We study structured nonsmooth convex finite-sum optimization that appears widely in machine learning applications, including support vector machines and least absolute deviation. For the primal-dual formulation of this problem, we propose a novel algorithm called \emph{Variance Reduction via Primal-Dual Accelerated Dual Averaging (\vrpda)}. In the nonsmooth and general convex setting, \vrpda~has the overall complexity $O(nd\log\min \{1/ε, n\} + d/ε)$ in terms of the primal-dual gap, where $n$ denotes the number of samples, $d$ the dimension of the primal variables, and $ε$ the desired accuracy. In the nonsmooth and strongly convex setting, the overall complexity of \vrpda~becomes $O(nd\log\min\{1/ε, n\} + d/\sqrtε)$ in terms of both the primal-dual gap and the distance between iterate and optimal solution. Both these results for \vrpda~improve significantly on state-of-the-art complexity estimates, which are $O(nd\log \min\{1/ε, n\} + \sqrt{n}d/ε)$ for the nonsmooth and general convex setting and $O(nd\log \min\{1/ε, n\} + \sqrt{n}d/\sqrtε)$ for the nonsmooth and strongly convex setting, in a much more simple and straightforward way. Moreover, both complexities are better than \emph{lower} bounds for general convex finite sums that lack the particular (common) structure that we consider. Our theoretical results are supported by numerical experiments, which confirm the competitive performance of \vrpda~compared to state-of-the-art.
Primal-dual multigrid methods for nonsmooth optimization
MODEMAT Research Center in Mathematical Modeling and Optimization（数学建模与优化模式研究中心）；Department of Mathematics, Escuela Politécnica Nacional (EPN)（国家理工大学数学系）；Department of Mathematics and Statistics, University of Helsinki（赫尔辛基大学数学与统计系）
We provide an overview of primal-dual algorithms for nonsmooth and non-convex-concave saddle-point problems. This flows around a new analysis of such methods, using Bregman divergences to formulate simplified conditions for convergence.
Primal-Dual Methods for Nonsmooth Nonconvex Optimization with Orthogonality Constraints
H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology（佐治亚理工学院H. Milton Stewart工业与系统工程系）；Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong（香港中文大学系统工程与工程管理系）
We study the extension of the Chambolle--Pock primal-dual algorithm to nonsmooth optimization problems involving nonlinear operators between function spaces. Local convergence is shown under technical conditions including metric regularity of the corresponding primal-dual optimality conditions. We also show convergence for a Nesterov-type accelerated variant provided one part of the functional is strongly convex. We show the applicability of the accelerated algorithm to examples of inverse problems with $L^1$- and $L^\infty$-fitting terms as well as of state-constrained optimal control problem…
In this paper, we consider a nonsmooth convex finite-sum problem with a conic constraint. To overcome the challenge of projecting onto the constraint set and computing the full (sub)gradient, we introduce a primal-dual incremental gradient scheme where only a component function and two constraints are used to update each primal-dual sub-iteration in a cyclic order. We demonstrate an asymptotic sublinear rate of convergence in terms of suboptimality and infeasibility which is an improvement over the state-of-the-art incremental gradient schemes in this setting. Numerical results suggest that th…
We consider the problem of minimizing a convex, separable, nonsmooth function subject to linear constraints. The numerical method we propose is a block-coordinate extension of the Chambolle-Pock primal-dual algorithm. We prove convergence of the method without resorting to assumptions like smoothness or strong convexity of the objective, full-rank condition on the matrix, strong duality or even consistency of the linear system. Freedom from imposing the latter assumption permits convergence guarantees for misspecified or noisy systems.
