Semi-proximal Mirror-Prox for Nonsmooth Composite Minimization
This work addresses optimization challenges in machine learning for high-dimensional data, offering a more practical alternative to proximal gradient methods, though it appears incremental as it builds on existing Mirror-Prox techniques.
The authors tackled high-dimensional nonsmooth composite minimization problems by proposing the Semi-Proximal Mirror-Prox algorithm, which achieves an optimal convergence rate of O(1/ε²) and shows promising experimental results compared to existing methods.
We propose a new first-order optimisation algorithm to solve high-dimensional non-smooth composite minimisation problems. Typical examples of such problems have an objective that decomposes into a non-smooth empirical risk part and a non-smooth regularisation penalty. The proposed algorithm, called Semi-Proximal Mirror-Prox, leverages the Fenchel-type representation of one part of the objective while handling the other part of the objective via linear minimization over the domain. The algorithm stands in contrast with more classical proximal gradient algorithms with smoothing, which require the computation of proximal operators at each iteration and can therefore be impractical for high-dimensional problems. We establish the theoretical convergence rate of Semi-Proximal Mirror-Prox, which exhibits the optimal complexity bounds, i.e. $O(1/ε^2)$, for the number of calls to linear minimization oracle. We present promising experimental results showing the interest of the approach in comparison to competing methods.