The Fast Proximal Gradient Method (FPGM) and the Monotone FPGM (MFPGM) for minimization of nonsmooth convex functions are introduced and applied to tomographic image reconstruction. Convergence properties of the sequence of objective function values are derived, including a $O\left(1/k^{2}\right)$ non-asymptotic bound. The presented theory broadens current knowledge and explains the convergence behavior of certain methods that are known to present good practical performance. Numerical experimentation involving computerized tomography image reconstruction shows the methods to be competitive in practical scenarios. Experimental comparison with Algebraic Reconstruction Techniques are performed uncovering certain behaviors of accelerated Proximal Gradient algorithms that apparently have not yet been noticed when these are applied to tomographic image reconstruction.
We propose BIBPA, a block inertial Bregman proximal algorithm for minimizing the sum of a block relatively smooth function (that is, relatively smooth concerning each block) and block separable nonsmooth nonconvex functions. We prove that the sequence generated by BIBPA subsequentially converges to critical points of the objective under standard assumptions, and globally converges when the objective function is additionally assumed to satisfy the Kurdyka-Łojasiewicz (KŁ) property. We also provide the convergence rate when the objective satisfies the Łojasiewicz inequality. We apply BIBPA to th…
We investigate the convergence of an application of a proximal gradient method to control problems with nonsmooth and nonconvex control cost. Here, we focus on control cost functionals that promote sparsity, which includes functionals of $L^p$-type for $p\in [0,1)$. We prove stationarity properties of weak limit points of the method. These properties are weaker than those provided by Pontryagin's maximum principle and weaker than $L$-stationarity.
In this paper we present a variant of the proximal forward-backward splitting iteration for solving nonsmooth optimization problems in Hilbert spaces, when the objective function is the sum of two nondifferentiable convex functions. The proposed iteration, which will be called Proximal Subgradient Splitting Method, extends the classical subgradient iteration for important classes of problems, exploiting the additive structure of the objective function. The weak convergence of the generated sequence was established using different stepsizes and under suitable assumptions. Moreover, we analyze t…
In the lines of our approach in \cite{Ouorou2019}, where we exploit Nesterov fast gradient concept \cite{Nesterov1983} to the Moreau-Yosida regularization of a convex function, we devise new proximal algorithms for nonsmooth convex optimization. These algorithms need no bundling mechanism to update the stability center while preserving the complexity estimates established in \cite{Ouorou2019}. We report some preliminary computational results on some academic test problem to give a first estimate of their performance in relation with the classical proximal bundle algorithm.
In this paper, two distributed multi-proximal primal-dual algorithms are proposed to deal with a class of distributed nonsmooth resource allocation problems. In these problems, the global cost function is the summation of local convex and nonsmooth cost functions, each of which consists of one twice differentiable function and multiple nonsmooth functions. Communication graphs of underling multi-agent systems are directed and strongly connected but not necessarily weighted-balanced. The multi-proximal splitting is designed to deal with the difficulty caused by the unproximable property of the …
Escaping Saddle Points for Nonsmooth Weakly Convex Functions via Perturbed Proximal Algorithms
Department of Electrical and Computer Engineering, University of California, Davis（加州大学戴维斯分校电气与计算机工程系）；Simon Business School, University of Rochester（罗切斯特大学西蒙商学院）
We propose new proximal bundle algorithms for minimizing a nonsmooth convex function. These algorithms are derived from the application of Nesterov fast gradient methods for smooth convex minimization to the so-called Moreau-Yosida regularization $F_μ$ of $f$ w.r.t. some $μ>0$. Since the exact values and gradients of $F_μ$ are difficult to evaluate, we use approximate proximal points thanks to a bundle strategy to get implementable algorithms. One of these algorithms appears as an implementable version of a special case of inertial proximal algorithm. We give their complexity estimates in term…
We propose a proximal algorithm for minimizing objective functions consisting of three summands: the composition of a nonsmooth function with a linear operator, another nonsmooth function, each of the nonsmooth summands depending on an independent block variable, and a smooth function which couples the two block variables. The algorithm is a full splitting method, which means that the nonsmooth functions are processed via their proximal operators, the smooth function via gradient steps, and the linear operator via matrix times vector multiplication. We provide sufficient conditions for the bou…
In this paper, we consider a proximal linearized alternating direction method of multipliers (PL-ADMM) for solving linearly constrained nonconvex and possibly nonsmooth optimization problems. The algorithm is generalized by using variable metric proximal terms in the primal updates and an over-relaxation stepsize in the multiplier update. We prove that the sequence generated by this method is bounded and its limit points are critical points. Under the powerful Kurdyka-{Łojasiewicz} properties we prove that the sequence has a finite length thus converges, and we drive its convergence rates.
