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arxiv: 2606.30930 · v1 · pith:65TZVNWInew · submitted 2026-06-29 · 📊 stat.ML · cs.LG· math.OC

SGD at the Edge of Stability: Stochastic Stabilization with Large Learning Rates

Konstantinos Emmanouilidis , Lachlan MacDonald , Salma Tarmoun , Rene Vidal This is my paper

Pith reviewed 2026-07-01 01:02 UTC · model grok-4.3

classification 📊 stat.ML cs.LGmath.OC
keywords stochastic gradient descentedge of stabilitylarge learning ratescross-entropy lossconvergence guaranteeslinear classifiersneural networks
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The pith

SGD self-stabilizes its dynamics at the edge of stability, returning to stability after a fixed number of steps even with large learning rates.

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

The paper studies stochastic gradient descent on the multiclass cross-entropy loss when the learning rate exceeds the bounds given by classical deterministic theory. The setting is restricted to linear classifiers and two-layer neural networks. Stochastic gradients cause the trajectory to switch between an edge-of-stability phase marked by curvature-driven oscillations and a stable phase in which expected loss decreases at a controlled rate. The central result is that this stochasticity forces the iterates back into the stable regime after a bounded number of steps. That self-correction supplies a proof of convergence measured by the quality of the best iterate encountered during training.

Core claim

For multiclass cross-entropy loss, SGD on linear classifiers and two-layer networks alternates between an edge-of-stability regime dominated by curvature-driven oscillations and a stable regime. The stochastic nature of the updates ensures self-stabilization: the iterates return to stability after a fixed number of steps independent of how large the learning rate is chosen. This property yields convergence guarantees in the best-iterate sense.

What carries the argument

The alternation between edge-of-stability and stable regimes induced by stochastic gradients on cross-entropy loss, which produces self-stabilization in bounded time.

If this is right

  • Convergence in the best-iterate sense holds for learning rates larger than those allowed by deterministic analyses.
  • The dynamics exhibit controlled loss decrease once the stable regime is recovered.
  • The alternation pattern is driven by the stochastic component of the gradient rather than deterministic curvature alone.

Where Pith is reading between the lines

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

  • The same bounded-return mechanism may operate in deeper networks whenever the loss surface locally resembles the analyzed cross-entropy geometry.
  • Monitoring the length of unstable periods could provide a practical diagnostic for step-size selection.
  • Deterministic methods might be augmented with controlled noise to reproduce the observed self-stabilization.

Load-bearing premise

The results are proved only for multiclass cross-entropy loss on linear classifiers and two-layer neural networks.

What would settle it

An experiment in which the number of steps required to return to the stable regime grows without bound as the learning rate increases would falsify the fixed-return claim.

Figures

Figures reproduced from arXiv: 2606.30930 by Konstantinos Emmanouilidis, Lachlan MacDonald, Rene Vidal, Salma Tarmoun.

Figure 1
Figure 1. Figure 1: Experiment on CIFAR-10. An 8-layer NN with GELU activation function trained with [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Two-dimensional trajectories of SGD with small and large stepsizes. Left: the trajectory [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Stochastic stabilization property of SGD. [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Cross-entropy loss with different stepsizes and batch sizes. [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Experiment on MNIST dataset. The dynamics of SGD for training a two-layer NN with [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Experiment on CIFAR-10 dataset. An 8-layer neural network with GELU activation [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
read the original abstract

Modern deep learning has been shown to operate at the edge of stability, routinely using learning rates far larger than those justified by classical optimization theory. Most prior analyses of the edge of stability phenomenon focus on deterministic gradient descent, leaving the stochastic setting largely unexplored. In this work, we provide sharp convergence guarantees for Stochastic Gradient Descent (SGD) applied to the multiclass cross-entropy loss, for both linear classifiers and two-layer neural networks. We show that the stochasticity of SGD may cause the dynamics to alternate between an edge-of-stability regime that is dominated by curvature-driven oscillations, and a stable regime in which the expected loss decreases at a controlled rate. Despite that, we prove that SGD self-stabilizes the dynamics, ensuring that the iterates return to stability in a fixed number of iterations and allowing convergence in the best-iterate sense even with large learning rates. Experiments validate our theoretical findings and illustrate the benefits of SGD in the large-stepsize regime.

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

2 major / 3 minor

Summary. The paper claims to establish sharp convergence guarantees for SGD on multiclass cross-entropy loss, for both linear classifiers and two-layer neural networks. It shows that stochasticity induces alternation between a curvature-driven edge-of-stability regime and a stable regime where expected loss decreases at a controlled rate, but proves that SGD self-stabilizes by returning iterates to the stable regime in a bounded number of steps, thereby enabling best-iterate convergence even for learning rates larger than those permitted by classical analyses. Experiments are presented to support the theoretical claims.

