Stochastic proximal splitting algorithm for composite minimization
This work addresses a gap in stochastic optimization for composite models with non-proximally tractable nonsmooth terms, which is incremental but relevant for large-scale or noisy contexts in machine learning.
The paper tackles composite optimization problems where only stochastic information is available for both smooth and nonsmooth components, proposing a stochastic proximal first-order scheme that achieves O(1/k) iteration complexity in expectation under strong convexity assumptions.
Supported by the recent contributions in multiple branches, the first-order splitting algorithms became central for structured nonsmooth optimization. In the large-scale or noisy contexts, when only stochastic information on the smooth part of the objective function is available, the extension of proximal gradient schemes to stochastic oracles is based on proximal tractability of the nonsmooth component and it has been deeply analyzed in the literature. However, there remained gaps illustrated by composite models where the nonsmooth term is not proximally tractable anymore. In this note we tackle composite optimization problems, where the access only to stochastic information on both smooth and nonsmooth components is assumed, using a stochastic proximal first-order scheme with stochastic proximal updates. We provide $\mathcal{O}\left( \frac{1}{k} \right)$ the iteration complexity (in expectation of squared distance to the optimal set) under the strong convexity assumption on the objective function. Empirical behavior is illustrated by numerical tests on parametric sparse representation models.