Optimal Asymptotic Rates for (Stochastic) Gradient Descent under the Local PL-Condition: A Geometric Approach

• URL: http://arxiv.org/abs/2605.14663v1
• Date: Thu, 14 May 2026 10:18:21 GMT
• Title: Optimal Asymptotic Rates for (Stochastic) Gradient Descent under the Local PL-Condition: A Geometric Approach
• Abstract summary: gradient descent (SGD) has been extensively studied the past decades due to its simplicity and broad applicability in machine learning.<n>We analyze the local behavior of gradient descent and minimize $C2$ of functions that satisfy the Polyak-Lojasiewicz (PL) setting and under a multiplicative setting.
• Score: 1.0026496861838448
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Stochastic gradient descent (SGD) has been studied extensively over the past decades due to its simplicity and broad applicability in machine learning. In this work, we analyze the local behavior of gradient descent and stochastic gradient descent for minimizing $C^2$-functions that satisfy the Polyak-Lojasiewicz (PL) inequality and under a multiplicative gradient noise model motivated by overparameterized neural networks. Using a geometric interpretation of the PL-condition, we prove a simple yet surprising fact: in this possibly non-convex setting, the asymptotic convergence rate of (S)GD matches the rate obtained for strongly convex quadratics.

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