Softmax Linear Units

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1. Introduction
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As Transformer generative models continue to gain real-world adoption , it becomes ever more important to ensure they behave predictably and safely, in both the short and long run.  Mechanistic interpretability – the project of attempting to reverse engineer neural networks into understandable computer programs – offers one possible avenue for addressing these safety issues: by understanding the internal structures that cause neural networks to produce the outputs they do, it may be possible to address current safety problems more systematically as well as anticipating future safety problems.

Until recently mechanistic interpretability has focused primarily on CNN vision models , but some recent efforts have begun to explore mechanistic interpretability for transformer language models .  Notably, we were able to reverse-engineer 1 and 2 layer attention-only transformers  and we used empirical evidence to draw indirect conclusions about in-context learning in arbitrarily large models .

Unfortunately, it has so far been difficult to mechanistically understand large models due to the difficulty of understanding their MLP (feedforward) layers.  This failure to understand and interpret MLP layers appears to be a major blocker to further progress. The underlying issue is that many neurons appear to be polysemantic , responding to multiple unrelated features. Polysemanticity has been observed before in vision models, but seems especially severe in standard transformer language models.  One plausible explanation for polysemanticity is the superposition hypothesis , which suggests that neural network layers have more features than neurons as part of a “sparse coding” strategy to simulate a much larger layer.  If true, this would make polysmenticity a functionally important property and thus especially difficult to remove without damaging ML performance.

In this paper, we report an architectural change which appears to substantially increase the fraction of MLP neurons which appear to be "interpretable" (i.e. respond to an articulable property of the input), at little to no cost to ML performance. Specifically, we replace the activation function with a softmax linear unit (which we term SoLU) and show that this significantly increases the fraction of neurons in the MLP layers which seem to correspond to readily human-understandable concepts, phrases, or categories on quick investigation, as measured by randomized and blinded experiments. We then study our SoLU models and use them to gain several new insights about how information is processed in transformers.  However, we also discover some evidence that the superposition hypothesis is true and there is no free lunch: SoLU may be making some features more interpretable by “hiding” others and thus making them even more deeply uninterpretable.  Despite this, SoLU still seems like a net win, as in practical terms it substantially increases the fraction of neurons we are able to understand.

Although preliminary, we argue that these results show the potential for a general approach of designing architectures for mechanistic interpretability: there may exist many different models or architectures which all achieve roughly state-of-the-art performance, but which differ greatly in how easy they are to reverse engineer. Put another way, we are in the curious position of being both reverse engineers trying to understand the algorithms neural network parameters implement, and also the hardware designers deciding the network architecture they must run on: perhaps we can exploit this second role to support the first. If so, it may be possible to move the field in a positive direction by discovering (and advocating for) those architectures which are most amenable to reverse engineering.

This paper is organized as follows. In [Section 2](#section-2), we give an overview of our key results. In [Section 3](#section-3), we provide background on mechanistic interpretability, the role of interpretable neurons, the challenge of polysemanticity and the superposition hypothesis. In [Section 4](#section-4) we motivate and introduce SoLU. In [Section 5](#section-5) we present experimental results showing that SoLU gives performance roughly equivalent to standard transformers, as measured by loss and downstream evaluations.  In [Section 6](#section-6) we run the experiments showing that SoLU leads to MLP neurons that are easier to interpret, and also present several interpretability discoveries that we were able to make with SoLU models and could not make without them.  [Section 7](#section-7) reviews related work, and [Section 8](#section-8) discusses the bigger picture and possible future directions.

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2. Key Results
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SoLU increases the fraction of MLP neurons which appear to have clear interpretations, while preserving performance. Specifically, SoLU increases the fraction of MLP neurons for which a human can quickly find a clear hypothesis explaining its activations from 35% to 60%, as measured by blinded experiments – although the gain is smaller for our largest models (see [Section 6.2](#section-6-2)).  This gain is achieved without any loss in performance: test loss and NLP evals are approximately the same for SoLU and non-SoLU models (see [Section 5](#section-5)) .

