Gao’s logit-diff ablation is not a circuit rule

Prerequisites for bounding the metric choice

Bound the circuit-test decision to these versions: Gao, Dupré la Tour, Tillman, Goh, Troll, Radford, Sutskever, Leike, and Wu, Scaling and evaluating sparse autoencoders, arXiv:2406.04093v1 (6 June 2024), and Muchane, Richardson, Park, and Veitch, Incorporating Hierarchical Semantics in Sparse Autoencoder Architectures, arXiv:2506.01197v1 (1 June 2025). Later revisions, code, and figure-only panels are not substitutes for claims these texts do not make.

Stuart Patching, On Divergence- and Gradient-Preserving Coarse-Graining for Finite Volume Primitive Equation Ocean Models, arXiv:2107.05577v2 (19 November 2021), derives averaging weights on a regular triangular Arakawa-B mesh that preserve divergence and, by the same construction, the vertical component of curl, and a different set of averaging weights that preserve the scalar gradient. On FESOM2 simulations, those particular coarse-grainings give an overall reduction in L1 error relative to other methods. Those L1 figures compare ocean fields under different averaging weights. They are not SAE latent ranks, and the paper does not specify activation patching or task accuracy as a language-model circuit-test target.

Keep four interventions distinct. Latent ablation, as used by Gao et al., changes one autoencoder latent and reads the resulting logits. Activation patching, in the usual circuit-discovery sense, substitutes activations taken from another forward pass; that protocol is not stated in either SAE paper. Reconstruction substitution replaces a residual-stream vector by an SAE reconstruction and then measures downstream loss. A probe fits a predictor from latent activations to labels; that is predictive association, not a causal patch.

Do not carry a width, layer, or loss from one model to another. Gao et al. state d_model = 768 for the GPT-2 small density discussion in this ablation section. Muchane et al. train on layer 20 of Gemma 2-2B. Those are different subject models, sites, and objectives.

What Gao et al. define for latent ablations

In Section 4, Gao et al. list four quality metrics. The fourth is ablation sparsity: whether ablating individual latents has a sparse effect on downstream logits. Section 4.5 is the only place the text defines a logit-difference procedure for that intervention. They motivate it with a hypothesis, cited to Olah et al. 2024, that natural features are sparse in downstream effects as well as in activations. That hypothesis is author interpretation. The intervention itself is the ablation-and-compare step.

The procedure, quoted in substance from Section 4.5, is as follows. At a particular token index, obtain the residual-stream latents and ablate each autoencoder latent one by one. Compare logits before and after ablation. That yields V logit differences per ablation and affected token, where V is the token-vocabulary size. Because a constant difference on every logit does not change post-softmax probabilities, subtract the median logit difference at each token. Concatenate across a set of T tokens at the ablated index or later, producing a vector d of length V·T. Sparsity is (‖d‖₁/‖d‖₂)², which they describe as an effective number of vocabulary tokens affected. Divide by V·T so the score lies between 0 and 1. Smaller values mean sparser effects.

If median subtraction leaves the zero vector, (‖d‖₁/‖d‖₂)² is undefined. Section 4.5 does not state a convention for that case, and it does not state whether “ablate” means set the latent to zero, to its mean, or to another baseline.

1. Fix the site and token. Take autoencoder latents at one token index on the residual stream. The published logit-diff comparison uses GPT-2 small, the post-MLP residual stream, layer 8.
2. Ablate one latent at a time. Gao et al. ablate each autoencoder latent separately and compare logits before and after. Do not fill in an unstated ablation value.
3. Form the raw logit-difference vector. For each affected token, keep all V logit differences, not only a designated correct-token logit.
4. Remove the per-token median. Subtract the median logit difference at that token so a uniform logit shift, which does not change softmax probabilities, does not count as an effect.
5. Concatenate the window. Stack the median-centered vectors for T tokens at the ablated index or later. In the reported runs, T = 16.
6. Score concentration, then normalize. Compute (‖d‖₁/‖d‖₂)² and divide by V·T. Report the fraction as sparsity of downstream logit effects. Smaller is sparser, which is the direction Gao et al. treat as better on this metric.
7. Stop at that estimand. Do not convert the fraction into circuit membership, task accuracy, or a ranking against activation patching. The paper does not define those conversions.

