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Error bound conditions (EBC) are properties that characterize the growth ofan objective function when a point is moved away from the optimal set. Theyhave recently received increasing attention in the field of optimization fordeveloping optimization algorithms with fast convergence. However, the studiesof EBC in statistical learning are hitherto still limited. The maincontributions of this paper are two-fold. First, we develop fast andintermediate rates of empirical risk minimization (ERM) under EBC for riskminimization with Lipschitz continuous, and smooth convex random functions.Second, we establish fast and intermediate rates of an efficient stochasticapproximation (SA) algorithm for risk minimization with Lipschitz continuousrandom functions, which requires only one pass of $n$ samples and adapts toEBC. For both approaches, the convergence rates span a full spectrum between$\widetilde O(1/\sqrt{n})$ and $\widetilde O(1/n)$ depending on the powerconstant in EBC, and could be even faster than $O(1/n)$ in special cases forERM. Moreover, these convergence rates are automatically adaptive without usingany knowledge of EBC. Overall, this work not only strengthens the understandingof ERM for statistical learning but also brings new fast stochastic algorithmsfor solving a broad range of statistical learning problems.

Fast Rates of ERM and Stochastic Approximation: Adaptive to Error Bound Conditions