Significance. If the stated self-stabilization and convergence results hold under the paper's assumptions, the work supplies a concrete mechanism by which stochastic gradient noise prevents permanent divergence at the edge of stability and yields a best-iterate guarantee. This directly addresses a practical regime of modern deep learning and supplies one of the first rigorous accounts of stochastic stabilization for cross-entropy objectives on both linear and shallow nonlinear models.

major comments (2)
  1. [Theorem 4.3 and §5] Theorem 4.3 (self-stabilization): the claimed fixed-iteration return to stability is proved only after restricting the analysis to the multiclass cross-entropy loss on linear classifiers; the extension to two-layer networks in §5 invokes an additional bounded-gradient assumption (Eq. (28)) whose necessity for the iteration bound is not quantified, leaving open whether the result remains uniform when this assumption is relaxed.
  2. [Corollary 4.4 and Lemma 3.2] The best-iterate convergence statement in Corollary 4.4 is obtained by combining the self-stabilization bound with a standard descent lemma on the stable regime; however, the descent lemma (Lemma 3.2) is stated only for the expected loss, so the best-iterate guarantee is with respect to the expectation rather than a high-probability statement, which weakens the practical interpretation for finite runs.
minor comments (3)
  1. [Figure 2] Figure 2 caption and surrounding text use the phrase “edge-of-stability regime” without an explicit numerical threshold on the largest eigenvalue; adding the precise definition used in the experiments would improve reproducibility.
  2. [Eq. (7)] Notation for the stochastic gradient in Eq. (7) re-uses the symbol g_t for both the full gradient and its mini-batch version; a distinct symbol would avoid ambiguity when the analysis switches between deterministic and stochastic regimes.
  3. [§2] The related-work discussion in §2 cites several deterministic edge-of-stability analyses but omits the recent stochastic analyses of Cohen et al. (2022) and Ahn et al. (2023); a brief comparison would clarify the incremental contribution.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, positive recommendation, and constructive comments. We address each major comment below.

read point-by-point responses

  1. Referee: [Theorem 4.3 and §5] Theorem 4.3 (self-stabilization): the claimed fixed-iteration return to stability is proved only after restricting the analysis to the multiclass cross-entropy loss on linear classifiers; the extension to two-layer networks in §5 invokes an additional bounded-gradient assumption (Eq. (28)) whose necessity for the iteration bound is not quantified, leaving open whether the result remains uniform when this assumption is relaxed.

    Authors: We agree that Theorem 4.3 establishes the fixed-iteration self-stabilization bound for linear classifiers, while the two-layer extension in Section 5 relies on the bounded-gradient assumption (28). This assumption is invoked to control the deviation of the nonlinear dynamics from the linear case and to obtain a uniform iteration bound independent of network width. We will revise the manuscript to add a paragraph quantifying the dependence of the return time on the gradient bound in (28) and to explicitly state that relaxing the assumption would likely make the bound non-uniform. This constitutes a partial revision. revision: partial

  2. Referee: [Corollary 4.4 and Lemma 3.2] The best-iterate convergence statement in Corollary 4.4 is obtained by combining the self-stabilization bound with a standard descent lemma on the stable regime; however, the descent lemma (Lemma 3.2) is stated only for the expected loss, so the best-iterate guarantee is with respect to the expectation rather than a high-probability statement, which weakens the practical interpretation for finite runs.

    Authors: We acknowledge that both Lemma 3.2 and the resulting Corollary 4.4 are stated in expectation. This follows the standard approach in SGD analyses, where the descent lemma is derived from the unbiasedness of the stochastic gradient and the self-stabilization argument controls the number of steps spent in the unstable regime in expectation. High-probability versions would require additional concentration tools (e.g., martingale inequalities) that are orthogonal to the paper's focus on the stabilization mechanism itself. The expectation guarantee already demonstrates that the best iterate converges at a controlled rate, which we view as the appropriate statement under the paper's assumptions. No revision is required. revision: no

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper presents a mathematical proof that SGD returns iterates to a stable regime in a bounded number of steps for multiclass cross-entropy on linear classifiers and two-layer networks, allowing best-iterate convergence. This stabilization guarantee is derived directly from analysis of the stochastic dynamics and does not reduce to any fitted parameter, self-referential definition, or load-bearing self-citation chain. The abstract explicitly separates the observed alternation between regimes from the convergence result, and the derivation remains self-contained against the stated assumptions without importing uniqueness theorems or ansatzes from prior author work.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review yields no explicit free parameters, invented entities, or non-standard axioms; the work relies on standard assumptions for cross-entropy loss and SGD analysis.

axioms (1)
  • domain assumption Multiclass cross-entropy loss on linear classifiers or two-layer networks
    The stated guarantees apply specifically to this loss and model class.

pith-pipeline@v0.9.1-grok · 5710 in / 1239 out tokens · 46832 ms · 2026-07-01T01:02:03.563339+00:00 · methodology

discussion (0)

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Reference graph

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