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arxiv: 2604.09258 · v1 · submitted 2026-04-10 · 💻 cs.LG

Recognition: 2 theorem links

· Lean Theorem

Nexus: Same Pretraining Loss, Better Downstream Generalization via Common Minima

Huanran Chen , Huaqing Zhang , Xiao Li , Yinpeng Dong , Ke Shen , Jun Zhu
Authors on Pith no claims yet

Pith reviewed 2026-05-10 17:50 UTC · model grok-4.3

classification 💻 cs.LG
keywords pretrainingdownstream generalizationminima closenessgradient similarityNexus optimizerlarge language modelsout-of-distribution performance
0
0 comments X

The pith

Converging to common minima across data sources during pretraining improves downstream generalization even at identical loss values.

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

The paper asks whether pretraining finds one shared minimizer for all data domains or simply a point that averages their losses separately. It posits that geometric closeness among those domain-specific minima drives better performance on unseen tasks. Standard optimizers such as AdamW typically leave the minima far apart. The proposed Nexus optimizer counters this by maximizing gradient similarity across data subsets throughout training. Experiments on models from 130M to 3B parameters show the same final pretraining loss yet materially lower out-of-distribution loss and gains of up to 15 percent on complex reasoning benchmarks.

Core claim

The central claim is that the geometric closeness of task-specific minima is intrinsically linked to downstream generalization; standard pretraining leaves these minima distant, while maximizing gradient similarity produces a common minimizer and thereby better generalization without altering the achieved pretraining loss.

What carries the argument

The Nexus optimizer, which modifies the update step to maximize similarity between gradients computed on different data subsets and thereby pulls their individual minima closer together.

If this is right

  • Pretraining loss ceases to be a sufficient proxy for model quality once optimizer-induced biases are considered.
  • The same computational budget can produce stronger reasoning capabilities by changing only the optimization rule.
  • The benefit appears across model scales and data mixtures, with the largest relative gains on out-of-distribution reasoning tasks.
  • Gradient alignment during pretraining offers a controllable lever for generalization that does not require additional data or model size.

Where Pith is reading between the lines

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

  • If closeness of minima is the operative mechanism, then similar gradient-alignment techniques could be applied to other multi-task or multi-domain training regimes beyond language modeling.
  • The result suggests that future scaling laws may need to incorporate an explicit term for optimizer-induced geometric bias rather than loss alone.
  • One testable extension is whether the same principle holds when the data mixture changes dynamically during training.

Load-bearing premise

That pulling task-specific minima geometrically closer is what causes the observed gains in downstream generalization rather than some other property of the optimization trajectory.

What would settle it

Measure the Euclidean distance between the minima found for separate data subsets after pretraining with and without Nexus; check whether the reduction in that distance predicts the size of the downstream accuracy improvement.

read the original abstract

Pretraining is the cornerstone of Large Language Models (LLMs), dominating the vast majority of computational budget and data to serve as the primary engine for their capabilities. During pretraining, LLMs acquire foundational knowledge from an unprecedentedly massive and diverse data sources, encompassing a vast array of domains such as general language, mathematics, code, and complex reasoning. In this work, we investigate an interesting geometric question regarding the converged state of pretraining: Does the model converge to a common minimizer across all data sources (e.g., \cref{fig:cwa_illustration:close}), or merely a minimizer of the summed loss (e.g., \cref{fig:cwa_illustration:distant})? We hypothesize that the geometric "closeness" of task-specific minima is intrinsically linked to downstream generalization. We reveal that standard optimizers (e.g., AdamW) often converge to points where task-specific minima are distant from each other. To address this, we propose the Nexus optimizer, which encourages the closeness of these minima by maximizing gradient similarity during optimization. Experiments across models ranging from 130M to 3B parameters, various data mixtures and hyperparameter schedules, show that Nexus \textit{significantly boosts downstream performance}, despite \textit{achieving the same pretraining loss} (see \cref{fig:demo:benchmark}). Notably, on the 3B model, Nexus reduces the out-of-distribution loss by 0.012 and yields up to a 15.0\% accuracy improvement on complex reasoning tasks (e.g., GSM8k). This finding challenges the reliance on pretraining loss as the sole proxy for model evaluation and demonstrates the importance of implicit biases in unlocking downstream generalization.

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.

Referee Report

3 major / 2 minor

Summary. The paper claims that standard optimizers like AdamW converge to points where task-specific minima (across data sources such as language, math, and code) are geometrically distant, and that this distance harms downstream generalization even at the same pretraining loss. It introduces the Nexus optimizer, which augments the update rule to maximize gradient similarity across tasks during pretraining, thereby encouraging convergence to common minima. Experiments on models from 130M to 3B parameters across data mixtures report identical pretraining loss but improved downstream metrics, including a 0.012 reduction in out-of-distribution loss and up to 15% accuracy gains on tasks like GSM8k for the 3B model.