SoLU’s benefits may come at the cost of “hiding” other features.  Despite the benefits mentioned above, SoLU is potentially a double-edged sword. We find theoretical and empirical evidence that it may “hide” some non-neuron-aligned features by decreasing their magnitude and then later recovering it with LayerNorm (see Sections [4.3](#section-4-3) and [Section 6.4](#section-6-4)) .  In other words, SoLU causes some previously non-interpretable features to become interpretable, but it may also make it even harder to interpret some already non-interpretable features.  On balance, however, it still seems like a win in that it pragmatically increases our understanding.

Architecture affects polysemanticity and MLP interpretability. Although it isn't a perfect solution, SoLU is a proof of concept that architectural decisions can dramatically affect polysemanticity, making it more tractable to understand transformer MLP layers. This suggests that exploring how other architectures affect polysemanticity could be a fruitful line of further attack. More generally, it suggests that designing models for mechanistic interpretability – picking architectures we expect to be easier to reverse engineer – may be a valuable direction.

An overview of the types of features which exist in MLP layers. SoLU seems to make some of the features in all layers easily interpretable. Prior to this, we'd found it very difficult to get traction on rigorously understanding features in MLP layers. In particular, despite significant effort, we made very little progress understanding the first MLP layer in any model. Simply having a sense of what kinds of features to expect in different layers was a powerful tool in reverse engineering models in the original circuits thread , and this moves us in a similar direction. We find that early features often deal with mapping raw tokens to semantic meaning (e.g. dealing with multi-token words, or tokens in different languages), more abstract features in middle layers, and features involved in mapping abstract concepts back to raw tokens in late layers. Detailed discussion can be found in [Section 6.3](#section-6-3).

Evidence for the superposition hypothesis. Very little is known about why polysemanticity occurs. In the mechanistic interpretability community, superposition is often treated as the default hypothesis simply because it seems intuitively more compelling than other explanations, but there is little evidence. Our SoLU results seem like moderate evidence for preferring the superposition hypothesis over alternatives.

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3. Background
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Before presenting the SoLU results, it is worth going through why understanding the MLP’s in transformer language models is hard, and specifically why the superposition hypothesis is plausible and thus why polysemanticity might be difficult to avoid.

### 3.1 The Importance of Understanding Activations

First of all, why is it even important to understand neurons/activations?  Previous work on language model mechanistic interpretability was (for example) able to discover induction heads without needing to understand activations.  And ultimately, don’t we only need to understand the parameters, which provide a complete description of the neural net?

A useful analogy might be to think of the parameters as a compiled computer program that we’re trying to understand, and the activations as variables in that program.  Just as a line of code in a computer program only makes sense if you understand what the variables represent, a parameter in a neural network can only be understood if you understand the activations it links together. This idea was originally articulated by Voss et al. , and is described in more depth in a [informal note on intuition](https://transformer-circuits.pub/2022/mech-interp-essay/index.html) accompanying this paper.  Concretely, there are many more parameters than activations, so the activations seem like a more likely “key” to what’s going on.

There are special cases where it's possible to side-step understanding activations, by rewriting a neural network into an equivalent model that doesn't make reference to intermediate activations. This is how we were able to reverse engineer attention-only transformers previously. However, the non-linear structure of MLP layers is not amenable to such tricks: if we want to understand transformers with MLP layers, it appears we must figure out how to understand what the activations of MLP layers encode.

### 3.2 Decomposing Activations and the Role of Bases

To get to polysemanticity and the superposition hypothesis, it’s first useful to talk about bases in neural network layers. The vector space of a neural network layer’s activations is called the "representation." For toy low-dimensional neural networks, it may be possible to explicitly visualize or analyze this space . But as the dimensionality increases, the curse of dimensionality takes hold and the volume of the space exponentially increases. The only path we see to fully understand such a representation is to decompose it into independently understandable components, which we'll call features. Finding such a decomposition is the difference between needing to understand N features and \exp(N) representational volume. (This might be seen as similar to how, in reverse engineering a computer program, we don't just think of the program's state space as a high-dimensional vector: we decompose it into a set of variables representing different things.)