This technical note studies a class of distributed nonsmooth convex consensus optimization problem. The cost function is a summation of local cost functions which are convex but nonsmooth. Each of the local cost functions consists of a twice differentiable convex function and two lower semi-continuous convex functions. We call this problem as single-smooth plus double-nonsmooth (SSDN) problem. Under mild conditions, we propose a distributed double proximal primal-dual optimization algorithm. The double proximal operator is designed to deal with the difficulty caused by the unproximable propert…
In this paper, we propose a novel primal-dual proximal splitting algorithm (PD-PSA), named BALPA, for the composite optimization problem with equality constraints, where the loss function consists of a smooth term and a nonsmooth term composed with a linear mapping. In BALPA, the dual update is designed as a proximal point for a time-varying quadratic function, which balances the implementation of primal and dual update and retains the proximity-induced feature of classic PD-PSAs. In addition, by this balance, BALPA eliminates the inefficiency of classic PD-PSAs for composite optimization prob…
In this paper, we investigate the continuous time partial primal-dual gradient dynamics (P-PDGD) for solving convex optimization problems with the form $ \min\limits_{x\in X,y\inΩ}\ f({x})+h(y),\ \textit{s.t.}\ A{x}+By=C $, where $ f({x}) $ is strongly convex and smooth, but $ h(y) $ is strongly convex and non-smooth. Affine equality and set constraints are included. We prove the exponential stability of P-PDGD, and bounds on decaying rates are provided. Moreover, it is also shown that the decaying rates can be regulated by setting the stepsize.
We propose a new randomized coordinate descent method for a convex optimization template with broad applications. Our analysis relies on a novel combination of four ideas applied to the primal-dual gap function: smoothing, acceleration, homotopy, and coordinate descent with non-uniform sampling. As a result, our method features the first convergence rate guarantees among the coordinate descent methods, that are the best-known under a variety of common structure assumptions on the template. We provide numerical evidence to support the theoretical results with a comparison to state-of-the-art al…
We prove convergence of a single time-scale stochastic subgradient method with subgradient averaging for constrained problems with a nonsmooth and nonconvex objective function having the property of generalized differentiability. As a tool of our analysis, we also prove a chain rule on a path for such functions.
We propose a new first-order primal-dual optimization framework for a convex optimization template with broad applications. Our optimization algorithms feature optimal convergence guarantees under a variety of common structure assumptions on the problem template. Our analysis relies on a novel combination of three classic ideas applied to the primal-dual gap function: smoothing, acceleration, and homotopy. The algorithms due to the new approach achieve the best known convergence rate results, in particular when the template consists of only non-smooth functions. We also outline a restart strat…
Institute of Evaluation and Assessment Research, Academy of Military Science（军事科学院评估论证研究中心）；Command and Control Engineering College, Army Engineering University（陆军工程大学指挥控制工程学院）；Institute of Automation, Chinese Academy of Sciences（中国科学院自动化研究所）
Averaging scheme has attracted extensive attention in deep learning as well as traditional machine learning. It achieves theoretically optimal convergence and also improves the empirical model performance. However, there is still a lack of sufficient convergence analysis for strongly convex optimization. Typically, the convergence about the last iterate of gradient descent methods, which is referred to as individual convergence, fails to attain its optimality due to the existence of logarithmic factor. In order to remove this factor, we first develop gradient descent averaging (GDA), which is …
In this paper we propose a primal-dual dynamical approach to the minimization of a structured convex function consisting of a smooth term, a nonsmooth term, and the composition of another nonsmooth term with a linear continuous operator. In this scope we introduce a dynamical system for which we prove that its trajectories asymptotically converge to a saddle point of the Lagrangian of the underlying convex minimization problem as time tends to infinity. In addition, we provide rates for both the violation of the feasibility condition by the ergodic trajectories and the convergence of the objec…
Many problems arising in image processing and signal recovery with multi-regularization can be formulated as minimization of a sum of three convex separable functions. Typically, the objective function involves a smooth function with Lipschitz continuous gradient, a linear composite nonsmooth function and a nonsmooth function. In this paper, we propose a primal-dual fixed-point (PDFP) scheme to solve the above class of problems. The proposed algorithm for three block problems is a fully splitting symmetric scheme, only involving explicit gradient and linear operators without inner iteration, w…
WeBank AI Group（微众银行AI集团）；Tencent Inc.（腾讯公司）；University of California, Davis（加州大学戴维斯分校）；The Hong Kong University of Science and Technology（香港科技大学）
Dual averaging-type methods are widely used in industrial machine learning applications due to their ability to promoting solution structure (e.g., sparsity) efficiently. In this paper, we propose a novel accelerated dual-averaging primal-dual algorithm for minimizing a composite convex function. We also derive a stochastic version of the proposed method which solves empirical risk minimization, and its advantages on handling sparse data are demonstrated both theoretically and empirically.
