Random Minibatch Subgradient Algorithms for Convex Problems with Functional Constraints

Abstract

In this paper we consider non-smooth convex optimization problems with (possibly) infinite intersection of constraints. In contrast to the classical approach, where the constraints are usually represented as intersection of simple sets, which are easy to project onto, in this paper we consider that each constraint set is given as the level set of a convex but not necessarily differentiable function. For these settings we propose subgradient iterative algorithms with random minibatch feasibility updates. At each iteration, our algorithms take a step aimed at only minimizing the objective function and then a subsequent step minimizing the feasibility violation of the observed minibatch of constraints. The feasibility updates are performed based on either parallel or sequential random observations of several constraint components. We analyze the convergence behavior of the proposed algorithms for the case when the objective function is restricted strongly convex and with bounded subgradients, while the functional constraints are endowed with a bounded first-order black-box oracle. For a diminishing stepsize, we prove sublinear convergence rates for the expected distances of the weighted averages of the iterates from the constraint set, as well as for the expected suboptimality of the function values along the weighted averages. Our convergence rates are known to be optimal for subgradient methods on this class of problems. Moreover, the rates depend explicitly on the minibatch size and show when minibatching helps a subgradient scheme with random feasibility updates.

Stochastic subgradient for composite convex optimization with functional constraints

“…Furthermore, in (Nedich, 2011) and (Nedich and Necoara, 2019) sublinear convergence rates are established either for convex or strongly convex deterministic objective functions, respectively, while in this paper we prove (sub)linear rates under an expected composite objective function which is either convex or satisfies relaxed strong convexity conditions. Moreover, (Nedich, 2011;Nedich and Necoara, 2019) present separately the convergence analysis for smooth and nonsmooth objective, while in this paper we present a unified convergence analysis covering both cases through the so-called stochastic bounded gradient condition. Hence, since we deal with stochastic composite objective functions, smooth or nonsmooth, and relaxed strong convexity assumptions, and since we consider a stochastic proximal gradient with new stepsize rules, our convergence analysis requires additional insights that differ from that of (Nedich, 2011;Nedich and Necoara, 2019).…”

“…However, the optimization problem, the algorithm and consequently the convergence analysis are different from the present paper. In particular, our algorithm is a stochastic proximal gradient extension of the algorithms proposed in (Nedich, 2011;Nedich and Necoara, 2019). Additionally, the stepsizes in (Nedich, 2011;Nedich and Necoara, 2019) are chosen decreasing, while in the present work for strongly like objective functions we derive insightful stepsize-switching rules which describe when one should switch from a constant to a decreasing stepsize regime.…”

“…In particular, our algorithm is a stochastic proximal gradient extension of the algorithms proposed in (Nedich, 2011;Nedich and Necoara, 2019). Additionally, the stepsizes in (Nedich, 2011;Nedich and Necoara, 2019) are chosen decreasing, while in the present work for strongly like objective functions we derive insightful stepsize-switching rules which describe when one should switch from a constant to a decreasing stepsize regime. Furthermore, in (Nedich, 2011) and (Nedich and Necoara, 2019) sublinear convergence rates are established either for convex or strongly convex deterministic objective functions, respectively, while in this paper we prove (sub)linear rates under an expected composite objective function which is either convex or satisfies relaxed strong convexity conditions.…”

“…The papers most related to our work are (Nedich, 2011;Nedich and Necoara, 2019), where subgradient methods with random feasibility steps are proposed for solving convex problems with deterministic objective and many functional constraints. However, the optimization problem, the algorithm and consequently the convergence analysis are different from the present paper.…”

“…Additionally, the stepsizes in (Nedich, 2011;Nedich and Necoara, 2019) are chosen decreasing, while in the present work for strongly like objective functions we derive insightful stepsize-switching rules which describe when one should switch from a constant to a decreasing stepsize regime. Furthermore, in (Nedich, 2011) and (Nedich and Necoara, 2019) sublinear convergence rates are established either for convex or strongly convex deterministic objective functions, respectively, while in this paper we prove (sub)linear rates under an expected composite objective function which is either convex or satisfies relaxed strong convexity conditions. Moreover, (Nedich, 2011;Nedich and Necoara, 2019) present separately the convergence analysis for smooth and nonsmooth objective, while in this paper we present a unified convergence analysis covering both cases through the so-called stochastic bounded gradient condition.…”