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REVIEW 4 major objections 7 minor 22 references

Skewed Memorization in Large Language Models: Quantification and Decomposition

T0 review · 4 major / 7 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read Memorization in fine-tuned LLMs is a skewed tail, amplified by data blending, and it decomposes into near-independent per-token recall probabilities.

desk verdict Dataset mixing and tail skewness results are worth publishing; the token-independence decomposition is a tautology and the extraction conclusion overreaches. read the letter →

arxiv 2502.01187 v1 pith:Z7XOP3B4 submitted 2025-02-03 cs.AI cs.CLcs.LG

Hao Li , Di Huang , Ziyu Wang , Amir M. Rahmani This is my paper · ORCID
classification cs.AIcs.CLcs.LG
keywords memorizationlargelanguagemodelssupervisedfine-tuningskeweddistributionprefixcontinuationtoken-leveldecompositionmutualinformationprivacyrisk
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Supervised fine-tuning makes large language models memorize a small subset of training data far more strongly than the average suggests, and this paper quantifies that skewed tail. The authors measure, for each training example, how many consecutive words a fine-tuned model can recall after a fixed prefix, and they show that worst-case recall grows with training time and dataset mixing even when average loss falls. Their central empirical claim is that the chance of recalling a token depends only weakly on whether earlier tokens in the same phrase were recalled, so the probability of a memorized run of length n factors roughly into a product of per-token probabilities. If that decomposition holds, a model that correctly recalls the beginning of a phrase gives almost no information about whether it will recall the next token, which undercuts a common assumption behind training-data extraction attacks. The practical stakes: a mixed dataset produced maximum memorized recall above 50 words while a homogeneous medical dataset stayed at 16.

What carries the argument

The load-bearing mechanism is the per-token correctness decomposition. For each training sample and token position j, the paper builds a binary correctness matrix c_ij marking whether the model recalls token j given the prefix and all earlier generated tokens. It defines the random variable N_pre, the number of consecutive tokens correctly recalled after a fixed prefix, and proposes that P(N_pre ≥ n) ≈ ∏_{j=1}^n p_j, where p_j is the marginal probability of correct recall at position j. The independence that makes the product valid is tested by estimating mutual information MI(C_{J_pre}, C_j) between prefix correctness and token j correctness; low MI is taken as evidence that earlier recall success does not raise the odds of later recall.

What would settle it

Compute the same mutual-information statistic for token positions beyond j = 3, or on a second model and dataset; if MI between prefix correctness and correctness at positions 4 through 10 grows systematically, the product approximation fails. A direct behavioral check: in a prefix-extraction attack, vary the length of the verified prefix and measure next-token accuracy; if accuracy improves with prefix length, the paper's independence conclusion is wrong for that setting.

Watch

Extended reading notes

Core claim

The paper claims that memorization in LLM supervised fine-tuning is not an average phenomenon: a small fraction of samples account for long verbatim recalls, and this skewed tail is missed by mean-based metrics and by small-sample estimates. Using a prefix-continuation setup with greedy decoding, it defines N_pre, the number of tokens recalled consecutively after a fixed prefix, and treats its distribution as the object of study. The token-level decomposition shows that the survival function P(N_pre ≥ n) is approximated by the product of per-token recall probabilities p_j, because empirical mutual information between correctness at position j and correctness at earlier positions stays low. From this the paper concludes that successful recall of initial tokens is not a reliable confidence indicator for subsequent extraction attempts. A separate empirical result is that dataset composition shifts memorization sharply: the same 200 samples show statistically significantly higher memorization when trained inside a mixed dataset than in a homogeneous medical dataset, with maximum memorization jumping from under 16 to more than 50 words by epoch 100.

Load-bearing premise

The entire decomposition rests on the assumption that per-token recall is nearly independent of earlier recall success, so that the probability of a long memorized run is the product of individual token probabilities.

Editorial extensions

If this is right

  • Small-sample estimates systematically miss the worst-memorized examples, so privacy auditing of SFT models should scan the full dataset or use extreme-value-aware sampling.
  • Dataset composition, not just size, drives worst-case memorization: the same 200 samples were memorized far more when trained amid mostly other-domain data than in a homogeneous medical dataset.
  • Reducing dataset size accelerates memorization growth: a 256-sample subset reached maximum memorization near 70 words within 100 training steps, while the 1024-sample subset stayed below 20.
  • The near-independence of per-token recall invalidates the assumption that verified-prefix extraction attempts gain confidence from earlier correct tokens.
  • The proposed prefix-match metric connects monotonically to ROUGE and Levenshtein distance, so it can be used alongside standard evaluation measures.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the independence result survives on other models and datasets, training-data extraction attacks that rely on a verified prefix become much weaker than the literature assumes; attackers would need per-token confidence measures, not prefix confidence.
  • The product decomposition implies memorization length should be predictable from the per-token loss curve alone, which is testable by comparing predicted and observed survival functions on held-out fine-tuning runs.
  • The embedding-similarity result suggests a concrete mitigation: removing or diversifying training samples with similar prefixes but divergent suffixes should reduce the worst-case tail, which the paper does not itself test.
  • A geometric or linear-trend per-token probability model could be used to extrapolate tail risk from short prefixes, enabling privacy audits on datasets too large to scan fully.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 7 minor

