Halting Time is Predictable for Large Models: A Universality Property and Average-case Analysis

TL;DR

This paper demonstrates that for large models trained with first-order methods, the halting time is universal and independent of input distribution, enabling explicit average-case convergence rates and revealing a new understanding of algorithm complexity.

Contribution

It establishes a universality property for halting time in large-scale optimization, allowing for explicit average-case analysis beyond worst-case bounds.

Findings

Halting time is distribution-independent for certain large-scale problems.

Explicit average-case convergence rates are derived.

Numerical simulations suggest broader applicability of universality.

Abstract

Average-case analysis computes the complexity of an algorithm averaged over all possible inputs. Compared to worst-case analysis, it is more representative of the typical behavior of an algorithm, but remains largely unexplored in optimization. One difficulty is that the analysis can depend on the probability distribution of the inputs to the model. However, we show that this is not the case for a class of large-scale problems trained with first-order methods including random least squares and one-hidden layer neural networks with random weights. In fact, the halting time exhibits a universality property: it is independent of the probability distribution. With this barrier for average-case analysis removed, we provide the first explicit average-case convergence rates showing a tighter complexity not captured by traditional worst-case analysis. Finally, numerical simulations suggest this universality property holds for a more general class of algorithms and problems.