Since the late 1950's when quasi-Newton methods first appeared, they have become one of the most widely used and efficient algorithmic paradigms for unconstrained optimization. Despite their immense practical success, there is little theory that shows why these methods are so efficient. We provide a semi-local rate of convergence for the randomized BFGS method which can be significantly better than that of gradient descent, finally giving theoretical evidence supporting the superior empirical performance of the method.
Non-asymptotic analysis of quasi-Newton methods have gained traction recently. In particular, several works have established a non-asymptotic superlinear rate of $\mathcal{O}((1/\sqrt{t})^t)$ for the (classic) BFGS method by exploiting the fact that its error of Newton direction approximation approaches zero. Moreover, a greedy variant of BFGS was recently proposed which accelerates its convergence by directly approximating the Hessian, instead of the Newton direction, and achieves a fast local quadratic convergence rate. Alas, the local quadratic convergence of Greedy-BFGS requires way more u…
Accelerated Stochastic Matrix Inversion: General Theory and Speeding up BFGS Rules for Faster Second-Order Optimization
Télécom ParisTech（巴黎电信学院）；King Abdullah University of Science and Technology（阿卜杜拉国王科技大学）；University of Edinburgh（爱丁堡大学）；Moscow Institute of Physics and Technology（莫斯科物理技术学院）；École polytechnique fédérale de Lausanne (EPFL)（洛桑联邦理工学院）
L-BFGS is the state-of-the-art optimization method for many large scale inverse problems. It has a small memory footprint and achieves superlinear convergence. The method approximates Hessian based on an initial approximation and an update rule that models current local curvature information. The initial approximation greatly affects the scaling of a search direction and the overall convergence of the method. We propose a novel, simple, and effective way to initialize the Hessian. Typically, the objective function is a sum of a data-fidelity term and a regularizer. Often, the Hessian of the da…
Quasi-Newton methods on Grassmannians and multilinear approximations of tensors
Linköping University（林雪平大学）；Institute for Computational Engineering and Sciences, The University of Texas at Austin（德克萨斯大学奥斯汀分校计算工程与科学研究所）；Department of Mathematics, University of California, Berkeley（加州大学伯克利分校数学系）
The standard L-BFGS method relies on gradient approximations that are not dominated by noise, so that search directions are descent directions, the line search is reliable, and quasi-Newton updating yields useful quadratic models of the objective function. All of this appears to call for a full batch approach, but since small batch sizes give rise to faster algorithms with better generalization properties, L-BFGS is currently not considered an algorithm of choice for large-scale machine learning applications. One need not, however, choose between the two extremes represented by the full batch …
Randomized Fast Subspace Descent (RFASD) Methods are developed and analyzed for smooth and non-constraint convex optimization problems. The efficiency of the method relies on a space decomposition which is stable in $A$-norm, and meanwhile, the condition number $κ_A$ measured in $A$-norm is small. At each iteration, the subspace is chosen randomly either uniformly or by a probability proportional to the local Lipschitz constants. Then in each chosen subspace, a preconditioned gradient descent method is applied. RFASD converges sublinearly for convex functions and linearly for strongly convex f…
We develop a practical approach to semidefinite programming (SDP) that includes the von Neumann entropy, or an appropriate variant, as a regularization term. In particular we solve the dual of the regularized program, demonstrating how a carefully chosen randomized trace estimator can be used to estimate dual gradients effectively. We also introduce specialized optimization approaches for common SDP, specifically SDP with diagonal constraint and the problem of the determining the spectral projector onto the span of extremal eigenvectors. We validate our approach on such problems with applicati…
The standard randomized sparse Kaczmarz (RSK) method is an algorithm to compute sparse solutions of linear systems of equations and uses sequential updates, and thus, does not take advantage of parallel computations. In this work, we introduce a parallel (mini batch) version of RSK based on averaging several Kaczmarz steps. Naturally, this method allows for parallelization and we show that it can also leverage large over-relaxation. We prove linear expected convergence and show that, given that parallel computations can be exploited, the method provably provides faster convergence than the sta…
Reinforcement Learning (RL) algorithms allow artificial agents to improve their action selections so as to increase rewarding experiences in their environments. Deep Reinforcement Learning algorithms require solving a nonconvex and nonlinear unconstrained optimization problem. Methods for solving the optimization problems in deep RL are restricted to the class of first-order algorithms, such as stochastic gradient descent (SGD). The major drawback of the SGD methods is that they have the undesirable effect of not escaping saddle points and their performance can be seriously obstructed by ill-c…
University of British Columbia（英属哥伦比亚大学）；Mila, Université de Montréal（蒙特利尔大学米拉研究所）；Element AI（元素人工智能公司）；CCAI Affiliate Chair (Amii)（加拿大先进人工智能研究所CCAI附属主席）
We consider stochastic second-order methods for minimizing smooth and strongly-convex functions under an interpolation condition satisfied by over-parameterized models. Under this condition, we show that the regularized subsampled Newton method (R-SSN) achieves global linear convergence with an adaptive step-size and a constant batch-size. By growing the batch size for both the subsampled gradient and Hessian, we show that R-SSN can converge at a quadratic rate in a local neighbourhood of the solution. We also show that R-SSN attains local linear convergence for the family of self-concordant f…
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.
