Scaling and evaluating sparse autoencoders
Thread (60 posts)
Is it correct to understand the sentence
We choose a layer near the end of the network, which should contain many features without being specialized for next-token predictions
as follows:
- computation occurs as data passesthrough the layers of the network, so later layers have more "refined" views of the relevant data
- on the other hand, since the final layers are specifically trained to do next-token prediction, if those layers were used for interpretability, the internal states of the model may be less accessible
Thinking Summary
The user is asking about a specific sentence from the paper about sparse autoencoders. They're trying to understand why the authors chose a layer near the end (but not at the very end) of the network for their experiments. The user's interpretation is basically correct. Let me confirm and expand on this: 1. Yes, later layers tend to have more refined/processed representations because computation occurs sequentially through layers 2. Yes, the very final layers are specialized for the specific task (next-token prediction) which might make them less useful for finding general interpretable features The key insight is that there's a trade-off: - Too early in the network: features might be too raw/low-level - Too late in the network: features might be too task-specific (overly specialized for predicting the next token) - Sweet spot near the end but not at the end: features are refined and high-level but still general-purpose This is a common pattern in interpretability work - the penultimate or near-final layers often contain the most useful representations for understanding what the model has learned.
Yes, your understanding is correct! The authors are describing a trade-off when selecting which layer to analyze:
-
Later layers have richer representations: As activations flow through the network, each layer builds upon previous computations, creating increasingly sophisticated and abstract features. Later layers have had more opportunity to combine and refine information.
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Final layers become task-specialized: The very last layers are heavily optimized for the specific output task (next-token prediction in this case). Their representations become increasingly tailored to producing the exact logits needed for vocabulary predictions, potentially at the cost of maintaining more general, interpretable features.
This creates a "sweet spot" near the end of the network where you get:
- Features that have undergone substantial processing and abstraction
- Features that haven't yet been fully transformed into task-specific prediction machinery
The paper mentions they chose layer 56 out of ~67 layers for GPT-4 (about 5/6 through) and layer 8 out of 12 for GPT-2 (3/4 through). This positioning aims to capture high-level semantic features while avoiding the extreme task specialization of the final layers.
This principle appears in other interpretability work too - for instance, when extracting features for downstream tasks, penultimate layers often work better than final layers precisely because they maintain more general-purpose representations.
How exactly does the Z term in section 2.2 promote sparsity?
Thinking Summary
The user is asking about section 2.2 of the paper on "Scaling and evaluating sparse autoencoders". Looking at section 2.2, they are discussing baseline ReLU autoencoders. The equation given is: z = ReLU(W_enc(x - b_pre) + b_enc) x̂ = W_dec z + b_pre And the training loss is: L = ||x - x̂||²₂ + λ||z||₁ The user is asking about how the ||z||₁ term promotes sparsity. This is referring to the L1 penalty term λ||z||₁ in the loss function. I should explain how L1 regularization promotes sparsity.
In section 2.2, the term you're referring to is the L1 penalty: in the loss function:
This L1 penalty promotes sparsity through the following mechanism:
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L1 norm penalizes the sum of absolute values: The L1 norm adds a cost proportional to the magnitude of each activation.
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Gradient behavior at zero: The key property is that the gradient of the L1 norm is constant (±λ) for all non-zero values:
- For :
- For :
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Drives activations to exactly zero: This constant gradient means there's always a fixed "force" pushing activations toward zero, regardless of how small they get. Combined with the ReLU (which already forces negative pre-activations to zero), this creates a strong incentive for the model to set many activations to exactly zero rather than keeping them at small positive values.
The paper actually identifies this as problematic because:
- It causes "activation shrinkage" - even activations that should be large get pulled toward zero
- It's an imperfect approximation of the true L0 norm (which counts non-zero elements)
- This is why they advocate for TopK activation functions instead, which directly control sparsity without needing the L1 penalty
The λ hyperparameter controls the strength of this sparsity pressure, requiring careful tuning to balance reconstruction quality against sparsity.
Can you elaborate on the gradient behavior at zero?
This is somehow important so the model is actually trainable right?
Thinking Summary
The user is asking about the gradient behavior of the L1 norm at zero, and they correctly intuit that this is important for trainability. This is actually a subtle but important point about the L1 norm. The L1 norm $||z||_1 = \sum_i |z_i|$ has a subgradient at zero rather than a proper gradient, because the absolute value function is not differentiable at 0. For a single component: - When $z_i > 0$: $\frac{\partial |z_i|}{\partial z_i} = 1$ - When $z_i < 0$: $\frac{\partial |z_i|}{\partial z_i} = -1$ - When $z_i = 0$: the gradient is undefined (technically it's a subgradient in the interval [-1, 1]) However, in the context of ReLU autoencoders, this is handled because: 1. The ReLU activation ensures $z_i \geq 0$ always 2. The gradient of ReLU(x) is 0 when x < 0 and 1 when x > 0 3. At exactly x = 0, implementations typically use either 0 or 1 as the gradient The key insight is that the L1 penalty creates a constant "pull" toward zero for any positive activation, which is different from L2 regularization where the pull gets weaker as values get smaller. For trainability, what matters is: - The gradient is well-defined almost everywhere - The subgradient at 0 can be handled by choosing any value in [-1, 1], typically 0 - In practice with ReLU, we only deal with non-negative values anyway
You're absolutely right - the gradient behavior at zero is crucial for trainability! Let me clarify this important but subtle point.
The challenge with L1 at zero:
The absolute value function is not differentiable at . Its derivative is:
- when
- when
- Undefined at (the function has a "corner")
This creates a potential problem: how do we compute gradients when ?
How it's handled in practice:
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Subgradient approach: At , we can use any value in the interval as a valid subgradient. Most implementations choose 0.
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ReLU interaction: Since the encoder uses ReLU activation:
The latents are always non-negative. This means:
- We only need to handle the case where
- The gradient of the L1 penalty is simply for all
- When , the ReLU gradient is already 0 (blocking backprop), so the L1 subgradient choice doesn't matter
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Smooth approximations: Some implementations use smooth approximations like for small to avoid the non-differentiability entirely.
Why this promotes sparsity effectively:
The key insight is that unlike L2 regularization (where the gradient vanishes as ), the L1 gradient remains constant at . This means:
- There's always a fixed "force" pushing positive activations toward zero
- Once an activation reaches zero, the ReLU keeps it there
- This creates true sparsity (exact zeros) rather than just small values
This is why L1 is so effective for sparsity, but also why it causes the "shrinkage" problem the paper discusses - it pulls all positive activations toward zero, even the ones that should be large!