One approach would be to search for a meaningful basis (or meaningful directions that might be part of a basis). This approach is often taken in the context of word embeddings (e.g. ), although also in other contexts (e.g. ). For word embeddings, there doesn't appear to be an alternative: word embeddings generally have what we call a [non-privileged basis](https://transformer-circuits.pub/2021/framework/index.html#def-privileged-basis) , since it can be freely rotated. If a representation, like a word embedding, is surrounded by purely linear operations such as matrix multiplies or addition, then we can “change basis” by applying any invertible matrix M with the matrix multiply before the layer and M^{-1} with the matrix multiply after, which leaves the final output invariant but changes the specific activations. As a result, such representations don't come with any "special basis" which might hint at how to understand them. The correct basis must be discovered. For example, in a word embedding, one might define a gender direction by subtracting "man" and "woman" .

In contrast, many neural networks have some representations with a [privileged basis](https://transformer-circuits.pub/2021/framework/index.html#def-privileged-basis) . In these representations, something about the network makes the default basis special. For example, if the layer has a coordinate-wise non-linear activation function (eg. ReLU), this “breaks the symmetry," distinguishing the specific basis of the activations as the unique basis in which the nonlinearity is applied. This doesn't guarantee that features will align with the basis, but it makes it plausible. In many ways, this is the ideal outcome if possible: not only does it allow us to side-step the difficult question of how to find a meaningful basis, but mechanistically reasoning about neural networks is easier when the basis one is reasoning in aligns cleanly with computation like activation functions.

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In transformers, the token embeddings, residual stream, and attention vectors are non-privileged, while MLP layer activations are privileged.

### 3.3 Neurons and Polysemanticity

We call the dimensions of a representation with a privileged basis "neurons." We often find neurons which map extremely cleanly to clear concepts. In the context of vision, these have ranged from low-level neurons like [curve detectors](https://distill.pub/2020/circuits/curve-detectors/) and [high-low frequency detectors](https://distill.pub/2020/circuits/frequency-edges/), to more complex neurons like [oriented dog-head detectors](https://distill.pub/2020/circuits/zoom-in/#claim-2-dog) or [car detectors](https://distill.pub/2020/circuits/zoom-in/#claim-2-superposition), to extremely abstract neurons corresponding to [famous people](https://distill.pub/2021/multimodal-neurons/#person-neurons), [emotions](https://distill.pub/2021/multimodal-neurons/#emotion-neurons), [geographic regions](https://distill.pub/2021/multimodal-neurons/#region-neurons), and [more](https://distill.pub/2021/multimodal-neurons/#guided-tour-of-neuron-families) . The claim that some neurons really do correspond to interpretable features is crucial to what kinds of interpretability research make sense, so it's worth noting that these interpretations aren't just casual claims made on superficial evidence. In some cases, these interpretations have held up to detailed investigation: Cammarata et al.  spend two papers investigating a handful of curve detector neurons and the circuits that implement them, using seven different lines of evidence to corroborate that the neurons really are curve detectors, with the goal of dispositively establishing that at least some neurons really are interpretable.

However, there are also many neurons which don't appear to correspond to understandable concepts – and we’ve found this to be especially true in transformer language models. One possibility is that these are in some sense alien features: they actually are the true features and they're just difficult for humans to understand (see  and discussion ). Sometimes features which are initially incomprehensible become obvious once the right hypothesis is proposed (e.g. ), so it's certainly possible! But many of these neurons appear to respond to several unrelated but individually understandable features, such as a neuron which responds to cat heads, fronts of cars, and paws. While we can't totally rule out that there isn't some deep commonality between a cat's paw and the front of a car, it seems like the simpler explanation is that the network has grouped several unrelated features together. We call these [polysemantic neurons](https://distill.pub/2020/circuits/zoom-in/#claim-1-polysemantic) .

Note that polysemanticity is what one would expect to observe if features weren't actually aligned with the privileged basis. But why wouldn't the features align with the neurons? While it could simply be chance, there's an alternative option: the superposition hypothesis .

### 3.4 The Superposition Hypothesis

Roughly, the idea behind the superposition hypothesis is that neural networks "want to represent more features than they have neurons," so they exploit a property of high-dimensional spaces to simulate a model with many more neurons. (Note that as a matter of terminology we use "polysemanticity" to refer to the empirical phenomenon of neurons responding to multiple features, and "superposition" to refer to the hypothesis described here.)