In this paper, we first study nonsmooth steepest descent method for nonsmooth functions defined on Hilbert space and establish the corresponding algorithm by proximal subgradients. Then, we use this algorithm to find stationary points for those functions satisfying prox-regularity and Lipschitzian continuity. As one application, the established algorithm is used to search the minimizer of lower semicontinuous and convex functions on finite-dimensional space. The convergent theorem, as one extension and improvement of the existing converging result for twice continuously differentiable convex f…
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 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…
In this paper we study nonconvex and nonsmooth optimization problems with semi-algebraic data, where the variables vector is split into several blocks of variables. The problem consists of one smooth function of the entire variables vector and the sum of nonsmooth functions for each block separately. We analyze an inertial version of the Proximal Alternating Linearized Minimization (PALM) algorithm and prove its global convergence to a critical point of the objective function at hand. We illustrate our theoretical findings by presenting numerical experiments on blind image deconvolution, on sp…
Nonsmoothness is often a curse for optimization; but it is sometimes a blessing, in particular for applications in machine learning. In this paper, we present the specific structure of nonsmooth optimization problems appearing in machine learning and illustrate how to leverage this structure in practice, for compression, acceleration, or dimension reduction. We pay a special attention to the presentation to make it concise and easily accessible, with both simple examples and general results.
We investigate the asymptotic properties of the trajectories generated by a second-order dynamical system of proximal-gradient type stated in connection with the minimization of the sum of a nonsmooth convex and a (possibly nonconvex) smooth function. The convergence of the generated trajectory to a critical point of the objective is ensured provided a regularization of the objective function satisfies the Kurdyka-Łojasiewicz property. We also provide convergence rates for the trajectory formulated in terms of the Łojasiewicz exponent.
The paper proposes and justifies a new algorithm of the proximal Newton type to solve a broad class of nonsmooth composite convex optimization problems without strong convexity assumptions. Based on advanced notions and techniques of variational analysis, we establish implementable results on the global convergence of the proposed algorithm as well as its local convergence with superlinear and quadratic rates. For certain structured problems, the obtained local convergence conditions do not require the local Lipschitz continuity of the corresponding Hessian mappings that is a crucial assumptio…
University of Minnesota（明尼苏达大学）；Johns Hopkins University（约翰斯·霍普金斯大学）；Northwestern University（西北大学）；Georgia Institute of Technology（佐治亚理工学院）
Many machine learning techniques sacrifice convenient computational structures to gain estimation robustness and modeling flexibility. However, by exploring the modeling structures, we find these "sacrifices" do not always require more computational efforts. To shed light on such a "free-lunch" phenomenon, we study the square-root-Lasso (SQRT-Lasso) type regression problem. Specifically, we show that the nonsmooth loss functions of SQRT-Lasso type regression ease tuning effort and gain adaptivity to inhomogeneous noise, but is not necessarily more challenging than Lasso in computation. We can …
Minimax optimization problems are an important class of optimization problems arising from modern machine learning and traditional research areas. While there have been many numerical algorithms for solving smooth convex-concave minimax problems, numerical algorithms for nonsmooth convex-concave minimax problems are very rare. This paper aims to develop an efficient numerical algorithm for a structured nonsmooth convex-concave minimax problem. A majorized semi-proximal alternating coordinate method (mspACM) is proposed, in which a majorized quadratic convex-concave function is adopted for appr…
Asymptotic Analysis of Gradient Mapping-type Stationarity Measure for the Sum of Nonconvex Nonsmooth Functions and Applications to Proximal Gradient-type Algorithms
For solving a wide class of nonconvex and nonsmooth problems, we propose a proximal linearized iteratively reweighted least squares (PL-IRLS) algorithm. We first approximate the original problem by smoothing methods, and second write the approximated problem into an auxiliary problem by introducing new variables. PL-IRLS is then built on solving the auxiliary problem by utilizing the proximal linearization technique and the iteratively reweighted least squares (IRLS) method, and has remarkable computation advantages. We show that PL-IRLS can be extended to solve more general nonconvex and nons…
Nonconvex and nonsmooth problems have recently attracted considerable attention in machine learning. However, developing efficient methods for the nonconvex and nonsmooth optimization problems with certain performance guarantee remains a challenge. Proximal coordinate descent (PCD) has been widely used for solving optimization problems, but the knowledge of PCD methods in the nonconvex setting is very limited. On the other hand, the asynchronous proximal coordinate descent (APCD) recently have received much attention in order to solve large-scale problems. However, the accelerated variants of …