Thinking Summary
The user is asking about the gradient behavior of the TopK activation function. This is a great question because TopK is not differentiable in the traditional sense - it's a discrete selection operation. Let me think through this carefully. TopK(x) returns the k largest values and zeros out the rest. This is a non-smooth operation with discontinuities. The paper doesn't explicitly discuss how they handle gradients through TopK, but this is a known issue in the literature. Common approaches for handling TopK gradients: 1. Straight-through estimator (STE) - pass gradients only through the selected top-k elements 2. Smooth approximations 3. Other techniques Looking at the paper, they don't explicitly state which method they use, but the most common and likely approach is the straight-through estimator for the selected elements.
Great question! The gradient behavior of TopK is quite different from L1 and presents an interesting challenge.
The TopK operation:
This keeps only the k largest values and sets all others to zero. This is fundamentally a non-differentiable operation because:
- It involves sorting/ranking (discrete operations)
- It has discontinuities when values cross the threshold for being in the top-k
How gradients are handled (likely using straight-through estimator):
The paper doesn't explicitly state their gradient method, but the standard approach is:
- Forward pass: Select the top-k activations, zero out the rest
- Backward pass:
- Pass gradients through only for the selected top-k elements
- Zero gradients for all non-selected elements
In pseudocode:
# Forward
mask = (z in top-k values) # Binary mask
z_sparse = z * mask
# Backward
grad_z = grad_output * mask # Only selected elements get gradients
Key differences from L1:
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No shrinkage bias: Selected activations receive gradients based only on reconstruction error, with no constant "pull" toward zero
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Binary selection: An activation either gets full gradient flow (if in top-k) or none at all (if not)
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Discrete switches: As training progresses, which latents are "active" can change discontinuously
Why this works better:
- No magnitude distortion: Active latents can learn their true optimal values without L1's shrinkage
- Clean sparsity: Exactly k activations, no hyperparameter tuning needed
- Better reconstruction: The paper shows TopK achieves better reconstruction-sparsity tradeoffs
The discrete nature might seem problematic, but in practice it works well because:
- The encoder still learns which features to activate through the pre-activation values
- The top-k selection provides a strong learning signal about feature importance
- The reconstruction loss guides the model to pick the most useful k features
What does the phrase
setting the L0 directly
mean? What is being set exactly and by whom at what stage
Thinking Summary
The user is asking about the phrase "setting the L0 directly" in the context of TopK autoencoders. They want to understand what is being set, by whom, and when. Looking at the paper, this is in section 2.3 about TopK activation functions. The L0 norm refers to the number of non-zero elements in a vector - in this case, the number of active latents in the sparse autoencoder. With TopK, you directly choose k (the number of top activations to keep), which directly determines the L0 norm of the activation vector. This is in contrast to ReLU autoencoders where you set λ (the L1 penalty coefficient) and the resulting L0 emerges indirectly from training.
Great question! The phrase "setting the L0 directly" refers to directly choosing the number of non-zero activations in the sparse autoencoder.
Here's what's happening:
What is being set: The parameter k
in the TopK activation function - this is the exact number of latents that will be active (non-zero) for each input.
By whom: The researcher/engineer designing the autoencoder architecture.
At what stage: This is set as a hyperparameter before training begins, and it remains fixed throughout training.
The key contrast is:
ReLU autoencoders (indirect):
- You set λ (L1 penalty strength)
- The actual sparsity (L0 = number of non-zero activations) emerges from training
- You don't know exactly how sparse it will be until you train it
- Need to tune λ through trial and error to achieve desired sparsity
TopK autoencoders (direct):
- You set k = 32 (for example)
- This guarantees exactly 32 activations will be non-zero for every input
- L0 = k always, by construction
- No tuning needed to achieve a specific sparsity level
As the paper states: "It enables setting the L0 directly, as opposed to tuning an L1 coefficient λ, enabling simpler model comparison and rapid iteration."
This is particularly valuable for research because you can directly compare autoencoders with identical sparsity levels, rather than trying to tune different λ values to achieve approximately the same L0.
In this literature is it correct that “parameter” generally means learned parameter and hyperparameter generally means selected (aside from the other implications of the terms)
Thinking Summary
The user is asking about the terminology in the context of the "Scaling and evaluating sparse autoencoders" paper, specifically about the distinction between "parameter" and "hyperparameter". In machine learning: - Parameters are typically the learned values during training (weights, biases, etc.) - Hyperparameters are typically set before training and control the learning process (learning rate, architecture choices, etc.) Looking at the paper, I can see: - Parameters mentioned: $W_{enc}$, $W_{dec}$, $b_{enc}$, $b_{pre}$ - these are learned during training - Hyperparameters mentioned: $\lambda$ (L1 penalty coefficient), $k$ (sparsity level in TopK), learning rate, etc. So yes, the user's understanding is correct in this context.
Yes, that's correct! In this paper (and in machine learning literature generally):
Parameters = learned values that are optimized during training:
- (encoder weights)
- (decoder weights)
- (encoder bias)
- (pre-encoder bias)
Hyperparameters = values selected before training that control the learning process:
- (L1 penalty coefficient for ReLU autoencoders)
- (number of active latents in TopK)
- (total number of latents)
- Learning rate
- Batch size
- Architecture choices
The paper explicitly mentions this distinction, for example:
- "λ is a hyperparameter that needs to be tuned" (Section 2.2)
- "We sweep learning rates at small scales and extrapolate the trend" (Section 2.1)
The key difference is that parameters are optimized by gradient descent during training, while hyperparameters are typically selected through experimentation, grid search, or based on scaling laws (as this paper does). The TopK approach is praised partly because it removes the need to tune the λ hyperparameter - instead directly setting the sparsity level k.
I've heard about the issue of dead latents before and have some idea of the shape of the problem but to be precise what exactly is the referent of "latent”? Latent parameter ? Terminology Q
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I've heard about the issue of dead latents before and have some idea of the shape of the problem but to be precise what exactly is the referent of "latent”? Latent parameter ? Terminology Q
Thinking Summary
The user is asking about what exactly a "latent" is in the context of sparse autoencoders and the "dead latent" problem. This is a good terminology question. In the context of sparse autoencoders (SAEs), a "latent" refers to one of the hidden units/dimensions in the bottleneck layer of the autoencoder. More specifically: - An autoencoder has an encoder that maps from input space (e.g., model activations) to a hidden representation - This hidden representation has multiple dimensions/units - each one is a "latent" - In sparse autoencoders, these latents are meant to represent interpretable features - The decoder then maps from these latents back to reconstruct the original input A "dead latent" is a latent that stops activating entirely during training - meaning it always outputs zero and thus doesn't contribute to the reconstruction. This is wasteful computationally and means we're not using the full capacity of the autoencoder. The paper mentions they use techniques to prevent dead latents, including initializing the encoder to the transpose of the decoder and using an auxiliary loss.
