arxiv: 2605.07282 · v1 · submitted 2026-05-08 · 💻 cs.LG

Recognition: no theorem link

The Convergence Gap: Instruction-Tuned Language Models Stabilize Later in the Forward Pass

Yifan Zhou

Authors on Pith no claims yet

Pith reviewed 2026-05-11 01:50 UTC · model grok-4.3

classification 💻 cs.LG

keywords convergence gapinstruction tuninglanguage model dynamicslayer-wise analysisnext-token predictionMLP layersmodel interpretabilitypost-training effects

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The pith

Instruction-tuned language models stay farther from their final next-token predictions deeper into the layer stack than their pretrained counterparts.

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

The paper introduces the convergence gap, a diagnostic that decodes the next-token distribution at every layer and measures its distance to the model's own final distribution. Across six paired pretrained and instruction-tuned checkpoints, the tuned versions show larger gaps, meaning they settle on their predictions later in the forward pass. This pattern holds under multiple controls that match endpoints, histories, and templates. The work identifies late MLP layers as the main site where the difference arises, shown by targeted swaps that increase or decrease the gap in the expected direction. A sympathetic reader would care because the finding points to a concrete change in internal dynamics caused by instruction tuning rather than just shifts in final behavior.

Core claim

Across paired checkpoints, instruction-tuned models exhibit larger convergence gaps, with their intermediate layer distributions remaining more distant from the final output distribution even in later layers. This delayed stabilization persists under endpoint-matched readouts, same-history checks, and fixed-template replay. Matched-prefix interventions demonstrate that late MLP modules are the strongest bidirectional control point: grafting late IT components into PT hosts raises late KL divergence by 0.34 nats while the reverse swap reduces it by 0.51 nats, far exceeding effects from random perturbations.

What carries the argument

The convergence gap, which decodes each layer's next-token distribution and computes its distance to the model's final distribution to track stabilization timing.

If this is right

Instruction-tuned checkpoints commit to next-token predictions later than pretrained ones under native prompting.
Late MLP computation provides the largest tested leverage for shifting the timing of stabilization.
Bidirectional grafting of late MLP modules transfers the convergence behavior between model types.
The delayed-settling pattern survives controls that match final outputs, input histories, and prompt templates.
This supplies a robust internal signature distinguishing post-training effects from pretraining dynamics.

Where Pith is reading between the lines

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

The result suggests developers could target late MLP windows when they want to alter how quickly a model locks in its output.
Similar layer-wise diagnostics might reveal whether other post-training methods such as preference tuning produce the same late-commitment signature.
If the pattern generalizes, early-layer interventions might be less effective at changing final behavior than late-layer ones in tuned models.
The diagnostic could be applied to track how architectural changes or different training objectives shift stabilization points.

Load-bearing premise

That the layer-wise distance between intermediate and final next-token distributions validly measures when a model has committed to its prediction, and that the paired checkpoints differ mainly because of instruction tuning rather than other training differences.

What would settle it

Measuring the convergence gap on a fresh pair of pretrained and instruction-tuned checkpoints and finding no consistent difference in stabilization timing, or finding that late MLP swaps no longer transfer the gap size, would falsify the central claim.

Figures

Figures reproduced from arXiv: 2605.07282 by Yifan Zhou.

**Figure 2.** Figure 2: Endpoint-matched convergence-gap controls. Matching endpoint confidence and entropy [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Figure 3: Fixed-history template audit. The native-chat effect survives forced continuation replay; [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

**Figure 4.** Figure 4: Depth-specific matched-prefix IT-MLP grafts into PT hosts. Late grafts most strongly [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

**Figure 5.** Figure 5: Symmetric matched-prefix graft/swap leverage. Late MLPs are the largest tested bidirec [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗

**Figure 6.** Figure 6: True late grafts versus matched random late residual-projection controls. Random perturba [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗

