Composite minimization involves a collection of functions which are aggregated in a nonsmooth manner. It covers, as a particular case, smooth approximation of minimax games, minimization of max-type functions, and simple composite minimization problems, where the objective function has a nonsmooth component. We design a higher-order majorization algorithmic framework for fully composite problems (possibly nonconvex). Our framework replaces each component with a higher-order surrogate such that the corresponding error function has a higher-order Lipschitz continuous derivative. We present convergence guarantees for our method for composite optimization problems with (non)convex and (non)smooth objective function. In particular, we prove stationary point convergence guarantees for general nonconvex (possibly nonsmooth) problems and under Kurdyka-Lojasiewicz (KL) property of the objective function we derive improved rates depending on the KL parameter. For convex (possibly nonsmooth) problems we also provide sublinear convergence rates.
This paper extends the algorithm schemes proposed in \cite{Nesterov2007a} and \cite{Nesterov2007b} to the minimization of the sum of a composite objective function and a convex function. Two proximal point-type schemes are provided and their global convergence is investigated. The worst case complexity bound is also estimated under certain Lipschitz conditions and nondegeneratedness. The algorithm is then accelerated to get a faster convergence rate for the strongly convex case.
We consider the problem of minimizing a composite convex function with two different access methods: an oracle, for which we can evaluate the value and gradient, and a structured function, which we access only by solving a convex optimization problem. We are motivated by two associated technological developments. For the oracle, systems like PyTorch or TensorFlow can automatically and efficiently compute gradients, given a computation graph description. For the structured function, systems like CVXPY accept a high level domain specific language description of the problem, and automatically tra…
We propose a first-order augmented Lagrangian algorithm (FALC) to solve the composite norm minimization problem min |sigma(F(X)-G)|_alpha + |C(X)- d|_beta subject to A(X)-b in Q; where sigma(X) denotes the vector of singular values of X, the matrix norm |sigma(X)|_alpha denotes either the Frobenius, the nuclear, or the L2-operator norm of X, the vector norm |.|_beta denotes either the L1-norm, L2-norm or the L infty-norm; Q is a closed convex set and A(.), C(.), F(.) are linear operators from matrices to vector spaces of appropriate dimensions. Basis pursuit, matrix completion, robust principa…
Proximal Newton-type methods for minimizing composite functions
Institute for Computational and Mathematical Engineering, Stanford University（斯坦福大学计算与数学工程研究所）；Department of Management Science and Engineering, Stanford University（斯坦福大学管理科学与工程系）
In this paper, we establish lower bounds for the oracle complexity of the first-order methods minimizing regularized convex functions. We consider the composite representation of the objective. The smooth part has Hölder continuous gradient of degree $ν\in [0, 1]$ and is accessible by a black-box local oracle. The composite part is a power of a norm. We prove that the best possible rate for the first-order methods in the large-scale setting for Euclidean norms is of the order $O(k^{- p(1 + 3ν) / (2(p - 1 - ν))})$ for the functional residual, where $k$ is the iteration counter and $p$ is the po…
In this paper we develop a randomized block-coordinate descent method for minimizing the sum of a smooth and a simple nonsmooth block-separable convex function and prove that it obtains an $ε$-accurate solution with probability at least $1-ρ$ in at most $O(\tfrac{n}ε \log \tfrac{1}ρ)$ iterations, where $n$ is the number of blocks. For strongly convex functions the method converges linearly. This extends recent results of Nesterov [Efficiency of coordinate descent methods on huge-scale optimization problems, CORE Discussion Paper #2010/2], which cover the smooth case, to composite minimization,…
We consider the problem of minimizing the composition of a smooth (nonconvex) function and a smooth vector mapping, where the inner mapping is in the form of an expectation over some random variable or a finite sum. We propose a stochastic composite gradient method that employs an incremental variance-reduced estimator for both the inner vector mapping and its Jacobian. We show that this method achieves the same orders of complexity as the best known first-order methods for minimizing expected-value and finite-sum nonconvex functions, despite the additional outer composition which renders the …
This article deals with multiobjective composite optimization problems that consist of simultaneously minimizing several objective functions, each of which is composed of a combination of smooth and non-smooth functions. To tackle these problems, we propose a generalized version of the conditional gradient method, also known as Frank-Wolfe method. The method is analyzed with three step size strategies, including Armijo-type, adaptive, and diminishing step sizes. We establish asymptotic convergence properties and iteration-complexity bounds, with and without convexity assumptions on the objecti…
A Unified Analysis on the Subgradient Upper Bounds for the Subgradient Methods Minimizing Composite Nonconvex, Nonsmooth and Non-Lipschitz Functions
