Semi-proximal Mirror-Prox for Nonsmooth Composite Minimization

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. $O(1/\epsilon^2)$, 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

Inexact Model: A Framework for Optimization and Variational Inequalities

In this paper we propose a general algorithmic framework for first-order methods in optimization in a broad sense, including minimization problems, saddle-point problems and variational inequalities. This framework allows to obtain many known methods as a special case, the list including accelerated gradient method, composite optimization methods, level-set methods, proximal methods. The idea of the framework is based on constructing an inexact model of the main problem component, i.e. objective function in optimization or operator in variational inequalities. Besides reproducing known results, our framework allows to construct new methods, which we illustrate by constructing a universal method for variational inequalities with composite structure. This method works for smooth and non-smooth problems with optimal complexity without a priori knowledge of the problem smoothness. We also generalize our framework for strongly convex objectives and strongly monotone variational inequalities.Comment: 41 page

Inexact relative smoothness and strong convexity for optimization and variational inequalities by inexact model

In this paper we propose a general algorithmic framework for first-order methods in optimization in a broad sense, including minimization problems, saddle-point problems and variational inequalities. This framework allows to obtain many known methods as a special case, the list including accelerated gradient method, composite optimization methods, level-set methods, Bregman proximal methods. The idea of the framework is based on constructing an inexact model of the main problem component, i.e. objective function in optimization or operator in variational inequalities. Besides reproducing known results, our framework allows to construct new methods, which we illustrate by constructing a universal conditional gradient method and universal method for variational inequalities with composite structure. These method works for smooth and non-smooth problems with optimal complexity without a priori knowledge of the problem smoothness. As a particular case of our general framework, we introduce relative smoothness for operators and propose an algorithm for VIs with such operator. We also generalize our framework for relatively strongly convex objectives and strongly monotone variational inequalities

Inexact relative smoothness and strong convexity for optimization and variational inequalities by inexact model

In this paper we propose a general algorithmic framework for first-order methods in optimization in a broad sense, including minimization problems, saddle-point problems and variational inequalities. This framework allows to obtain many known methods as a special case, the list including accelerated gradient method, composite optimization methods, level-set methods, Bregman proximal methods. The idea of the framework is based on constructing an inexact model of the main problem component, i.e. objective function in optimization or operator in variational inequalities. Besides reproducing known results, our framework allows to construct new methods, which we illustrate by constructing a universal conditional gradient method and universal method for variational inequalities with composite structure. These method works for smooth and non-smooth problems with optimal complexity without a priori knowledge of the problem smoothness. As a particular case of our general framework, we introduce relative smoothness for operators and propose an algorithm for VIs with such operator. We also generalize our framework for relatively strongly convex objectives and strongly monotone variational inequalities

An Inexact Primal-Dual Smoothing Framework for Large-Scale Non-Bilinear Saddle Point Problems

We develop an inexact primal-dual first-order smoothing framework to solve a class of non-bilinear saddle point problems with primal strong convexity. Compared with existing methods, our framework yields a significant improvement over the primal oracle complexity, while it has competitive dual oracle complexity. In addition, we consider the situation where the primal-dual coupling term has a large number of component functions. To efficiently handle this situation, we develop a randomized version of our smoothing framework, which allows the primal and dual sub-problems in each iteration to be solved by randomized algorithms inexactly in expectation. The convergence of this framework is analyzed both in expectation and with high probability. In terms of the primal and dual oracle complexities, this framework significantly improves over its deterministic counterpart. As an important application, we adapt both frameworks for solving convex optimization problems with many functional constraints. To obtain an $\varepsilon$-optimal and $\varepsilon$-feasible solution, both frameworks achieve the best-known oracle complexities (in terms of their dependence on $\varepsilon$)