose a Fully First-order Stochastic Approximation (F2SA) method, and study its non-asymptotic convergence properties. Specifically, we show that F2SA converges to an epsilon-stationary solution of the bilevel problem after epsilon^{-7/2}, epsilon^{-5/2}, and epsilon^{-3/2} iterations (each iteration using O(1) samples) when stochastic noises are in both level objectives, only in the upper-level objective, and not present (deterministic settings), respectively. We further show that if we employ momentum-assisted gradient estimators, the iteration complexities can be improved to epsilon^{-5/2}, epsilon^{-4/2}, and epsilon^{-3/2}, respectively. We demonstrate even superior practical performance of the proposed method over existing second-order based approaches on MNIST data-hypercleaning experiments.

While it is widely known that neural networks are universal approximators of continuous functions, a less known and perhaps more powerful result is that a neural network with a single hidden layer can approximate accurately any nonlinear continuous operator. This universal approximation theorem is suggestive of the potential application of neural networks in learning nonlinear operators from data. However, the theorem guarantees only a small approximation error for a sufficient large network, and does not consider the important optimization and generalization errors. To realize this theorem in practice, we propose deep operator networks (DeepONets) to learn operators accurately and efficiently from a relatively small dataset. A DeepONet consists of two sub-networks, one for encoding the input function at a fixed number of sensors x_i, i=1,dots,m (branch net), and another for encoding the locations for the output functions (trunk net). We perform systematic simulations for identifying two types of operators, i.e., dynamic systems and partial differential equations, and demonstrate that DeepONet significantly reduces the generalization error compared to the fully-connected networks. We also derive theoretically the dependence of the approximation error in terms of the number of sensors (where the input function is defined) as well as the input function type, and we verify the theorem with computational results. More importantly, we observe high-order error convergence in our computational tests, namely polynomial rates (from half order to fourth order) and even exponential convergence with respect to the training dataset size.

Causal discovery with latent confounders is an important but challenging task in many scientific areas. Despite the success of some overcomplete independent component analysis (OICA) based methods in certain domains, they are computationally expensive and can easily get stuck into local optima. We notice that interestingly, by making use of higher-order cumulants, there exists a closed-form solution to OICA in specific cases, e.g., when the mixing procedure follows the One-Latent-Component structure. In light of the power of the closed-form solution to OICA corresponding to the One-Latent-Component structure, we formulate a way to estimate the mixing matrix using the higher-order cumulants, and further propose the testable One-Latent-Component condition to identify the latent variables and determine causal orders. By iteratively removing the share identified latent components, we successfully extend the results on the One-Latent-Component structure to the Multi-Latent-Component structure and finally provide a practical and asymptotically correct algorithm to learn the causal structure with latent variables. Experimental results illustrate the asymptotic correctness and effectiveness of the proposed method.

Efficiently approximating local curvature information of the loss function is a key tool for optimization and compression of deep neural networks. Yet, most existing methods to approximate second-order information have high computational or storage costs, which can limit their practicality. In this work, we investigate matrix-free, linear-time approaches for estimating Inverse-Hessian Vector Products (IHVPs) for the case when the Hessian can be approximated as a sum of rank-one matrices, as in the classic approximation of the Hessian by the empirical Fisher matrix. We propose two new algorithms as part of a framework called M-FAC: the first algorithm is tailored towards network compression and can compute the IHVP for dimension d, if the Hessian is given as a sum of m rank-one matrices, using O(dm^2) precomputation, O(dm) cost for computing the IHVP, and query cost O(m) for any single element of the inverse Hessian. The second algorithm targets an optimization setting, where we wish to compute the product between the inverse Hessian, estimated over a sliding window of optimization steps, and a given gradient direction, as required for preconditioned SGD. We give an algorithm with cost O(dm + m^2) for computing the IHVP and O(dm + m^3) for adding or removing any gradient from the sliding window. These two algorithms yield state-of-the-art results for network pruning and optimization with lower computational overhead relative to existing second-order methods. Implementations are available at [9] and [17].

The commitment to single-precision floating-point arithmetic is widespread in the deep learning community. To evaluate whether this commitment is justified, the influence of computing precision (single and double precision) on the optimization performance of the Conjugate Gradient (CG) method (a second-order optimization algorithm) and RMSprop (a first-order algorithm) has been investigated. Tests of neural networks with one to five fully connected hidden layers and moderate or strong nonlinearity with up to 4 million network parameters have been optimized for Mean Square Error (MSE). The training tasks have been set up so that their MSE minimum was known to be zero. Computing experiments have disclosed that single-precision can keep up (with superlinear convergence) with double-precision as long as line search finds an improvement. First-order methods such as RMSprop do not benefit from double precision. However, for moderately nonlinear tasks, CG is clearly superior. For strongly nonlinear tasks, both algorithm classes find only solutions fairly poor in terms of mean square error as related to the output variance. CG with double floating-point precision is superior whenever the solutions have the potential to be useful for the application goal.

Singly-TASE operators for the numerical solution of stiff differential equations were proposed by Calvo et al. in J.Sci. Comput. 2023 to reduce the computational cost of Runge-Kutta-TASE (RKTASE) methods when the involved linear systems are solved by some LU factorization. In this paper we propose a modification of these methods to improve the efficiency by considering different TASE operators for each stage of the Runge-Kutta. We prove that the resulting RKTASE methods are equivalent to W-methods (Steihaug and Wolfbrandt, Mathematics of Computation,1979) and this allows us to obtain the order conditions of the proposed Modified Singly-RKTASE methods (MSRKTASE) through the theory developed for the W-methods. We construct new MSRKTASE methods of order two and three and demonstrate their effectiveness through numerical experiments on both linear and nonlinear stiff systems. The results show that the MSRKTASE schemes significantly enhance efficiency and accuracy compared to previous Singly-RKTASE schemes.

Leveraging second-order information about the loss at the scale of deep networks is one of the main lines of approach for improving the performance of current optimizers for deep learning. Yet, existing approaches for accurate full-matrix preconditioning, such as Full-Matrix Adagrad (GGT) or Matrix-Free Approximate Curvature (M-FAC) suffer from massive storage costs when applied even to small-scale models, as they must store a sliding window of gradients, whose memory requirements are multiplicative in the model dimension. In this paper, we address this issue via a novel and efficient error-feedback technique that can be applied to compress preconditioners by up to two orders of magnitude in practice, without loss of convergence. Specifically, our approach compresses the gradient information via sparsification or low-rank compression before it is fed into the preconditioner, feeding the compression error back into future iterations. Experiments on deep neural networks show that this approach can compress full-matrix preconditioners to up to 99\% sparsity without accuracy loss, effectively removing the memory overhead of full-matrix preconditioners such as GGT and M-FAC. Our code is available at https://github.com/IST-DASLab/EFCP.

