Only Tails Matter: Average-Case Universality and Robustness in the Convex Regime

TL;DR

This paper advances average-case analysis of optimization algorithms by linking eigenvalue concentration near spectral edges to problem complexity, introducing a new method, and demonstrating Nesterov's near-optimality in the average case.

Contribution

It introduces the Generalized Chebyshev method and establishes its asymptotic optimality under spectral concentration assumptions, enhancing average-case optimization analysis.

Findings

Eigenvalue concentration near spectral edges determines average complexity.

The Generalized Chebyshev method is asymptotically optimal under certain spectral hypotheses.

Nesterov's method is nearly optimal in the average-case setting.

Abstract

The recently developed average-case analysis of optimization methods allows a more fine-grained and representative convergence analysis than usual worst-case results. In exchange, this analysis requires a more precise hypothesis over the data generating process, namely assuming knowledge of the expected spectral distribution (ESD) of the random matrix associated with the problem. This work shows that the concentration of eigenvalues near the edges of the ESD determines a problem's asymptotic average complexity. This a priori information on this concentration is a more grounded assumption than complete knowledge of the ESD. This approximate concentration is effectively a middle ground between the coarseness of the worst-case scenario convergence and the restrictive previous average-case analysis. We also introduce the Generalized Chebyshev method, asymptotically optimal under a hypothesis on this concentration and globally optimal when the ESD follows a Beta distribution. We compare its performance to classical optimization algorithms, such as gradient descent or Nesterov's scheme, and we show that, in the average-case context, Nesterov's method is universally nearly optimal asymptotically.