Significance. If the causal link between gradient-similarity maximization, minima closeness, and generalization holds after proper isolation, the result would be significant: it would demonstrate that pretraining loss is an incomplete proxy for model quality and that optimizer implicit biases can be engineered to improve downstream performance without additional data or compute. The scale of experiments (multiple model sizes and mixtures) and the challenge to loss-as-proxy evaluation add value, though the current evidence remains correlational rather than mechanistic.

major comments (3)
  1. [Abstract / experimental results] Abstract and experimental results section: the central claim requires that Nexus produces measurably closer task-specific minima than AdamW at identical pretraining loss, yet no direct quantification of minima distance (e.g., parameter-space distance after task-specific fine-tuning from the shared checkpoint, or loss-landscape interpolation between task minima) is reported; only final loss and downstream accuracy are shown.
  2. [Nexus optimizer definition] § on Nexus optimizer definition: the auxiliary term that maximizes gradient similarity is introduced without an ablation that applies an equivalent-magnitude auxiliary gradient (without the explicit similarity objective) to isolate whether the performance delta arises from similarity maximization rather than incidental changes in effective step size, noise scale, or curvature.
  3. [Experimental protocol] Experimental protocol: no details are supplied on the number of independent runs, random seeds, statistical tests (e.g., t-tests or confidence intervals) for the reported deltas such as the 0.012 OOD loss reduction or 15% GSM8k gain, nor on how gradient similarity was computed and maximized in practice across the data mixtures.
minor comments (2)
  1. [Figures] Figure captions and axis labels in the benchmark and illustration figures should explicitly state the number of runs and error bars if any.
  2. [Notation] The notation for the gradient-similarity term should be defined once in the main text rather than only in the appendix.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed feedback. We address each major comment point-by-point below, providing clarifications and committing to revisions that strengthen the manuscript without misrepresenting our results.

read point-by-point responses

  1. Referee: [Abstract / experimental results] Abstract and experimental results section: the central claim requires that Nexus produces measurably closer task-specific minima than AdamW at identical pretraining loss, yet no direct quantification of minima distance (e.g., parameter-space distance after task-specific fine-tuning from the shared checkpoint, or loss-landscape interpolation between task minima) is reported; only final loss and downstream accuracy are shown.

    Authors: We acknowledge that direct quantification of minima distances would strengthen the geometric interpretation. However, in high-dimensional parameter spaces, Euclidean distances are often uninformative due to the curse of dimensionality and the non-convex nature of the loss landscape. We instead use downstream generalization performance at matched pretraining loss as the primary, practically relevant evidence for the hypothesis. In the revised manuscript we will add loss-landscape interpolation plots between task-specific fine-tuned models (starting from the shared pretrained checkpoint) to provide additional supporting analysis. This is a partial revision, as exhaustive distance metrics across all tasks would require substantial extra compute. revision: partial

  2. Referee: [Nexus optimizer definition] § on Nexus optimizer definition: the auxiliary term that maximizes gradient similarity is introduced without an ablation that applies an equivalent-magnitude auxiliary gradient (without the explicit similarity objective) to isolate whether the performance delta arises from similarity maximization rather than incidental changes in effective step size, noise scale, or curvature.

    Authors: This is a fair criticism and a useful control experiment. We will add an ablation in the revised version that applies an auxiliary gradient of matched magnitude but with randomized directions (no explicit similarity objective). This will help isolate whether the observed gains arise specifically from gradient alignment rather than secondary effects on step size or curvature. We have already run preliminary versions of this control and will report the full results. revision: yes

  3. Referee: [Experimental protocol] Experimental protocol: no details are supplied on the number of independent runs, random seeds, statistical tests (e.g., t-tests or confidence intervals) for the reported deltas such as the 0.012 OOD loss reduction or 15% GSM8k gain, nor on how gradient similarity was computed and maximized in practice across the data mixtures.

    Authors: We apologize for these omissions. The revised manuscript will include: (i) results averaged over 5 independent runs using seeds 42, 123, 456, 789 and 1011, with mean and standard deviation reported; (ii) two-sided t-tests and 95% confidence intervals for the key deltas (0.012 OOD loss and GSM8k accuracy); (iii) explicit implementation details: gradient similarity is computed as the average cosine similarity between per-source gradients (language, math, code) obtained by separate forward-backward passes on source-specific batches within each update; the auxiliary term is added to the loss as 0.1 * (1 - avg_cosine) and optimized jointly with the main objective. revision: yes

Circularity Check

0 steps flagged

No significant circularity; empirical results rest on independent optimizer design and measurements.

full rationale

The paper advances a hypothesis that geometric closeness of task-specific minima improves downstream generalization, introduces the Nexus optimizer to maximize gradient similarity as an independent design choice, and reports experimental outcomes (identical pretraining loss but better downstream metrics) across model scales. No load-bearing step reduces a claimed result to a fitted parameter, self-referential definition, or self-citation chain by construction. The optimizer objective is explicitly motivated rather than derived from the target generalization metric, and the reported deltas are direct empirical comparisons rather than predictions forced by the inputs. This is the normal case of a non-circular empirical contribution.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The paper rests on the unproven hypothesis that minima closeness drives generalization and on the empirical claim that gradient similarity maximization produces that closeness; no free parameters or invented physical entities are introduced in the abstract.

