In this paper, we study a family of non-convex and possibly non-smooth inf-projection minimization problems, where the target objective function is equal to minimization of a joint function over another variable. This problem includes difference of convex (DC) functions and a family of bi-convex functions as special cases. We develop stochastic algorithms and establish their first-order convergence for finding a (nearly) stationary solution of the target non-convex function under different conditions of the component functions. To the best of our knowledge, this is the first work that comprehensively studies stochastic optimization of non-convex inf-projection minimization problems with provable convergence guarantee. Our algorithms enable efficient stochastic optimization of a family of non-decomposable DC functions and a family of bi-convex functions. To demonstrate the power of the proposed algorithms we consider an important application in variance-based regularization, and experiments verify the effectiveness of our inf-projection based formulation and the proposed stochastic algorithm in comparison with previous stochastic algorithms based on the min-max formulation for achieving the same effect.
To solve the above min-max formulation, @cite_6 proposed stochastic primal-dual algorithms based on the stochastic mirror prox methods proposed in for addressing convex-concave problems. When the loss function @math is non-convex (e.g., the hypothesis class is defined by deep neural networks), the resulting min-max problem is non-convex in terms of @math and but is concave in terms of @math . Recently, @cite_12 proposed new stochastic algorithms for solving the non-convex concave min-max problem when the objective function is weakly convex with respect to the minimization variable given the maximization variable. They proved the convergence to a nearly stationary point of the minimization objective function. However, the stochastic algorithms proposed in are not scalable due to updating and maintaining of the dual variable @math .
@cite_12 @cite_6
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Min-max saddle-point problems have broad applications in many tasks in machine learning, e.g., distributionally robust learning, learning with non-decomposable loss, or learning with uncertain data. Although convex-concave saddle-point problems have been broadly studied with efficient algorithms and solid theories available, it remains a challenge to design provably efficient algorithms for non-convex saddle-point problems, especially when the objective function involves an expectation or a large-scale finite sum. Motivated by recent literature on non-convex non-smooth minimization, this paper studies a family of non-convex min-max problems where the minimization component is non-convex (weakly convex) and the maximization component is concave. We propose a proximally guided stochastic subgradient method and a proximally guided stochastic variance-reduced method for expected and finite-sum saddle-point problems, respectively. We establish the computation complexities of both methods for finding a nearly stationary point of the corresponding minimization problem. Policy evaluation is a crucial step in many reinforcement-learning procedures, which estimates a value function that predicts states' long-term value under a given policy. In this paper, we focus on policy evaluation with linear function approximation over a fixed dataset. We first transform the empirical policy evaluation problem into a (quadratic) convex-concave saddle point problem, and then present a primal-dual batch gradient method, as well as two stochastic variance reduction methods for solving the problem. These algorithms scale linearly in both sample size and feature dimension. Moreover, they achieve linear convergence even when the saddle-point problem has only strong concavity in the dual variables but no strong convexity in the primal variables. Numerical experiments on benchmark problems demonstrate the effectiveness of our methods. In this paper, we investigate the non-asymptotic stationary convergence behavior of Stochastic Mirror Descent (SMD) for nonconvex optimization. We focus on a general class of nonconvex nonsmooth stochastic optimization problems, in which the objective can be decomposed into a relatively weakly convex function (possibly non-Lipschitz) and a simple non-smooth convex regularizer. We prove that SMD, without the use of mini-batch, is guaranteed to converge to a stationary point in a convergence rate of @math . The efficiency estimate matches with existing results for stochastic subgradient method, but is evaluated under a stronger stationarity measure. Our convergence analysis applies to both the original SMD and its proximal version, as well as the deterministic variants, for solving relatively weakly convex problems. Classical stochastic gradient methods are well suited for minimizing expected-value objective functions. However, they do not apply to the minimization of a nonlinear function involving expected values or a composition of two expected-value functions, i.e., the problem @math minxEvfv(Ew[gw(x)]). In order to solve this stochastic composition problem, we propose a class of stochastic compositional gradient descent (SCGD) algorithms that can be viewed as stochastic versions of quasi-gradient method. SCGD update the solutions based on noisy sample gradients of @math fv,gw and use an auxiliary variable to track the unknown quantity @math