**Figure 7.** Figure 7: Gemma behavioral case study. Native IT is strongly preferred over a late PT MLP swap [PITH_FULL_IMAGE:figures/full_fig_p006_7.png] view at source ↗

read the original abstract

Final outputs hide when a checkpoint commits to its next-token prediction. We introduce the convergence gap, a model-diffing diagnostic that decodes each layer's next-token distribution and measures its distance to the model's own final distribution. Across six paired pretrained and instruction-tuned checkpoints in native prompting regimes, instruction-tuned checkpoints remain farther from their final predictions later into the stack. The effect persists under endpoint-matched raw and tuned readouts, endpoint-free same-history checks, and fixed-history template replay. Matched-prefix interventions identify late MLP windows as the largest tested leverage point: late IT grafts into PT hosts increase late KL by +0.34 nats, while PT-late swaps into IT hosts reduce it by -0.51 nats; matched random late perturbations give only +0.003 versus +0.327 for the true late graft. A preselected Gemma case study provides behavior-facing plausibility for the same late swap, without serving as a benchmark claim. These results identify a robust predictiondynamics signature of post-training: released instruction-following checkpoints tend to settle later, and late MLP computation is the strongest tested bidirectional handle on that delay under matched histories.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper shows instruction-tuned models stabilize next-token predictions later via a new convergence gap metric, with late MLPs as a modifiable handle, but the link to tuning itself rests on imperfectly matched checkpoint pairs.

read the letter

The main observation is that instruction-tuned checkpoints stay farther from their final next-token distributions deeper into the layers than their pretrained versions do. They measure this with a convergence gap that decodes each layer and compares its distribution to the model's own final output using standard divergence. Across the six pairs the gap is larger and persists later for the tuned models, and grafting late MLPs from one into the other moves the gap in the expected direction while random swaps do not.

Referee Report

2 major / 3 minor

Summary. The paper introduces the 'convergence gap' diagnostic, which decodes next-token distributions from each layer and measures their KL divergence to the model's final distribution. It reports that, across six paired pretrained and instruction-tuned checkpoints under native prompting, instruction-tuned models exhibit larger gaps (i.e., stabilize later in the forward pass). The effect survives endpoint-matched readouts, same-history controls, and fixed-history template replay. Bidirectional late-MLP grafting experiments show that inserting IT late layers into PT hosts increases late KL by +0.34 nats while the reverse reduces it by -0.51 nats, with matched random perturbations producing negligible change (+0.003). A Gemma case study illustrates behavioral plausibility of the late swaps.

Significance. If the central empirical pattern holds, the work supplies a concrete, measurable signature of how post-training alters internal prediction dynamics in LLMs. The bidirectional grafting results provide a stronger handle than pure correlation and point to late MLP computation as a high-leverage locus. This could inform future mechanistic interpretability, model editing, and analysis of alignment effects, provided the metric is shown to be robust to alternative explanations.

major comments (2)

[checkpoint selection and pairing description] The attribution of the convergence gap to instruction tuning itself rests on the six PT/IT pairs differing primarily in the post-training phase. The manuscript does not supply a side-by-side accounting of training data mixtures, total tokens, learning-rate schedules, or any intermediate alignment steps between the members of each pair (see the checkpoint description and experimental setup). Without such matching details, systematic differences in pre-instruction-tuning regimes remain a plausible confound for the observed later stabilization.
[grafting experiments] The intervention results report specific KL shifts (+0.34 nats for IT-to-PT grafts, -0.51 nats for PT-to-IT grafts) but do not include per-layer variance, number of tokens or prompts over which the averages are taken, or statistical tests. Because these numbers are presented as the strongest evidence for late-MLP causality, the absence of uncertainty quantification weakens the load-bearing claim (see the grafting experiments section).

minor comments (3)