Zeroth-order (ZO) optimization has become a popular technique for solving machine learning (ML) problems when first-order (FO) information is difficult or impossible to obtain. However, the scalability of ZO optimization remains an open problem: Its use has primarily been limited to relatively small-scale ML problems, such as sample-wise adversarial attack generation. To our best knowledge, no prior work has demonstrated the effectiveness of ZO optimization in training deep neural networks (DNNs) without a significant decrease in performance. To overcome this roadblock, we develop DeepZero, a principled ZO deep learning (DL) framework that can scale ZO optimization to DNN training from scratch through three primary innovations. First, we demonstrate the advantages of coordinatewise gradient estimation (CGE) over randomized vector-wise gradient estimation in training accuracy and computational efficiency. Second, we propose a sparsityinduced ZO training protocol that extends the model pruning methodology using only finite differences to explore and exploit the sparse DL prior in CGE. Third, we develop the methods of feature reuse and forward parallelization to advance the practical implementations of ZO training. Our extensive experiments show that DeepZero achieves state-of-the-art (SOTA) accuracy on ResNet-20 trained on CIFAR-10, approaching FO training performance for the first time. Furthermore, we show the practical utility of DeepZero in applications of certified adversarial defense and DL-based partial differential equation error correction, achieving 10-20% improvement over SOTA. We believe our results will inspire future research on scalable ZO optimization and contribute to advancing DL with black box. Codes are available at https://github.com/OPTML-Group/DeepZero.

Neural fields have become widely used in various fields, from shape representation to neural rendering, and for solving partial differential equations (PDEs). With the advent of hybrid neural field representations like Instant NGP that leverage small MLPs and explicit representations, these models train quickly and can fit large scenes. Yet in many applications like rendering and simulation, hybrid neural fields can cause noticeable and unreasonable artifacts. This is because they do not yield accurate spatial derivatives needed for these downstream applications. In this work, we propose two ways to circumvent these challenges. Our first approach is a post hoc operator that uses local polynomial fitting to obtain more accurate derivatives from pre-trained hybrid neural fields. Additionally, we also propose a self-supervised fine-tuning approach that refines the hybrid neural field to yield accurate derivatives directly while preserving the initial signal. We show applications of our method to rendering, collision simulation, and solving PDEs. We observe that using our approach yields more accurate derivatives, reducing artifacts and leading to more accurate simulations in downstream applications.

We consider in this work quantities that can be obtained as limits of powers of parametrized matrices, for instance the inverse matrix or the logarithm of the determinant. Under the assumption of affine dependence in the parameters, we use the Empirical Interpolation Method (EIM) to derive an approximation for powers of these matrices, from which we derive a nonintrusive approximation for the aforementioned limits. We derive upper bounds of the error made by the obtained formula. Finally, numerical comparisons with classical intrusive and nonintrusive approximation techniques are provided: in the considered test-cases, our algorithm performs well compared to the nonintrusive ones.

We provide a new theoretical framework for the variable-step deferred correction (DC) methods based on the well-known BDF2 formula. By using the discrete orthogonal convolution kernels, some high-order BDF2-DC methods are proven to be stable on arbitrary time grids according to the recent definition of stability (SINUM, 60: 2253-2272). It significantly relaxes the existing step-ratio restrictions for the BDF2-DC methods (BIT, 62: 1789-1822). The associated sharp error estimates are established by taking the numerical effects of the starting approximations into account, and they suggest that the BDF2-DC methods have no aftereffect, that is, the lower-order starting scheme for the BDF2 scheme will not cause a loss in the accuracy of the high-order BDF2-DC methods. Extensive tests on the graded and random time meshes are presented to support the new theory.

Fine-tuning large pre-trained LLMs generally demands extensive GPU memory. Traditional first-order optimizers like SGD encounter substantial difficulties due to increased memory requirements from storing activations and gradients during both the forward and backward phases as the model size expands. Alternatively, zeroth-order (ZO) techniques can compute gradients using just forward operations, eliminating the need to store activations. Furthermore, by leveraging CPU capabilities, it's feasible to enhance both the memory and processing power available to a single GPU. We propose a novel framework, ZO2 (Zeroth-Order Offloading), for efficient zeroth-order fine-tuning of LLMs with only limited GPU memory. Our framework dynamically shifts model parameters between the CPU and GPU as required, optimizing computation flow and maximizing GPU usage by minimizing downtime. This integration of parameter adjustments with ZO's double forward operations reduces unnecessary data movement, enhancing the fine-tuning efficacy. Additionally, our framework supports an innovative low-bit precision approach in AMP mode to streamline data exchanges between the CPU and GPU. Employing this approach allows us to fine-tune extraordinarily large models, such as the OPT-175B with more than 175 billion parameters, on a mere 18GB GPU--achievements beyond the reach of traditional methods. Moreover, our framework achieves these results with almost no additional time overhead and absolutely no accuracy loss compared to standard zeroth-order methods. ZO2's code has been open-sourced in https://github.com/liangyuwang/zo2.

Score-based diffusion models, while achieving remarkable empirical performance, often suffer from low sampling speed, due to extensive function evaluations needed during the sampling phase. Despite a flurry of recent activities towards speeding up diffusion generative modeling in practice, theoretical underpinnings for acceleration techniques remain severely limited. In this paper, we design novel training-free algorithms to accelerate popular deterministic (i.e., DDIM) and stochastic (i.e., DDPM) samplers. Our accelerated deterministic sampler converges at a rate O(1/{T}^2) with T the number of steps, improving upon the O(1/T) rate for the DDIM sampler; and our accelerated stochastic sampler converges at a rate O(1/T), outperforming the rate O(1/T) for the DDPM sampler. The design of our algorithms leverages insights from higher-order approximation, and shares similar intuitions as popular high-order ODE solvers like the DPM-Solver-2. Our theory accommodates ell_2-accurate score estimates, and does not require log-concavity or smoothness on the target distribution.

Lattices are architected metamaterials whose properties strongly depend on their geometrical design. The analogy between lattices and graphs enables the use of graph neural networks (GNNs) as a faster surrogate model compared to traditional methods such as finite element modelling. In this work, we generate a big dataset of structure-property relationships for strut-based lattices. The dataset is made available to the community which can fuel the development of methods anchored in physical principles for the fitting of fourth-order tensors. In addition, we present a higher-order GNN model trained on this dataset. The key features of the model are (i) SE(3) equivariance, and (ii) consistency with the thermodynamic law of conservation of energy. We compare the model to non-equivariant models based on a number of error metrics and demonstrate its benefits in terms of predictive performance and reduced training requirements. Finally, we demonstrate an example application of the model to an architected material design task. The methods which we developed are applicable to fourth-order tensors beyond elasticity such as piezo-optical tensor etc.

Autoregressive patch-based image generation has recently shown competitive results in terms of image quality and scalability. It can also be easily integrated and scaled within Vision-Language models. Nevertheless, autoregressive models require a defined order for patch generation. While a natural order based on the dictation of the words makes sense for text generation, there is no inherent generation order that exists for image generation. Traditionally, a raster-scan order (from top-left to bottom-right) guides autoregressive image generation models. In this paper, we argue that this order is suboptimal, as it fails to respect the causality of the image content: for instance, when conditioned on a visual description of a sunset, an autoregressive model may generate clouds before the sun, even though the color of clouds should depend on the color of the sun and not the inverse. In this work, we show that first by training a model to generate patches in any-given-order, we can infer both the content and the location (order) of each patch during generation. Secondly, we use these extracted orders to finetune the any-given-order model to produce better-quality images. Through our experiments, we show on two datasets that this new generation method produces better images than the traditional raster-scan approach, with similar training costs and no extra annotations.