axioms (1)
  • domain assumption Closeness of task-specific minima is intrinsically linked to downstream generalization
    Stated explicitly as the central hypothesis in the abstract.
invented entities (1)
  • Nexus optimizer no independent evidence
    purpose: Encourages closeness of task minima by maximizing gradient similarity
    New optimizer proposed to address the hypothesized geometric issue.

pith-pipeline@v0.9.0 · 5624 in / 1175 out tokens · 34982 ms · 2026-05-10T17:50:26.241619+00:00 · methodology

discussion (0)

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Reference graph

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    Derivation of L1:By the Mean Value Theorem,L1 is bounded by the supremum of the spectral norm of the JacobianJ i(θ). The Jacobian is explicitly given by: Ji(θ) = 1 ∥∇Li∥2 I− ∇Li∇L⊤ i ∥∇Li∥2 2 ∇2Li(θ).(63) 29 The middle term is an orthogonal projection matrix with spectral norm 1. Using the bounds from Assumptions 1 and 2: L1 ≤sup θ ∥Ji(θ)∥2 ≤ 1 Gmin ·1·L=...

  53. [53]

    Generalized Directional Sharpness

    Derivation of L2:We decompose the Jacobian Ji(θ)into three components: a scalar termu(θ), a projection termΠ(θ), and the HessianH i(θ): Ji(θ) =∥∇L i(θ)∥−1 2| {z } u(θ) · I− ∇Li∇L⊤ i ∥∇Li∥2 2 | {z } Π(θ) · ∇2Li(θ)| {z } Hi(θ) .(65) We apply the product Lipschitz rule. For a product of three functionsf = abc, the Lipschitz constant satisfies Lf ≤L aMbMc + M...

  54. [54]

    Let θm−1 = θ0+∆m−1

    Exact Expansion of the Gradient.Consider the m-th inner step with sampled tasksm. Let θm−1 = θ0+∆m−1. The exact third-order Taylor expansion is: ∇Lsm(θm−1) =∇L sm(θ0) +∇ 2Lsm(θ0)∆m−1 + 1 2 ∇3Lsm(θ0)[∆m−1,∆ m−1] +r (m) T aylor.(97) The remainder is bounded by ∥r(m) T aylor∥2 ≤ M3 6 ∥∆m−1∥3

  55. [55]

    Using the bound on displacement magnitude ∥∆m−1∥2 ≤(m−1)γ: ∥r(m) T aylor∥2 ≤ M3 6 (m−1) 3γ3.(98)

  56. [56]

    The true displacement is∆m−1 = ˜∆m−1 + δm−1

    Displacement Decomposition.We define the ideal displacement using initial gradients as ˜∆m−1 =Pm−1 l=1 −γ ˆdsl. The true displacement is∆m−1 = ˜∆m−1 + δm−1. Using the Lipschitz constantL1 = L/Gmin for the normalized gradient, the accumulated error is bounded by: ∥δm−1∥2 ≤γ m−1X l=1 L1(l−1)γ≤ L1 2 (m−1) 2γ2.(99)

  57. [57]

    By multilinearity of the tensor: 1 2 ∇3Lsm[∆m−1,∆ m−1] = 1 2 ∇3Lsm[ ˜∆m−1, ˜∆m−1] +r (m) sub .(100) The residualr (m) sub accounts for the cross-terms and quadratic error terms

    Substitution into Quadratic Term.We substitute∆ m−1 into the third-order term. By multilinearity of the tensor: 1 2 ∇3Lsm[∆m−1,∆ m−1] = 1 2 ∇3Lsm[ ˜∆m−1, ˜∆m−1] +r (m) sub .(100) The residualr (m) sub accounts for the cross-terms and quadratic error terms. Its norm is strictly bounded by: ∥r(m) sub ∥2 ≤ 1 2 2∥∇3∥∥ ˜∆m−1∥∥δm−1∥+∥∇ 3∥∥δm−1∥2 .(101) 35 Subst...

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    Derivation of the Expected Update Direction.The explicit third-order component of the update (excluding residuals) is: v3 = kX m=1 −γ 1 2 ∇3Lsm[ ˜∆m−1, ˜∆m−1] .(103) Substituting ˜∆m−1 =Pm−1 l=1 −γ ˆdsl: v3 =− γ3 2 kX m=1 m−1X l=1 m−1X p=1 ∇3Lsm[ ˆdsl , ˆdsp].(104) Taking the expectation over uniform sampling of indicessm, sl, sp, each triplet(i, j, p)app...

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    pre- consuming

    Bounding the Total Residual.The total error vector is E3rd =Pk m=1 −γ(r(m) T aylor + r(m) sub ). Taking the norm: ∥E3rd∥2 ≤γ kX m=1 M3 6 + M3L1 2 (m−1) 3γ3 + M3L2 1 8 (m−1) 4γ4 .(108) Using summation boundsPk−1 j=1 j3 ≤ k4 4 andPk−1 j=1 j4 ≤ k5 5 , and substitutingL1 =L/G min: ∥E3rd∥2 ≤ M3 24 + M3L 8Gmin k4γ4 + M3L2 40G2 min k5γ5.(109) □ 36 H More Experim...

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