Very recently, the papers "Point Convergence of Nesterov's Accelerated Gradient Method: An AI-Assisted Proof" by Jang and Ryu, and "The Iterates of Nesterov's Accelerated Algorithm Converge in the Critical Regimes" by Bot, Fadili, and Nguyen have simultaneously resolved a long-standing open problem concerning Nesterov's accelerated gradient method. These works show that the iterates of the algorithm (known in its composite form as FISTA) indeed converge to an optimal solution. In this work, we extend these results and prove that, in infinite dimensional Hilbert spaces, the iterates of such an algorithm still converge (in the weak sense) even when the proximity operator and the gradient are computed inexactly, with the latter possibly stochastic.
We consider a variable metric and inexact version of the FISTA-type algorithm considered in (Chambolle, Pock, 2016, Calatroni, Chambolle, 2019) for the minimization of the sum of two (possibly strongly) convex functions. The proposed algorithm is combined with an adaptive (non-monotone) backtracking strategy, which allows for the adjustment of the algorithmic step-size along the iterations in order to improve the convergence speed. We prove a linear convergence result for the function values, which depends on both the strong convexity moduli of the two functions and the upper and lower bounds …
One of the most popular and important first-order iterations that provides optimal complexity of the classical proximal gradient method (PGM) is the "Fast Iterative Shrinkage/Thresholding Algorithm" (FISTA). In this paper, two inexact versions of FISTA for minimizing the sum of two convex functions are studied. The proposed schemes inexactly solve their subproblems by using relative error criteria instead of exogenous and diminishing error rules. When the evaluation of the proximal operator is difficult, inexact versions of FISTA are necessary and the relative error rules proposed here may hav…
In this note, we consider a special instance of the scaled, inexact and adaptive generalised Fast Iterative Soft-Thresholding Algorithm (SAGE-FISTA) recently proposed in (Rebegoldi, Calatroni, '21) for the efficient solution of strongly convex composite optimisation problems. In particular, we address here the sole (non-strongly) convex optimisation scenario, which is frequently encountered in many imaging applications. The proposed inexact S-FISTA algorithm shows analogies to the variable metric and inexact version of FISTA studied in (Bonettini, Rebegoldi, Ruggiero, '19), the main difference…
In this paper we propose an adaptively extrapolated proximal gradient method, which is based on the accelerated proximal gradient method (also known as FISTA), however we locally optimize the extrapolation parameter by carrying out an exact (or inexact) line search. It turns out that in some situations, the proposed algorithm is equivalent to a class of SR1 (identity minus rank 1) proximal quasi-Newton methods. Convergence is proved in a general non-convex setting, and hence, as a byproduct, we also obtain new convergence guarantees for proximal quasi-Newton methods. The efficiency of the new …
This paper presents a multilevel FISTA algorithm, based on the use of the Moreau envelope to build the correction brought by the coarse models, which is easy to compute when the explicit form of the proximal operator of the considered functions is known. This approach is supported by strong theoretical guarantees: we prove both the rate of convergence and the convergence of the iterates to a minimum in the convex case, an important result for ill-posed problems. We evaluate our approach on image restoration problems and we show that it outperforms classical FISTA for large-scale images.
