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This paper proves near-optimal query bounds for estimating the total variation distance between two autoregressive models, and validates the method by measuring how far two inference engines serving identical weights drift apart.

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2026-08-01 12:33 UTC pith:YYDKIPEC

load-bearing objection Solid theory, honest but imperfect empirical bridge: the theorems hold under the stated noisy-oracle model, and the paper deserves serious refereeing despite the independence assumption being violated on real engines. the 2 major comments →

arxiv 2607.19510 v1 pith:YYDKIPEC submitted 2026-07-21 cs.LG cs.DSstat.MEstat.ML

Total Variation Distance Estimation in Autoregressive Models

Eric Price , Kevin Tian , Zhiyang Xun , Yusong Zhu This is my paper
classification cs.LG cs.DSstat.MEstat.ML
keywords total variation distanceautoregressive modelsdistribution testingprefix query oraclesmultilevel Monte Carlonoisy logitsLLM inference enginessample complexity
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The authors aim to show that total variation distance — the probability that an optimal detector could tell two models' outputs apart — can be estimated efficiently for autoregressive LLM distributions, despite TV having no chain rule. They give an O(n/ε²)-query algorithm when full prefix logits are available and prove this is tight; under weaker sample access they give O(n²K/ε²) queries, and under noisy logits they interpolate with O((n+n²σ²)/ε²) up to log factors. The key move is to treat serving engines as noisy prefix distribution oracles and use a multilevel Monte Carlo telescoping sum to average away oracle noise. The authors validate the method by measuring vLLM vs. SGLang serving identical Qwen3-0.6B weights, finding TV≈0.59 at length 500 even where KL divergence is infinite.

Core claim

The paper's central claim is that the total variation distance between two length-n autoregressive distributions can be estimated to additive ε with query counts that are near-optimal under three natural API models. Under prefix logit access, a likelihood-ratio estimator — average of (1−µ(X)/π(X))₊ over trajectories sampled from π — uses O(n/ε²) queries, and the paper proves a matching lower bound by embedding a biased-coin estimation problem into blocks that require n queries to reveal. Under noisy prefix distribution access, where each query returns an unbiased random sketch of the next-token distribution with relative variance σ², the paper uses the mixture representation TV(π,µ)=E_{X∼M}[

What carries the argument

The load-bearing objects are: (1) the mixture-representation identity TV(π,µ)=E_{X∼M}[|π(X)−µ(X)|/(π(X)+µ(X))] with M=(π+µ)/2, which turns TV into an expectation over a single mixture distribution; (2) the noisy prefix distribution oracle (Definition 3), which returns unbiased estimates of each prefix-conditional next-token distribution with relative variance σ² and subsumes both logit access (σ=0) and sample access (σ²<K); and (3) multilevel Monte Carlo applied to a telescoping sequence of averaging depths rₗ=1+cnσ²2ˡ. Coupled increments Yₗ=Zₗ(X)−Z_{l−1}(X) share the same outer draw X, giving variance that decays as 2⁻ˡ while query cost grows as n²σ²2ˡ, so balancing across levels yields the

Load-bearing premise

The load-bearing premise is that every repeated query to the same prefix returns an independent, unbiased estimate of a fixed next-token distribution with bounded relative variance σ²; the paper itself measures short-lag autocorrelations up to ±0.08 and slow drift in production engines, so independence is violated and is patched only to exchangeability by permutation.

What would settle it

Run the noisy-oracle estimator on a synthetic pair with known TV, feeding an oracle whose repeated per-prefix outputs are correlated (e.g., block-correlated with lag-1 autocorrelation 0.5) at fixed σ; if the query count needed to reach error ε grows faster than O((n+n²σ²)/ε² log²(1/ε)) or the estimate plateaus at a biased value, the independence premise behind the main theorem is falsified.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • A user can compare two 'same-model' serving engines, or a provider can validate a faster inference stack, by estimating TV to any additive ε with query counts that scale linearly in sequence length under logit access, independent of vocabulary size.
  • Under sample-only access from closed APIs, the improved O(n²K/ε²) bound removes the earlier n³/ε⁵ dependence and the alphabet-size penalty m when the next-token support K is sparse.
  • The noisy-logit guarantee quantifies how per-query nondeterminism degrades estimation: TV remains estimable even when σ is large, with cost growing as n²σ²/ε², and it recovers the exact logit bound as σ→0.
  • Because TV directly bounds the error of any black-box distinguishing test, a small estimated TV implies no single-output detector can reliably tell the two engines apart, while a large TV quantifies the gap in an interpretable 0-to-1 scale.
  • The case study shows cross-engine distances should be reported with their length, truncation, and serving configuration, since measured TV grows with n and with stricter top-k truncation.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • A direct extension the authors leave implicit: the calibrated noisy-oracle model could serve as a practical regression test for inference stacks — if a kernel or batching change moves the estimated TV above a chosen threshold, it should be flagged before deployment.
  • Because real per-call noise is heavy-tailed (excess kurtosis ≈9) and concentrated on low-probability tokens, using a single conservative σ may inflate the practical budget; an adaptive per-prefix variance estimate would likely tighten the query count.
  • The exchangeability patch for correlated repeats suggests a testable hypothesis: if batch-scheduling dependencies do not average out, the estimator would plateau above the true TV; measuring TV against a ground-truth pair across batch sizes would expose the threshold.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 3 minor

