Non-Asymptotic Analysis of Excess Risk via Empirical Risk Landscape

Mingyang Yi, Ruoyu Wang, Zhi-Ming Ma

In this paper, we provide a unified analysis of the excess risk of the model trained by some proper algorithms in both convex and non-convex regime. In contrary to the existing results in the literature that depends on iteration steps, our bounds to the excess risk do not diverge with the number of iterations. This underscores that, at least for loss functions of certain types, the excess risk on it can be guaranteed after a period of training. Our technique relies on a non-asymptotic characterization of the empirical risk landscape. To be rigorous, under the condition that the local minima of population risk are non-degenerate, each local minimum of the smooth empirical risk is guaranteed to generalize well. The conclusion is independent of the convexity. Combining this with the classical optimization result, we derive converged upper bounds to the excess risk in both convex and non-convex regime.

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