If true, the superposition hypothesis means there is no basis in which activations are interpretable: searching for an interpretable basis is fundamentally the wrong framing. Especially important features might get dedicated neurons, but most features don't align with neurons because they need to share and can't have a dedicated one.

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This section isn't a formal argument for the superposition hypothesis, but it's worth trying to sketch out the intuition for why it might be plausible. We start with the following intuitions about neural networks and features:

* Neural networks represent features as directions in activation space. Since neural networks are primarily built from matrix multiplications, it is both easier to construct a feature by embedding it as a direction and easier to use it (rather than some non-linear representation).
* There are far more possible features than neurons. These features vary in importance. For example, in the context of language, every person who lives or has lived could theoretically come up in text, and our models don't have billions of neurons. But there's a wide variety in how famous and influential people are, and how much they influence the text around them (the feature importance).
* Since there are a large number of parameters associated with every neuron, the most efficient way to encode many facts in parameters may not align with neurons. Language models in particular are incentivized to store large amounts of information in their parameters. Having a neuron corresponding to a single feature may "waste" parameters that could be used to store additional facts.
* Features are sparse.  Continuing the example of people as potential features from above, most tokens in text or positions in an image don't contain any given person (or even a person at all). This is the same observation as the sparsity of features in natural image statistics that motivates classic work on sparse coding (see e.g. ).

We can further combine these intuitions with the following ideas from mathematics:

* Almost Orthogonal Vectors. Although it's only possible to have n orthogonal vectors in an n-dimensional space, it's possible to have \exp(n) many "almost orthogonal" (<\epsilon cosine similarity) vectors in high-dimensional spaces. See the [Johnson–Lindenstrauss lemma](https://en.wikipedia.org/wiki/Johnson%E2%80%93Lindenstrauss_lemma) .
* Compressed sensing. In general, if one projects a vector into a lower-dimensional space, one can't reconstruct the original vector. However, this changes if one knows that the original vector is sparse. In this case, it is often possible to recover the original vector. This situation is studied by a line of research called [compressed sensing](https://en.wikipedia.org/wiki/Compressed_sensing).

Together, these give us the basic ingredients for the superposition hypothesis. Ideally, networks could achieve a lower loss if they could represent more features. The number of features they can represent as orthogonal direction is limited by the number of neurons. However, it may be the case that representing more features is worth the cost of having "interference" between them because they aren't exactly orthogonal, especially if sparsity means that this interference is uncommon.

That is, a small neural network may be able to approximately "simulate" a sparse larger model, at the cost of some "interference" (figure below). And if it’s the case that the underlying data the model is trying to represent genuinely has a lot of sparse features, then this may be the best thing for the model to do.

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To be clear, the presence of nonlinear activation functions (the “privileged basis”) does create an incentive for features to align with this basis and not get superposed.  But if the gains to sparse coding are large enough, this incentive will get overwhelmed.  And when there isn’t a privileged basis (as in word embeddings and residual streams), we should expect the pressure for superposition to be even stronger.

### 3.5 What Can We Do About Superposition?

If we believe the superposition hypothesis, what should we do if we want to understand models? Broadly, there are two approaches:

* Create models with less superposition. Perhaps by making the right architectural decisions, regularizations, or making other design choices, we can reduce the need for superposition in neural networks.
* Find a way to understand representations with superposition.  An alternative to avoiding polysemanticity is to accept that models will have it and try to infer the higher-dimensional structure we believe exists in activation vectors, possibly using ideas from compressed sensing.

This paper will focus on the first approach, creating models with less superposition. Our intuition is that if it's possible to avoid superposition at training time, that would be easier than trying to deal with superposition after the fact. In the next section, we will introduce SoLU, an activation function designed to reduce polysemanticity and superposition in models.

---

4. SoLU: Designing for Interpretability
---------------------------------------

The goal of mechanistic interpretability is to reverse engineer neural networks. But we aren't just the reverse engineers – we're also the hardware designers. Just as a computer program might be easier to reverse engineer if it makes use of special CPU instructions designed for a particular use case, the right neural network architecture may make neural networks easier to reverse engineer.