Summary. The paper studies the query complexity of estimating the total variation (TV) distance between two length-n autoregressive distributions under three oracle models: prefix sample access, exact prefix logit access, and a noisy prefix distribution oracle with bounded relative variance. The main theoretical results are: (i) an O~(n^2 K/ε^2) sample-access estimator improving the prior O~(n^3 m/ε^5) bound of Meel et al.; (ii) an O(n/ε^2) logit-access estimator with a matching Ω(n/ε^2) lower bound; and (iii) an O~((n+n^2 σ^2)/ε^2) noisy-logit estimator that smoothly interpolates between the two, with a corresponding lower bound. The upper bound uses a likelihood-ratio estimator for logit access and a multilevel Monte Carlo scheme for noisy access, based on a χ² bound for repeated oracle averages. The paper also presents an empirical case study measuring TV between vLLM and SGLang serving identical Qwen weights, including a denoising effect of repeated queries and a multilevel schedule selection protocol.

Significance. If the technical claims hold, this is a substantial contribution to distribution testing for autoregressive models. The logit-access bound is tight and apparently new; the noisy-oracle interpolation is natural and practically motivated; the sample-access bound improves over the prior state of the art in both n and ε. The proofs of Lemmas 6–9 and Theorems 4–7 are self-contained and internally consistent: the χ² bound, the Lipschitz bound, the MLMC variance balance, and the rare-escape lower-bound embedding are all carefully argued. The paper also ships reproducible code and gives a detailed empirical protocol. The main weakness is that the empirical validation on real serving engines relies on an independence assumption that the paper itself measures as violated; the exchangeability patch does not fully restore the concentration guarantees for the nonlinear estimators used. The theoretical results remain valid under the stated model, but the practical claims need to be framed more carefully.

major comments (2)
  1. [Section 4.1 and Appendix B.1 / Section 5, Figures 2–4] Definition 3 and Theorem 6 require that repeated oracle calls at a fixed prefix be independent. Appendix B.1 reports short-lag autocorrelations up to ±0.08 and slow drift in vLLM and SGLang, so the real engines violate this premise. The repeat-axis permutation makes the repeats exchangeable, not independent. For linear averages this removes a common per-cell component to leading order, but the estimators bπ_r, bμ_r, and hence Ẑ are nonlinear functions (products and ratios) of those averages, so the variance and concentration guarantees of Lemma 6 and Theorem 6 do not provably transfer. The paper's own diagnostic that raw-order variance ratios inflate by roughly 2× confirms that the dependence is non-negligible. Consequently the empirical RMSE curves and multilevel savings in Figures 4 and Table 2 are an approximate-iid heuristic rather than a consequence of Theorem 6. Please state this
  2. [Appendix B.2] The synthetic noisy oracle is said to 'satisfy the unbiasedness and relative-variance conditions of Definition 3 exactly while relaxing simplex membership.' But Definition 3 requires the oracle to return a vector in P(Σ), and the construction leaves individual draws unclipped so they may leave [0,1]. A vector with negative components is not a probability distribution, so the χ² expression in Definition 3 is not well defined for such outputs. The later clipping of the r-averaged probability to [10^{-12},1] changes the estimator and may introduce bias. To make the synthetic validation a genuine instantiation of Definition 3, the authors should use a proper noisy distribution oracle (e.g., the Gaussian logit perturbation of Lemma 5 followed by softmax) or explicitly label the synthetic experiment as a heuristic stress test outside the model.
minor comments (3)
  1. [Lemma 4] The statement and proof use a lowercase 'k' for the support size (e.g., 'σ² = k' and '≤ |supp(π_s)| ≤ k'), but the paper otherwise reserves K for the maximum support size. Please use K consistently or define k locally.
  2. [Algorithm 1] The query count '3nN' is conservative: the sampling phase already reveals the π log-probabilities along the sampled trajectories, so the additional evaluation of L_{π,i} may be redundant. This does not affect the stated O(n/ε² log(1/δ)) bound, but the constant could be clarified.
  3. [Figure 2 caption] The caption uses 'TV(π, μ)' in text such as 'estimated TV( , )' without defining which distribution is π and which is μ in the figure. Please make the notation explicit.