Summary. The paper introduces a prefix-continuation memorization statistic Npre (the number of consecutive words correctly recalled after a fixed 100-character prefix) and studies its distribution when fine-tuning Llama-3.1-8B-Instruct with LoRA on a medical QA dataset, a mixed medical/open-domain dataset, and small subsets. The experiments report that memorization distributions are highly skewed; that maximum and upper-tail memorization grow with training epochs even while loss decreases; that mixing 200 out-of-domain samples into 9,723 medical samples increases maximum memorization from about 16 to about 50 words; that smaller datasets memorize earlier; and that small samples systematically underestimate worst-case memorization. The paper also proposes a theoretical decomposition of memorization length into per-token recall probabilities, claims these are nearly independent based on low mutual information and a product check, and concludes that successful recall of initial tokens cannot serve as a confidence indicator for subsequent extraction attempts. Additional analyses connect memorization to embedding similarity and to standard text metrics.

Significance. The empirical phenomena measured here, including skew, epoch growth, small-dataset effects, and context-dependent memorization of the same samples, are relevant and, if confirmed across seeds and datasets, would be a useful contribution. The paper has concrete strengths: it scans full training sets rather than relying on small samples; it uses non-parametric distribution analysis; the comparison of identical samples in two dataset contexts is a clean design; and the appendix relations to ROUGE and edit distance could help connect memorization metrics to standard evaluation. The central theoretical claim, however, is not currently established: the independence of per-token recall is supported only by early-position mutual information and by a product check that is partly a chain-rule identity, both on one run. Because the extraction-methodology conclusion depends on this claim, the manuscript needs substantially more evidence or a more modest framing before it can be accepted.

major comments (4)
  1. [Section 3.5.2 and Eq. (2.3.1)] The product check in Figure 8 is circular. Because p_j is defined, immediately after Eq. (2.3.1), as the probability that token j is correct given all previous tokens are correct, the identity P(Npre >= n) = product of p_j holds by the chain rule; a close match in Figure 8 is therefore guaranteed by construction and does not validate independence. Remark C.1 does not repair this; it only exhibits a small constructed pattern in which conditional and unconditional probabilities coincide, not an argument that the actual correctness matrix is independent. Please provide non-tautological evidence: compare conditional and unconditional per-position probabilities, report MI or other dependence measures for all positions with non-negligible frequency, and test whether the product form predicts a held-out dataset or a second model. Without such evidence, the conclusion in Section 3.5.2 that prefix recall cannot serve as a confidence indicator for extraction is unsupported.
  2. [Section 3.5.2, Figure 7] The MI evidence is too thin for the load-bearing independence claim. MI is reported only for predictor combinations of columns j=1 and j=2, with the product check extending to j=3, for one model, one dataset, one run, and no seeds or error bars. Pairwise MI at the first two positions cannot rule out dependencies at later positions or dependencies involving combinations of earlier tokens. Report MI curves across all j with non-negligible empirical mass, over multiple seeds and checkpoints and at least one additional dataset; a simple conditional-independence test, such as regressing c_j on the full vector of previous correctness indicators, would be more convincing.
  3. [Section 2.3.1 vs. Section 3.1] There is a units mismatch between the theory and the experiments. The theoretical factorization is derived for tokens, while Npre is measured word-by-word in the experiments, and Figures 7 and 8 plot word-position columns. The empirical survival function is therefore not in the units of the theory, and the approximation P(Npre >= n) approximately equal to the product of p_j in Section 3.5.2 cannot be read as validating the token-level derivation. Either restate the decomposition at the word level or measure Npre in tokens throughout.
  4. [Section 3.2] The headline mixing result rests on one randomly selected set of 200 replacement samples and one training run. Since Section 3.4 shows that the sample maximum is highly sensitive to sampling, the magnitude of the effect, about 16 versus about 50 words, needs a distribution over random swaps and seeds, and the signed-rank comparison in Figure 3 needs a test statistic and p-value. The qualitative direction is plausible, but the quantitative claim is not yet supported by the reported evidence.
minor comments (7)
  1. [Section 2.3.1] The displayed equation after the sentence beginning 'Autoregressive models generate token j + 1' contains 'P(npre + 1c|...)' with an unexplained '1c'; please fix the typo and define all symbols.
  2. [Section 2.3.1] The notation for per-token probabilities is inconsistent: p^o_k, p_k, p_j, and p_0, the linear-trend intercept, are introduced without clear definitions, and p_0 is overloaded as both a base probability and the intercept of the linear trend.
  3. [Section 3.1] The text contains the typo 'onsecutive word match'; it should be 'consecutive word match'.
  4. [Figure 6] The x-axis label 'Sparcity Gap' should be 'Sparsity Gap', and the caption should state explicitly whether the plotted quantity is Sfull minus Sinput as defined in Section 2.3.2.
  5. [Appendix C.1] The proof is garbled in several places, including 'q' with a subscript 'bar arg max[i]p[i]' and an undefined 'bar arg max' expression; please rewrite the proof for readability.
  6. [Table 1] The notation Comb(j) is listed in Table 1 but is not used later in the paper; either remove it or define its role in the analysis.
  7. [References] The citation 'Sehanobish et al.' appears without a year and appears to concern scalable neural network kernels rather than memorization; please verify that this reference is correct and relevant.