Efficient numerical solvers for partial differential equations empower science and engineering. One of the commonly employed numerical solvers is the preconditioned conjugate gradient (PCG) algorithm which can solve large systems to a given precision level. One challenge in PCG solvers is the selection of preconditioners, as different problem-dependent systems can benefit from different preconditioners. We present a new method to introduce inductive bias in preconditioning conjugate gradient algorithm. Given a system matrix and a set of solution vectors arise from an underlying distribution, we train a graph neural network to obtain an approximate decomposition to the system matrix to be used as a preconditioner in the context of PCG solvers. We conduct extensive experiments to demonstrate the efficacy and generalizability of our proposed approach in solving various 2D and 3D linear second-order PDEs.

Gradient-free/zeroth-order methods for black-box convex optimization have been extensively studied in the last decade with the main focus on oracle calls complexity. In this paper, besides the oracle complexity, we focus also on iteration complexity, and propose a generic approach that, based on optimal first-order methods, allows to obtain in a black-box fashion new zeroth-order algorithms for non-smooth convex optimization problems. Our approach not only leads to optimal oracle complexity, but also allows to obtain iteration complexity similar to first-order methods, which, in turn, allows to exploit parallel computations to accelerate the convergence of our algorithms. We also elaborate on extensions for stochastic optimization problems, saddle-point problems, and distributed optimization.

A popular approach to sample a diffusion-based generative model is to solve an ordinary differential equation (ODE). In existing samplers, the coefficients of the ODE solvers are pre-determined by the ODE formulation, the reverse discrete timesteps, and the employed ODE methods. In this paper, we consider accelerating several popular ODE-based sampling processes (including EDM, DDIM, and DPM-Solver) by optimizing certain coefficients via improved integration approximation (IIA). We propose to minimize, for each time step, a mean squared error (MSE) function with respect to the selected coefficients. The MSE is constructed by applying the original ODE solver for a set of fine-grained timesteps, which in principle provides a more accurate integration approximation in predicting the next diffusion state. The proposed IIA technique does not require any change of a pre-trained model, and only introduces a very small computational overhead for solving a number of quadratic optimization problems. Extensive experiments show that considerably better FID scores can be achieved by using IIA-EDM, IIA-DDIM, and IIA-DPM-Solver than the original counterparts when the neural function evaluation (NFE) is small (i.e., less than 25).

We propose a new definition of higher-order DisCoCat (categorical compositional distributional) models where the meaning of a word is not a diagram, but a diagram-valued higher-order function. Our models can be seen as a variant of Montague semantics based on a lambda calculus where the primitives act on string diagrams rather than logical formulae. As a special case, we show how to translate from the Lambek calculus into Peirce's system beta for first-order logic. This allows us to give a purely diagrammatic treatment of higher-order and non-linear processes in natural language semantics: adverbs, prepositions, negation and quantifiers. The theoretical definition presented in this article comes with a proof-of-concept implementation in DisCoPy, the Python library for string diagrams.

Computers calculate transcendental functions by approximating them through the composition of a few limited-precision instructions. For example, an exponential can be calculated with a Taylor series. These approximation methods were developed over the centuries by mathematicians, who emphasized the attainability of arbitrary precision. Computers, however, operate on few limited precision types, such as the popular float32. In this study, we show that when aiming for limited precision, existing approximation methods can be outperformed by programs automatically discovered from scratch by a simple evolutionary algorithm. In particular, over real numbers, our method can approximate the exponential function reaching orders of magnitude more precision for a given number of operations when compared to previous approaches. More practically, over float32 numbers and constrained to less than 1 ULP of error, the same method attains a speedup over baselines by generating code that triggers better XLA/LLVM compilation paths. In other words, in both cases, evolution searched a vast space of possible programs, without knowledge of mathematics, to discover previously unknown optimized approximations to high precision, for the first time. We also give evidence that these results extend beyond the exponential. The ubiquity of transcendental functions suggests that our method has the potential to reduce the cost of scientific computing applications.

To overcome the overparameterized problem in Pre-trained Language Models (PLMs), pruning is widely used as a simple and straightforward compression method by directly removing unimportant weights. Previous first-order methods successfully compress PLMs to extremely high sparsity with little performance drop. These methods, such as movement pruning, use first-order information to prune PLMs while fine-tuning the remaining weights. In this work, we argue fine-tuning is redundant for first-order pruning, since first-order pruning is sufficient to converge PLMs to downstream tasks without fine-tuning. Under this motivation, we propose Static Model Pruning (SMP), which only uses first-order pruning to adapt PLMs to downstream tasks while achieving the target sparsity level. In addition, we also design a new masking function and training objective to further improve SMP. Extensive experiments at various sparsity levels show SMP has significant improvements over first-order and zero-order methods. Unlike previous first-order methods, SMP is also applicable to low sparsity and outperforms zero-order methods. Meanwhile, SMP is more parameter efficient than other methods due to it does not require fine-tuning.

This paper advances the computational efficiency of Deep Hedging frameworks through the novel integration of Kronecker-Factored Approximate Curvature (K-FAC) optimization. While recent literature has established Deep Hedging as a data-driven alternative to traditional risk management strategies, the computational burden of training neural networks with first-order methods remains a significant impediment to practical implementation. The proposed architecture couples Long Short-Term Memory (LSTM) networks with K-FAC second-order optimization, specifically addressing the challenges of sequential financial data and curvature estimation in recurrent networks. Empirical validation using simulated paths from a calibrated Heston stochastic volatility model demonstrates that the K-FAC implementation achieves marked improvements in convergence dynamics and hedging efficacy. The methodology yields a 78.3% reduction in transaction costs (t = 56.88, p < 0.001) and a 34.4% decrease in profit and loss (P&L) variance compared to Adam optimization. Moreover, the K-FAC-enhanced model exhibits superior risk-adjusted performance with a Sharpe ratio of 0.0401, contrasting with -0.0025 for the baseline model. These results provide compelling evidence that second-order optimization methods can materially enhance the tractability of Deep Hedging implementations. The findings contribute to the growing literature on computational methods in quantitative finance while highlighting the potential for advanced optimization techniques to bridge the gap between theoretical frameworks and practical applications in financial markets.

Discrete diffusion models have emerged as a powerful generative modeling framework for discrete data with successful applications spanning from text generation to image synthesis. However, their deployment faces challenges due to the high dimensionality of the state space, necessitating the development of efficient inference algorithms. Current inference approaches mainly fall into two categories: exact simulation and approximate methods such as tau-leaping. While exact methods suffer from unpredictable inference time and redundant function evaluations, tau-leaping is limited by its first-order accuracy. In this work, we advance the latter category by tailoring the first extension of high-order numerical inference schemes to discrete diffusion models, enabling larger step sizes while reducing error. We rigorously analyze the proposed schemes and establish the second-order accuracy of the theta-trapezoidal method in KL divergence. Empirical evaluations on GPT-2 level text and ImageNet-level image generation tasks demonstrate that our method achieves superior sample quality compared to existing approaches under equivalent computational constraints.

We present semantic correctness proofs of automatic differentiation (AD). We consider a forward-mode AD method on a higher order language with algebraic data types, and we characterise it as the unique structure preserving macro given a choice of derivatives for basic operations. We describe a rich semantics for differentiable programming, based on diffeological spaces. We show that it interprets our language, and we phrase what it means for the AD method to be correct with respect to this semantics. We show that our characterisation of AD gives rise to an elegant semantic proof of its correctness based on a gluing construction on diffeological spaces. We explain how this is, in essence, a logical relations argument. Throughout, we show how the analysis extends to AD methods for computing higher order derivatives using a Taylor approximation.