Ewgw(x). We prove that the SCGD converge almost surely to an optimal solution for convex optimization problems, as long as such a solution exists. The convergence involves the interplay of two iterations with different time scales. For nonsmooth convex problems, the SCGD achieves a convergence rate of @math O(k-1 4) in the general case and @math O(k-2 3) in the strongly convex case, after taking k samples. For smooth convex problems, the SCGD can be accelerated to converge at a rate of @math O(k-2 7) in the general case and @math O(k-4 5) in the strongly convex case. For nonconvex problems, we prove that any limit point generated by SCGD is a stationary point, for which we also provide the convergence rate analysis. Indeed, the stochastic setting where one wants to optimize compositions of expected-value functions is very common in practice. The proposed SCGD methods find wide applications in learning, estimation, dynamic programming, etc. The usual approach to developing and analyzing first-order methods for non-smooth (stochastic or deterministic) convex optimization assumes that the objective function is uniformly Lipschitz continuous with parameter @math . However, in many settings the non-differentiable convex function @math is not uniformly Lipschitz continuous -- for example (i) the classical support vector machine (SVM) problem, (ii) the problem of minimizing the maximum of convex quadratic functions, and even (iii) the univariate setting with @math . Herein we develop a notion of "relative continuity" that is determined relative to a user-specified "reference function" @math (that should be computationally tractable for algorithms), and we show that many non-differentiable convex functions are relatively continuous with respect to a correspondingly fairly-simple reference function @math . We also similarly develop a notion of "relative stochastic continuity" for the stochastic setting. We analysis two standard algorithms -- the (deterministic) mirror descent algorithm and the stochastic mirror descent algorithm -- for solving optimization problems in these two new settings, and we develop for the first time computational guarantees for instances where the objective function is not uniformly Lipschitz continuous. This paper is a companion paper for non-differentiable convex optimization to the recent paper by Lu, Freund, and Nesterov, which developed similar sorts of results for differentiable convex optimization. We consider the fundamental problem in non-convex optimization of efficiently reaching a stationary point. In contrast to the convex case, in the long history of this basic problem, the only known theoretical results on first-order non-convex optimization remain to be full gradient descent that converges in @math iterations for smooth objectives, and stochastic gradient descent that converges in @math iterations for objectives that are sum of smooth functions. We provide the first improvement in this line of research. Our result is based on the variance reduction trick recently introduced to convex optimization, as well as a brand new analysis of variance reduction that is suitable for non-convex optimization. For objectives that are sum of smooth functions, our first-order minibatch stochastic method converges with an @math rate, and is faster than full gradient descent by @math . We demonstrate the effectiveness of our methods on empirical risk minimizations with non-convex loss functions and training neural nets. We consider an algorithm that successively samples and minimizes stochastic models of the objective function. We show that under weak-convexity and Lipschitz conditions, the algorithm drives the expected norm of the gradient of the Moreau envelope to zero at the rate @math . Our result yields new complexity guarantees for the stochastic proximal point algorithm on weakly convex problems and for the stochastic prox-linear algorithm for minimizing compositions of convex functions with smooth maps. Moreover, our result also recovers the recently obtained complexity estimate for the stochastic proximal subgradient method on weakly convex problems. An algorithm for solving the problem: minimize @math (a convex function) subject to @math , @math , each @math a concave function, is presented. Specifically, the function [ P [ x,t,r_k ] f( x ) + r_k^ - 1 [ g_i ( x ) - t_i ] ^2 ] is minimized over all x, nonnegative t, for a strictly decreasing null sequence @math . This extends the work of T. Pietrzykowski [5]. It is proved that for every @math , there exists a finite point @math which minimizes P, and which solves the convex programming problem as @math . This algorithm is similar to the Sequential Unconstrained Minimization Technique (SUMT) [1] in that it solves the (Wolfe) dual programming problem [6]. It differs from SUMT in that (1) it approaches the optimum from the region of infeasibility (i.e., it is a relaxation technique), (2) it does not require a nonempty interior to the nonlinearly constrained region, (3) no separate feasibilit... We consider a generic convex optimization problem associated with regularized empirical risk minimization of linear predictors. The problem structure allows us to reformulate it as a convex-concave saddle point problem. We propose a stochastic primal-dual coordinate (SPDC) method, which alternates between maximizing over a randomly chosen dual variable and minimizing over the primal variables. An extrapolation step on the primal variables is performed to obtain accelerated convergence