In the context of sparse autoencoders (SAEs), a "latent" refers to a single hidden unit or dimension in the autoencoder's bottleneck layer.
More precisely:
- The encoder maps from the input space (e.g., GPT-4 residual stream activations of dimension ) to a hidden representation of dimension
- Each of the dimensions in this hidden representation is a "latent"
- Each latent is associated with one row of the encoder weight matrix and one column of the decoder weight matrix
So when they train a "16 million latent autoencoder," they mean million hidden units.
A dead latent is one that stops activating (always outputs zero) during training. This is problematic because:
- It wastes computational resources - you're storing and computing with parameters that don't contribute
- It reduces the effective capacity of the autoencoder
- As the paper notes, this problem gets worse at scale - they mention finding up to 90% dead latents in some configurations
The paper uses the definition that a latent is "dead" if it hasn't activated in 10 million tokens. Their mitigation strategies (transpose initialization and auxiliary loss) reduce this to only 7% dead latents even in their largest model.
Like I mentioned I understand the issue during training I am asking about precision in terminology
Thinking Summary
The user is asking for precise terminology clarification. They understand the dead latent problem but want to know exactly what "latent" refers to. Let me be very precise about the terminology. In the context of sparse autoencoders: - "Latent" typically refers to the activation value (the scalar output) of a hidden unit - But colloquially, people often use "latent" to refer to the entire hidden unit itself - The hidden unit consists of parameters (weights and biases) that produce the latent activation So to be maximally precise: - Latent activation/value: The scalar value $z_i$ produced by hidden unit $i$ - Latent unit/dimension: The $i$-th hidden unit, including its associated parameters - When people say "dead latent" they mean a latent unit whose activation is always zero I should clarify this ambiguity in terminology.
You're right to ask for precision. The terminology is indeed ambiguous in common usage.
Strictly speaking:
- "Latent" should refer to the activation value (a scalar) produced by the -th hidden unit
- The latent unit or latent dimension refers to the hidden unit itself, including its associated parameters (row of , bias , and column of )
In practice: People often use "latent" as shorthand for "latent unit" - conflating the activation with the unit that produces it.
So when the paper says:
- "16 million latent autoencoder" → 16 million latent units
- "dead latents" → latent units whose activations are always zero
- "latent activations " → the actual activation values
The imprecision comes from the fact that in most contexts, it's clear from usage whether we mean the unit or its activation. But you're correct that this conflation can be confusing, especially when discussing both the parameters and their outputs in the same context.
We find two important ingredients for preventing dead latents: we initialize the encoder to the transpose of the decoder, and we use an auxiliary loss that models reconstruction error using the top- k aux dead latents (see Section A.2 for more details). Using these techniques, even in our largest (16 million latent) autoencoder only 7% of latents are dead
Can you give intuition for why these methods world prevent dead latents?
Thinking Summary
The user is asking about two methods for preventing dead latents in sparse autoencoders: 1. Initializing the encoder to the transpose of the decoder 2. Using an auxiliary loss that models reconstruction error using the top-k_aux dead latents I should explain the intuition behind why these methods help prevent dead latents. Dead latents are neurons/features that stop activating entirely during training. This is a common problem in sparse autoencoders because: - The sparsity constraint encourages most latents to be zero - Once a latent stops being used, it gets no gradient signal - Without gradient signal, it can't learn to become useful again Let me think about each method: 1. Initializing encoder as transpose of decoder: - This creates a symmetric initialization where encoding and decoding are initially inverses - This means at initialization, features have a natural correspondence - It likely prevents the initial "winner take all" dynamics where some features dominate early 2. Auxiliary loss on dead latents: - This gives gradient signal to latents that aren't naturally activating - By forcing the top k_aux dead latents to help with reconstruction, they get gradient updates - This prevents them from getting stuck in a permanently dead state I should explain this clearly with the intuition about gradient flow and the dynamics of training.
Great question! The intuition behind these methods relates to the fundamental problem of gradient flow in sparse autoencoders.
Why Dead Latents Occur
Dead latents happen because of a vicious cycle:
- A latent stops activating (due to competition with other latents)
- No activation → no gradient signal through that latent
- No gradient → no parameter updates → latent can't learn to become useful
- The latent remains permanently "dead"
Method 1: Encoder-Decoder Transpose Initialization
Initializing creates a symmetric starting point where:
- At initialization, the encoder and decoder form approximate inverses
- Each latent starts with a "natural" direction in activation space
- This prevents early "winner-take-all" dynamics where a few latents dominate
Think of it like giving each latent a fair starting position rather than random initialization where some latents might start in much better positions to capture activations.
Method 2: Auxiliary Loss on Dead Latents
This directly addresses the "no gradient" problem:
- The auxiliary loss forces the top dead latents to participate in reconstruction
- Even if they wouldn't naturally activate, they now receive gradient signal
- This gradient helps them learn useful features and potentially "come back to life"
It's like a "revival mechanism" - instead of letting dead latents remain permanently unusable, you periodically give them a chance to contribute and learn.
The combination is powerful: good initialization reduces early deaths, while the auxiliary loss provides a recovery mechanism for any latents that do die during training.
The auxiliary loss part doesn't make sense to me, can you explain in detail without using analogies
Thinking Summary
The user wants a detailed, technical explanation of the auxiliary loss mechanism without analogies. Let me break down exactly how this works mechanistically. Looking at the paper, the auxiliary loss is described in Section A.2. The key idea is that they identify dead latents (those that haven't activated in some number of tokens), and then force them to participate in reconstruction through an additional loss term. Let me explain the mechanism step by step: 1. They track which latents are "dead" (haven't activated recently) 2. They compute a separate reconstruction using only the top k_aux dead latents 3. They add this as an auxiliary loss term to the main loss This ensures dead latents receive gradients even when they wouldn't naturally activate.
You're right to push for clarity here. Let me explain the auxiliary loss mechanism precisely:
The Auxiliary Loss Mechanism
During training, the system tracks which latents haven't activated in the last N tokens (typically 10 million). These are marked as "dead."
For each training batch:
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Normal forward pass: Input → TopK activation → only k latents activate → reconstruction loss
-
Auxiliary forward pass:
- Take the same input
- Look at the pre-activation values for all dead latents
- Select the top dead latents (highest pre-activation values among dead latents)
- Force these latents to activate
- Compute a separate reconstruction using only these dead latents
- Calculate MSE between this reconstruction and the original input
-
Combined loss:
Why This Creates Gradients
In the normal TopK operation, dead latents have pre-activation values below the threshold, so they output zero. When a latent outputs zero, its gradient is zero (due to the chain rule).