A Proximal Decomposition Method for Solving Convex Variational Inverse Problems
UPMC Université Paris 06（巴黎第六大学）；Laboratoire Jacques-Louis Lions – UMR 7598（雅克-路易·利翁斯实验室）；Université Paris-Est（巴黎东大学）；Institut Gaspard Monge and UMR CNRS 8049（加斯帕尔·蒙日研究所）
In this paper, we consider an accelerated method for solving nonconvex and nonsmooth minimization problems. We propose a Bregman Proximal Gradient algorithm with extrapolation(BPGe). This algorithm extends and accelerates the Bregman Proximal Gradient algorithm (BPG), which circumvents the restrictive global Lipschitz gradient continuity assumption needed in Proximal Gradient algorithms (PG). The BPGe algorithm has higher generality than the recently introduced Proximal Gradient algorithm with extrapolation(PGe), and besides, due to the extrapolation step, BPGe converges faster than BPG algori…
The problem of minimizing the sum of nonsmooth, convex objective functions defined on a real Hilbert space over the intersection of fixed point sets of nonexpansive mappings, onto which the projections cannot be efficiently computed, is considered. The use of proximal point algorithms that use the proximity operators of the objective functions and incremental optimization techniques is proposed for solving the problem. With the focus on fixed point approximation techniques, two algorithms are devised for solving the problem. One blends an incremental subgradient method, which is a useful algor…
Projective Proximal Gradient Descent for A Class of Nonconvex Nonsmooth Optimization Problems: Fast Convergence Without Kurdyka-Lojasiewicz (KL) Property
We propose a new first-order optimisation algorithm to solve high-dimensional non-smooth composite minimisation problems. Typical examples of such problems have an objective that decomposes into a non-smooth empirical risk part and a non-smooth regularisation penalty. The proposed algorithm, called Semi-Proximal Mirror-Prox, leverages the Fenchel-type representation of one part of the objective while handling the other part of the objective via linear minimization over the domain. The algorithm stands in contrast with more classical proximal gradient algorithms with smoothing, which require th…
We consider optimization problems over the Stiefel manifold whose objective function is the summation of a smooth function and a nonsmooth function. Existing methods for solving this kind of problems can be classified into three classes. Algorithms in the first class rely on information of the subgradients of the objective function and thus tend to converge slowly in practice. Algorithms in the second class are proximal point algorithms, which involve subproblems that can be as difficult as the original problem. Algorithms in the third class are based on operator-splitting techniques, but they…
Research Center for Logic, Optimization and Security (LOS)（逻辑、优化与安全研究中心（LOS））；University of Bucharest（布加勒斯特大学）
Supported by the recent contributions in multiple branches, the first-order splitting algorithms became central for structured nonsmooth optimization. In the large-scale or noisy contexts, when only stochastic information on the smooth part of the objective function is available, the extension of proximal gradient schemes to stochastic oracles is based on proximal tractability of the nonsmooth component and it has been deeply analyzed in the literature. However, there remained gaps illustrated by composite models where the nonsmooth term is not proximally tractable anymore. In this note we tac…
In this paper, we introduce a stochastic projected subgradient method for weakly convex (i.e., uniformly prox-regular) nonsmooth, nonconvex functions---a wide class of functions which includes the additive and convex composite classes. At a high-level, the method is an inexact proximal point iteration in which the strongly convex proximal subproblems are quickly solved with a specialized stochastic projected subgradient method. The primary contribution of this paper is a simple proof that the proposed algorithm converges at the same rate as the stochastic gradient method for smooth nonconvex p…
The University of Iowa（爱荷华大学）；University of California（加利福尼亚大学）；Pennsylvania State University（宾夕法尼亚州立大学）
Riemannian optimization has drawn a lot of attention due to its wide applications in practice. Riemannian stochastic first-order algorithms have been studied in the literature to solve large-scale machine learning problems over Riemannian manifolds. However, most of the existing Riemannian stochastic algorithms require the objective function to be differentiable, and they do not apply to the case where the objective function is nonsmooth. In this paper, we present two Riemannian stochastic proximal gradient methods for minimizing nonsmooth function over the Stiefel manifold. The two methods, n…
We develop a Levenberg-Marquardt method for minimizing the sum of a smooth nonlinear least-squar es term $f(x) = \tfrac{1}{2} \|F(x)\|_2^2$ and a nonsmooth term $h$. Both $f$ and $h$ may be nonconvex. Steps are computed by minimizing the sum of a regularized linear least-squares model and a model of $h$ using a first-order method such as the proximal gradient method. We establish global convergence to a first-order stationary point of both a trust-region and a regularization variant of the Levenberg-Marquardt method under the assumptions that $F$ and its Jacobian are Lipschitz continuous and $…