Optimizing deformation energies over a mesh, in two or three dimensions, is a common and critical problem in physical simulation and geometry processing. We present three new improvements to the state of the art: a barrier-aware line-search filter that cures blocked descent steps due to element barrier terms and so enables rapid progress; an energy proxy model that adaptively blends the Sobolev (inverse-Laplacian-processed) gradient and L-BFGS descent to gain the advantages of both, while avoiding L-BFGS's current limitations in geometry optimization tasks; and a characteristic gradient norm p…
We propose an inexact variable-metric proximal point algorithm to accelerate gradient-based optimization algorithms. The proposed scheme, called QNing can be notably applied to incremental first-order methods such as the stochastic variance-reduced gradient descent algorithm (SVRG) and other randomized incremental optimization algorithms. QNing is also compatible with composite objectives, meaning that it has the ability to provide exactly sparse solutions when the objective involves a sparsity-inducing regularization. When combined with limited-memory BFGS rules, QNing is particularly effecti…
In Part I of this work, we have proposed a general framework of decentralized stochastic quasi-Newton methods, which converge linearly to the optimal solution under the assumption that the local Hessian inverse approximations have bounded positive eigenvalues. In Part II, we specify two fully decentralized stochastic quasi-Newton methods, damped regularized limited-memory DFP (Davidon-Fletcher-Powell) and damped limited-memory BFGS (Broyden-Fletcher-Goldfarb-Shanno), to locally construct such Hessian inverse approximations without extra sampling or communication. Both of the methods use a fixe…
The problem of minimizing a separable convex function under linearly coupled constraints arises from various application domains such as economic systems, distributed control, and network flow. The main challenge for solving this problem is that the size of data is very large, which makes usual gradient-based methods infeasible. Recently, Necoara, Nesterov, and Glineur [Journal of Optimization Theory and Applications, 173 (2017) 227-2254] proposed an efficient randomized coordinate descent method to solve this type of optimization problems and presented an appealing convergence analysis. In th…
In this paper we study stochastic quasi-Newton methods for nonconvex stochastic optimization, where we assume that only stochastic information of the gradients of the objective function is available via a stochastic first-order oracle (SFO). Firstly, we propose a general framework of stochastic quasi-Newton methods for solving nonconvex stochastic optimization. The proposed framework extends the classic quasi-Newton methods working in deterministic settings to stochastic settings, and we prove its almost sure convergence to stationary points. Secondly, we propose a general framework for a clas…
The Kaczmarz algorithm is a simple iterative scheme for solving consistent linear systems. At each step, the method projects the current iterate onto the solution space of a single constraint. Hence, it requires very low cost per iteration and storage, and it has a linear rate of convergence. Distributed implementations of Kaczmarz have become, in recent years, the de facto architectural choice for large-scale linear systems. Therefore, in this paper we develop a family of randomized block Kaczmarz algorithms that uses at each step a subset of the constraints and extrapolated stepsizes, and ca…
We study decentralized non-convex finite-sum minimization problems described over a network of nodes, where each node possesses a local batch of data samples. In this context, we analyze a single-timescale randomized incremental gradient method, called GT-SAGA. GT-SAGA is computationally efficient as it evaluates one component gradient per node per iteration and achieves provably fast and robust performance by leveraging node-level variance reduction and network-level gradient tracking. For general smooth non-convex problems, we show the almost sure and mean-squared convergence of GT-SAGA to a…
We provide an exact analysis of a class of randomized algorithms for solving overdetermined least-squares problems. We consider first-order methods, where the gradients are pre-conditioned by an approximation of the Hessian, based on a subspace embedding of the data matrix. This class of algorithms encompasses several randomized methods among the fastest solvers for least-squares problems. We focus on two classical embeddings, namely, Gaussian projections and subsampled randomized Hadamard transforms (SRHT). Our key technical innovation is the derivation of the limiting spectral density of SRH…