Circularity Check

0 steps flagged

No circularity: theorems are parameter-free from the stated noisy-oracle model; lower bounds reduce to external classical results; experiments estimate σ and TV rather than fitting predictions.

full rationale

No circular step is present. Theorems 4 and 6 derive query bounds directly from Definition 3's relative-variance assumption via Lemma 1 (likelihood-ratio representation), Lemma 6 (χ² accumulation), Lemma 8 (squared-error bound), and Lemma 9 (MLMC variance decay); σ is an input parameter, not fitted to the target TV. The sample-access result instantiates Definition 3 with one-hot vectors and σ²≤K through Lemma 4, an explicit simulation reduction rather than a renaming. Lower bounds (Theorems 5, 7) embed the problem into finite-domain TV estimation and invoke the external minimax bound of [JHW18] (Proposition 1), not any result of the present authors. Experiments determine σ from repeated queries and measure TV from trajectories; Figure 4's pilot-chosen schedules include pilot costs and are compared against baselines, so no fitted value is relabeled as a prediction. The paper's own caveat, "The main caveat is that we need independence across samples" (Section 4.1), and the measured autocorrelations in Appendix B.1 are a threat-model limitation on real serving engines; they affect whether Theorem 6's assumptions hold empirically, but they are not a circularity in the derivation itself.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 0 invented entities

The theoretical results are parameter-free: the only quantities that enter are inputs (n, ε, δ, σ², K, m) and universal constants. No number is fitted to data in any theorem. In the experiments, σ is estimated by calibration and the MLMC schedule is chosen by a pilot; these are data-dependent choices, not fitted parameters of the claims. No new physical or conceptual entities are postulated.

axioms (4)
  • domain assumption Noisy prefix distribution oracle (Definition 3): each query returns an unbiased estimate p̂ of the true conditional p with E_{a∼p}[Var(p̂(a)/p(a))] ≤ σ², and outputs are independent across queries.
    Central to Theorem 3 and to interpreting Figures 1-2. The paper's own caveat in §4.1: 'The main caveat is that we need independence across samples.' Appendix B.1 finds short-lag autocorrelations up to ±0.08 in real engines, so this premise is only approximately true in practice.
  • domain assumption Exact prefix logit oracle (Definition 1), with one vector query returning log-probabilities for all tokens.
    Used in Theorem 2 and Algorithm 1; in practice logits are noisy, which motivates Definition 3 (Section 4.1).
  • domain assumption Autoregressive factorization π(x) = ∏_i π(x_i | x_<i) and chain-rule computation of sequence log-probabilities in n prefix queries (Equation 7).
    Defines the problem and the cost model; all algorithms rely on n queries per trajectory.
  • standard math Standard inequalities (Hoeffding, Chebyshev, mutual-information bounds, Gaussian Poincaré) and the finite-domain minimax l1 lower bound of [JHW18].
    Invoked in proofs of Theorems 4-7 and in Proposition 1; standard background.

pith-pipeline@v1.3.0-alltime-deepseek · 31137 in / 21032 out tokens · 190823 ms · 2026-08-01T12:33:07.668709+00:00 · methodology