The auxiliary loss bypasses this by:
- Selecting dead latents based on their pre-activation values (even though these are below the TopK threshold)
- Computing a reconstruction loss specifically through these latents
- This reconstruction error backpropagates through these forced-active latents, updating their parameters
The dead latents now receive gradient signal proportional to how well they can help reconstruct the input, allowing them to potentially learn useful features and eventually activate naturally in the main forward pass.
Thinking Summary
The user is right - the auxiliary loss does seem like a hack without clear theoretical justification. Let me think about this more carefully. The fundamental issue is that TopK creates a hard threshold - latents below the threshold get exactly zero gradient. This is different from L1-regularized ReLU autoencoders where even small activations get some gradient (though reduced by L1). The auxiliary loss is essentially saying "let's give dead latents a chance to learn by forcing them to participate occasionally." But why should this lead to good features? Why not just increase k? Why should dead latents that haven't been useful suddenly become useful? I should acknowledge this criticism is valid while also considering what theoretical justification might exist.
You're absolutely right - it is a hack. The auxiliary loss lacks a principled theoretical foundation.
The Core Problem
TopK creates a hard optimization landscape:
- Latents above the threshold get full gradients
- Latents below get exactly zero gradient
- No smooth transition or "pressure" to improve
This is fundamentally different from L1-regularized autoencoders where small activations still receive (penalized) gradients.
Why Not Just Increase k?
A more principled approach might be:
- Use larger k (more active latents)
- Use a softer activation function (like sparsemax)
- Use better initialization to prevent deaths in the first place
What the Auxiliary Loss Actually Does
It's essentially importance sampling with a heuristic: "latents with highest pre-activation values among dead latents are most likely to be useful if given a chance." But there's no guarantee these latents represent meaningful features rather than noise.
The fact that it empirically works (reducing dead latents from 90% to 7%) suggests there's some underlying phenomenon we don't fully understand about the optimization landscape of sparse autoencoders. But you're correct that it's an engineering solution rather than a theoretically motivated one.
A more principled approach might involve understanding why latents die in the first place and addressing that root cause, rather than applying this post-hoc revival mechanism.
However, latents are the important artifact of training (not reconstruction predictions), whereas for language models we typically care only about token predictions. Comparing MSE across different n is thus not a fair comparison — the latents have a looser information bottleneck with larger n , so lower MSE is more easily achieved. Thus, this approach is arguably unprincipled for autoencoder training.
I thought I understood the distinction between latents in traing and next token prediction during inference but I don't understand the "unfair comparison" or the reason for it being unprincipled so I must be missing something.
Thinking Summary
The user is asking about a specific passage from the "Scaling and evaluating sparse autoencoders" paper. They're confused about why comparing MSE across different n (number of latents) is considered an "unfair comparison" and why this approach is "arguably unprincipled for autoencoder training." Let me break down what's happening here: 1. The paper is discussing two approaches to choosing autoencoder size and token budget 2. The first approach (L(C)) trains autoencoders to optimal MSE given available compute 3. The authors criticize this approach The key insight is about what we're actually trying to optimize for: - In language models: We care about the final output (token predictions). The internal representations are just a means to an end. - In sparse autoencoders: We care about the latents themselves - they are the interpretable features we're trying to extract. The reconstruction is just a way to ensure the latents capture meaningful information. When you increase n (number of latents), you're giving the autoencoder more capacity. With more latents, it's easier to achieve lower MSE because you have a "looser information bottleneck" - more degrees of freedom to represent the input. So comparing MSE across different n values is like comparing the performance of two students on exams where one student gets to use more notes than the other. The one with more notes will likely do better, but that doesn't mean they understand the material better. The "unprincipled" part comes from the fact that if you're optimizing for MSE across different n values, you're not really optimizing for what you care about (quality of individual latents), you're just optimizing for reconstruction quality, which naturally improves with more latents.
The key insight here is about what we're actually optimizing for in sparse autoencoders versus language models.
In language models, the internal representations are just a means to an end - we ultimately care about the quality of token predictions. But in sparse autoencoders, the latents themselves are the primary output - they're the interpretable features we're trying to discover.
The "unfair comparison" arises because:
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Capacity increases with n: When you have more latents (larger n), you have more degrees of freedom to represent the input. It's like having a larger vocabulary to describe something - naturally you can be more precise.
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Information bottleneck loosens: With more latents, the autoencoder doesn't need to compress information as tightly. Each latent can be more specialized and capture narrower features, making reconstruction easier.
-
MSE naturally decreases: This means that a larger autoencoder will almost always achieve lower MSE than a smaller one, but this doesn't necessarily mean the individual latents are better or more interpretable.
The comparison is "unprincipled" because optimizing for MSE across different n values incentivizes simply making n larger, rather than finding the best interpretable features at a given scale. It's like judging compression algorithms by how well they preserve data while ignoring the compression ratio - you'd always choose the one that barely compresses at all.
This is why the authors prefer training to convergence at fixed n (their L(N) approach) - it gives a fairer comparison of what quality of features you can extract at different scales, rather than just rewarding models for having more capacity.
When they say "we consider downstream kl divergence” downstream of what?
Thinking Summary
The user is asking about the meaning of "downstream KL divergence" in the context of the sparse autoencoders paper. Let me find the relevant section in the paper. Looking at section 4.1 "Downstream loss", the paper says: "To measure whether we model features relevant to language modeling, we follow prior work (Bills et al., 2023; Cunningham et al., 2023; Bricken et al., 2023; Braun et al., 2024) and consider downstream Kullback-Leibler (KL) divergence and cross-entropy loss. In both cases, we test an autoencoder by replacing the residual stream by the reconstructed value during the forward pass, and seeing how it affects downstream predictions." So "downstream" refers to the predictions that come after the layer where the autoencoder is applied. They're measuring how much the language model's output predictions change when they replace the original residual stream activations with the autoencoder's reconstruction. The specific setup: - They train autoencoders on residual stream activations at a specific layer (e.g., layer 8 in GPT-2 small) - During evaluation, they replace the residual stream at that layer with the autoencoder reconstruction - They measure how this affects the model's final output predictions (the "downstream" predictions) - The KL divergence measures how different the probability distributions are between the original model and the model with reconstructed activations
In this paper, "downstream" refers to the language model's predictions that come after the layer where they apply the sparse autoencoder.
Specifically, they:
- Train autoencoders on residual stream activations at a particular layer (e.g., layer 8 of 12 in GPT-2 small)
- During evaluation, replace the residual stream at that layer with the autoencoder's reconstruction
- Let the model continue processing through the remaining layers
- Measure how much the final output predictions change
The "downstream KL divergence" measures the difference between:
- The probability distribution over next tokens from the original model
- The probability distribution when using reconstructed activations
This tests whether the autoencoder preserves the information that matters for the model's actual task (predicting tokens), not just whether it can reconstruct activations accurately. A good autoencoder should have low downstream KL divergence, meaning the model's behavior is largely preserved even when using the sparse reconstructed features.