In this paper we present the solver DuQuad specialized for solving general convex quadratic problems arising in many engineering applications. When it is difficult to project on the primal feasible set, we use the (augmented) Lagrangian relaxation to handle the complicated constraints and then, we apply dual first order algorithms based on inexact dual gradient information for solving the corresponding dual problem. The iteration complexity analysis is based on two types of approximate primal solutions: the primal last iterate and an average of primal iterates. We provide computational complex…
Successive quadratic approximations (SQA) are numerically efficient for minimizing the sum of a smooth function and a convex function. The iteration complexity of inexact SQA methods has been analyzed recently. In this paper, we present an algorithmic framework of inexact SQA methods with four types of line searches, and analyze its global complexity under milder assumptions. First, we show its well-definedness and some decreasing properties. Second, under the quadratic growth condition and a uniform positive lower bound condition on stepsizes, we show that the function value sequence and the …
We consider the proximal-gradient method for minimizing an objective function that is the sum of a smooth function and a non-smooth convex function. A feature that distinguishes our work from most in the literature is that we assume that the associated proximal operator does not admit a closed-form solution. To address this challenge, we study two adaptive and implementable termination conditions that dictate how accurately the proximal-gradient subproblem is solved. We prove that the number of iterations required for the inexact proximal-gradient method to reach a $τ> 0$ approximate first-ord…
An inexact accelerated stochastic Alternating Direction Method of Multipliers (AS-ADMM) scheme is developed for solving structured separable convex optimization problems with linear constraints. The objective function is the sum of a possibly nonsmooth convex function and a smooth function which is an average of many component convex functions. Problems having this structure often arise in machine learning and data mining applications. AS-ADMM combines the ideas of both ADMM and the stochastic gradient methods using variance reduction techniques. One of the ADMM subproblems employs a lineariza…
This paper shows how, in a quasi metric space, an inexact proximal algorithm with a generalized perturbation term appears to be a nice tool for Behavioral Sciences (Psychology, Economics, Management, Game theory,...). More precisely, the new perturbation term represents an index of resistance to change, defined as a "curved enough" function of the quasi distance between two successive iterates. Using this behavioral point of view, the present paper shows how such a generalized inexact proximal algorithm can modelize the formation of habits and routines in a striking way. This idea comes from a…
University of Colorado Boulder（科罗拉多大学博尔德分校）；IBM Research Ireland（IBM爱尔兰研究院）
This paper considers an online proximal-gradient method to track the minimizers of a composite convex function that may continuously evolve over time. The online proximal-gradient method is inexact, in the sense that: (i) it relies on an approximate first-order information of the smooth component of the cost; and, (ii) the proximal operator (with respect to the non-smooth term) may be computed only up to a certain precision. Under suitable assumptions, convergence of the error iterates is established for strongly convex cost functions. On the other hand, the dynamic regret is investigated when…
We consider optimization algorithms that successively minimize simple Taylor-like models of the objective function. Methods of Gauss-Newton type for minimizing the composition of a convex function and a smooth map are common examples. Our main result is an explicit relationship between the step-size of any such algorithm and the slope of the function at a nearby point. Consequently, we (1) show that the step-sizes can be reliably used to terminate the algorithm, (2) prove that as long as the step-sizes tend to zero, every limit point of the iterates is stationary, and (3) show that conditions,…
University of California, Los Angeles（加州大学洛杉矶分校）；University of Texas at Arlington（德克萨斯大学阿灵顿分校）；Sun Yat-sen University（中山大学）
Robust principal component analysis (RPCA) is a widely used tool for dimension reduction. In this work, we propose a novel non-convex algorithm, coined Iterated Robust CUR (IRCUR), for solving RPCA problems, which dramatically improves the computational efficiency in comparison with the existing algorithms. IRCUR achieves this acceleration by employing CUR decomposition when updating the low rank component, which allows us to obtain an accurate low rank approximation via only three small submatrices. Consequently, IRCUR is able to process only the small submatrices and avoid expensive computin…
This paper considers the problem of distributed optimization over time-varying graphs. For the case of undirected graphs, we introduce a distributed algorithm, referred to as DIGing, based on a combination of a distributed inexact gradient method and a gradient tracking technique. The DIGing algorithm uses doubly stochastic mixing matrices and employs fixed step-sizes and, yet, drives all the agents' iterates to a global and consensual minimizer. When the graphs are directed, in which case the implementation of doubly stochastic mixing matrices is unrealistic, we construct an algorithm that in…
Derivatives of Stochastic Gradient Descent in parametric optimization
Université Paul Sabatier（保罗·萨巴蒂埃大学）；Toulouse School of Economics（图卢兹经济学院）；Université Toulouse Capitole（图卢兹第一大学）；CNRS（法国国家科学研究中心）；Laboratoire J. A. Dieudonné（迪厄多内实验室）
Samsung AI Center（三星人工智能中心）；King Abdullah University of Science and Technology（阿卜杜拉国王科技大学）
In this work, we consider the problem of minimizing the sum of Moreau envelopes of given functions, which has previously appeared in the context of meta-learning and personalized federated learning. In contrast to the existing theory that requires running subsolvers until a certain precision is reached, we only assume that a finite number of gradient steps is taken at each iteration. As a special case, our theory allows us to show the convergence of First-Order Model-Agnostic Meta-Learning (FO-MAML) to the vicinity of a solution of Moreau objective. We also study a more general family of first…
