# Phantom Gains: Auditing Self-Improvement Against a Measured Null > Transition-level auditing of LLM self-improvement requires measured nulls for every statistic; without them, a frozen model falsely appears to expand at 0.280. - **Source:** [arXiv](https://arxiv.org/abs/2608.20290) - **Published:** 2026-08-22 - **Permalink:** https://picx.dev/p/Inam8u - **Whiteboard:** https://picx.dev/p/Inam8u/image ## Summary # PHANTOM GAINS: AUDITING SELF-IMPROVEMENT AGAINST A MEASURED NULL ## Summary (Overview) - **Seven measurement failures** are identified in transition-level auditing of LLM self-improvement, each demonstrated to invert a reported conclusion when its control is absent—including two that are standard practice in the field. - **The expansion statistic (ER₁) assigns a frozen, untrained model a rate of 0.280**, meaning a model that never trains appears to "expand" over a quarter of unreached problems; the natural threshold repair (ER₂) still yields a non-zero null of 0.058, not zero. - **A threshold-free replacement** using per-problem exact tests against a pooled baseline under FDR control detects nothing on any held-out replicate and remains stable under all multiple-testing rules, error rates, and pool sizes tested. - **External distillation improves 8–11 of 22 rarely-reached AIME problems** versus 0–2 for three forms of self-training, a dissociation that survives matching in stream, volume, and evaluation and is not explained by the teacher's larger overall gain (β = 1.91, p < 10⁻⁸). - **Three training seeds are the minimum** for trustworthy comparisons—a policy-gradient arm's three seeds disagree about when the model collapses, but not that it does. ## Introduction and Theoretical Foundation ### The Problem with Aggregate Metrics The paper addresses a critical question in AI self-improvement: how to determine whether a language model has genuinely improved itself. Traditional evaluation relied on mean accuracy—accuracy before versus after training. However, aggregate metrics cannot distinguish between problems a model newly learned to solve versus those it could already solve marginally, nor do they reveal what capability was lost while the mean rose. ### The Transition-Level Paradigm A second generation of analyses moved to tracking individual problems, monitoring which enter and leave a model's "reachable set." This transition-level analysis creates a new challenge: **a transition is a difference between two noisy estimates**, and the events of interest (learning and corruption) occur exactly where sampling error is most severe. ### The Sharpening Hypothesis The paper positions itself within the theoretical framework of "sharpening"—the idea that self-improvement concentrates probability mass on sequences the model already assigns high likelihood, improving first-sample accuracy without introducing genuinely new capabilities. This contrasts with true "expansion," where the model acquires capabilities beyond its base distribution. ### The Need for Measured Nulls The central theoretical contribution is the argument that **every statistic in a transition-level audit requires a separately measured null hypothesis**. The paper demonstrates that assuming null behavior—rather than measuring it—systematically produces phantom gains and failures across all seven failure modes identified. ## Methodology ### The Ledger Framework The paper fixes an evaluation set $\mathcal{P}$ and checkpoints $t = 0, \ldots, T$, drawing $k$ samples per problem at each checkpoint. The core object is the **corruption-to-learning ratio (CLR)**: $$ \mathrm{CLR} = \frac{\left|\{p : s_{0}(p) = \mathbf{solved}, s_{T}(p) = \mathbf{unsolved}\}\right|}{\left|\{p : s_{0}(p) = \mathbf{unsolved}, s_{T}(p) = \mathbf{solved}\}\right|} \tag{1} $$ ### The Hysteresis-Based State Estimator Rather than relying on single greedy decodes (which are shown to be non-deterministic), the paper treats the solve rate $\hat{\pi}_t(p)$ as an estimator of the model's pass@1 probability, discretizing with hysteresis: $$ s_{t}(p) = \begin{cases} \mathbf{solved} & \hat{\pi}_{t}(p) \geq \theta_{\mathrm{hi}}, \\ \mathbf{unsolved} & \hat{\pi}_{t}(p) < \theta_{\mathrm{lo}}, \\ s_{t-1}(p) & \text{otherwise}, \end{cases} \quad \theta_{\mathrm{hi/lo}} = \frac{1}{2} \pm z\sqrt{\frac{1}{4k}}, \tag{2} $$ with $z = 2$, $k = 128$, giving a band of [0.41, 0.59]. ### The Expansion Statistic and Its Replacement The expansion rate with explicit evidence threshold $m$ is: $$ \mathrm{ER}_{m} = \frac{|\{p : \hat{\pi}_{0}(p) = 0 \land k\hat{\pi}_{T}(p) \geq m\}|}{|\{p : \hat{\pi}_{0}(p) = 0\}|}. \tag{3} $$ The paper demonstrates this statistic is fundamentally broken and replaces it with a per-problem Fisher exact test against a pooled baseline under false-discovery-rate control. ### Experimental Design - **Backbone**: Qwen3-8B with rank-32 LoRA, three rounds of 256 problems - **Arms**: STaR, majority-vote self-training (TTRL-style), distillation from gpt-oss-120b (positive control), and a policy-gradient arm - **Evaluation sets**: MATH-500 (200 problems, corruption-focused), AIME 2025/2026 (60 problems, expansion-focused), and a 1,163-problem difficulty band - **Matched design**: All arms share stream, retained volume, and evaluation; only the correctness filter varies - **Frozen control**: Every arm's checkpoint 0 is an independent