The general training setup in the same paper uses context length 64 for all experiments, subtracts the mean over the model dimension, and normalizes inputs to unit norm before the autoencoder. Section 4.5 does not restate the token sample used for the ablation statistic beyond the model, site, layer, and T = 16. Their code release is linked from the paper as openai/sparse_autoencoder; the paper text, not an unread code path, is the support for the procedure above.

On those GPT-2 small layer-8 autoencoders, Gao et al. report that latents are sparse in an absolute sense, with this score at 10–14%, whereas ablating residual-stream channels gives 60%, slightly better than the random-vector reference of about 2/π. Figure 6 plots probe loss and logit-diff sparsity against number of latents n and active latents k; higher n generally improves the metrics, and both are worse at L₀ = 512, which Section E.5 identifies as a dense-solution regime as k approaches d_model = 768. Those statements are about SAE quality under this ablation score. They are not a ranking of named circuit components.

What neither paper sets as a circuit-test target

Gao et al. also measure downstream KL and delta cross-entropy after replacing the residual stream by the full autoencoder reconstruction, 1d probe cross-entropy on 61 binary tasks, and n-gram explanation precision and recall. Increasing L₀ makes token-pattern explanations worse but probe loss and ablation sparsity better, until L₀ nears d_model. That is evidence that SAE-quality metrics disagree about which autoencoder looks better. It is not an experiment on whether patching metrics reorder component-level causal attributions.

They cite prior work on circuits and say applications include identifying circuits. The contribution statement mentions early exploration of circuit sparsity. None of that text specifies activation patching, names task accuracy as the circuit objective, or reports a reordering of causal components. Clamping latents found by attribution, mentioned qualitatively for a GPT-2 autoencoder with n = 131072 and k = 32, is a separate intervention from the Section 4.5 sparsity score.

Muchane et al. cite circuit discovery as a downstream use of SAEs and cite Gao et al. for the TopK architecture and for an auxiliary dead-latent loss they chose not to use. Their own experiments do not run Gao’s logit-difference ablation. They train on 1 billion residual-stream vectors from layer 20 of Gemma 2-2B, taken from the first 256 tokens of Wikipedia articles, normalized to unit norm, without the mean subtraction or bias Gao et al. use. The training objective is reconstruction, plus auxiliary terms, for 4 epochs. Reconstruction is scored by Euclidean error, 1 minus explained variance, and SAEBench 1 − cross-entropy loss after replacing activations with reconstructions. Interpretability checks are feature visualizations, a first-letter absorption probe, and cross-lingual overlap of top-activated features. The first-letter benchmark measures how much of a classification probe is left unexplained by one feature. That is probing and absorption, not a circuit test scored by task accuracy, and not activation patching.

A labeled toy, not a published reordering

Neither SAE paper supplies a public component ranking that can be resorted under two patching metrics. The following arithmetic is hypothetical. It is not a Claude, GPT-2, GPT-4, or Gemma result, and it is not an estimate from either PDF. It is also not a result from Patching’s ocean-model coarse-graining.

Assume one token, T = 1, and a toy vocabulary of V = 3 so the median is the middle order statistic. Coordinate 1 is an unofficial “target” logit. Also assume an unofficial accuracy drop, chosen only to show rank conflict. Gao et al. do not rank circuits by either unofficial score.

Component A has raw logit differences (2.0, 0.0, 0.0) and accuracy drop 0.02. Component B has (0.5, 1.5, 1.0) and accuracy drop 0.25. Component C has (0.4, 0.4, −0.2) and accuracy drop 0.10.