Antai College of Economics and Management, Shanghai Jiao Tong University（上海交通大学安泰经济管理学院）；School of Data Science, The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳）数据科学学院）；Institute of Translational Medicine and National Center for Translational Medicine, Shanghai Jiao Tong University（上海交通大学转化医学研究院和转化医学中心）；Department of Industrial and Systems Engineering, University of Minnesota（明尼苏达大学工业与系统工程系）
We consider global efficiency of algorithms for minimizing a sum of a convex function and a composition of a Lipschitz convex function with a smooth map. The basic algorithm we rely on is the prox-linear method, which in each iteration solves a regularized subproblem formed by linearizing the smooth map. When the subproblems are solved exactly, the method has efficiency $\mathcal{O}(\varepsilon^{-2})$, akin to gradient descent for smooth minimization. We show that when the subproblems can only be solved by first-order methods, a simple combination of smoothing, the prox-linear method, and a fa…
We consider the problem of minimizing a finite sum of convex functions subject to the set of minimizers of a convex differentiable function. In order to solve the problem, an algorithm combining the incremental proximal gradient method with smooth penalization technique is proposed. We show the convergence of the generated sequence of iterates to an optimal solution of the optimization problems, provided that a condition expressed via the Fenchel conjugate of the constraint function is fulfilled. Finally, the functionality of the method is illustrated by some numerical experiments addressing i…
In this paper we propose a proximal algorithm for minimizing an objective function of two block variables consisting of three terms: 1) a smooth function, 2) a nonsmooth function which is a composition between a strictly increasing, concave, differentiable function and a convex nonsmooth function, and 3) a smooth function which couples the two block variables. We propose a variable metric composite proximal alternating linearized minimization (CPALM) to solve this class of problems. Building on the powerful Kurdyka-Łojasiewicz property, we derive the convergence analysis and establish that eac…
Composite function minimization captures a wide spectrum of applications in both computer vision and machine learning. It includes bound constrained optimization and cardinality regularized optimization as special cases. This paper proposes and analyzes a new Matrix Splitting Method (MSM) for minimizing composite functions. It can be viewed as a generalization of the classical Gauss-Seidel method and the Successive Over-Relaxation method for solving linear systems in the literature. Incorporating a new Gaussian elimination procedure, the matrix splitting method achieves state-of-the-art perfor…
In this paper we propose a second--order method for solving \emph{linear composite sparse optimization problems} consisting of minimizing the sum of a differentiable (possibly nonconvex function) and a nondifferentiable convex term. The composite nondifferentiable convex penalizer is given by $\ell_1$--norm of a matrix multiplied with the coefficient vector. The algorithm that we propose for the case of the linear composite $\ell_1$ problem relies on the three main ingredients that power the OESOM algorithm \cite{dlrlm07}: the minimum norm subgradient, a projection step and, in particular, the…
We present two approximate versions of the proximal subgradient method for minimizing the sum of two convex functions (not necessarily differentiable). The algorithms involve, at each iteration, inexact evaluations of the proximal operator and approximate subgradients of the functions (namely: the $ε$-subgradients). The methods use different error criteria for approximating the proximal operators. We provide an analysis of the convergence and rate of convergence properties of these methods, considering various stepsize rules, including both, diminishing and constant stepsizes. For the case whe…
Decentralized optimization is a promising parallel computation paradigm for large-scale data analytics and machine learning problems defined over a network of nodes. This paper is concerned with decentralized non-convex composite problems with population or empirical risk. In particular, the networked nodes are tasked to find an approximate stationary point of the average of local, smooth, possibly non-convex risk functions plus a possibly non-differentiable extended valued convex regularizer. Under this general formulation, we propose the first provably efficient, stochastic proximal gradient…
In this paper, we study the iteration complexity of cubic regularization of Newton method for solving composite minimization problems with uniformly convex objective. We introduce the notion of second-order condition number of a certain degree and justify the linear rate of convergence in a nondegenerate case for the method with an adaptive estimate of the regularization parameter. The algorithm automatically achieves the best possible global complexity bound among different problem classes of uniformly convex objective functions with Hölder continuous Hessian of the smooth part of the objecti…
In this paper we propose a distributed version of a randomized block-coordinate descent method for minimizing the sum of a partially separable smooth convex function and a fully separable non-smooth convex function. Under the assumption of block Lipschitz continuity of the gradient of the smooth function, this method is shown to have a sublinear convergence rate. Linear convergence rate of the method is obtained for the newly introduced class of generalized error bound functions. We prove that the new class of generalized error bound functions encompasses both global/local error bound function…
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.
