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Average-case analysis computes the complexity of an algorithm averaged overall possible inputs. Compared to worst-case analysis, it is more representativeof the typical behavior of an algorithm, but remains largely unexplored inoptimization. One difficulty is that the analysis can depend on the probabilitydistribution of the inputs to the model. However, we show that this is not thecase for a class of large-scale problems trained with first-order methodsincluding random least squares and one-hidden layer neural networks with randomweights. In fact, the halting time exhibits a universality property: it isindependent of the probability distribution. With this barrier for average-caseanalysis removed, we provide the first explicit average-case convergence ratesshowing a tighter complexity not captured by traditional worst-case analysis.Finally, numerical simulations suggest this universality property holds for amore general class of algorithms and problems.

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