In the classical transformer attention scheme, we are given three n times d size matrices Q, K, V (the query, key, and value tokens), and the goal is to compute a new n times d size matrix D^{-1} exp(QK^top) V where D = diag( exp(QK^top) {bf 1}_n ). In this work, we study a generalization of attention which captures triple-wise correlations. This generalization is able to solve problems about detecting triple-wise connections that were shown to be impossible for transformers. The potential downside of this generalization is that it appears as though computations are even more difficult, since the straightforward algorithm requires cubic time in n. However, we show that in the bounded-entry setting (which arises in practice, and which is well-studied in both theory and practice), there is actually a near-linear time algorithm. More precisely, we show that bounded entries are both necessary and sufficient for quickly performing generalized computations: bullet On the positive side, if all entries of the input matrices are bounded above by o(sqrt[3]{log n}) then we show how to approximate the ``tensor-type'' attention matrix in n^{1+o(1)} time. bullet On the negative side, we show that if the entries of the input matrices may be as large as Omega(sqrt[3]{log n}), then there is no algorithm that runs faster than n^{3-o(1)} (assuming the Strong Exponential Time Hypothesis from fine-grained complexity theory). We also show that our construction, algorithms, and lower bounds naturally generalize to higher-order tensors and correlations. Interestingly, the higher the order of the tensors, the lower the bound on the entries needs to be for an efficient algorithm. Our results thus yield a natural tradeoff between the boundedness of the entries, and order of the tensor one may use for more expressive, efficient attention computation.

As first-order optimization methods become the method of choice for solving large-scale optimization problems, optimization solvers based on first-order algorithms are being built. Such general-purpose solvers must robustly detect infeasible or misspecified problem instances, but the computational complexity of first-order methods for doing so has yet to be formally studied. In this work, we characterize the optimal accelerated rate of infeasibility detection. We show that the standard fixed-point iteration achieves a O(1/k^2) and O(1/k) rates, respectively, on the normalized iterates and the fixed-point residual converging to the infimal displacement vector, while the accelerated fixed-point iteration achieves O(1/k^2) and mathcal{O}(1/k^2) rates. We then provide a matching complexity lower bound to establish that Theta(1/k^2) is indeed the optimal accelerated rate.

This paper investigates the global convergence of stepsized Newton methods for convex functions. We propose several simple stepsize schedules with fast global convergence guarantees, up to O (k^{-3}), nearly matching lower complexity bounds Omega (k^{-3.5}) of second-order methods. For cases with multiple plausible smoothness parameterizations or an unknown smoothness constant, we introduce a stepsize backtracking procedure that ensures convergence as if the optimal smoothness parameters were known.

This paper provides a non-asymptotic analysis of linear stochastic approximation (LSA) algorithms with fixed stepsize. This family of methods arises in many machine learning tasks and is used to obtain approximate solutions of a linear system Atheta = b for which A and b can only be accessed through random estimates {({bf A}_n, {bf b}_n): n in N^*}. Our analysis is based on new results regarding moments and high probability bounds for products of matrices which are shown to be tight. We derive high probability bounds on the performance of LSA under weaker conditions on the sequence {({bf A}_n, {bf b}_n): n in N^*} than previous works. However, in contrast, we establish polynomial concentration bounds with order depending on the stepsize. We show that our conclusions cannot be improved without additional assumptions on the sequence of random matrices {{bf A}_n: n in N^*}, and in particular that no Gaussian or exponential high probability bounds can hold. Finally, we pay a particular attention to establishing bounds with sharp order with respect to the number of iterations and the stepsize and whose leading terms contain the covariance matrices appearing in the central limit theorems.

Diffusion models have demonstrated remarkable success in generative tasks, yet their iterative denoising process results in slow inference, limiting their practicality. While existing acceleration methods exploit the well-known U-shaped similarity pattern between adjacent steps through caching mechanisms, they lack theoretical foundation and rely on simplistic computation reuse, often leading to performance degradation. In this work, we provide a theoretical understanding by analyzing the denoising process through the second-order Adams-Bashforth method, revealing a linear relationship between the outputs of consecutive steps. This analysis explains why the outputs of adjacent steps exhibit a U-shaped pattern. Furthermore, extending Adams-Bashforth method to higher order, we propose a novel caching-based acceleration approach for diffusion models, instead of directly reusing cached results, with a truncation error bound of only \(O(h^k)\) where h is the step size. Extensive validation across diverse image and video diffusion models (including HunyuanVideo and FLUX.1-dev) with various schedulers demonstrates our method's effectiveness in achieving nearly 3times speedup while maintaining original performance levels, offering a practical real-time solution without compromising generation quality.

We study the problem of efficiently generating differentially private synthetic data that approximate the statistical properties of an underlying sensitive dataset. In recent years, there has been a growing line of work that approaches this problem using first-order optimization techniques. However, such techniques are restricted to optimizing differentiable objectives only, severely limiting the types of analyses that can be conducted. For example, first-order mechanisms have been primarily successful in approximating statistical queries only in the form of marginals for discrete data domains. In some cases, one can circumvent such issues by relaxing the task's objective to maintain differentiability. However, even when possible, these approaches impose a fundamental limitation in which modifications to the minimization problem become additional sources of error. Therefore, we propose Private-GSD, a private genetic algorithm based on zeroth-order optimization heuristics that do not require modifying the original objective. As a result, it avoids the aforementioned limitations of first-order optimization. We empirically evaluate Private-GSD against baseline algorithms on data derived from the American Community Survey across a variety of statistics--otherwise known as statistical queries--both for discrete and real-valued attributes. We show that Private-GSD outperforms the state-of-the-art methods on non-differential queries while matching accuracy in approximating differentiable ones.

In solving partial differential equations (PDEs), Fourier Neural Operators (FNOs) have exhibited notable effectiveness compared to Convolutional Neural Networks (CNNs). This paper presents clear empirical evidence through spectral analysis to elucidate the superiority of FNO over CNNs: FNO is significantly more capable of learning low-frequencies. This empirical evidence also unveils FNO's distinct low-frequency bias, which limits FNO's effectiveness in learning high-frequency information from PDE data. To tackle this challenge, we introduce SpecBoost, an ensemble learning framework that employs multiple FNOs to better capture high-frequency information. Specifically, a secondary FNO is utilized to learn the overlooked high-frequency information from the prediction residual of the initial FNO. Experiments demonstrate that SpecBoost noticeably enhances FNO's prediction accuracy on diverse PDE applications, achieving an up to 71% improvement.

Numerical simulations in climate, chemistry, or astrophysics are computationally too expensive for uncertainty quantification or parameter-exploration at high-resolution. Reduced-order or surrogate models are multiple orders of magnitude faster, but traditional surrogates are inflexible or inaccurate and pure machine learning (ML)-based surrogates too data-hungry. We propose a hybrid, flexible surrogate model that exploits known physics for simulating large-scale dynamics and limits learning to the hard-to-model term, which is called parametrization or closure and captures the effect of fine- onto large-scale dynamics. Leveraging neural operators, we are the first to learn grid-independent, non-local, and flexible parametrizations. Our multiscale neural operator is motivated by a rich literature in multiscale modeling, has quasilinear runtime complexity, is more accurate or flexible than state-of-the-art parametrizations and demonstrated on the chaotic equation multiscale Lorenz96.

