Stochastic Frank-Wolfe for Composite Convex Minimization

A broad class of convex optimization problems can be formulated as a semidefinite program (SDP), minimization of a convex function over the positive-semidefinite cone subject to some affine constraints. The majority of classical SDP solvers are designed for the deterministic setting where problem data is readily available. In this setting, generalized conditional gradient methods (aka Frank-Wolfe-type methods) provide scalable solutions by leveraging the so-called linear minimization oracle instead of the projection onto the semidefinite cone. Most problems in machine learning and modern engineering applications, however, contain some degree of stochasticity. In this work, we propose the first conditional-gradient-type method for solving stochastic optimization problems under affine constraints. Our method guarantees $\mathcal{O}(k^{-1/3})$ convergence rate in expectation on the objective residual and $\mathcal{O}(k^{-5/12})$ on the feasibility gap

Approximate Frank-Wolfe Algorithms over Graph-structured Support Sets

In this paper, we propose approximate Frank-Wolfe (FW) algorithms to solve convex optimization problems over graph-structured support sets where the \textit{linear minimization oracle} (LMO) cannot be efficiently obtained in general. We first demonstrate that two popular approximation assumptions (\textit{additive} and \textit{multiplicative gap errors)}, are not valid for our problem, in that no cheap gap-approximate LMO oracle exists in general. Instead, a new \textit{approximate dual maximization oracle} (DMO) is proposed, which approximates the inner product rather than the gap. When the objective is $L$-smooth, we prove that the standard FW method using a $\delta$-approximate DMO converges as $\mathcal{O}(L / \delta t + (1-\delta)(\delta^{-1} + \delta^{-2}))$ in general, and as $\mathcal{O}(L/(\delta^2(t+2)))$ over a $\delta$-relaxation of the constraint set. Additionally, when the objective is $\mu$-strongly convex and the solution is unique, a variant of FW converges to $\mathcal{O}(L^2\log(t)/(\mu \delta^6 t^2))$ with the same per-iteration complexity. Our empirical results suggest that even these improved bounds are pessimistic, with significant improvement in recovering real-world images with graph-structured sparsity.Comment: 30 pages, 8 figure

A Stochastic Sequential Quadratic Optimization Algorithm for Nonlinear Equality Constrained Optimization with Rank-Deficient Jacobians

A sequential quadratic optimization algorithm is proposed for solving smooth nonlinear equality constrained optimization problems in which the objective function is defined by an expectation of a stochastic function. The algorithmic structure of the proposed method is based on a step decomposition strategy that is known in the literature to be widely effective in practice, wherein each search direction is computed as the sum of a normal step (toward linearized feasibility) and a tangential step (toward objective decrease in the null space of the constraint Jacobian). However, the proposed method is unique from others in the literature in that it both allows the use of stochastic objective gradient estimates and possesses convergence guarantees even in the setting in which the constraint Jacobians may be rank deficient. The results of numerical experiments demonstrate that the algorithm offers superior performance when compared to popular alternatives

Stochastic Frank-Wolfe for Composite Convex Minimization

A broad class of convex optimization problems can be formulated as a semidefinite program (SDP), minimization of a convex function over the positive-semidefinite cone subject to some affine constraints. The majority of classical SDP solvers are designed for the deterministic setting where problem data is readily available. In this setting, generalized conditional gradient methods (aka Frank-Wolfe-type methods) provide scalable solutions by leveraging the so-called linear minimization oracle instead of the projection onto the semidefinite cone. Most problems in machine learning and modern engineering applications, however, contain some degree of stochasticity. In this work, we propose the first conditional-gradient-type method for solving stochastic optimization problems under affine constraints. Our method guarantees O(k(-1/3)) convergence rate in expectation on the objective residual and O(k (5/12)) on the feasibility gap