Zeroth-order (gradient-free) method is a class of powerful optimization tool for many machine learning problems because it only needs function values (not gradient) in the optimization. In particular, zeroth-order method is very suitable for many complex problems such as black-box attacks and bandit feedback, whose explicit gradients are difficult or infeasible to obtain. Recently, although many zeroth-order methods have been developed, these approaches still exist two main drawbacks: 1) high function query complexity; 2) not being well suitable for solving the problems with complex penalties and constraints. To address these challenging drawbacks, in this paper, we propose a novel fast zeroth-order stochastic alternating direction method of multipliers (ADMM) method (, ZO-SPIDER-ADMM) with lower function query complexity for solving nonconvex problems with multiple nonsmooth penalties. Moreover, we prove that our ZO-SPIDER-ADMM has the optimal function query complexity of @math for finding an @math -approximate local solution, where @math and @math denote the sample size and dimension of data, respectively. In particular, the ZO-SPIDER-ADMM improves the existing best nonconvex zeroth-order ADMM methods by a factor of @math . Moreover, we propose a fast online ZO-SPIDER-ADMM ( ZOO-SPIDER-ADMM). Our theoretical analysis shows that the ZOO-SPIDER-ADMM has the function query complexity of @math , which improves the existing best result by a factor of @math . Finally, we utilize a task of structured adversarial attack on black-box deep neural networks to demonstrate the efficiency of our algorithms.
So far, the above ADMM methods need to repeatedly calculate gradients of the loss function over the iterations. However, in many machine learning problems, the gradients of objective functions are difficult or infeasible to obtain. For example, in adversarial attack to black-box DNNs @cite_4 @cite_23 , only evaluation values (, function values) are provided. Thus, @cite_18 @cite_16 have proposed the zeroth-order online and stochastic ADMM methods for solving some convex problems. More recently, @cite_13 has proposed the nonconvex ZO-SVRG-ADMM and ZO-SAGA-ADMM methods. * -8pt
@cite_18 @cite_4 @cite_23 @cite_16 @cite_13
2946968605 2531036984 1523661875 2798170643
Alternating direction method of multipliers (ADMM) is a popular optimization tool for the composite and constrained problems in machine learning. However, in many machine learning problems such as black-box attacks and bandit feedback, ADMM could fail because the explicit gradients of these problems are difficult or infeasible to obtain. Zeroth-order (gradient-free) methods can effectively solve these problems due to that the objective function values are only required in the optimization. Recently, though there exist a few zeroth-order ADMM methods, they build on the convexity of objective function. Clearly, these existing zeroth-order methods are limited in many applications. In the paper, thus, we propose a class of fast zeroth-order stochastic ADMM methods (i.e., ZO-SVRG-ADMM and ZO-SAGA-ADMM) for solving nonconvex problems with multiple nonsmooth penalties, based on the coordinate smoothing gradient estimator. Moreover, we prove that both the ZO-SVRG-ADMM and ZO-SAGA-ADMM have convergence rate of @math , where @math denotes the number of iterations. In particular, our methods not only reach the best convergence rate @math for the nonconvex optimization, but also are able to effectively solve many complex machine learning problems with multiple regularized penalties and constraints. Finally, we conduct the experiments of black-box binary classification and structured adversarial attack on black-box deep neural network to validate the efficiency of our algorithms. In the paper, we study the stochastic alternating direction method of multipliers (ADMM) for the nonconvex optimizations, and propose three classes of the nonconvex stochastic ADMM with variance reduction, based on different reduced variance stochastic gradients. Specifically, the first class called the nonconvex stochastic variance reduced gradient ADMM (SVRG-ADMM), uses a multi-stage scheme to progressively reduce the variance of stochastic gradients. The second is the nonconvex stochastic average gradient ADMM (SAG-ADMM), which additionally uses the old gradients estimated in the previous iteration. The third called SAGA-ADMM is an extension of the SAG-ADMM method. Moreover, under some mild conditions, we establish the iteration complexity bound of @math of the proposed methods to obtain an @math -stationary solution of the nonconvex optimizations. In particular, we provide a general framework to analyze the iteration complexity of these nonconvex stochastic ADMM methods with variance reduction. Finally, some numerical experiments demonstrate the effectiveness of our methods. In this paper we study the problem of minimizing the average of a large number ( @math ) of smooth convex loss functions. We propose a new method, S2GD (Semi-Stochastic Gradient Descent), which runs for one or several epochs in each of which a single full gradient and a random number of stochastic gradients is computed, following a geometric law. The total work needed for the method to output an @math -accurate solution in expectation, measured in the number of passes over data, or equivalently, in units equivalent to the computation of a single gradient of the loss, is @math , where @math is the condition number. This is achieved by running the method for @math epochs, with a single gradient evaluation and @math stochastic gradient evaluations in each. The SVRG method of Johnson and Zhang arises as a special case. If our method is limited to a single epoch only, it needs to evaluate at most @math stochastic gradients. In contrast, SVRG requires @math stochastic gradients. To illustrate our theoretical results, S2GD only needs the workload equivalent to about 2.1 full gradient evaluations to find an @math -accurate solution for a problem with @math and @math . Weight pruning methods for deep neural networks (DNNs) have been investigated recently, but prior work in this area is mainly heuristic, iterative pruning, thereby lacking guarantees on the weight reduction ratio and convergence time. To mitigate these limitations, we present a systematic weight pruning framework of DNNs using the alternating direction method of multipliers (ADMM). We first formulate the weight pruning problem of DNNs as a nonconvex optimization problem with combinatorial constraints specifying the sparsity requirements, and then adopt the ADMM framework for systematic weight pruning. By using ADMM, the original nonconvex optimization problem is decomposed into two subproblems that are solved iteratively. One of these subproblems can be solved using stochastic gradient descent, the other can be solved analytically. Besides, our method achieves a fast convergence rate.