This paper investigates two accelerated primal-dual mirror dynamical approaches for smooth and nonsmooth convex optimization problems with affine and closed, convex set constraints. In the smooth case, an accelerated primal-dual mirror dynamical approach (APDMD) based on accelerated mirror descent and primal-dual framework is proposed and accelerated convergence properties of primal-dual gap, feasibility measure and the objective function value along with trajectories of APDMD are derived by the Lyapunov analysis method. Then, we extend APDMD into two distributed dynamical approaches to deal w…
The Stochastic Primal-Dual Hybrid Gradient (SPDHG) was proposed by Chambolle et al. (2018) and is an efficient algorithm to solve some nonsmooth large-scale optimization problems. In this paper we prove its almost sure convergence for convex but not necessarily strongly convex functionals. We also look into its application to parallel Magnetic Resonance Imaging reconstruction in order to test performance of SPDHG. Our numerical results show that for a range of settings SPDHG converges significantly faster than its deterministic counterpart.
Momentum methods are now used pervasively within the machine learning community for training non-convex models such as deep neural networks. Empirically, they out perform traditional stochastic gradient descent (SGD) approaches. In this work we develop a Lyapunov analysis of SGD with momentum (SGD+M), by utilizing a equivalent rewriting of the method known as the stochastic primal averaging (SPA) form. This analysis is much tighter than previous theory in the non-convex case, and due to this we are able to give precise insights into when SGD+M may out-perform SGD, and what hyper-parameter sche…
We propose a new self-adaptive, double-loop smoothing algorithm to solve composite, nonsmooth, and constrained convex optimization problems. Our algorithm is based on Nesterov's smoothing technique via general Bregman distance functions. It self-adaptively selects the number of iterations in the inner loop to achieve a desired complexity bound without requiring the accuracy a priori as in variants of Augmented Lagrangian methods (ALM). We prove $\BigO{\frac{1}{k}}$-convergence rate on the last iterate of the outer sequence for both unconstrained and constrained settings in contrast to ergodic …
Block coordinate descent methods and stochastic subgradient methods have been extensively studied in optimization and machine learning. By combining randomized block sampling with stochastic subgradient methods based on dual averaging, we present stochastic block dual averaging (SBDA)---a novel class of block subgradient methods for convex nonsmooth and stochastic optimization. SBDA requires only a block of subgradients and updates blocks of variables and hence has significantly lower iteration cost than traditional subgradient methods. We show that the SBDA-based methods exhibit the optimal c…
King Abdullah University of Science and Technology（阿卜杜拉国王科技大学）
We consider minimizing the sum of three convex functions, where the first one F is smooth, the second one is nonsmooth and proximable and the third one is the composition of a nonsmooth proximable function with a linear operator L. This template problem has many applications, for instance, in image processing and machine learning. First, we propose a new primal-dual algorithm, which we call PDDY, for this problem. It is constructed by applying Davis-Yin splitting to a monotone inclusion in a primal-dual product space, where the operators are monotone under a specific metric depending on L. We …
Chinese University of Hong Kong（香港中文大学）；Princeton University（普林斯顿大学）；Northwestern University（西北大学）；University of Minnesota（明尼苏达大学）
Despite the success of single-agent reinforcement learning, multi-agent reinforcement learning (MARL) remains challenging due to complex interactions between agents. Motivated by decentralized applications such as sensor networks, swarm robotics, and power grids, we study policy evaluation in MARL, where agents with jointly observed state-action pairs and private local rewards collaborate to learn the value of a given policy. In this paper, we propose a double averaging scheme, where each agent iteratively performs averaging over both space and time to incorporate neighboring gradient informat…
University of Edinburgh（爱丁堡大学）；Toyota Technological Institute at Chicago（芝加哥丰田技术研究所）
We address the issue of using mini-batches in stochastic optimization of SVMs. We show that the same quantity, the spectral norm of the data, controls the parallelization speedup obtained for both primal stochastic subgradient descent (SGD) and stochastic dual coordinate ascent (SCDA) methods and use it to derive novel variants of mini-batched SDCA. Our guarantees for both methods are expressed in terms of the original nonsmooth primal problem based on the hinge-loss.