They note this is important because an autoencoder with low reconstruction error might still fail to capture the features most relevant for the model's behavior - the downstream metrics help verify that the learned features actually matter for the model's predictions.
Thinking Summary
The user wants me to write out the formulas for the downstream KL divergence metric. Let me look at what the paper says about this. From the paper, they mention: 1. They replace the residual stream with the reconstructed value during the forward pass 2. They measure KL divergence and cross-entropy loss 3. They also mention "delta cross-entropy" - the difference to the perfect-autoencoder cross-entropy Let me write out the mathematical formulas for these concepts.
Sure! Here are the key formulas for the downstream metrics:
Downstream KL Divergence
Let:
- = original residual stream activations at layer
- = autoencoder reconstruction of
- = original model's output distribution over tokens
- = model's output distribution when using reconstructed activations
The downstream KL divergence is:
where is the vocabulary.
Downstream Cross-Entropy Loss
The delta cross-entropy they mention is:
where:
This measures how much worse the model's predictions become when using reconstructed activations.
The Forward Pass
During evaluation, for a model with layers:
h_0 &= \text{embed}(\text{tokens}) \\ h_i &= \text{layer}_i(h_{i-1}) \quad \text{for } i < \ell \\ h_\ell &= \text{layer}_\ell(h_{\ell-1}) \\ \hat{h}_\ell &= \text{SAE}_{\text{decode}}(\text{SAE}_{\text{encode}}(h_\ell)) \\ h_i &= \text{layer}_i(\hat{h}_{i-1}) \quad \text{for } i > \ell \\ p(y|\hat{x}) &= \text{softmax}(W_{out} \cdot h_L) \end{align}$$ The key insight is that these metrics test whether the sparse features capture what's actually important for the model's task, not just whether they can reconstruct activations accurately.Oh the "stream" in "downstream" is referring to the latent stream?
Thinking Summary
The user is asking about the terminology "downstream" and whether it relates to the "residual stream" mentioned in the paper. Let me clarify this. In the context of transformers and this paper: - "Residual stream" refers to the main information pathway through the transformer layers - it's the vector that gets updated at each layer and carries information forward - "Downstream" is a more general term meaning "later in the processing pipeline" or "after a certain point" So "downstream" doesn't specifically refer to the residual stream itself, but rather to what happens after/later than the point where they intervene. The residual stream is the thing they're modifying, and "downstream" refers to the subsequent processing and outputs that depend on that stream.
Yes, exactly! The "stream" being referred to is the residual stream that flows through the transformer layers.
In transformer architecture:
- The residual stream is the main information pathway that gets updated at each layer
- Information flows from early layers → later layers → final output
So when they say "downstream KL divergence," they mean:
- They intervene at a specific layer's residual stream (e.g., layer 8)
- They replace it with the autoencoder reconstruction
- This modified residual stream then flows downstream through layers 9, 10, 11, 12...
- They measure how much the final outputs change
The terminology comes from thinking of computation as a flow:
- Upstream = earlier layers (before the intervention point)
- Downstream = later layers (after the intervention point)
It's like a river where you add dye at one point and measure how it affects everything downstream. Here they're modifying the residual stream at one layer and measuring how it affects all the subsequent computations that depend on it.
Unfortunately, precision is extremely expensive to evaluate when the simulations are using GPT-4 as in Bills et al. (2023)
Why is this the case?
Thinking Summary
The user is asking about why precision is extremely expensive to evaluate when simulations are using GPT-4 as in Bills et al. (2023). Looking at the paper, they mention this in section 4.3 "Finding simple explanations for features". The context is that they're trying to evaluate explanations for what features (latents) in their sparse autoencoders represent. The Bills et al. (2023) approach (from the linked paper "Language models can explain neurons in language models") uses GPT-4 to generate natural language explanations of what neurons/features are doing, and then uses GPT-4 again to simulate whether those explanations would predict when the neuron/feature activates. To evaluate precision, you need to check: of all the times the explanation predicts the feature should activate, how often does it actually activate? This requires: 1. Running GPT-4 on many text samples to see if the explanation predicts activation 2. Checking against ground truth activations The expense comes from needing to make many GPT-4 API calls for the simulation step - you need to test the explanation on a large corpus of text to get reliable precision estimates. Each test requires a GPT-4 call to simulate whether the explanation would predict activation on that text. In contrast, the N2G (Neuron to Graph) approach they use instead is much cheaper because it just uses n-gram patterns with wildcards rather than natural language explanations that need to be simulated with a language model.
The expense comes from the evaluation methodology used by Bills et al. (2023). To measure precision of a natural language explanation for a neuron/feature, they need to:
-
Use GPT-4 to simulate predictions: For each text sample in a large evaluation corpus, they call GPT-4 to determine whether the explanation predicts the neuron should activate on that text.
-
Compare against ground truth: Check whether the neuron actually activated on each sample.
-
Calculate precision: Of all the samples where the explanation predicted activation, what fraction actually had the neuron activate?
The computational cost is high because:
- Volume: You need thousands of text samples to get statistically reliable precision estimates
- API calls: Each sample requires a GPT-4 API call to simulate the explanation
- Token costs: Each simulation involves sending the explanation + context to GPT-4
For example, if you want to evaluate 10,000 text samples and each requires ~500 tokens for the prompt, that's 5 million tokens just for one neuron's precision score.
This is why the authors switched to N2G (Neuron to Graph) - it uses simple n-gram pattern matching with wildcards instead of natural language explanations, making precision evaluation orders of magnitude cheaper since it doesn't require any LLM calls.
When our goal is for a model’s activations to be interpretable, one question we can ask is: how much performance do we sacrifice if we use only the parts of the model that we can interpret?
Our downstream loss metric measures how much of the performance we’re capturing (but our features could be uninterpretable), and our explanation based metric measures how monosemantic our features are (but they might not explain most of the model). This suggests combining our downstream loss and explanation metrics, by using our explanations to simulate autoencoder latents, and then checking downstream loss after decoding. This metric also has the advantage that it values both recall and precision in a way that is principled, and also values recall more for latents that activate more densely.