In the paper, we develop a composite version of Mirror Prox algorithm for solving convex-concave saddle point problems and monotone variational inequalities of special structure, allowing to cover saddle point/variational analogies of what is usually called "composite minimization" (minimizing a sum of an easy-to-handle nonsmooth and a general-type smooth convex functions "as if" there were no nonsmooth component at all). We demonstrate that the composite Mirror Prox inherits the favourable (and unimprovable already in the large-scale bilinear saddle point case) $O(1/ε)$ efficiency estimate of…
We introduce new global and local inexact oracle concepts for a wide class of convex functions in composite convex minimization. Such inexact oracles naturally come from primal-dual framework, barrier smoothing, inexact computations of gradients and Hessian, and many other situations. We also provide examples showing that the class of convex functions equipped with the newly inexact second-order oracles is larger than standard self-concordant as well as Lipschitz gradient function classes. Further, we investigate several properties of convex and/or self-concordant functions under the inexact s…
This paper proposes an efficient adaptive variant of a quadratic penalty accelerated inexact proximal point (QP-AIPP) method proposed earlier by the authors. Both the QP-AIPP method and its variant solve linearly set constrained nonconvex composite optimization problems using a quadratic penalty approach where the generated penalized subproblems are solved by a variant of the underlying AIPP method. The variant, in turn, solves a given penalized subproblem by generating a sequence of proximal subproblems which are then solved by an accelerated composite gradient algorithm. The main difference …
This paper considers the optimization problem of minimizing a rational function. We reformulate this problem as polynomial optimization by the technique of homogenization. These two problems are shown to be equivalent under some generic conditions. The exact Jacobian SDP relaxation method proposed by Nie is used to solve the resulting polynomial optimization. We also prove that the assumption of nonsingularity in Nie's method can be weakened as the finiteness of singularities. Some numerical examples are given to illustrate the efficiency of our method.