Proximal splitting algorithms are well suited to solving large-scale nonsmooth optimization problems, in particular those arising in machine learning. We propose a new primal-dual algorithm, in which the dual update is randomized; equivalently, the proximity operator of one of the function in the problem is replaced by a stochastic oracle. For instance, some randomly chosen dual variables, instead of all, are updated at each iteration. Or, the proximity operator of a function is called with some small probability only. A nonsmooth variance-reduction technique is implemented so that the algorit…
Nonlinearly constrained nonconvex and nonsmooth optimization models play an increasingly important role in machine learning, statistics and data analytics. In this paper, based on the augmented Lagrangian function we introduce a flexible first-order primal-dual method, to be called nonconvex auxiliary problem principle of augmented Lagrangian (NAPP-AL), for solving a class of nonlinearly constrained nonconvex and nonsmooth optimization problems. We demonstrate that NAPP-AL converges to a stationary solution at the rate of o(1/\sqrt{k}), where k is the number of iterations. Moreover, under an a…
For a class of nonsmooth composite optimization problems with linear equality constraints, we utilize a Lyapunov-based approach to establish the global exponential stability of the primal-dual gradient flow dynamics based on the proximal augmented Lagrangian. The result holds when the differentiable part of the objective function is strongly convex with a Lipschitz continuous gradient; the non-differentiable part is proper, lower semi-continuous, and convex; and the matrix in the linear constraint is full row rank. Our quadratic Lyapunov function generalizes recent result from strongly convex …
In this paper, we propose an inertial accelerated primal-dual method for the linear equality constrained convex optimization problem. When the objective function has a ``nonsmooth + smooth'' composite structure, we further propose an inexact inertial primal-dual method by linearizing the smooth individual function and solving the subproblem inexactly. Assuming merely convexity, we prove that the proposed methods enjoy $\mathcal{O}(1/k^2)$ convergence rate on the objective residual and the feasibility violation in the primal model. Numerical results are reported to demonstrate the validity of t…
Federated Learning (FL) is a distributed learning paradigm that scales on-device learning collaboratively and privately. Standard FL algorithms such as FedAvg are primarily geared towards smooth unconstrained settings. In this paper, we study the Federated Composite Optimization (FCO) problem, in which the loss function contains a non-smooth regularizer. Such problems arise naturally in FL applications that involve sparsity, low-rank, monotonicity, or more general constraints. We first show that straightforward extensions of primal algorithms such as FedAvg are not well-suited for FCO since th…
"Classical" First Order (FO) algorithms of convex optimization, such as Mirror Descent algorithm or Nesterov's optimal algorithm of smooth convex optimization, are well known to have optimal (theoretical) complexity estimates which do not depend on the problem dimension. However, to attain the optimality, the domain of the problem should admit a "good proximal setup". The latter essentially means that 1) the problem domain should satisfy certain geometric conditions of "favorable geometry", and 2) the practical use of these methods is conditioned by our ability to compute at a moderate cost {\…
This work is concerned with the optimization of nonconvex, nonsmooth composite optimization problems, whose objective is a composition of a nonlinear mapping and a nonsmooth nonconvex function, that can be written as an infimal convolution (inf-conv). To tackle this problem class we propose to reformulate the problem exploiting its inf-conv structure and derive a block coordinate descent scheme on a Bregman augmented Lagrangian, that can be implemented asynchronously. We prove convergence of our scheme to stationary points of the original model for a specific choice of the penalty parameter. O…
Differentiating Nonsmooth Solutions to Parametric Monotone Inclusion Problems
Toulouse School of Economics（图卢兹经济学院）；University of Toulouse（图卢兹大学）；IRIT（图卢兹计算机科学研究所）；CNRS（法国国家科学研究中心）；Université Toulouse III Paul Sabatier（图卢兹第三大学保罗·萨巴蒂埃大学）；Institut Universitaire de France（法兰西大学研究院）；CVN（认知与神经计算研究中心）；CentraleSupélec（巴黎中央理工-高等电力学院）；Université Paris-Saclay（巴黎萨克雷大学）
Iowa State University（爱荷华州立大学）；Johns Hopkins University（约翰斯·霍普金斯大学）；Princeton University（普林斯顿大学）