Thinking Summary
The user is asking about the formula $E[sa]/E[s^2]$ in section 4.4 of the paper. Let me re-read that section to understand the context. In section 4.4 "Explanation reconstruction", they're discussing combining downstream loss and explanation metrics. They use N2G (Neuron to Graph) explanations to simulate autoencoder latents, then check downstream loss after decoding. The specific text says: "We tried this with N2G explanations. N2G produces a simulated value based on the node in the trie, but we scale this value to minimize variance explained. Specifically, we compute E[sa]/E[s^2], where s is the simulated value and a is the true value, and we estimate this quantity over a training set of tokens." So: - s = simulated value (from N2G explanation) - a = true value (actual autoencoder latent activation) - They're computing E[sa]/E[s^2] This looks like they're finding an optimal scaling factor for the simulated values. Let me think about why this formula makes sense. If we want to find a scaling factor α such that αs best approximates a in some sense, we might want to minimize E[(αs - a)^2]. Taking the derivative with respect to α: d/dα E[(αs - a)^2] = d/dα E[α^2s^2 - 2αsa + a^2] = E[2αs^2 - 2sa] = 2αE[s^2] - 2E[sa] Setting this to 0: 2αE[s^2] - 2E[sa] = 0 α = E[sa]/E[s^2] So E[sa]/E[s^2] is the optimal scaling factor that minimizes the mean squared error between the scaled simulation αs and the true activation a. I'm identifying this as a linear regression technique without an intercept. It's a way to find the best linear relationship between the simulated and actual values, essentially finding the most accurate scaling coefficient.
The computation is finding the optimal scaling factor to apply to the simulated values to best match the true activations.
Here's the intuition:
What they're doing: N2G produces simulated activation values based on pattern matching, but these raw simulated values might be systematically too large or too small compared to the true latent activations . They want to find a scaling factor such that best approximates .
The optimization: To find this scaling factor, they minimize the mean squared error:
Taking the derivative and setting to zero:
Solving for :
Interpretation: This is essentially the regression coefficient for predicting from without an intercept. It answers: "By what factor should we scale our simulated values to best match the true activations on average?"
Why this matters: N2G might correctly identify when a feature activates (getting the pattern right) but be off on the magnitude. This scaling correction ensures that when they decode these simulated activations back through the decoder, the reconstruction is as faithful as possible to what would happen with the true activations.
I don't understand the setup in the second paragraph of §4.5. When they say "at a particular token index" what is the actual setup there? What is the index set? When they say "each autoencoder latent" what precisely is the referent of latent? The description seems densely packed and comprehensible for practitioners but is a little hard to be sure I am following precisely. cf. linked posts
Thinking Summary
The user is asking for clarification about Section 4.5 on "Sparsity of ablation effects". They find the description dense and hard to follow precisely. Let me reread that section carefully. The key paragraph is: "At a particular token index, we obtain the latents at the residual stream, and proceed to ablate each autoencoder latent one by one, and compare the resulting logits before and after ablation. This process leads to V logit differences per ablation and affected token, where V is the size of the token vocabulary." Breaking this down: 1. "At a particular token index" - This seems to mean they pick a specific position in the sequence (e.g., the 5th token in a sequence) 2. "we obtain the latents at the residual stream" - They run the model forward and extract the residual stream activations at that position, then encode them with the SAE to get the latent activations z 3. "ablate each autoencoder latent one by one" - For each of the k active latents (since they use TopK), they set that latent to 0 and decode back to residual stream 4. "compare the resulting logits before and after ablation" - They continue the forward pass with the modified residual stream and see how the output logits change The confusion might be: - What exactly is the "index set"? It seems like they're doing this for multiple token positions - When they say "each autoencoder latent", do they mean each of the k active ones, or all n latents? - The dense packing of technical details Let me provide a clearer explanation.
Looking at §4.5, I can help clarify the setup. The description is indeed quite dense. Let me break it down:
The Setup:
-
"At a particular token index" - They pick a specific position in a sequence (e.g., position 5 in a 64-token context). The index set appears to be multiple sampled positions across their evaluation data.
-
"We obtain the latents at the residual stream" - At that position, they:
- Extract the residual stream activation vector
- Pass it through the SAE encoder to get latent activations
- For TopK, only of these will be non-zero
-
"Ablate each autoencoder latent one by one" - Given the context of TopK, "each autoencoder latent" likely means each of the active latents (those with non-zero activations). For each active latent :
- Set (while keeping other latents unchanged)
- Decode back to get modified
- Replace the original residual stream with this modified version
- Complete the forward pass to get new logits
-
Measuring effects - They track how this ablation affects logits at:
- The current position
- The next positions
The resulting metric measures whether ablating individual SAE features has sparse, localized effects on model outputs (good) or dense, widespread effects (bad).
The terminology confusion you noted in your linked posts is relevant here - "latent" is being used to mean both the latent unit and its activation value, which adds to the density of the description.
Thinking Summary
The user is asking about V and T in the context of section 4.5. Let me re-read that section carefully. From the text: "This process leads to V logit differences per ablation and affected token, where V is the size of the token vocabulary." "Finally, we concatenate these vectors together across some set of T future tokens (at the ablated index or later) to obtain a vector of V⋅T total numbers." "We perform this for various autoencoders trained on the post-MLP residual stream at layer 8 in GPT-2 small, with T=16." So: - V = vocabulary size (number of possible tokens) - T = number of future positions they track (16 in their experiments) The measurement process: 1. For each ablation of an active latent 2. At each of T positions (current + next T-1) 3. They get V logit differences (one per vocabulary item) 4. This gives V×T total numbers per ablation
V and T define the dimensions of their measurement:
- V = vocabulary size (e.g., 50,257 for GPT-2)
- T = 16 = number of token positions they track
For each ablation experiment:
- When you ablate one active latent at position
- You measure logit changes at positions
- At each position, you get logit differences (one per vocabulary token)
- Total: numbers per ablation
They then measure sparsity of this dimensional vector using , which gives an "effective number of vocab tokens affected."
The key insight: If ablating a single SAE feature only affects a few specific tokens at a few specific positions (sparse effect), that's interpretable. If it changes many logits across many positions (dense effect), that suggests the feature isn't capturing a clean, modular computation.
For example:
- Sparse/good: Ablating a "plural noun" feature mainly affects "-s" endings at the next position
- Dense/bad: Ablating a feature changes logits broadly across vocabulary and positions
Can you unpack
Because a constant difference at every logit does not affect the post-softmax probabilities
Thinking Summary
The user is asking about why a constant difference at every logit doesn't affect post-softmax probabilities. This is a fundamental property of the softmax function. If we have logits $l_1, l_2, ..., l_V$ and we add a constant $c$ to all of them, the softmax probabilities remain unchanged. Let me work through the math: - Original softmax: $p_i = \frac{e^{l_i}}{\sum_j e^{l_j}}$ - After adding constant $c$: $p_i' = \frac{e^{l_i + c}}{\sum_j e^{l_j + c}} = \frac{e^{l_i} \cdot e^c}{\sum_j e^{l_j} \cdot e^c} = \frac{e^{l_i} \cdot e^c}{e^c \cdot \sum_j e^{l_j}} = \frac{e^{l_i}}{\sum_j e^{l_j}} = p_i$ So the probabilities are invariant to adding a constant to all logits. This is why they subtract the median - to remove any constant shift component from the logit differences.
