Mirror Descent Under Generalized Smoothness

Smoothness is crucial for attaining fast rates in first-order optimization. However, many optimization problems in modern machine learning involve non-smooth objectives. Recent studies relax the smoothness assumption by allowing the Lipschitz constant of the gradient to grow with respect to the gradient norm, which accommodates a broad range of objectives in practice. Despite this progress, existing generalizations of smoothness are restricted to Euclidean geometry with $\ell_2$-norm and only have theoretical guarantees for optimization in the Euclidean space. In this paper, we address this limitation by introducing a new $\ell*$-smoothness concept that measures the norm of Hessian in terms of a general norm and its dual, and establish convergence for mirror-descent-type algorithms, matching the rates under the classic smoothness. Notably, we propose a generalized self-bounding property that facilitates bounding the gradients via controlling suboptimality gaps, serving as a principal component for convergence analysis. Beyond deterministic optimization, we establish an anytime convergence for stochastic mirror descent based on a new bounded noise condition that encompasses the widely adopted bounded or affine noise assumptions.

Efficient Algorithms for Empirical Group Distributional Robust Optimization and Beyond

We investigate the empirical counterpart of group distributionally robust optimization (GDRO), which aims to minimize the maximal empirical risk across $m$ distinct groups. We formulate empirical GDRO as a $\textit{two-level}$ finite-sum convex-concave minimax optimization problem and develop a stochastic variance reduced mirror prox algorithm. Unlike existing methods, we construct the stochastic gradient by per-group sampling technique and perform variance reduction for all groups, which fully exploits the $\textit{two-level}$ finite-sum structure of empirical GDRO. Furthermore, we compute the snapshot and mirror snapshot point by a one-index-shifted weighted average, which distinguishes us from the naive ergodic average. Our algorithm also supports non-constant learning rates, which is different from existing literature. We establish convergence guarantees both in expectation and with high probability, demonstrating a complexity of $\mathcal{O}\left(\frac{m\sqrt{\bar{n}\ln{m}}}{\varepsilon}\right)$, where $\bar n$ is the average number of samples among $m$ groups. Remarkably, our approach outperforms the state-of-the-art method by a factor of $\sqrt{m}$. Furthermore, we extend our methodology to deal with the empirical minimax excess risk optimization (MERO) problem and manage to give the expectation bound and the high probability bound, accordingly. The complexity of our empirical MERO algorithm matches that of empirical GDRO at $\mathcal{O}\left(\frac{m\sqrt{\bar{n}\ln{m}}}{\varepsilon}\right)$, significantly surpassing the bounds of existing methods.