Recently, efficient fine-tuning of large-scale pre-trained models has attracted increasing research interests, where linear probing (LP) as a fundamental module is involved in exploiting the final representations for task-dependent classification. However, most of the existing methods focus on how to effectively introduce a few of learnable parameters, and little work pays attention to the commonly used LP module. In this paper, we propose a novel Moment Probing (MP) method to further explore the potential of LP. Distinguished from LP which builds a linear classification head based on the mean of final features (e.g., word tokens for ViT) or classification tokens, our MP performs a linear classifier on feature distribution, which provides the stronger representation ability by exploiting richer statistical information inherent in features. Specifically, we represent feature distribution by its characteristic function, which is efficiently approximated by using first- and second-order moments of features. Furthermore, we propose a multi-head convolutional cross-covariance (MHC^3) to compute second-order moments in an efficient and effective manner. By considering that MP could affect feature learning, we introduce a partially shared module to learn two recalibrating parameters (PSRP) for backbones based on MP, namely MP_{+}. Extensive experiments on ten benchmarks using various models show that our MP significantly outperforms LP and is competitive with counterparts at less training cost, while our MP_{+} achieves state-of-the-art performance.

We study the bi-parameter local linearization of the one-dimensional nonlinear stochastic wave equation driven by a Gaussian noise, which is white in time and has a spatially homogeneous covariance structure of Riesz-kernel type. We establish that the second-order increments of the solution can be approximated by those of the corresponding linearized wave equation, modulated by the diffusion coefficient. These findings extend the previous results of Huang et al. HOO2024, which addressed the case of space-time white noise. As applications, we analyze the quadratic variation of the solution and construct a consistent estimator for the diffusion parameter.

We examine the connections between deterministic, complete, and general global optimisation of continuous functions and a general concept of regression from the perspective of constructive type theory via the concept of 'searchability'. We see how the property of convergence of global optimisation is a straightforward consequence of searchability. The abstract setting allows us to generalise searchability and continuity to higher-order functions, so that we can formulate novel convergence criteria for regression, derived from the convergence of global optimisation. All the theory and the motivating examples are fully formalised in the proof assistant Agda.

Starting from the Mellin-Barnes integral representation of a Feynman integral depending on set of kinematic variables z_i, we derive a system of partial differential equations w.r.t.\ new variables x_j, which parameterize the differentiable constraints z_i=y_i(x_j). In our algorithm, the powers of propagators can be considered as arbitrary parameters. Our algorithm can also be used for the reduction of multiple hypergeometric sums to sums of lower dimension, finding special values and reduction equations of hypergeometric functions in a singular locus of continuous variables, or finding systems of partial differential equations for master integrals with arbitrary powers of propagators. As an illustration, we produce a differential equation of fourth order in one variable for the one-loop two-point Feynman diagram with two different masses and arbitrary propagator powers.

Physics-informed Neural Networks (PINNs) have recently achieved remarkable progress in solving Partial Differential Equations (PDEs) in various fields by minimizing a weighted sum of PDE loss and boundary loss. However, there are several critical challenges in the training of PINNs, including the lack of theoretical frameworks and the imbalance between PDE loss and boundary loss. In this paper, we present an analysis of second-order non-homogeneous PDEs, which are classified into three categories and applicable to various common problems. We also characterize the connections between the training loss and actual error, guaranteeing convergence under mild conditions. The theoretical analysis inspires us to further propose MultiAdam, a scale-invariant optimizer that leverages gradient momentum to parameter-wisely balance the loss terms. Extensive experiment results on multiple problems from different physical domains demonstrate that our MultiAdam solver can improve the predictive accuracy by 1-2 orders of magnitude compared with strong baselines.

We introduce a general formulation for automatic differentiation through direct form filters, yielding a closed-form backpropagation that includes initial condition gradients. The result is a single expression that can represent both the filter and its gradients computation while supporting parallelism. C++/CUDA implementations in PyTorch achieve at least 1000x speedup over naive Python implementations and consistently run fastest on the GPU. For the low-order filters commonly used in practice, exact time-domain filtering with analytical gradients outperforms the frequency-domain method in terms of speed. The source code is available at https://github.com/yoyolicoris/philtorch.

We discuss the numerical solution of initial value problems for varepsilon^2,varphi''+a(x),varphi=0 in the highly oscillatory regime, i.e., with a(x)>0 and 0<varepsilonll 1. We analyze and implement an approximate solution based on the well-known WKB-ansatz. The resulting approximation error is of magnitude O(varepsilon^{N}) where N refers to the truncation order of the underlying asymptotic series. When the optimal truncation order N_{opt} is chosen, the error behaves like O(varepsilon^{-2}exp(-cvarepsilon^{-1})) with some c>0.

The higher-order correlation clustering problem is an expressive model, and recently, local search heuristics have been proposed for several applications. Certifying optimality, however, is NP-hard and practically hampered already by the complexity of the problem statement. Here, we focus on establishing partial optimality conditions for the special case of complete graphs and cubic objective functions. In addition, we define and implement algorithms for testing these conditions and examine their effect numerically, on two datasets.

In this work, we focus on a fractional differential equation in Riesz form discretized by a polynomial B-spline collocation method. For an arbitrary polynomial degree p, we show that the resulting coefficient matrices possess a Toeplitz-like structure. We investigate their spectral properties via their symbol and we prove that, like for second order differential problems, also in this case the given matrices are ill-conditioned both in the low and high frequencies for large p. More precisely, in the fractional scenario the symbol has a single zero at 0 of order α, with α the fractional derivative order that ranges from 1 to 2, and it presents an exponential decay to zero at π for increasing p that becomes faster as α approaches 1. This translates in a mitigated conditioning in the low frequencies and in a deterioration in the high frequencies when compared to second order problems. Furthermore, the derivation of the symbol reveals another similarity of our problem with a classical diffusion problem. Since the entries of the coefficient matrices are defined as evaluations of fractional derivatives of the B-spline basis at the collocation points, we are able to express the central entries of the coefficient matrix as inner products of two fractional derivatives of cardinal B-splines. Finally, we perform a numerical study of the approximation behavior of polynomial B-spline collocation. This study suggests that, in line with non-fractional diffusion problems, the approximation order for smooth solutions in the fractional case is p+2-α for even p, and p+1-α for odd p.

In recent years, a variety of gradient-based first-order methods have been developed to solve bi-level optimization problems for learning applications. However, theoretical guarantees of these existing approaches heavily rely on the simplification that for each fixed upper-level variable, the lower-level solution must be a singleton (a.k.a., Lower-Level Singleton, LLS). In this work, we first design a counter-example to illustrate the invalidation of such LLS condition. Then by formulating BLPs from the view point of optimistic bi-level and aggregating hierarchical objective information, we establish Bi-level Descent Aggregation (BDA), a flexible and modularized algorithmic framework for generic bi-level optimization. Theoretically, we derive a new methodology to prove the convergence of BDA without the LLS condition. Our investigations also demonstrate that BDA is indeed compatible to a verify of particular first-order computation modules. Additionally, as an interesting byproduct, we also improve these conventional first-order bi-level schemes (under the LLS simplification). Particularly, we establish their convergences with weaker assumptions. Extensive experiments justify our theoretical results and demonstrate the superiority of the proposed BDA for different tasks, including hyper-parameter optimization and meta learning.

