The Optimality of (Accelerated) SGD for High-Dimensional Quadratic Optimization

TL;DR

This paper analyzes the conditions under which stochastic gradient descent (SGD) and its accelerated variants are optimal for high-dimensional quadratic problems, providing convergence bounds and insights into their effectiveness based on problem complexity.

Contribution

The paper establishes convergence bounds for momentum-accelerated SGD and characterizes problem classes where SGD and ASGD are min-max optimal, revealing new insights into their learning biases.

Findings

SGD is effective for dense features with norm constraints.

SGD performs well on easy problems without saturation.

Momentum accelerates convergence in harder problems.

Abstract

Stochastic gradient descent (SGD) is a widely used algorithm in machine learning, particularly for neural network training. Recent studies on SGD for canonical quadratic optimization or linear regression show it attains well generalization under suitable high-dimensional settings. However, a fundamental question -- for what kinds of high-dimensional learning problems SGD and its accelerated variants can achieve optimality has yet to be well studied. This paper investigates SGD with two essential components in practice: exponentially decaying step size schedule and momentum. We establish the convergence upper bound for momentum accelerated SGD (ASGD) and propose concrete classes of learning problems under which SGD or ASGD achieves min-max optimal convergence rates. The characterization of the target function is based on standard power-law decays in (functional) linear regression. Our results unveil new insights for understanding the learning bias of SGD: (i) SGD is efficient in learning ``dense'' features where the corresponding weights are subject to an infinity norm constraint; (ii) SGD is efficient for easy problem without suffering from the saturation effect; (iii) momentum can accelerate the convergence rate by order when the learning problem is relatively hard. To our knowledge, this is the first work to clearly identify the optimal boundary of SGD versus ASGD for the problem under mild settings.