We study a stochastic and distributed algorithm for nonconvex problems whose objective consists of a sum of $N$ nonconvex $L_i/N$-smooth functions, plus a nonsmooth regularizer. The proposed NonconvEx primal-dual SpliTTing (NESTT) algorithm splits the problem into $N$ subproblems, and utilizes an augmented Lagrangian based primal-dual scheme to solve it in a distributed and stochastic manner. With a special non-uniform sampling, a version of NESTT achieves $ε$-stationary solution using $\mathcal{O}((\sum_{i=1}^N\sqrt{L_i/N})^2/ε)$ gradient evaluations, which can be up to $\mathcal{O}(N)$ times…
Dual first-order methods are essential techniques for large-scale constrained convex optimization. However, when recovering the primal solutions, we need $T(ε^{-2})$ iterations to achieve an $ε$-optimal primal solution when we apply an algorithm to the non-strongly convex dual problem with $T(ε^{-1})$ iterations to achieve an $ε$-optimal dual solution, where $T(x)$ can be $x$ or $\sqrt{x}$. In this paper, we prove that the iteration complexity of the primal solutions and dual solutions have the same $O\left(\frac{1}{\sqrtε}\right)$ order of magnitude for the accelerated randomized dual coordin…
Stochastic approximation techniques have been used in various contexts in data science. We propose a stochastic version of the forward-backward algorithm for minimizing the sum of two convex functions, one of which is not necessarily smooth. Our framework can handle stochastic approximations of the gradient of the smooth function and allows for stochastic errors in the evaluation of the proximity operator of the nonsmooth function. The almost sure convergence of the iterates generated by the algorithm to a minimizer is established under relatively mild assumptions. We also propose a stochastic…
On the monotone and primal-dual active set schemes for $\ell^p$-type problems, $p \in (0,1]$
University of Graz（格拉茨大学）；Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences（奥地利科学院约翰·拉德纳计算与应用数学研究所）
We study the \emph{Proximal Alternating Predictor-Corrector} (PAPC) algorithm introduced recently by Drori, Sabach and Teboulle to solve nonsmooth structured convex-concave saddle point problems consisting of the sum of a smooth convex function, a finite collection of nonsmooth convex functions and bilinear terms. We introduce the notion of pointwise quadratic supportability, which is a relaxation of a standard strong convexity assumption and allows us to show that the primal sequence is R-linearly convergent to an optimal solution and the primal-dual sequence is globally Q-linearly convergent…
A class of non-smooth and non-convex optimization problems with penalty constraints linked to variational inequalities (VI) is studied with respect to its shape differentiability. The specific problem stemming from quasi-brittle fracture describes an elastic body with a Barenblatt cohesive crack under the inequality condition of non-penetration at the crack faces. Based on the Lagrange approach and using smooth penalization with the Lavrentiev regularization, a formula for the shape derivative is derived. The explicit formula contains both primal and adjoint states and is useful for finding de…
A distributed nonsmooth robust resource allocation problem with cardinality constrained uncertainty is investigated in this paper. The global objective is consisted of local objectives, which are convex but nonsmooth. Each agent is constrained in its private convex set and has only the information of its corresponding local objective. The resource allocation condition is subject to the cardinality constrained uncertainty sets. By employing the duality theory of convex optimization, a dual problem of the robust resource allocation problem is presented. For solving this dual problem, a distribut…
Preconditioned primal-dual algorithms for saddle point problems: non-ergodic convergence rates
Bernoulli Institute for Mathematics, Computer Science and Artificial Intelligence, University of Groningen（格罗宁根大学伯努利数学、计算机科学与人工智能研究所）；Instituto de Ciencias de la Ingeniería, Universidad de O’Higgins（奥希金斯大学工程科学研究所）；Centro de Modelamiento Matemático (CNRS IRL2807), Universidad de Chile（智利大学数学建模中心）
This paper investigates the input-output performance of secondary frequency controllers through the control-theoretic notion of $H_2$ norms. We consider a quadratic objective accounting for the cost of reserve procurement and provide exact analytical formulae for the performance of continuous-time aggregated averaging controllers. Then, we contrast it with distributed averaging controllers -- seeking optimality conditions such as identical marginal costs -- and primal-dual controllers which have gained attention as systematic techniques to design distributed algorithms solving convex optimizat…