The lack of adversarial robustness has been recognized as an important issue for state-of-the-art machine learning (ML) models, e.g., deep neural networks (DNNs). Thereby, robustifying ML models against adversarial attacks is now a major focus of research. However, nearly all existing defense methods, particularly for robust training, made the white-box assumption that the defender has the access to the details of an ML model (or its surrogate alternatives if available), e.g., its architectures and parameters. Beyond existing works, in this paper we aim to address the problem of black-box defense: How to robustify a black-box model using just input queries and output feedback? Such a problem arises in practical scenarios, where the owner of the predictive model is reluctant to share model information in order to preserve privacy. To this end, we propose a general notion of defensive operation that can be applied to black-box models, and design it through the lens of denoised smoothing (DS), a first-order (FO) certified defense technique. To allow the design of merely using model queries, we further integrate DS with the zeroth-order (gradient-free) optimization. However, a direct implementation of zeroth-order (ZO) optimization suffers a high variance of gradient estimates, and thus leads to ineffective defense. To tackle this problem, we next propose to prepend an autoencoder (AE) to a given (black-box) model so that DS can be trained using variance-reduced ZO optimization. We term the eventual defense as ZO-AE-DS. In practice, we empirically show that ZO-AE- DS can achieve improved accuracy, certified robustness, and query complexity over existing baselines. And the effectiveness of our approach is justified under both image classification and image reconstruction tasks. Codes are available at https://github.com/damon-demon/Black-Box-Defense.

New geometric methods for fast evaluation of derivatives of polynomial and rational B\'{e}zier curves are proposed. They apply an algorithm for evaluating polynomial or rational B\'{e}zier curves, which was recently given by the authors. Numerical tests show that the new approach is more efficient than the methods which use the famous de Casteljau algorithm. The algorithms work well even for high-order derivatives of rational B\'{e}zier curves of high degrees.

This paper makes a formal connection between two families of widely used matrix factorization algorithms in numerical linear algebra. One family consists of the Jacobi eigenvalue algorithm and its variants for computing the Hermitian eigendecomposition and singular value decomposition. The other consists of Gaussian elimination and the Gram-Schmidt procedure with various pivoting rules for computing the Cholesky decomposition and QR decomposition respectively. Both families are cast as special cases of a more general class of factorization algorithms. We provide a randomized pivoting rule that applies to this general class (which differs substantially from the usual pivoting rules for Gaussian elimination / Gram-Schmidt) which results in the same linear rate of convergence for each algorithm, irrespective of which factorization it computes. A second important consequence of this randomized pivoting rule is a provable, effective bound on the numerical stability of the Jacobi eigenvalue algorithm, which addresses a longstanding open problem of Demmel and Veseli\'c `92.

Automatic differentiation through digital signal processing algorithms for virtual analogue modelling has recently gained popularity. These algorithms are typically more computationally efficient than black-box neural networks that rely on dense matrix multiplications. Due to their differentiable nature, they can be integrated with neural networks and jointly trained using gradient descent algorithms, resulting in more efficient systems. Furthermore, signal processing algorithms have significantly fewer parameters than neural networks, allowing the application of the Newton-Raphson method. This method offers faster and more robust convergence than gradient descent at the cost of quadratic storage. This paper presents a method to emulate analogue levelling amplifiers using a feed-forward digital compressor with parameters optimised via the Newton-Raphson method. We demonstrate that a digital compressor can successfully approximate the behaviour of our target unit, the Teletronix LA-2A. Different strategies for computing the Hessian matrix are benchmarked. We leverage parallel algorithms for recursive filters to achieve efficient training on modern GPUs. The resulting model is made into a VST plugin and is open-sourced at https://github.com/aim-qmul/4a2a.

Language models have made significant progress in generating coherent text and predicting next tokens based on input prompts. This study compares the next-token prediction performance of two well-known models: OpenAI's GPT-2 and Meta's Llama-2-7b-chat-hf on Theory of Mind (ToM) tasks. To evaluate their capabilities, we built a dataset from 10 short stories sourced from the Explore ToM Dataset. We enhanced these stories by programmatically inserting additional sentences (infills) using GPT-4, creating variations that introduce different levels of contextual complexity. This setup enables analysis of how increasing context affects model performance. We tested both models under four temperature settings (0.01, 0.5, 1.0, 2.0) and evaluated their ability to predict the next token across three reasoning levels. Zero-order reasoning involves tracking the state, either current (ground truth) or past (memory). First-order reasoning concerns understanding another's mental state (e.g., "Does Anne know the apple is salted?"). Second-order reasoning adds recursion (e.g., "Does Anne think that Charles knows the apple is salted?"). Our results show that adding more infill sentences slightly reduces prediction accuracy, as added context increases complexity and ambiguity. Llama-2 consistently outperforms GPT-2 in prediction accuracy, especially at lower temperatures, demonstrating greater confidence in selecting the most probable token. As reasoning complexity rises, model responses diverge more. Notably, GPT-2 and Llama-2 display greater variability in predictions during first- and second-order reasoning tasks. These findings illustrate how model architecture, temperature, and contextual complexity influence next-token prediction, contributing to a better understanding of the strengths and limitations of current language models.

As an adaptive method, Shampoo employs a structured second-moment estimation, and its effectiveness has attracted growing attention. Prior work has primarily analyzed its estimation scheme through the Frobenius norm. Motivated by the natural connection between the second moment and a covariance matrix, we propose studying Shampoo's estimation as covariance estimation through the lens of Kullback-Leibler (KL) minimization. This alternative perspective reveals a previously hidden limitation, motivating improvements to Shampoo's design. Building on this insight, we develop a practical estimation scheme, termed KL-Shampoo, that eliminates Shampoo's reliance on Adam for stabilization, thereby removing the additional memory overhead introduced by Adam. Preliminary results show that KL-Shampoo improves Shampoo's performance, enabling it to stabilize without Adam and even outperform its Adam-stabilized variant, SOAP, in neural network pretraining.

Important classes of active matter systems can be modeled using kinetic theories. However, kinetic theories can be high dimensional and challenging to simulate. Reduced-order representations based on tracking only low-order moments of the kinetic model serve as an efficient alternative, but typically require closure assumptions to model unrepresented higher-order moments. In this study, we present a learning framework based on neural networks that exploit rotational symmetries in the closure terms to learn accurate closure models directly from kinetic simulations. The data-driven closures demonstrate excellent a-priori predictions comparable to the state-of-the-art Bingham closure. We provide a systematic comparison between different neural network architectures and demonstrate that nonlocal effects can be safely ignored to model the closure terms. We develop an active learning strategy that enables accurate prediction of the closure terms across the entire parameter space using a single neural network without the need for retraining. We also propose a data-efficient training procedure based on time-stepping constraints and a differentiable pseudo-spectral solver, which enables the learning of stable closures suitable for a-posteriori inference. The coarse-grained simulations equipped with data-driven closure models faithfully reproduce the mean velocity statistics, scalar order parameters, and velocity power spectra observed in simulations of the kinetic theory. Our differentiable framework also facilitates the estimation of parameters in coarse-grained descriptions conditioned on data.