0 comments
read the original abstract

Modern LLM deployments use a number of implementation choices and inference optimizations (e.g., batching, custom kernels, and quantization) on top of fixed weights, so two engines serving "the same model" can produce meaningfully different distributions. We study the problem of estimating the total variation (TV) distance between two length-$n$ autoregressive distributions to additive error $\varepsilon$, under three access models. (1) Under sample access, we use $\widetilde{O}(n^2 K/\varepsilon^2)$ queries, where $K$ is the maximum support of the next-token distribution. This improves upon the $\widetilde{O}(n^3 m/\varepsilon^5)$-query estimator of Meel et al. (2025), where $m \geq K$ is the total size of the token alphabet. (2) Under logit access, we use $O(n/\varepsilon^2)$ queries, and this is tight. (3) Under noisy logit access, we smoothly interpolate between the above two guarantees: if probability values are given to relative error $\sigma$, we use $\widetilde{O}((n+n^2\sigma^2)/\varepsilon^2)$ queries. We complement our theoretical results with an empirical evaluation of our algorithms, for example measuring the distance between SGLang and vLLM serving identical weights. Our experiments highlight the robustness and practicality of estimating the total variation distance, which remains estimable where the KL divergence is infinite. Our code is available at https://github.com/XunZhiyang/llm-tv-estimation.

Figures

Figures reproduced from arXiv: 2607.19510 by Eric Price, Kevin Tian, Yusong Zhu, Zhiyang Xun.

Figure 1
Figure 1. Figure 1: Inference engines as noisy logit oracles. A served model is treated as a per-token log￾probability oracle: querying a fixed prefix repeatedly, we measure the relative standard deviation σ of the returned probabilities (σ 2 is the relative-variance parameter of Definition 3; five Qwen models, bf16, top-k = 20; calibration in Appendix B.3). (a) σ by attention backend under two batching conditions: isolated (… view at source ↗
Figure 2
Figure 2. Figure 2: Denoising separates true distance from oracle noise. (a) Per-token log-probability differences against a fixed reference (cuDNN, served): the oracle against itself and auto (SDPA’s au￾tomatic backend selection) against cuDNN yield the same zero-centered spike, while pairs that gen￾uinely differ (a changed batching condition; FlashAttention-2; math) yield broadened distributions. (b) The same pairs under th… view at source ↗
Figure 3
Figure 3. Figure 3: The total variation distance between two inference engines. vllm and sglang serving identical Qwen3-0.6B weights (bf16, top-k = 20), measured at the sampling condition (B = N = 512, replay scoring; Appendix B.5). The estimate decomposes at the trajectory level into a shared-support log-probability difference (both engines retain every sampled token and only the probabilities differ; purple) and support mis… view at source ↗
Figure 4
Figure 4. Figure 4: Multilevel variance reduction on two noisy conditions from the case study. (a) Engine pair, n = 2000: dashed curves are fixed single-level configurations (repetition count r), each flattening at its bias floor; the green curve is the pilot-chosen schedule at each budget, all pilot costs included (protocol in Appendix B.7). (b) Engine pair, B = 256 rescoring: schedules sharing the top level r = 16 converge … view at source ↗
Figure 5
Figure 5. Figure 5: Mean absolute error of Algorithm 1 on the synthetic hard instance at σ = 0 (exact logits), against 1/ √ N, with TV (π, µ) = 0.388 known by enumeration. The dashed line is the Gaussian￾asymptotic reference p 2/π p Var(R)/N. The fitted slope is −0.493 against the theoretical −0.500. 10 6 10 7 oracle-call budget 0.00 0.05 0.10 0.15 0.20 mean absolute error = 0.04 1 = 0.013, 2 = 0.002, 3 = 0.000 10 6 10 7 orac… view at source ↗
Figure 6
Figure 6. Figure 6: Mean absolute error against oracle-call budget for [PITH_FULL_IMAGE:figures/full_fig_p032_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Mean empirical top-k support union | S i≤r top-k(i)|/k against the number of repeated queries r at a fixed prefix (k = 20; 1,500 (trajectory, position) cells per model). Across five models the union exceeds the stated k by at most ∼10%, far below 2k. the step’s top-k log-softmax before the forcing is applied, since log-probabilities returned by the engine are computed after the processor and are contaminat… view at source ↗
Figure 8
Figure 8. Figure 8: Self-scoring faithfulness of two scoring interfaces. Because oracle noise at this condition is [PITH_FULL_IMAGE:figures/full_fig_p035_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: The cross-engine TV depends strongly on the API top- [PITH_FULL_IMAGE:figures/full_fig_p037_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Why the cross-engine distance varies with the prompt (story vs. code; positions where [PITH_FULL_IMAGE:figures/full_fig_p038_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Logit versus sample access: the same estimator run with different oracles on the syn [PITH_FULL_IMAGE:figures/full_fig_p039_11.png] view at source ↗

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