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Register a new userPrimal-Dual Frank-Wolfe for Constrained Stochastic Programs with Convex and Non-convex Objectives
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We study constrained stochastic programs where the decision vector at each time slot cannot be chosen freely but is tied to the realization of an underlying random state vector. The goal is to minimize a general objective function subject to linear constraints. A typical scenario where such programs appear is opportunistic scheduling over a network of time-varying channels, where the random state vector is the channel state observed, and the control vector is the transmission decision which depends on the current channel state. We consider a primal-dual type Frank-Wolfe algorithm that has a low complexity update during each slot and that learns to make efficient decisions without prior knowledge of the probability distribution of the random state vector. We establish convergence time guarantees for the case of both convex and non-convex objective functions. We also emphasize application of the algorithm to non-convex opportunistic scheduling and distributed non-convex stochastic optimization over a connected graph.
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Stochastic gradient methods (SGMs) have been widely used for solving stochastic optimization problems. A majority of existing works assume no constraints or easy-to-project constraints. In this paper, we consider convex stochastic optimization problems with expectation constraints. For these problems, it is often extremely expensive to perform projection onto the feasible set. Several SGMs in the literature can be applied to solve the expectation-constrained stochastic problems. We propose a novel primal-dual type SGM based on the Lagrangian function. Different from existing methods, our method incorporates an adaptiveness technique to speed up convergence. At each iteration, our method inquires an unbiased stochastic subgradient of the Lagrangian function, and then it renews the primal variables by an adaptive-SGM update and the dual variables by a vanilla-SGM update. We show that the proposed method has a convergence rate of $O(1/sqrt{k})$ in terms of the objective error and the constraint violation. Although the convergence rate is the same as those of existing SGMs, we observe its significantly faster convergence than an existing non-adaptive primal-dual SGM and a primal SGM on solving the Neyman-Pearson classification and quadratically constrained quadratic programs. Furthermore, we modify the proposed method to solve convex-concave stochastic minimax problems, for which we perform adaptive-SGM updates to both primal and dual variables. A convergence rate of $O(1/sqrt{k})$ is also established to the modified method for solving minimax problems in terms of primal-dual gap.
The stochastic Frank-Wolfe method has recently attracted much general interest in the context of optimization for statistical and machine learning due to its ability to work with a more general feasible region. However, there has been a complexity gap in the guaranteed convergence rate for stochastic Frank-Wolfe compared to its deterministic counterpart. In this work, we present a new generalized stochastic Frank-Wolfe method which closes this gap for the class of structured optimization problems encountered in statistical and machine learning characterized by empirical loss minimization with a certain type of ``linear prediction property (formally defined in the paper), which is typically present loss minimization problems in practice. Our method also introduces the notion of a ``substitute gradient that is a not-necessarily-unbiased sample of the gradient. We show that our new method is equivalent to a particular randomized coordinate mirror descent algorithm applied to the dual problem, which in turn provides a new interpretation of randomized dual coordinate descent in the primal space. Also, in the special case of a strongly convex regularizer our generalized stochastic Frank-Wolfe method (as well as the randomized dual coordinate descent method) exhibits linear convergence. Furthermore, we present computational experiments that indicate that our method outperforms other stochastic Frank-Wolfe methods consistent with the theory developed herein.
We introduce a novel primal-dual flow for affine constrained convex optimization problem. As a modification of the standard saddle-point system, our primal-dual flow is proved to possesses the exponential decay property, in terms of a tailored Lyapunov function. Then a class of primal-dual methods for the original optimization problem are obtained from numerical discretizations of the continuous flow, and with a unified discrete Lyapunov function, nonergodic convergence rates are established. Among those algorithms, we can recover the (linearized) augmented Lagrangian method and the quadratic penalty method with continuation technique. Also, new methods with a special inner problem, that is a linear symmetric positive definite system or a nonlinear equation which may be solved efficiently via the semi-smooth Newton method, have been proposed as well. Especially, numerical tests on the linearly constrained $l_1$-$l_2$ minimization show that our method outperforms the accelerated linearized Bregman method.
This paper considers a general convex constrained problem setting where functions are not assumed to be differentiable nor Lipschitz continuous. Our motivation is in finding a simple first-order method for solving a wide range of convex optimization problems with minimal requirements. We study the method of weighted dual averages (Nesterov, 2009) in this setting and prove that it is an optimal method.
The Frank-Wolfe method and its extensions are well-suited for delivering solutions with desirable structural properties, such as sparsity or low-rank structure. We introduce a new variant of the Frank-Wolfe method that combines Frank-Wolfe steps and steepest descent steps, as well as a novel modification of the Frank-Wolfe gap to measure convergence in the non-convex case. We further extend this method to incorporate in-face directions for preserving structured solutions as well as block coordinate steps, and we demonstrate computational guarantees in terms of the modified Frank-Wolfe gap for all of these variants. We are particularly motivated by the application of this methodology to the training of neural networks with sparse properties, and we apply our block coordinate method to the problem of $ell_1$ regularized neural network training. We present the results of several numerical experiments on both artificial and real datasets demonstrating significant improvements of our method in training sparse neural networks.
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