If “more causal” means larger absolute target-logit change, the values are A = 2.0, B = 0.5, C = 0.4, so the order is A, B, C. If “more causal” means larger accuracy drop, the values are B = 0.25, C = 0.10, A = 0.02, so the order is B, C, A. A moves from first to last. That is a reordering under two decision rules these papers never compare.

Applying Gao et al.’s formula to the same toy vectors answers a third question, concentration after median removal. For A, the median is 0, d = (2, 0, 0), ‖d‖₁ = 2, ‖d‖₂ = 2, (‖d‖₁/‖d‖₂)² = 1, and division by V·T = 3 gives 1/3. For B, the ordered values are 0.5, 1.0, 1.5, so the median is 1.0 and d = (−0.5, 0.5, 0.0). Then ‖d‖₁ = 1, ‖d‖₂² = 0.25 + 0.25 + 0 = 0.5, and (‖d‖₁/‖d‖₂)² = 1/0.5 = 2. Division by 3 gives 2/3. For C, the ordered values are −0.2, 0.4, 0.4, so the median is 0.4 and d = (−0.6, 0, 0). Then (‖d‖₁/‖d‖₂)² = 1 and the normalized score is 1/3. Smaller-is-sparser order is A tied with C, then B. That matches neither unofficial causal order. The toy uses V = 3, not GPT-2’s vocabulary, and it does not implement an unstated ablation baseline.

A real reordering test would have to be run and reported as new work: fix the model, checkpoint, layer, intervention, and component set; pre-specify at least two scores; and compare rank order. These two papers do not contain that test.

Checks and failure cases

Check that a claimed “Gao logit difference” still has every Section 4.5 operation: per-latent ablation, all V logits, median subtraction, concatenation over T, and normalization by V·T. A single target-logit delta, a probability delta, or an accuracy delta is a different metric even if it begins from the same ablation.

Do not treat downstream delta cross-entropy as this procedure. In Section 4.1, Gao et al. replace the whole residual stream with the reconstruction and measure KL and cross-entropy gap. That tests whether the reconstruction preserves language-model loss. It does not score one latent’s logit-difference sparsity. Their 16 million latent GPT-4 reconstruction results, including the stated correspondence to 10% of GPT-4 pretraining compute and 98.2% zero-ablation fidelity, belong to that reconstruction metric, not to Figure 6’s GPT-2 small logit-diff comparison.

Do not import Muchane et al.’s SAEBench cross-entropy, explained variance, or first-letter absorption into a circuit ranking. Those scores use Gemma 2-2B layer-20 reconstructions or probes. Muchane et al. also drop Gao et al.’s mean subtraction because they found it hurt training stability, and they replace Gao et al.’s dead-latent auxiliary with an orthogonality penalty. Chaining the papers does not transfer the 10–14% or 60% ablation figures.

Do not import Patching’s FESOM2 L1 reductions either. arXiv:2107.05577v2 reports those errors for coarse-grained ocean velocity and pressure-gradient fields under operator-specific averaging weights, not for language-model components under a patching metric.

Do not read “smaller sparsity” as “keep this component.” Gao et al. use the small fraction as evidence of sparse downstream effects, under an explicit hypothesis, and they report that the metric worsens when L₀ = 512 and solutions become dense. A latent with a large, diffuse logit effect can look important for a task and poor on this sparsity score, as the toy is constructed to allow.

Absence of a comparison is not evidence of invariance. Gao et al. and Muchane et al. do not report an experiment that measured activation patching and task accuracy and found the same causal order. They omit that comparison. Their stated results therefore do not establish that a logit-difference score, activation patching, and task accuracy would select the same components.

1. Prerequisites for bounding the metric choice
2. What Gao et al. define for latent ablations
3. What neither paper sets as a circuit-test target
4. A labeled toy, not a published reordering
5. Checks and failure cases