Operator splitting schemes have been successfully used in computational sciences to reduce complex problems into a series of simpler subproblems. Since 1950s, these schemes have been widely used to solve problems in PDE and control. Recently, large-scale optimization problems in machine learning, signal processing, and imaging have created a resurgence of interest in operator-splitting based algorithms because they often have simple descriptions, are easy to code, and have (nearly) state-of-the-art performance for large-scale optimization problems. Although operator splitting techniques were introduced over 60 years ago, their importance has significantly increased in the past decade. This paper introduces a new operator-splitting scheme for solving a variety of problems that are reduced to a monotone inclusion of three operators, one of which is cocoercive. Our scheme is very simple, and it does not reduce to any existing splitting schemes. Our scheme recovers the existing forward-backward, Douglas-Rachford, and forward-Douglas-Rachford splitting schemes as special cases.Our new splitting scheme leads to a set of new and simple algorithms for a variety of other problems, including the 3-set split feasibility problems, 3-objective minimization problems, and doubly and multiple regularization problems, as well as the simplest extension of the classic ADMM from 2 to 3 blocks of variables. In addition to the basic scheme, we introduce several modifications and enhancements that can improve the convergence rate in practice, including an acceleration that achieves the optimal rate of convergence for strongly monotone inclusions. Finally, we evaluate the algorithm on several applications.

We consider a family of algorithms that successively sample and minimize simple stochastic models of the objective function. We show that under reasonable conditions on approximation quality and regularity of the models, any such algorithm drives a natural stationarity measure to zero at the rate O(k −1/4 ). As a consequence, we obtain the first complexity guarantees for the stochastic proximal point, proximal subgradient, and regularized Gauss-Newton methods for minimizing compositions of convex functions with smooth maps. The guiding principle, underlying the complexity guarantees, is that all algorithms under consideration can be interpreted as approximate descent methods on an implicit smoothing of the problem, given by the Moreau envelope. Specializing to classical circumstances, we obtain the long-sought convergence rate of the stochastic projected gradient method, without batching, for minimizing a smooth function on a closed convex set.

This work considers the question: what convergence guarantees does the stochastic subgradient method have in the absence of smoothness and convexity? We prove that the stochastic subgradient method, on any semialgebraic locally Lipschitz function, produces limit points that are all first-order stationary. More generally, our result applies to any function with a Whitney stratifiable graph. In particular, this work endows the stochastic subgradient method, and its proximal extension, with rigorous convergence guarantees for a wide class of problems arising in data science-including all popular deep learning architectures.

Splitting schemes are a class of powerful algorithms that solve complicated monotone inclusions and convex optimization problems that are built from many simpler pieces. They give rise to algorithms in which the simple pieces of the decomposition are processed individually. This leads to easily implementable and highly parallelizable algorithms, which often obtain nearly state-of-the-art performance.In the first part of this paper, we analyze the convergence rates of several general splitting algorithms and provide examples to prove the tightness of our results. The most general rates are proved for the fixed-point residual (FPR) of the Krasnosel'skiȋ-Mann (KM) iteration of nonexpansive operators, where we improve the known big-O rate to little-o. We show the tightness of this result and improve it in several special cases. In the second part of this paper, we use the convergence rates derived for the KM iteration to analyze the objective error convergence rates for the Douglas-Rachford (DRS), Peaceman-Rachford (PRS), and ADMM splitting algorithms under general convexity assumptions. We show, by way of example, that the rates obtained for these algorithms are tight in all cases and obtain the surprising statement: The DRS algorithm is nearly as fast as the proximal point algorithm (PPA) in the ergodic sense and nearly as slow as the subgradient method in the nonergodic sense. Finally, we provide several applications of our result to feasibility problems, model fitting, and distributed optimization. Our analysis is self-contained, and most results are deduced from a basic lemma that derives convergence rates for summable sequences, a simple diagram that decomposes each relaxed PRS iteration, and fundamental inequalities that relate the FPR to objective error.Keywords Krasnosel'skiȋ-Mann algorithm · Douglas-Rachford Splitting · Peaceman-Rachford Splitting · Alternating Direction Method of Multipliers · nonexpansive operator · averaged operator · fixed-point algorithm · little-o convergence Mathematics Subject Classification (2000) 47H05 · 65K05 · 65K15 · 90C25

Splitting schemes are a class of powerful algorithms that solve complicated monotone inclusion and convex optimization problems that are built from many simpler pieces. They give rise to algorithms in which the simple pieces of the decomposition are processed individually. This leads to easily implementable and highly parallelizable algorithms, which often obtain nearly state-of-the-art performance.In this paper, we provide a comprehensive convergence rate analysis of the Douglas-Rachford splitting (DRS), Peaceman-Rachford splitting (PRS), and alternating direction method of multipliers (ADMM) algorithms under various regularity assumptions including strong convexity, Lipschitz differentiability, and bounded linear regularity. The main consequence of this work is that relaxed PRS and ADMM automatically adapt to the regularity of the problem and achieve convergence rates that improve upon the (tight) worst-case rates that hold in the absence of such regularity. All of the results are obtained using simple techniques.Keywords Douglas-Rachford Splitting · Peaceman-Rachford Splitting · Alternating Direction Method of Multipliers · nonexpansive operator · averaged operator · fixed-point algorithm Mathematics Subject Classification (2000) 47H05 · 65K05 · 65K15 · 90C25