Circularity Check

1 steps flagged · score 6.0 of 10

Per-token product check in §3.5.2 is a chain-rule tautology; independence conclusion rests on thin MI for j=1,2 only.

  1. self definitional [Section 3.5.2 (Decomposing Memorization at the Token Level), with definitions in Section 2.3.1 and Figure 8]
    "Empirically, we find that M I(CJpre , Cj) is consistently low across multiple training checkpoints in the baseline case (Figure 7). This suggests a weak dependency between correctness at j and correctness at earlier positions, supporting the approximation: P (Npre ≥ n) = 1 − F (n − 1) ≈ Qn j=1 pj. Since the cumulative product of per-token probabilities aligns closely with the observed memorization distribution (Figure 8), this validates our method section’s assumption that term-wise correctness probabilities approximate the memorization distribution."

    In §2.3.1, p_k is defined as the probability that the model memorizes the k-th block given the prefix and given all previous memorization correct. §3.5.2 estimates exactly this conditional probability from the binary correctness matrix c_ij. By the chain rule of conditional probability, P(Npre ≥ n) = ∏_{j=1}^n P(C_j=1 | C_1,...,C_{j-1}=1) = ∏ p_j, identically, regardless of whether token correctness is independent across positions. Therefore, comparing the cumulative product of these conditional probabilities with the observed survival function is a definitional identity, not an empirical test of independence.

full rationale

The paper's central derivation is the factorization of memorization length into per-token conditional probabilities and the claim that these are nearly independent. The load-bearing validation is the product check in §3.5.2. Since p_j is defined as P(token j correct | all previous tokens correct), the product ∏ p_j equals the survival function P(Npre ≥ n) exactly by the chain rule, so the 'alignment' in Figure 8 is forced by construction and cannot distinguish independence from dependence. The only genuinely non-tautological evidence for independence is the mutual information analysis, but it is presented only for early token positions (j=1,2 in Figure 7; j=1..3 in Figure 8), for a single model/dataset/run, with no seeds or error bars. Additionally, §3.1 measures Npre on words while the decomposition in §2.3.1 is derived for tokens, creating a units mismatch that weakens the empirical link but is not itself circularity. Self-citations such as Wang et al. 2024 are not load-bearing. The score is 6 because one of the paper's key 'validations' reduces to a definitional identity, although the MI analysis provides some independent, if thin, support for the central claim.

Assumptions & free parameters 4 free parameters · 5 assumptions · 2 invented entities

The central claim rests on the chain-rule decomposition (a standard identity, but presented as empirical validation), on an untested conditional-independence assumption covering only two token positions, and on several hand-chosen measurement choices (100-character prefix, 200-sample swap, word-level matching). The fitted quantities are per-token probabilities estimated from the same outputs used for evaluation. No new physical or architectural entities are postulated; the two invented objects are a measurement statistic and a similarity-gap predictor, both externally measurable.