We can apply this line of thinking to our present challenge. We need to understand MLP layer activations, but this is difficult because transformer MLP neurons are often very polysemantic, possibly due to feature superposition. And so the question is, how can we create a neural network architecture which will encourage features to align with neurons, and discourage polysemanticity?

### 4.1 Properties that May Reduce Polysemanticity

Transformer MLP layers are not designed to avoid polysemanticity. As a result, there are quite a few architectural properties that could plausibly reduce polysemanticity and haven't really been explored. We’re aware that decreasing polysemanticity might harm performance (due to the superposition hypothesis), but tactically speaking it makes sense to look for ways to decrease polysemanticity, and then see if we can find any that don’t harm performance.  Although we won't try all of these in this paper, here are a few potential ways to decrease polysemanticity, along with argument for why they may help:

* Activation Sparsity: Polysemantic neurons must fire more often, since they represent multiple things. Additionally, for a polysemantic neuron to be useful, the activations of other neurons must disambiguate it. This means that polysemanticity requires more neurons to fire at the same time. Put another way, superposition requires a gap between the sparsity of the underlying features and the sparsity of the neurons representing them. This suggests that encouraging activation sparsity may make it harder for neurons to be polysemantic. This sparsity could be achieved by using a different activation function, or by using a regularizer. (It's also worth noting that sparsity needs to be defined with respect to a basis; this is a very general argument for why it's plausible that any kind of sparsity might encourage a privileged basis.)
* Lateral Inhibition / Co-Occurrence Sparsity: It's possible to design activation functions (e.g. softmax), where one neuron firing reduces the amount other neurons fire. This encourages a very specific kind of (approximate) activation sparsity which one might call "co-occurrence sparsity." It isn't just that on average the neurons are sparse, it's that in any particular instance, relatively few neurons fire simultaneously. This may be an even stronger way to discourage polysemanticity, by making it hard for other neurons to disambiguate a neuron's meaning. It's important to think through the exact mechanism of lateral inhibition to ensure that the architecture is truly creating a situation where a few neurons "win" and sparsity emerges, rather than all neurons inhibiting each other resulting in highly diffuse small activations.
* Weight Sparsity: If one imagines two features in adjacent layers of a neural network, it seems natural to think there's some "ideal weight" between them. If every feature corresponds to a neuron (as in the imagined ideal model in [Section 3.4](#section-3-4)), this ideal weight would simply correspond to an entry in the weight matrix. But if one or both of the features is represented in a non-neuron aligned way, the ideal weight will start to be "smeared out" across many entries in the weight matrix. As a result, we should expect layers with polysemantic neurons to have more small weights, and layers where features are aligned with the basis to have a smaller number of "more concentrated" large sparse weights. This means that weight sparsity may be a way to discourage polysemanticity and encourage a privileged basis. However, it seems to us that weight sparsity only makes sense when both the input and output are privileged bases. In a transformer, all weights have the residual stream as either an input or output, and in our conceptualization of transformers, the residual stream is necessarily polysemantic and not basis aligned. As a result, we don't believe weight sparsity is suitable for a standard transformer architecture.
* Superlinear Activation Functions: A superlinear activation function is one where increasing the pre-activation value of a neuron increases its activation at a greater than linear rate while in the activating regime, holding other neurons fixed. Examples include \text{ReLU}(x)^2, \text{softmax}(x)\*x, and \exp(x). Consider what happens if you spread a feature over N neurons with this activation function. Since f\left(\frac{x}{\sqrt{N}}\right) < \frac{f(x)}{\sqrt{N}}, spreading a feature across neurons will automatically "shrink" it, requiring either larger activations or larger output weights. This makes it harder for non-basis aligned features to co-exist with basis-aligned ones. It also makes it worse for a polysemantic neural network if two features overlapping in the same neuron co-occur, since f(a+b) > f(a)+f(b) means the "interfrence" between the features would be greater than the sum of its parts.
* Change Neurons per FLOP / param: If one accepts the superposition hypothesis, the reason we have polysemanticity is that there aren't enough neurons for all the features the model would ideally like to represent. Unfortunately, naively making models bigger may not fix this, since more capable models may want to represent more features. Instead, we want to create more neurons without making models larger. Some architectural approaches may allow for this. (In some ways this seems like the most attractive way to reduce polysemanticity, if the hypothesis is right, since it's "giving the neural net what it wants" rather than "forcing it to do something it doesn't want.")