rate. We also develop a mini-batch version of the SPDC method which facilitates parallel computing, and an extension with weighted sampling probabilities on the dual variables, which has a better complexity than uniform sampling on unnormalized data. Both theoretically and empirically, we show that the SPDC method has comparable or better performance than several state-of-the-art optimization methods. We consider a generic convex optimization problem associated with regularized empirical risk minimization of linear predictors. The problem structure allows us to reformulate it as a convex-concave saddle point problem. We propose a stochastic primal-dual coordinate (SPDC) method, which alternates between maximizing over a randomly chosen dual variable and minimizing over the primal variable. An extrapolation step on the primal variable is performed to obtain accelerated convergence rate. We also develop a mini-batch version of the SPDC method which facilitates parallel computing, and an extension with weighted sampling probabilities on the dual variables, which has a better complexity than uniform sampling on unnormalized data. Both theoretically and empirically, we show that the SPDC method has comparable or better performance than several state-of-the-art optimization methods. We consider convex-concave saddle-point problems where the objective functions may be split in many components, and extend recent stochastic variance reduction methods (such as SVRG or SAGA) to provide the first large-scale linearly convergent algorithms for this class of problems which are common in machine learning. While the algorithmic extension is straightforward, it comes with challenges and opportunities: (a) the convex minimization analysis does not apply and we use the notion of monotone operators to prove convergence, showing in particular that the same algorithm applies to a larger class of problems, such as variational inequalities, (b) there are two notions of splits, in terms of functions, or in terms of partial derivatives, (c) the split does need to be done with convex-concave terms, (d) non-uniform sampling is key to an efficient algorithm, both in theory and practice, and (e) these incremental algorithms can be easily accelerated using a simple extension of the "catalyst" framework, leading to an algorithm which is always superior to accelerated batch algorithms. We propose a new algorithm for minimizing regularized empirical loss: Stochastic Dual Newton Ascent (SDNA). Our method is dual in nature: in each iteration we update a random subset of the dual variables. However, unlike existing methods such as stochastic dual coordinate ascent, SDNA is capable of utilizing all local curvature information contained in the examples, which leads to striking improvements in both theory and practice -- sometimes by orders of magnitude. In the special case when an L2-regularizer is used in the primal, the dual problem is a concave quadratic maximization problem plus a separable term. In this regime, SDNA in each step solves a proximal subproblem involving a random principal submatrix of the Hessian of the quadratic function; whence the name of the method. We propose a new algorithm for minimizing regularized empirical loss: Stochastic Dual Newton Ascent (SDNA). Our method is dual in nature: in each iteration we update a random subset of the dual variables. However, unlike existing methods such as stochastic dual coordinate ascent, SDNA is capable of utilizing all curvature information contained in the examples, which leads to striking improvements in both theory and practice - sometimes by orders of magnitude. In the special case when an L2-regularizer is used in the primal, the dual problem is a concave quadratic maximization problem plus a separable term. In this regime, SDNA in each step solves a proximal subproblem involving a random principal submatrix of the Hessian of the quadratic function; whence the name of the method. If, in addition, the loss functions are quadratic, our method can be interpreted as a novel variant of the recently introduced Iterative Hessian Sketch. We consider supervised learning problems over training sets in which both the number of training examples and the dimension of the feature vectors are large. We focus on the case where the loss function defining the quality of the parameter we wish to estimate may be non-convex, but also has a convex regularization. We propose a Doubly Stochastic Successive Convex approximation scheme (DSSC) able to handle non-convex regularized expected risk minimization. The method operates by decomposing the decision variable into blocks and operating on random subsets of blocks at each step. The algorithm belongs to the family of successive convex approximation methods since we replace the original non-convex stochastic objective by a strongly convex sample surrogate function, and solve the resulting convex program, for each randomly selected block in parallel. The method operates on subsets of features (block coordinate methods) and training examples (stochastic approximation) at each step. In contrast to many stochastic convex methods whose almost sure behavior is not guaranteed in non-convex settings, DSSC attains almost sure convergence to a stationary solution of the problem. Numerical experiments on a non-convex variant of a lasso regression problem show that DSSC performs favorably in this setting. We investigate the minimization of Newton's functional for the problem of the body of minimal resistance