Nonsmooth optimization problems arising in practice tend to exhibit beneficial smooth substructure: their domains stratify into active manifolds of smooth variation, which common proximal algorithms identify in finite time. Identification then entails a transition to smooth dynamics, and accommodates second-order acceleration techniques. While identification is clearly useful algorithmically, empirical evidence suggests that even those algorithms that do not identify the active manifold in finite time -- notably the subgradient method -- are nonetheless affected by it. This work seeks to explain this phenomenon, asking: how do active manifolds impact the subgradient method in nonsmooth optimization? In this work, we answer this question by introducing two algorithmically useful properties -- aiming and subgradient approximation -- that fully expose the smooth substructure of the problem. We show that these properties imply that the shadow of the (stochastic) subgradient method along the active manifold is precisely an inexact Riemannian gradient method with an implicit retraction. We prove that these properties hold for a wide class of problems, including cone reducible/decomposable functions and generic semialgebraic problems. Moreover, we develop a thorough calculus, proving such properties are preserved under smooth deformations and spectral lifts. This viewpoint then leads to several algorithmic consequences that parallel results in smooth optimization, despite the nonsmoothness of the problem: local rates of convergence, asymptotic normality, and saddle point avoidance. The asymptotic normality results appear to be new even in the most classical setting of stochastic nonlinear programming. The results culminate in the following observation: the perturbed subgradient method on generic, Clarke regular semialgebraic problems, converges only to local minimizers.
تحميل البحث