### 4.2 The SoLU Activation Function

It turns out that several of these properties – lateral inhibition, as well as approximate sparsity and superlinearity – can be achieved with a relatively simple change to the MLP activation function.

Modern transformers often use the GeLU activation function. Recall that GeLU is approximated closely by \text{sigmoid}(1.7x)\*x. What if we replaced sigmoid with softmax, its natural extension from binary to multivariate probabilities? We call this activation function a "softmax linear unit" or SoLU:

\text{SoLU}(x) = x \* \text{softmax}(x)

To see why this may discourage polysemanticity and superposition, it's helpful to consider a few examples. Firstly, when SoLU is applied to a vector of large and small values, the large values will suppress smaller values:

\text{SoLU}(4,1,4,1) ~\approx~ (2,0,2,0)

Perhaps more importantly, large basis aligned vectors are preserved, while a feature spread across many dimensions will be suppressed to a smaller magnitude:

\text{SoLU}(4,0,0,0) ~\approx~ (4,0,0,0) \text{SoLU}(1,1,1,1) ~\approx~ \left(\frac{1}{4},\frac{1}{4},\frac{1}{4},\frac{1}{4}\right)

### 4.3 LayerNorm

Our preliminary experiments found that simply using a SoLU activation function seemed to make neurons much more interpretable, but came at a major performance cost. Generally, SoLU models without any other changes had performance equivalent to a model 30-50% smaller than their actual size, with larger models being affected more.  This is exactly what we’d expect to see if the superposition hypothesis was true – we can decrease polysemanticity, but doing so harms the network’s ML performance.

However, we found empirically that this performance penalty can be fixed, while also preserving the interpretability gains, by applying an extra LayerNorm after the SoLU, similar to .  This greatly improves ML performance, so for the majority of our experiments the function we actually apply isNote however that the activations we try to interpret are those before the extra layer norm, not after.:

f(x) = \text{LN}(\text{SoLU}(x)) = \text{LN}(x \* \text{softmax}(x))

We originally added LayerNorm on the intuition that it might fix issues with activation scale and improve optimization. Unfortunately, we now believe that at least part of the reason for the performance improvement is the extra LayerNorm may allow superposition to be smuggled through in smaller activations.  However, under this theory, the combined operation would still tend push at least some features to single neurons with large activations, potentially allowing increased interpretability to coexist with superposition.

We'll discuss this empirically later, but for now note that LayerNorm is invariant to scaling the input, since \text{LN}(x') divides by \sigma(x') and \sigma(\alpha x') = \alpha \sigma(x'). This means that if an entire vector is small because it was very distributed and SoLU suppressed it, it will be rescaled to be larger.

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)
More generally, it means that the denominator of softmax has no effect on the final behavior of the model (although it does change the activations we observe pre-LayerNorm). Training a model with an exponential activation would be identical if we ignored intermediate activations:

\text{LN}(\text{SoLU}(x)) ~=~ \text{LN}\left(x\frac{\exp(x)}{\sum\_i \exp(x\_i)}\right) ~=~ \text{LN}\left(x \* \exp(x)\right)

### 4.4 Parallelism Implementation Details

Our larger models are trained using tensor parallelism, such that MLP activations are never present on a single accelerator. For those models, we split both the softmax and the layer norm to act over a subset of dimensions, allowing each processor to operate locally without additional communication. We report results for these "blocked" models, but in our informal experiments, this blocking does not appear to have a substantial effect on either ML performance or our interpretability results.

---

5. Results on Performance
-------------------------

In this section we confirm that SoLU (the version with LayerNorm) has comparable ML performance to a baseline model.  This is important because interpretability changes are unlikely to be widely adopted if they significantly hurt model performance.Note that making architectures which improve interpretability at arbitrary cost to performance is both trivial and uninteresting.  As a reductio ad absurdum, we could replace any neural network with a linear regression, which is highly interpretable but likely achieves very poor performance.  Of course, architecture changes which result in minor performance decreases but major interpretability improvements may still be worth pursuing.  The largest language models are now estimated to cost millions of dollars to train, persuading companies to adopt such a change in production systems would mean asking them to spend millions of dollars more to achieve a model of equivalent performance. This seems like a tough sell, even if the interpretability improvements were dramatic.  Thus, it seems important to confirm competitiveness.