This is a key property of the softmax function. Let me explain:
The Math: If you have logits and compute softmax probabilities:
Now if you add a constant to every logit:
The terms cancel out! The probabilities remain unchanged.
Why this matters for their metric: When they ablate a latent and measure logit differences, they might get something like:
- Token A: +2.5
- Token B: +2.5
- Token C: +2.5
- ... (all tokens shifted by same amount)
This uniform shift doesn't change which token is most likely or by how much. It's essentially a "null effect" on the actual distribution.
By subtracting the median logit difference, they remove this constant component and focus on the relative changes between different tokens - which are what actually matter for the model's behavior. This makes their sparsity metric more meaningful, as it now measures whether the ablation has selective effects on specific tokens rather than just shifting everything equally.
Can you elaborate on the refinement method described in the first paragraph of § 5.1 and its use in other applications as cited
Thinking Summary
The user is asking about the refinement method described in section 5.1 of the paper "Scaling and evaluating sparse autoencoders". Let me find that section and understand what they're referring to. In section 5.1 "TopK prevents activation shrinkage", they describe a refinement procedure to measure activation shrinkage. The key passage is: "To empirically measure the magnitude of activation shrinkage, we consider whether different (and potentially larger) activations would result in better reconstruction given a fixed decoder. We first run the encoder to obtain a set of activated latents, save the sparsity mask, and then optimize only the nonzero values to minimize MSE. This refinement method has been proposed multiple times such as in k-SVD (Aharon et al., 2006), the relaxed Lasso (Meinshausen, 2007), or ITI (Maleki, 2009). We solve for the optimal activations with a positivity constraint using projected gradient descent." So the refinement method is: 1. Run the encoder to get activated latents and their sparsity pattern 2. Fix the sparsity mask (which latents are active) 3. Optimize only the values of the active latents to minimize reconstruction error 4. Use positivity constraint and projected gradient descent They cite three prior works that have used similar refinement approaches: - k-SVD (Aharon et al., 2006) - Relaxed Lasso (Meinshausen, 2007) - ITI (Maleki, 2009) The user wants me to elaborate on this method and its use in other applications as cited.
The refinement method described in §5.1 is a two-stage optimization procedure designed to measure and correct for activation shrinkage in sparse autoencoders. Here's how it works:
The Refinement Procedure
-
First stage: Run the encoder normally to obtain:
- The set of activated latents (sparsity pattern)
- Initial activation values
-
Second stage: Fix the sparsity mask and optimize only the magnitudes:
- Keep the same set of active latents (don't change which ones are "on")
- Optimize their values to minimize reconstruction MSE
- Apply positivity constraint using projected gradient descent
The key insight is that if the refined activations are systematically larger than the original ones, it indicates the encoder was suffering from activation shrinkage (typically due to L1 penalties).
Historical Applications
The paper cites three prior works that used similar refinement approaches:
k-SVD (Aharon et al., 2006)
In dictionary learning for sparse representation, k-SVD alternates between:
- Sparse coding: Finding sparse coefficients given a fixed dictionary
- Dictionary update: Updating the dictionary given fixed sparse patterns
- The refinement step updates coefficient values while maintaining sparsity structure
Relaxed Lasso (Meinshausen, 2007)
This addresses the bias introduced by L1 regularization:
- First stage: Standard Lasso to select variables
- Second stage: Refit coefficients using only selected variables without L1 penalty
- This "relaxes" the shrinkage while maintaining variable selection
ITI - Iterative Thresholding with Inversion (Maleki, 2009)
A sparse recovery algorithm that:
- Applies thresholding to select active components
- Then inverts (refits) on the selected support set
- Iterates between selection and refinement
Common Theme
All these methods recognize that sparsity-inducing penalties (like L1) create two effects:
- Selection: Choosing which components are active
- Shrinkage: Reducing the magnitude of active components
The refinement step separates these effects, keeping the selection but removing the shrinkage bias. This leads to better reconstruction quality while maintaining the same sparsity level.
In the context of this paper, the refinement procedure serves as a diagnostic tool - TopK autoencoders show minimal change during refinement (indicating no shrinkage), while ReLU autoencoders with L1 penalties show systematic increases in activation values during refinement.
JumpReLU mentioned
Another approach is to replace the ReLU activation function with a ProLU (Taggart, 2024) (also known as TRec (Konda et al., 2014), or JumpReLU (Erichson et al., 2019)), which sets all values below a positive threshold to zero … Because the parameter θ is non-differentiable, it requires a approximate gradient such as a ReLU equivalent (ProLU-ReLU) or a straight-through estimator (ProLU-STE) (Taggart, 2024).
In the section on progressive codes, they mention that using Multi-TopK mitigates overfitting. I don't see an explicit discussion on how they measure how progressive the code is, what am I missing?
Thinking Summary
The user is asking about the progressive codes section of the paper and how the authors measure how progressive a code is. Let me look at section 5.3 "Progressive recovery" carefully. Looking at the text: "In a progressive code, a partial transmission still allows reconstructing the signal with reasonable fidelity (Skodras et al., 2001). For autoencoders, learning a progressive code means that ordering latents by activation magnitude gives a way to progressively recover the original vector. To study this property, we replace the autoencoder activation function (after training) by a TopK(k′) activation function where k′ is different than during training. We then evaluate each value of k′ by placing it in the L0-MSE plane (Figure 10)." So they measure progressiveness by: 1. Training with a specific k value 2. At test time, changing the activation function to use different k' values 3. Plotting how the MSE changes as k' varies (shown in Figure 10) A good progressive code would show smooth improvement in MSE as k' increases. They find that: - Regular TopK "only gives a progressive code up to the value of k used during training" - meaning MSE improves up to k, then gets worse - Multi-TopK (training with multiple k values like L(k) + L(4k)/8) gives "a progressive code over all k′" So the measurement is essentially: does the reconstruction quality improve smoothly as you allow more latents to activate at test time? They visualize this in the L0-MSE plane shown in Figure 10.
Looking at Section 5.3, they measure how progressive a code is by testing how well the autoencoder performs when they change the number of active latents at test time.
Specifically, they:
- Train with a fixed k value (e.g., k=32)
- At test time, replace the activation function with TopK(k′) where k′ is different from training
- Plot the results in the L0-MSE plane (Figure 10) to see how reconstruction quality changes as k′ varies
A truly progressive code would show monotonic improvement in MSE as k′ increases - meaning you can smoothly trade off between sparsity and reconstruction quality by just changing how many top activations you keep.
