On the Stability of Nonlinear Dynamics in GD and SGD: Beyond Quadratic Potentials

2026-02-16 · Source: stat.ML updates on arXiv.org · Field: Technology & Digital — Artificial Intelligence & Machine Learning · Depth: Expert, quick

Summary

Rotem Mulayoff and Sebastian U. Stich, in a paper accepted to COLT 2026, investigate the dynamical stability of iterates during training for Gradient Descent (GD) and Stochastic Gradient Descent (SGD), moving beyond traditional linearization methods. Their work addresses the limitation that linear analysis can misrepresent full nonlinear behavior, where GD might stably oscillate near a linearly unstable minimum. The authors derive an exact criterion for stable oscillations of GD near multivariate minima, which incorporates high-order derivatives. For SGD, they demonstrate that nonlinear dynamics can diverge in expectation if even a single batch is unstable, suggesting stability is not solely an average effect. Conversely, they prove that if all batches are linearly stable, SGD's nonlinear dynamics remain stable in expectation.

Key takeaway

For AI scientists optimizing models, you should recognize that traditional linear stability analyses of GD and SGD may be insufficient. Your understanding of convergence and minima should account for nonlinear dynamics, especially considering high-order derivatives for GD and the potential for single batch instability to cause SGD divergence. This implies a need to move beyond simple linear models when diagnosing training issues or designing robust optimization strategies, potentially exploring methods that explicitly manage nonlinear effects.

Key insights

Nonlinear dynamics significantly impact GD/SGD stability, often contradicting linear approximations, especially with high-order derivatives and single batch instability.

Principles

• Linear stability analysis can be misleading for GD/SGD.
• High-order derivatives are crucial for GD stability criteria.
• Single batch instability can drive SGD divergence.

Method

The paper derives an exact criterion for stable GD oscillations near multivariate minima using high-order derivatives and extends this analysis to SGD's nonlinear dynamics.

In practice

• Re-evaluate GD/SGD stability assumptions.
• Consider high-order derivatives in optimization analysis.
• Monitor individual batch stability in SGD.

Topics

• Gradient Descent
• Stochastic Gradient Descent
• Optimization Stability
• Nonlinear Dynamics
• Machine Learning Theory
• High-Order Derivatives

Best for: Research Scientist, AI Scientist

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Editorial summary, takeaway, and curation by AIssential. Original article published by stat.ML updates on arXiv.org.