free parameters (4)
  • per-token memorisation probability p_j = estimated from the empirical correctness matrix; early columns roughly 0.075-0.2 in Figure 8
    The geometric approximation N_pre ~ Geom(1-p) and the product decomposition require per-token probabilities, estimated from the same generated outputs that define N_pre; they are not externally grounded.
  • prefix length c = 100 characters = 100 characters
    The entire memorisation measure depends on this hand-chosen cut point (Section 3.1); no sensitivity analysis is reported, and a different cut would change measured lengths.
  • swap count for mixed dataset = 200 samples out of 9,723
    The headline mixing result replaces 200 randomly selected samples (Section 3.2); the number and the choice of which samples are swapped are hand-picked and unseeded.
  • linear-trend parameters alpha and p_0 = not reported
    The model p_j = alpha*j + p_0 is proposed in Section 2.3.1 to generalise the geometric case but is never fitted, tested, or reported.
assumptions (5)
  • standard math Autoregressive factorisation of the sequence likelihood
    Section 2.3.1 uses P(N_pre = n) = (product of per-position probabilities) * (1 - p_n), the chain rule for first-failure times; not itself an empirical claim.
  • domain assumption Conditional independence of per-token recall given local context
    Remark 2.1 asserts MI(C_Jpre, C_j) ~= 0, which underwrites the geometric approximation and the extraction-methodology conclusion; tested only for token columns 1-2 in one model and dataset.
  • domain assumption Greedy LLM decoding approximates the term-wise Bayes optimal classifier
    Theorem 2.6 requires argmax of the model distribution to equal argmax of the true distribution, used to argue per-token likelihood optimisation drives memorisation; stated as a condition, not verified.
  • domain assumption Cosine-similarity gap Delta_S = S_full - S_input captures suffix diversity relevant to memorisation
    Remark 2.7 and Figure 6 assume the embedding gap measures the entropy that suppresses memorisation; no correlation coefficient, p-value, or controlled test is reported.
  • domain assumption Word-level matching is a faithful proxy for token-level memorisation
    Section 3.1 measures consecutive word matches while Section 2.3 decomposes token probabilities; the paper never reconciles tokens and words.
invented entities (2)
  • Prefix-continuation memorisation statistic N_pre independent evidence
    purpose: The central measurement: length of consecutive correct tokens or words recalled beyond a fixed prefix of a training example.
    Directly computable from model outputs and the training set, so it is falsifiable and reproducible; it is a new metric rather than a hidden mechanism.
  • Embedding similarity gap Delta_S independent evidence
    purpose: A predictor of memorisation likelihood from local data density in embedding space (suffix variability).
    Computable from any embedding function on any dataset, so it can be tested outside the paper; the paper's own evidence for its predictive value is a scattered plot without a correlation statistic.

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Cite this review

Pith. "Pith review of Skewed Memorization in Large Language Models: Quantification and Decomposition." pith.science (2026). https://pith.science/paper/Z7XOP3B4

@misc{pith2026250201187,
  author       = {Pith},
  title        = {Pith review of: Skewed Memorization in Large Language Models: Quantification and Decomposition},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Z7XOP3B4}},
  note         = {Machine review of arXiv:2502.01187}
}
read the original abstract

Memorization in Large Language Models (LLMs) poses privacy and security risks, as models may unintentionally reproduce sensitive or copyrighted data. Existing analyses focus on average-case scenarios, often neglecting the highly skewed distribution of memorization. This paper examines memorization in LLM supervised fine-tuning (SFT), exploring its relationships with training duration, dataset size, and inter-sample similarity. By analyzing memorization probabilities over sequence lengths, we link this skewness to the token generation process, offering insights for estimating memorization and comparing it to established metrics. Through theoretical analysis and empirical evaluation, we provide a comprehensive understanding of memorization behaviors and propose strategies to detect and mitigate risks, contributing to more privacy-preserving LLMs.

Figures

Figures reproduced from arXiv: 2502.01187 by the authors.

Figure 1
Figure 1. Memorization trends across training epochs in Lavita. Maximum and high-percentile memorization increase as loss de￾creases, but extreme cases appear early in training. 1.0. Similarly, [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Memorization trends in the mixed dataset. While mem￾orization patterns remain skewed, the overall distribution differs from Lavita, highlighting dataset-dependent memorization effects. 0 25 50 75 100 125 150 175 200 Data Index 0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0 Word Similarity Length Word Similarity Length Comparison (same subsample data within Original vs data within Mixed) Mixed LaVita Overlapping [PITH_FUL… view at source ↗
Figure 3
Figure 3. Memorization comparison for the same subset of sam￾ples trained in Lavita vs. the mixed dataset. Certain data points exhibit large memorization differences across contexts, emphasiz￾ing dataset-dependent effects. 3.3. Scaling Laws in Memorization Expanding on prior research on scaling laws in deep learn￾ing (Kaplan et al., 2020; Hoffmann et al., 2022), we investi￾gate how dataset size influences memorization pattern… view at source ↗
Figures from the paper (2 more)
Figure 5
Figure 5. Figure 5: Probability of missing high-memorization instances as a function of sample size. Smaller samples fail to capture the distribution’s upper tail, leading to systematic underestimation of extreme cases. tion scales with sample size, reinforcing the necessity of full-datas…
Figure 7
Figure 7. Figure 7: Mutual information between term-wise correctness and prefix correctness remains low across checkpoints, suggesting weak dependence. implications for the reliability of model extraction attacks. The finding fundamentally undermines a key assumption in sequence extractio…

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Reviewed August 9, 2026 · model on record in the stance chip above.