To demonstrate this, we train transformer language models with and without SoLU for a range of different sizes, and evaluate both the loss and the performance on the following downstream NLP tasks: Lambada , ARC , OpenBookQA , TriviaQA , arithmetic, MMLU , and HellaSwag.

Our baseline model uses an architecture similar to GPT-3  and Gopher , and identical to what is described in our own previous language model baselines . We train models ranging from 1 layer to 64 layers (approximately 50 billion parameters), in successive factors of roughly 4 in parameters.  Our SoLU models have all the same hyperparameters and architectural details as our baselines and differ only in using the SoLU activation function.

Training curves for the models are shown in Figure 1.  We plot both the loss (Figure 1 top) and a measure of performance difference that converts loss differences into an effective multiplier on model size (Figure 1 bottom), which allows us to zoom in on small differences in performance.  As shown in the plots, SoLU is roughly equivalent to the baseline for all model sizes, always falling between a 1.05x and a 0.95x multiplier in model size (roughly equivalent to a change in loss of ±0.01 nats in most cases, compared to a total loss of 1.6-3 nats).  There is potentially a trend towards SoLU performing slightly better relative to the baseline at large model sizes, though all differences are small and more likely than not to be random noise (on the 50B model, SoLU is equivalent to increasing the model size by 1.01x).

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Figure 1: Loss curves for baseline (dotted line) and SoLU (solid line) models ranging from 10 million parameters to 50 billion parameters.  Top plot shows learning curves, bottom plot shows a “model size equivalent” version of the same data, with the baseline model set to 1.0x and SoLU models measured in terms of the baseline model size they perform equivalently to, as predicted from the scale laws.  For example, if a 1B baseline model achieved loss 2.3, a 2B baseline model achieved loss 2.1, and a 1B SoLU model also achieved loss 2.1, the SoLU model would be said to perform at 2x model size relative to the baseline.
Although downstream tasks often correlate well with the loss on a sufficiently broad training set , it’s possible for the macroscopic loss to hide deficiencies in particular tasks or areas, so we run several representative downstream evaluations to confirm the picture suggested by the loss curves.  We evaluate on the Lambada, OpenBookQA, ARC, HellaSwag, MMLU, TriviaQA, and arithmetic datasets, and the results are shown in Figure 2.  We see similar overall performance on baseline vs SoLU at all model sizes, with significant differences on a couple tasks (arithmetic seems better with SoLU, whereas TriviaQA seems better with the baseline) but similar performance on most and no systematic trend one way or the other.

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)
Figure 2: Performance on downstream tasks.
It is worth noting that we do not scan a range of hyperparameters (we scan only model size) for either SoLU or the baseline, and the optimal hyperparameters for SoLU might be different from those for the baseline model.  However, the baseline model’s hyperparameters were used in  and are similar to those in , while SoLU has not been tuned at all, so even if this effect is present, it likely underestimates the performance of SoLU, suggesting SoLU is at least as good as the baseline.

Finally there is another sense of “performance” worth mentioning – the efficiency of model training.  SoLU involves a softmax over the feedforward activations and thus adds a small amount of additional computation, but it is tiny compared to the main matrix multiplies, and with proper GPU kernels, we have found that it slows model training by only an insignificant amount (a less than 1% difference in speed).In principle, one could sidestep this small cost by training an isomorphic model with exponential activation functions and then switching to SoLU after training, ignoring concerns about different numerics.

Overall, then, we conclude that SoLU with LayerNorm appears to achieve competitive ML and training performance compared to a standard transformer.