Their key findings:
- Regular TopK overfits: It only gives good progressive behavior up to the training value of k. Beyond that, MSE actually gets worse as you add more latents
- Multi-TopK fixes this: By training with multiple k values (e.g., ), they get smooth progressive behavior across all k′ values
So the "progressiveness" is measured by whether the L0-MSE curve shows smooth improvement rather than degradation when you vary the number of active latents at test time. Figure 10 visualizes this directly - the Multi-TopK curves are smooth across the entire range while regular TopK curves bend upward (worse MSE) after the training k value.
How exactly doesreplacing the activation function at test time work ( as in eg $5.3.2 )? Which parts stay the same and where do the jump ReLU parameters come from?
When replacing the activation function at test time (as described in Section 5.3.2), here's what happens:
What stays the same:
- All trained weights: encoder weights , decoder weights
- All biases: ,
- The overall architecture: encoder → activation → decoder
What changes:
Only the activation function itself is swapped out. For example:
- If trained with TopK(k=32), you might test with TopK(k'=64) or JumpReLU(θ)
- If trained with ReLU, you might test with TopK(k') or JumpReLU(θ)
Where JumpReLU parameters come from:
The threshold parameter for JumpReLU is chosen at test time as a hyperparameter. The authors sweep across different values of to generate the curves in Figure 10. Specifically:
For JumpReLU:
The process is:
- Train autoencoder with original
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In § 6, they mention hat they only use a context length of 64. Most of the really interesting behavior I've seen in language models happens in long context where the model can do in-context learning. Why would we believe or not believe that these interpretability methods carry over to longer context situations, i.e. Why do we expect or not expect that autoencodos trained on of short sequences would carry over to longer ones?
Thinking Summary
The user is asking about the context length limitation in the sparse autoencoder paper (64 tokens) and whether the interpretability methods would carry over to longer contexts. This is a great question about the generalization of these methods. Key points to consider: 1. Why they used short context (computational efficiency) 2. What changes with longer context (in-context learning, more complex behaviors) 3. Reasons to expect generalization 4. Reasons to be skeptical 5. What kinds of features might be missed I should give a balanced view that acknowledges both potential for generalization and important limitations.
This is an excellent question that gets at a fundamental limitation of the paper. The 64-token context length is indeed quite restrictive compared to the interesting emergent behaviors we see in longer contexts. Let me think through both sides:
Reasons to expect some generalization:
-
Basic features are context-invariant: Many low-level features (syntax, basic semantics, entity recognition) likely activate similarly regardless of context length. A "Python code" feature or "discussing economics" feature probably works similarly at token 64 vs token 1024.
-
Compositional structure: If the model builds complex behaviors by composing simpler features, and SAEs capture those atomic features well, then longer-context behaviors might still decompose into these same building blocks.
-
Local attention patterns dominate: Even in long contexts, much of the computation is still local - most tokens attend strongly to nearby tokens. Features learned from 64-token windows capture these local patterns.
Reasons to be skeptical:
-
In-context learning is fundamentally different: ICL involves the model learning new input-output mappings on the fly. The features that implement "store this example," "retrieve relevant examples," and "apply learned pattern" might be completely invisible in short contexts.
-
Long-range dependencies create new features: Features like "callback to paragraph 3's main claim" or "maintaining consistency with character traits established 500 tokens ago" simply cannot exist in 64-token windows.
-
Attention head specialization: Some attention heads might specialize for long-range retrieval in ways that are never activated in short contexts. The SAEs would miss these entirely.
-
Different computational strategies: Models might use entirely different algorithms for tasks depending on available context. A 64-token window might use shallow pattern matching while 2000 tokens enables more sophisticated reasoning.
I think the most likely outcome is that these SAEs capture a subset of features that are necessary but not sufficient for understanding long-context behavior. They'd miss the architectural innovations that make transformers powerful for long sequences - the very features that distinguish them from models that can only handle short contexts.
Initialization for optimizer mostly seems to follow Towards Monosemanticity
Using the referenced paper, can you unpack each of the bullet points in §A.1 into a few-paragraphs explanation?
Thinking Summary
The user is asking me to look at Section A.1 of the "Scaling and evaluating sparse autoencoders" paper and unpack each bullet point with a few paragraphs of explanation. Let me find Section A.1 in the paper. Looking through the content, I can see references to Appendix A for optimization details. Let me search for Section A.1 specifically. From the paper extract, I can see mentions of Section A.1 and Section A.2, but the full content of these sections isn't included in the extract. However, there are several references to initialization schemes and optimization details throughout the paper. The paper mentions: - Section A.1 appears to be about initialization schemes - Section A.2 is about auxiliary losses for preventing dead latents - There's a mention that "in the infinite width limit, TopK autoencoders with our initialization scheme are actually optimal at init using our init scheme (Section A.1)" Since the full Section A.1 isn't provided in the extract, I should inform the user that I don't have access to the complete content of that section. However, I can discuss what the paper says about initialization based on the references throughout the document.
I don't have access to the complete Section A.1 from the paper in the provided extract. The content appears to be truncated and doesn't include the full appendix sections. However, based on references throughout the paper, I can provide context about what Section A.1 likely covers:
Initialization Schemes (Section A.1 Context)
From the paper's references to Section A.1, it appears to focus on initialization methods for sparse autoencoders. The paper mentions several key aspects:
Encoder-Decoder Transpose Initialization
The paper states they "initialize the encoder to the transpose of the decoder" as one of their key techniques for preventing dead latents. This initialization strategy likely helps ensure that the encoder and decoder start with compatible representations, potentially reducing the likelihood that certain latents will fail to activate during training. In traditional autoencoders, this transpose initialization can help maintain a reasonable reconstruction quality from the start of training.
Optimal Initialization for TopK in Infinite Width
The paper makes an intriguing claim that "in the infinite width limit, TopK autoencoders with our initialization scheme are actually optimal at init." This suggests that their initialization method has special theoretical properties when the autoencoder has infinite latents. This could mean that in the limit of infinite features, their initialization already provides perfect reconstruction before any training, which would be a remarkable property.
Connection to Dead Latent Prevention
The initialization scheme appears to be crucial for preventing dead latents - features that stop activating entirely during training. The paper reports achieving only 7% dead latents even in their 16 million latent autoencoder, which they attribute partly to their initialization strategy combined with auxiliary losses.
Without the full Section A.1 text, I cannot provide the specific bullet points and their detailed explanations. To get the complete information, you would need access to the full paper including all appendices. The archived version or full PDF would contain these details that aren't included in this extract.
continuing here since this post suffered from a truncation bug in context import that caused the appendices to be cut off