evaluation of the untrained model, providing 110 frozen comparisons at no additional experimental cost ## Empirical Validation / Results ### The Seven Measurement Failures | | Failure | Would Have Concluded | Control Cost | |---|---|---|---| | **F1** | Single-decode ledger | "A frozen model has CLR = 1.5" | $27 | | **F2** | One-success expansion threshold | "Majority-vote expands 0.143 of AIME" | $0 | | **F3** | Fixed token cap | "Distillation is the most destructive" | $0 | | **F4** | Underpowered ledger | "The two methods do not differ" | $88 | | **F5** | Single training seed | "Majority-vote corrupts less than STaR" | $282 | | **F6** | Underpowered probe | "Self-training erodes refusal by 10 points" | $2 | | **F7** | A null measured once | "ER₂ has a null of zero" | $0 | ### Key Quantitative Results **F1 — Single Decode Artifacts**: Under greedy decoding, a frozen model shows 9 corruptions vs. 6 learnings (CLR = 1.5). Serializing requests reduces flips from 16/200 to 4/200 (Fisher p = 0.005), confirming batching as the dominant mechanism, but 2% of verdicts still flip even when fully serialized. **F2 — Expansion Statistic Failure**: The frozen control "expands" 7/25 = 0.280 of unreached problems at m=1. The ER₂ repair still yields a null of 0.058 [0.038, 0.078] over 110 frozen comparisons, and only ER₅ approaches zero at 0.003. **F5 — Seed Instability**: Majority-vote self-training shows CLR values of 0.55, 1.53, and 1.33 across three seeds with accuracy changes of +4.0, −2.8, and −1.0 points respectively—same method, same data, landing on either side of CLR = 1. **Matched Ladder Results**: The teacher (distillation) detects 8–11 of 22 low-base problems per seed versus 0–2 for self-training arms. A logistic model rejects the explanation that this follows from the teacher's larger overall gain (β = 1.91, p < 10⁻⁸). **Corruption Quantification**: Both methods destroy substantial capability: 106 problems for STaR and 88 for majority-vote SFT against a design-matched floor of 8, with over half exceeding the maximum frozen-model solve-rate movement of 0.234. **Policy-Gradient Collapse**: All three seeds collapse into unbounded repetition, but at different rounds (after 1, 2, and 3 updates). On seed 2, the majority vote became more accurate (77.9% to 85.1%) and more unanimous while the model was dying—a supervision signal that reads healthy while the model it supervises fails. ## Theoretical and Practical Implications ### For Evaluation Methodology The paper establishes that **transition-level auditing is itself a measurement problem requiring its own controls**. The finding that the expansion statistic assigns a frozen model a rate of 0.280—and that the natural repair still yields a non-zero null—has direct implications for any study reporting capability expansion in self-improving models. ### The Measured Null Paradigm The insight that **every arm's checkpoint 0 is an independent evaluation of the untrained model** makes measured nulls nearly free. However, the paper's analysis of replicate counts reveals that four frozen replicates are insufficient—the ER₂ null still ranges 0.022–0.098 at G=4, and only at G=9 does it stabilize within ±0.02 of the value eleven replicates give. ### On Self-Improvement Claims The results support the sharpening hypothesis over expansion: all newly solved problems were already reachable at pass@k. The null result on expansion is bounded—no self-training arm lifted a base-0 problem above roughly a 2.4% pass rate—rather than an absence of evidence. ### Practical Recommendations Seven practices follow, each caught an inverted result here: (1) report a no-op control for every statistic with an interval; (2) match the control's design to the arm's; (3) define correctness through a solve-rate estimator; (4) prefer per-problem tests against pooled baselines; (5) log generation length per checkpoint; (6) size transition comparisons from prior power analysis; and (7) report at least three training seeds. ## Conclusion This paper demonstrates that transition-level auditing, while the correct response to the inadequacy of aggregate metrics, introduces its own measurement challenges. The seven identified failure modes—each demonstrated to invert a reported conclusion—include two that are standard practice and one that arises inside the correction for another. The expansion statistic in current use assigns a model that never trained a rate of 0.280; the natural threshold repair takes that null to 0.058 rather than zero. The key limitation is the modest training regime: three rounds of rank-32 LoRA, far short of production schedules. The paper establishes that this regime does not expand, not that self-training cannot. The expansion analysis also rests on only 22 low-base problems on one benchmark, and on the 10 problems the base model never reaches, the comparison is not significant. Future work should focus on scaling the controlled audit to longer training schedules, expanding the diversity of backbones and evaluation sets, and developing more sensitive probes for the boundary between sharpening and expansion. The replicates needed to bound a null already exist in any multi-arm study—provided there are enough of them. --- _Markdown view of https://picx.dev/p/Inam8u, served by PicX — AI-generated visual whiteboard summaries of research papers._