---

6. Results on Interpretability
------------------------------

Having shown that SoLU is competitive in ML performance, we now demonstrate our main point: that it makes model neurons easier to interpret.  [Section 6.1](#section-6-1) describes the quantitative experiments we perform, [Section 6.2](#section-6-2) goes through the results of those experiments, [Section 6.3](#section-6-3) explores some discoveries we are able to make in the SoLU models that we weren’t able to make previously in baseline models, and [Section 6.4](#section-6-4) discusses how the post-activation LayerNorm may complicate the picture.

### 6.1 Setup of Experiments

We are interested in whether neurons are "interpretable" – that is, do their activations reliably correspond to a coherent, articulable property of the input? Determining that a neuron is interpretable in this sense is not straightforward. While one can often develop a theory of neuron behavior quite rapidly, verifying that theory (or correcting it if the original theory is mistaken) can take a large amount of human effort.  For example, Cammarata et al.  dedicated an entire two papers to rigorously investigating a handful of curve detector neurons in a vision model using seven different lines of evidence.

In order to make it practically feasible to study a large number of neurons across several different models, we therefore settle for measuring something less ambitious: whether a given neuron suggests a plausible interpretation given a small amount of human attention.  This will lead to both some false positives (neuron appears to have a plausible explanation that on closer inspection would turn out to be wrong) and false negatives (there is a simple correct theory of the neuron’s firings but we don’t succeed in finding it quickly).  Nevertheless it is still likely correlated with neurons being interpetable on closer investigation. Additionally, it seems related to the property of being easily interpretable, which would be valuable in its own right: if more neurons are interpretable with low-effort, it makes it more likely that large assemblages of them can be reverse-engineered.

To measure whether a neuron is “interpretable at first glance," we asked human evaluators (some of the authors) to examine a series of text snippets (typically 20 snippets of length a few paragraphs each) that include tokens where the neuron fires heavily.  The firings are highlighted in different shades of red (corresponding to activation magnitude), allowing the evaluator to quickly skim the snippets for a common theme.  An example of the dataset examples evaluators see is shown in Figure 3.

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)
Figure 3: Evaluators are shown dataset examples a neuron fires on, highlighted by activation magnitude, as seen above.  Neurons are selected randomly from one of a SoLU model or its corresponding baseline, the human evaluator (one of the authors) spends 1-2 minutes evaluating whether a single hypothesis or concept explains 80% of the strongest firings, and marks the neuron INTERPRETABLE if so and NOT INTERPRETABLE otherwise.
The evaluator is instructed to examine the firings for 1-2 minutes per neuron, and then indicate whether they have found a plausible theory to explain the firings.  The specific instructions were to mark INTERPRETABLE if “80% or more of the strongest firings can be explained by a single rule or category (e.g. the word “apple," or any phrase relating to music)," and NOT INTERPRETABLE otherwise.

We performed experiments on the 1 layer, 16 layer, 24 layer, 40 layer, and 64 layer (50 billion parameter) models.   For each size of model, evaluators were presented with 60 neurons from the baseline model (without SoLU activation) and 60 neurons from the corresponding SoLU model – for a total of 60\*2\*5=600 neurons across all experiments.  To prevent us from being biased in favor of our models, the neurons were presented to evaluators in a randomized and blinded manner (evaluators did not know which neurons came from which model).

Finally, since our SoLU models include both the SoLU itself and an extra layer norm, we did one experiment to disambiguate the effect of the SoLU and the layer norm.  Namely, we trained a 16 layer model with the extra layer norm but not the SoLU, and evaluated 60 neurons from this model as well, bringing the grand total to 660 neurons.

### 6.2 Quantitative Results

The results of our experiment on what fraction of neurons are preliminarily interpretable are shown below in Figure 4.  For models from 1 layer to 40 layers, the SoLU model’s neurons are substantially more interpretable than the baseline’s neurons, with increases of roughly 25 absolute percentage points, from ~35% interpretable to ~60% interpretable.  This increases the fraction of interpretable neurons by 1.7x.  Although the effect is moderate in size, the sample size, consistent gap, and consistent absolute rates of interpretable neurons suggest a real and persistent effect of the SoLU models.

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Nelson Elhage
Tristan Hume
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Tom Henighan
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Sheer El Showk
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Nova DasSarma
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Amanda Askell
Kamal Ndousse
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Dawn Drain
Anna Chen
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Deep Ganguli
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