Uniform Stability for First-Order Empirical Risk Minimization

We consider the problem of designing uniformly stable first-order optimization algorithms for empirical risk minimization. Uniform stability is often used to obtain generalization error bounds for optimization algorithms, and we are interested in a general approach to achieve it. For Euclidean geometry, we suggest a black-box conversion which given a smooth optimization algorithm, produces a uniformly stable version of the algorithm while maintaining its convergence rate up to logarithmic factors. Using this reduction we obtain a (nearly) optimal algorithm for smooth optimization with convergence rate O^e(1/T²) and uniform stability O(T²/n), resolving an open problem of Chen et al. (2018); Attia and Koren (2021). For more general geometries, we develop a variant of Mirror Descent for smooth optimization with convergence rate O^e(1/T) and uniform stability O(T/n), leaving open the question of devising a general conversion method as in the Euclidean case.