Ongoing efforts that span over decades show a rise of AI methods for accelerating scientific discovery, yet accelerating discovery in mathematics remains a persistent challenge for AI. Specifically, AI methods were not effective in creation of formulas for mathematical constants because each such formula must be correct for infinite digits of precision, with "near-true" formulas providing no insight toward the correct ones. Consequently, formula discovery lacks a clear distance metric needed to guide automated discovery in this realm. In this work, we propose a systematic methodology for categorization, characterization, and pattern identification of such formulas. The key to our methodology is introducing metrics based on the convergence dynamics of the formulas, rather than on the numerical value of the formula. These metrics enable the first automated clustering of mathematical formulas. We demonstrate this methodology on Polynomial Continued Fraction formulas, which are ubiquitous in their intrinsic connections to mathematical constants, and generalize many mathematical functions and structures. We test our methodology on a set of 1,768,900 such formulas, identifying many known formulas for mathematical constants, and discover previously unknown formulas for pi, ln(2), Gauss', and Lemniscate's constants. The uncovered patterns enable a direct generalization of individual formulas to infinite families, unveiling rich mathematical structures. This success paves the way towards a generative model that creates formulas fulfilling specified mathematical properties, accelerating the rate of discovery of useful formulas.

We consider the problem of modeling high-speed flows using machine learning methods. While most prior studies focus on low-speed fluid flows in which uniform time-stepping is practical, flows approaching and exceeding the speed of sound exhibit sudden changes such as shock waves. In such cases, it is essential to use adaptive time-stepping methods to allow a temporal resolution sufficient to resolve these phenomena while simultaneously balancing computational costs. Here, we propose a two-phase machine learning method, known as ShockCast, to model high-speed flows with adaptive time-stepping. In the first phase, we propose to employ a machine learning model to predict the timestep size. In the second phase, the predicted timestep is used as an input along with the current fluid fields to advance the system state by the predicted timestep. We explore several physically-motivated components for timestep prediction and introduce timestep conditioning strategies inspired by neural ODE and Mixture of Experts. As ShockCast is the first framework for learning high-speed flows, we evaluate our methods by generating two supersonic flow datasets, available at https://huggingface.co/datasets/divelab. Our code is publicly available as part of the AIRS library (https://github.com/divelab/AIRS).

We study first-order methods with preconditioning for solving structured nonlinear convex optimization problems. We propose a new family of preconditioners generated by symmetric polynomials. They provide first-order optimization methods with a provable improvement of the condition number, cutting the gaps between highest eigenvalues, without explicit knowledge of the actual spectrum. We give a stochastic interpretation of this preconditioning in terms of coordinate volume sampling and compare it with other classical approaches, including the Chebyshev polynomials. We show how to incorporate a polynomial preconditioning into the Gradient and Fast Gradient Methods and establish the corresponding global complexity bounds. Finally, we propose a simple adaptive search procedure that automatically chooses the best possible polynomial preconditioning for the Gradient Method, minimizing the objective along a low-dimensional Krylov subspace. Numerical experiments confirm the efficiency of our preconditioning strategies for solving various machine learning problems.

This paper focuses on the problem of constrained stochastic optimization. A zeroth order Frank-Wolfe algorithm is proposed, which in addition to the projection-free nature of the vanilla Frank-Wolfe algorithm makes it gradient free. Under convexity and smoothness assumption, we show that the proposed algorithm converges to the optimal objective function at a rate Oleft(1/T^{1/3}right), where T denotes the iteration count. In particular, the primal sub-optimality gap is shown to have a dimension dependence of Oleft(d^{1/3}right), which is the best known dimension dependence among all zeroth order optimization algorithms with one directional derivative per iteration. For non-convex functions, we obtain the Frank-Wolfe gap to be Oleft(d^{1/3}T^{-1/4}right). Experiments on black-box optimization setups demonstrate the efficacy of the proposed algorithm.

Graph Neural Networks (GNNs) with equivariant properties have emerged as powerful tools for modeling complex dynamics of multi-object physical systems. However, their generalization ability is limited by the inadequate consideration of physical inductive biases: (1) Existing studies overlook the continuity of transitions among system states, opting to employ several discrete transformation layers to learn the direct mapping between two adjacent states; (2) Most models only account for first-order velocity information, despite the fact that many physical systems are governed by second-order motion laws. To incorporate these inductive biases, we propose the Second-order Equivariant Graph Neural Ordinary Differential Equation (SEGNO). Specifically, we show how the second-order continuity can be incorporated into GNNs while maintaining the equivariant property. Furthermore, we offer theoretical insights into SEGNO, highlighting that it can learn a unique trajectory between adjacent states, which is crucial for model generalization. Additionally, we prove that the discrepancy between this learned trajectory of SEGNO and the true trajectory is bounded. Extensive experiments on complex dynamical systems including molecular dynamics and motion capture demonstrate that our model yields a significant improvement over the state-of-the-art baselines.

We study the effect of mini-batching on the loss landscape of deep neural networks using spiked, field-dependent random matrix theory. We demonstrate that the magnitude of the extremal values of the batch Hessian are larger than those of the empirical Hessian. We also derive similar results for the Generalised Gauss-Newton matrix approximation of the Hessian. As a consequence of our theorems we derive an analytical expressions for the maximal learning rates as a function of batch size, informing practical training regimens for both stochastic gradient descent (linear scaling) and adaptive algorithms, such as Adam (square root scaling), for smooth, non-convex deep neural networks. Whilst the linear scaling for stochastic gradient descent has been derived under more restrictive conditions, which we generalise, the square root scaling rule for adaptive optimisers is, to our knowledge, completely novel. %For stochastic second-order methods and adaptive methods, we derive that the minimal damping coefficient is proportional to the ratio of the learning rate to batch size. We validate our claims on the VGG/WideResNet architectures on the CIFAR-100 and ImageNet datasets. Based on our investigations of the sub-sampled Hessian we develop a stochastic Lanczos quadrature based on the fly learning rate and momentum learner, which avoids the need for expensive multiple evaluations for these key hyper-parameters and shows good preliminary results on the Pre-Residual Architecure for CIFAR-100.

Deep Neural Networks (DNNs) have been a large driver and enabler for AI breakthroughs in recent years. These models have been getting larger in their attempt to become more accurate and tackle new upcoming use-cases, including AR/VR and intelligent assistants. However, the training process of such large models is a costly and time-consuming process, which typically yields a single model to fit all targets. To mitigate this, various techniques have been proposed in the literature, including pruning, sparsification or quantization of the model weights and updates. While able to achieve high compression rates, they often incur computational overheads or accuracy penalties. Alternatively, factorization methods have been leveraged to incorporate low-rank compression in the training process. Similarly, such techniques (e.g.,~SVD) frequently rely on the computationally expensive decomposition of layers and are potentially sub-optimal for non-linear models, such as DNNs. In this work, we take a further step in designing efficient low-rank models and propose Maestro, a framework for trainable low-rank layers. Instead of regularly applying a priori decompositions such as SVD, the low-rank structure is built into the training process through a generalized variant of Ordered Dropout. This method imposes an importance ordering via sampling on the decomposed DNN structure. Our theoretical analysis demonstrates that our method recovers the SVD decomposition of linear mapping on uniformly distributed data and PCA for linear autoencoders. We further apply our technique on DNNs and empirically illustrate that Maestro enables the extraction of lower footprint models that preserve model performance while allowing for graceful accuracy-latency tradeoff for the deployment to devices of different capabilities.

