Empirical risk minimization: probabilistic complexity and stepsize strategy

Empirical risk minimization is recognized as a special form in standard convex optimization. When using a first order method, the Lipschitz constant of the empirical risk plays a crucial role in the convergence analysis and stepsize strategies for these problems. We derive the probabilistic bounds for such Lipschitz constants using random matrix theory. We show that, on average, the Lipschitz constant is bounded by the ratio of the dimension of the problem to the amount of training data. We use our results to develop a new stepsize strategy for first order methods. The proposed algorithm, Probabilistic Upper-bound Guided stepsize strategy, outperforms the regular stepsize strategies with strong theoretical guarantee on its performance.