The self-concordant-like property of a smooth convex function is a new analytical structure that generalizes the self-concordant notion. While a wide variety of important applications feature the self-concordant-like property, this concept has heretofore remained unexploited in convex optimization. To this end, we develop a variable metric framework of minimizing the sum of a "simple" convex function and a self-concordant-like function. We introduce a new analytic step-size selection procedure and prove that the basic gradient algorithm has improved convergence guarantees as compared to "fast"…
We consider the problem of minimizing the sum of a smooth function and a composition of a zero-one loss function with a linear operator, namely zero-one composite optimization problem (0/1-COP). It is a versatile model including the support vector machine (SVM), multi-label classification (MLC), maximum rank correlation (MRC) and so on. However, due to the nonconvexity, discontinuity and NP-hardness of the 0/1-COP, it is intractable to design a globally convergent algorithm and the work attempting to solve it directly is scarce. In this paper, we first define and characterize the proximal stat…
École Polytechnique Fédérale de Lausanne (EPFL)（洛桑联邦理工学院）
We propose a variable metric framework for minimizing the sum of a self-concordant function and a possibly non-smooth convex function, endowed with an easily computable proximal operator. We theoretically establish the convergence of our framework without relying on the usual Lipschitz gradient assumption on the smooth part. An important highlight of our work is a new set of analytic step-size selection and correction procedures based on the structure of the problem. We describe concrete algorithmic instances of our framework for several interesting applications and demonstrate them numericall…
This paper provides a new way of developing the fast iterative shrinkage/thresholding algorithm (FISTA) that is widely used for minimizing composite convex functions with a nonsmooth term such as the $\ell_1$ regularizer. In particular, this paper shows that FISTA corresponds to an optimized approach to accelerating the proximal gradient method with respect to a worst-case bound of the cost function. This paper then proposes a new algorithm that is derived by instead optimizing the step coefficients of the proximal gradient method with respect to a worst-case bound of the composite gradient ma…
This paper computes composite indicators of corporate social responsibility (CSR) from an efficiency perspective for food and beverage manufacturing firms in various world regions over the period from 2011 to 2018. From a methodological perspective, we extend the measurement of composite indicators within data envelopment analysis, allowing for the non-convexities of the production set and for the appropriate comparison of indicators between groups of firms. From an empirical point of view, we contribute by comparing the efficiency in CSR practices of food and beverage companies across regions…
This paper proposes two proximal Newton-CG methods for convex nonsmooth optimization problems in composite form. The algorithms are based on a a reformulation of the original nonsmooth problem as the unconstrained minimization of a continuously differentiable function, namely the forward-backward envelope (FBE). The first algorithm is based on a standard line search strategy, whereas the second one combines the global efficiency estimates of the corresponding first-order methods, while achieving fast asymptotic convergence rates. Furthermore, they are computationally attractive since each Newt…
Toyota Technological Institute at Chicago（芝加哥丰田理工学院）
We provide tight upper and lower bounds on the complexity of minimizing the average of $m$ convex functions using gradient and prox oracles of the component functions. We show a significant gap between the complexity of deterministic vs randomized optimization. For smooth functions, we show that accelerated gradient descent (AGD) and an accelerated variant of SVRG are optimal in the deterministic and randomized settings respectively, and that a gradient oracle is sufficient for the optimal rate. For non-smooth functions, having access to prox oracles reduces the complexity and we present optim…
We study the worst-case convergence rates of the proximal gradient method for minimizing the sum of a smooth strongly convex function and a non-smooth convex function whose proximal operator is available. We establish the exact worst-case convergence rates of the proximal gradient method in this setting for any step size and for different standard performance measures: objective function accuracy, distance to optimality and residual gradient norm. The proof methodology relies on recent developments in performance estimation of first-order methods based on semidefinite programming. In the case …
This paper presents an accelerated composite gradient (ACG) variant, referred to as the AC-ACG method, for solving nonconvex smooth composite minimization problems. As opposed to well-known ACG variants that are either based on a known Lipschitz gradient constant or a sequence of maximum observed curvatures, the current one is based on the average of all past observed curvatures. More specifically, AC-ACG uses a positive multiple of the average of all observed curvatures until the previous iteration as a way to estimate the "function curvature" at the current point and then two resolvent evalu…
This paper considers a class of constrained convex stochastic composite optimization problems whose objective function is given by the summation of a differentiable convex component, together with a nonsmooth but convex component. The nonsmooth component has an explicit max structure that may not easy to compute its proximal mapping. In order to solve these problems, we propose a mini-batch stochastic Nesterov's smoothing (MSNS) method. Convergence and the optimal iteration complexity of the method are established. Numerical results are provided to illustrate the efficiency of the proposed MSN…
The article is devoted to the development of algorithmic methods ensuring efficient complexity bounds for strongly convex-concave saddle point problems in the case when one of the groups of variables is high-dimensional, and the other is relatively low-dimensional (up to a hundred). The proposed technique is based on reducing problems of this type to a problem of minimizing a convex (maximizing a concave) functional in one of the variables, for which it is possible to find an approximate gradient at an arbitrary point with the required accuracy using an auxiliary optimization subproblem with a…