The correct use of model evaluation, model selection, and algorithm selection techniques is vital in academic machine learning research as well as in many industrial settings. This article reviews different techniques that can be used for each of these three subtasks and discusses the main advantages and disadvantages of each technique with references to theoretical and empirical studies. Further, recommendations are given to encourage best yet feasible practices in research and applications of machine learning. Common methods such as the holdout method for model evaluation and selection are covered, which are not recommended when working with small datasets. Different flavors of the bootstrap technique are introduced for estimating the uncertainty of performance estimates, as an alternative to confidence intervals via normal approximation if bootstrapping is computationally feasible. Common cross-validation techniques such as leave-one-out cross-validation and k-fold cross-validation are reviewed, the bias-variance trade-off for choosing k is discussed, and practical tips for the optimal choice of k are given based on empirical evidence. Different statistical tests for algorithm comparisons are presented, and strategies for dealing with multiple comparisons such as omnibus tests and multiple-comparison corrections are discussed. Finally, alternative methods for algorithm selection, such as the combined F-test 5x2 cross-validation and nested cross-validation, are recommended for comparing machine learning algorithms when datasets are small.

Foundations of a new projection-based model reduction approach for convection dominated nonlinear fluid flows are summarized. In this method the evolution of the flow is approximated in the Lagrangian frame of reference. Global basis functions are used to approximate both the state and the position of the Lagrangian computational domain. It is demonstrated that in this framework, certain wave-like solutions exhibit low-rank structure and thus, can be efficiently compressed using relatively few global basis. The proposed approach is successfully demonstrated for the reduction of several simple but representative problems.

Several first order stochastic optimization methods commonly used in the Euclidean domain such as stochastic gradient descent (SGD), accelerated gradient descent or variance reduced methods have already been adapted to certain Riemannian settings. However, some of the most popular of these optimization tools - namely Adam , Adagrad and the more recent Amsgrad - remain to be generalized to Riemannian manifolds. We discuss the difficulty of generalizing such adaptive schemes to the most agnostic Riemannian setting, and then provide algorithms and convergence proofs for geodesically convex objectives in the particular case of a product of Riemannian manifolds, in which adaptivity is implemented across manifolds in the cartesian product. Our generalization is tight in the sense that choosing the Euclidean space as Riemannian manifold yields the same algorithms and regret bounds as those that were already known for the standard algorithms. Experimentally, we show faster convergence and to a lower train loss value for Riemannian adaptive methods over their corresponding baselines on the realistic task of embedding the WordNet taxonomy in the Poincare ball.

The Reduced Basis Method can be exploited in an efficient way only if the so-called affine dependence assumption on the operator and right-hand side of the considered problem with respect to the parameters is satisfied. When it is not, the Empirical Interpolation Method is usually used to recover this assumption approximately. In both cases, the Reduced Basis Method requires to access and modify the assembly routines of the corresponding computational code, leading to an intrusive procedure. In this work, we derive variants of the EIM algorithm and explain how they can be used to turn the Reduced Basis Method into a nonintrusive procedure. We present examples of aeroacoustic problems solved by integral equations and show how our algorithms can benefit from the linear algebra tools available in the considered code.

Convolution is a broadly useful operation with applications including signal processing, machine learning, probability, optics, polynomial multiplication, and efficient parsing. Usually, however, this operation is understood and implemented in more specialized forms, hiding commonalities and limiting usefulness. This paper formulates convolution in the common algebraic framework of semirings and semimodules and populates that framework with various representation types. One of those types is the grand abstract template and itself generalizes to the free semimodule monad. Other representations serve varied uses and performance trade-offs, with implementations calculated from simple and regular specifications. Of particular interest is Brzozowski's method for regular expression matching. Uncovering the method's essence frees it from syntactic manipulations, while generalizing from boolean to weighted membership (such as multisets and probability distributions) and from sets to n-ary relations. The classic trie data structure then provides an elegant and efficient alternative to syntax. Pleasantly, polynomial arithmetic requires no additional implementation effort, works correctly with a variety of representations, and handles multivariate polynomials and power series with ease. Image convolution also falls out as a special case.

We made a comparative analysis of numerical methods for multidimensional optimization. The main parameter is a number of computations of the test function to reach necessary accuracy, as it is computationally "slow". For complex functions, analytic differentiation by many parameters can cause problems associated with a significant complication of the program and thus slowing its operation. For comparison, we used the methods: "brute force" (or minimization on a regular grid), Monte Carlo, steepest descent, conjugate gradients, Brent's method (golden section search), parabolic interpolation etc. The Monte-Carlo method was applied to the eclipsing binary system AM Leo.

Optimization with matrix gradient orthogonalization has recently demonstrated impressive results in the training of deep neural networks (Jordan et al., 2024; Liu et al., 2025). In this paper, we provide a theoretical analysis of this approach. In particular, we show that the orthogonalized gradient method can be seen as a first-order trust-region optimization method, where the trust-region is defined in terms of the matrix spectral norm. Motivated by this observation, we develop the stochastic non-Euclidean trust-region gradient method with momentum, which recovers the Muon optimizer (Jordan et al., 2024) as a special case, along with normalized SGD and signSGD with momentum (Cutkosky and Mehta, 2020; Sun et al., 2023). In addition, we prove state-of-the-art convergence results for the proposed algorithm in a range of scenarios, which involve arbitrary non-Euclidean norms, constrained and composite problems, and non-convex, star-convex, first- and second-order smooth functions. Finally, our theoretical findings provide an explanation for several practical observations, including the practical superiority of Muon compared to the Orthogonal-SGDM algorithm of Tuddenham et al. (2022) and the importance of weight decay in the training of large-scale language models.

Approximating nonlinear differential equations using a neural network provides a robust and efficient tool for various scientific computing tasks, including real-time predictions, inverse problems, optimal controls, and surrogate modeling. Previous works have focused on embedding dynamical systems into networks through two approaches: learning a single solution operator (i.e., the mapping from input parametrized functions to solutions) or learning the governing system of equations (i.e., the constitutive model relative to the state variables). Both of these approaches yield different representations for the same underlying data or function. Additionally, observing that families of differential equations often share key characteristics, we seek one network representation across a wide range of equations. Our method, called Predicting Operators and Symbolic Expressions (PROSE), learns maps from multimodal inputs to multimodal outputs, capable of generating both numerical predictions and mathematical equations. By using a transformer structure and a feature fusion approach, our network can simultaneously embed sets of solution operators for various parametric differential equations using a single trained network. Detailed experiments demonstrate that the network benefits from its multimodal nature, resulting in improved prediction accuracy and better generalization. The network is shown to be able to handle noise in the data and errors in the symbolic representation, including noisy numerical values, model misspecification, and erroneous addition or deletion of terms. PROSE provides a new neural network framework for differential equations which allows for more flexibility and generality in learning operators and governing equations from data.