[methods / metric definition] The convergence gap is introduced via prose description; a compact equation (e.g., defining D_l as the KL between layer-l and final distributions) would improve precision and allow readers to verify the exact divergence used.
[figures] Figure panels that overlay multiple controls would benefit from explicit legends or captions stating which lines correspond to endpoint-matched versus same-history conditions.
[controls subsection] The abstract states that the effect 'persists under ... fixed-history template replay,' yet the main text could add one sentence clarifying the exact template construction and whether it is identical across PT and IT models.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback on potential confounds in checkpoint pairing and the need for stronger statistical support in the grafting experiments. We address each major comment below and describe the revisions we will incorporate.

read point-by-point responses

Referee: [checkpoint selection and pairing description] The attribution of the convergence gap to instruction tuning itself rests on the six PT/IT pairs differing primarily in the post-training phase. The manuscript does not supply a side-by-side accounting of training data mixtures, total tokens, learning-rate schedules, or any intermediate alignment steps between the members of each pair (see the checkpoint description and experimental setup). Without such matching details, systematic differences in pre-instruction-tuning regimes remain a plausible confound for the observed later stabilization.

Authors: The six pairs consist of publicly released checkpoints from the same model families and organizations (e.g., Llama-2/3 PT vs. IT, Gemma PT vs. IT), where the instruction-tuned versions are the official post-trained releases built on the corresponding pretrained bases. Official technical reports confirm that differences are concentrated in the post-training phase, with shared architectures and pretraining data up to that point. While exact proprietary data mixtures and schedules are not always disclosed by providers, the pattern holds consistently across all six independent pairs, making a single pre-training confound unlikely. In revision we will add an expanded checkpoint table listing model sizes, release sources, and all publicly documented post-training details. revision: partial
Referee: [grafting experiments] The intervention results report specific KL shifts (+0.34 nats for IT-to-PT grafts, -0.51 nats for PT-to-IT grafts) but do not include per-layer variance, number of tokens or prompts over which the averages are taken, or statistical tests. Because these numbers are presented as the strongest evidence for late-MLP causality, the absence of uncertainty quantification weakens the load-bearing claim (see the grafting experiments section).

Authors: We agree that uncertainty quantification strengthens the grafting claims. The reported KL shifts are averages computed over a held-out evaluation set of 512 prompts (each 128 tokens long). In the revised version we will report per-layer standard errors, the precise token/prompt counts, and results from paired statistical tests (e.g., bootstrap confidence intervals) to assess the reliability of the +0.34 nats and -0.51 nats effects. The matched random-perturbation control already shows that non-specific late-layer changes produce negligible shifts (+0.003). revision: yes

Circularity Check

0 steps flagged

No circularity: purely empirical diagnostic with direct measurements

full rationale

The paper defines the convergence gap directly as the distributional distance (e.g., KL) between each layer's decoded next-token distribution and the model's own final distribution. All results consist of empirical comparisons across six paired checkpoints under multiple controls (endpoint-matched readouts, same-history checks, template replay, and graft interventions). No equations, fitted parameters, or derivations reduce any reported quantity to its inputs by construction; the gap is computed from the model's forward pass outputs without self-reference or renaming of known results. Self-citations, if present, are not load-bearing for the central empirical claim, which remains falsifiable via independent replication on the same or new checkpoints.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the assumption that KL divergence between layer-wise and final next-token distributions measures stabilization progress, plus standard transformer architecture assumptions. No free parameters are fitted to produce the gap itself; the gap is a direct measurement.

axioms (1)

domain assumption KL divergence between a layer's next-token distribution and the final distribution quantifies how much the layer has not yet converged to the model's committed prediction.
This is the definitional basis for the convergence gap introduced in the abstract.

invented entities (1)

convergence gap no independent evidence
purpose: Diagnostic metric to track layer-wise stabilization to final next-token prediction
Newly defined quantity based on existing divergence measures but applied to per-layer readouts.

pith-pipeline@v0.9.0 · 5497 in / 1311 out tokens · 38868 ms · 2026-05-11T01:50:58.843857+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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