with maximal height @math cite butt in the class of convex developable functions defined in a disc. This class is a natural candidate to find a (non-radial) minimizer in accordance with the results of cite lrp2. We prove that the minimizer in this class has a minimal set in the form of a regular polygon with @math sides centered in the disc, and numerical experiments indicate that the natural number @math is a non-decreasing function of @math . The corresponding functions all achieve a lower value of the functional than the optimal radially symmetric function with the same height @math . In stochastic convex optimization the goal is to minimize a convex function @math over a convex set @math where @math is some unknown distribution and each @math in the support of @math is convex over @math . The optimization is commonly based on i.i.d. samples @math from @math . A standard approach to such problems is empirical risk minimization (ERM) that optimizes @math . Here we consider the question of how many samples are necessary for ERM to succeed and the closely related question of uniform convergence of @math to @math over @math . We demonstrate that in the standard @math setting of Lipschitz-bounded functions over a @math of bounded radius, ERM requires sample size that scales linearly with the dimension @math . This nearly matches standard upper bounds and improves on @math dependence proved for @math setting by Shalev- (2009). In stark contrast, these problems can be solved using dimension-independent number of samples for @math setting and @math dependence for @math setting using other approaches. We further show that our lower bound applies even if the functions in the support of @math are smooth and efficiently computable and even if an @math regularization term is added. Finally, we demonstrate that for a more general class of bounded-range (but not Lipschitz-bounded) stochastic convex programs an infinite gap appears already in dimension 2. We consider regularized stochastic learning and online optimization problems, where the objective function is the sum of two convex terms: one is the loss function of the learning task, and the other is a simple regularization term such as l1-norm for promoting sparsity. We develop extensions of Nesterov's dual averaging method, that can exploit the regularization structure in an online setting. At each iteration of these methods, the learning variables are adjusted by solving a simple minimization problem that involves the running average of all past subgradients of the loss function and the whole regularization term, not just its subgradient. In the case of l1-regularization, our method is particularly effective in obtaining sparse solutions. We show that these methods achieve the optimal convergence rates or regret bounds that are standard in the literature on stochastic and online convex optimization. For stochastic learning problems in which the loss functions have Lipschitz continuous gradients, we also present an accelerated version of the dual averaging method. We prove novel convergence results for a stochastic proximal gradient algorithm suitable for solving a large class of convex optimization problems, where a convex objective function is given by the sum of a smooth and a possibly non-smooth component. We consider the iterates convergence and derive @math non asymptotic bounds in expectation in the strongly convex case, as well as almost sure convergence results under weaker assumptions. Our approach allows to avoid averaging and weaken boundedness assumptions which are often considered in theoretical studies and might not be satisfied in practice. The Alternating Direction Method of Multipliers (ADMM) has received lots of attention recently due to the tremendous demand from large-scale and data-distributed machine learning applications. In this paper, we present a stochastic setting for optimization problems with non-smooth composite objective functions. To solve this problem, we propose a stochastic ADMM algorithm. Our algorithm applies to a more general class of convex and nonsmooth objective functions, beyond the smooth and separable least squares loss used in lasso. We also demonstrate the rates of convergence for our algorithm under various structural assumptions of the stochastic function: O(1 √t)) for convex functions and O(log t t) for strongly convex functions. Compared to previous literature, we establish the convergence rate of ADMM for convex problems in terms of both the objective value and the feasibility violation. A novel application named Graph-Guided SVM is proposed to demonstrate the usefulness of our algorithm. Abstract We consider the problem of linear transceiver design to achieve max–min fairness in a downlink MIMO multicell network. This problem can be formulated as maximizing the minimum rate among all the users in an interfering broadcast channel (IBC). In this paper we show that when the number of antennas is at least two at each of the transmitters and the receivers, the min rate maximization problem is NP-hard in the number of users. Moreover, we develop a low-complexity algorithm for this problem by iteratively solving a sequence of convex subproblems. We theoretically establish the global convergence of the proposed algorithm to the set of stationary points, which may be suboptimal due to the non-convexity of the original minimum rate maximization problem. Numerical simulations show that this algorithm is efficient in achieving fairness among all the users.