In stochastic optimization problems using noisy zeroth-order (ZO) oracles only, the randomized counterpart of the Kiefer-Wolfowitz-type method is widely used to estimate the gradient. Existing algorithms generate randomized perturbation from a zero-mean and unit-covariance distribution. In contrast, this work considers the generalization where the perturbations have a possibly non-identity covariance constructed from the history of the ZO queries. We propose to feed the second-order approximation into the covariance matrix of the random perturbation, so it is dubbed as Hessian-aided random perturbation (HARP). HARP collects four zeroth-order queries per iteration to form approximations for both the gradient and the Hessian. We show the convergence (in an almost surely sense) and derive the convergence rate for HARP under mild assumptions. We demonstrate, with theoretical guarantees and numerical experiments, that HARP is less sensitive to ill-conditioning and more query-efficient than other gradient approximation schemes whose random perturbation has an identity covariance.
When the available information is noisy zeroth-order (ZO) oracle, stochastic approximation methods are popular for estimating the root of the multivariate gradient equation. Inspired by the Stein's identity, this work establishes a novel Hessian approximation scheme. We compare it alongside with second-order simultaneous perturbation stochastic approximation (2SPSA). On the basis of the almost sure convergence and the same convergence rate, 2SPSA requires four ZO queries, while ours requires three ZO queries. Moreover, 2SPSA requires two statistically independent perturbations and two differen…
The class of algorithms called Hessian Estimation Evolution Strategies (HE-ESs) update the covariance matrix of their sampling distribution by directly estimating the curvature of the objective function. The approach is practically efficient, as attested by respectable performance on the BBOB testbed, even on rather irregular functions. In this paper we formally prove two strong guarantees for the (1+4)-HE-ES, a minimal elitist member of the family: stability of the covariance matrix update, and as a consequence, linear convergence on all convex quadratic problems at a rate that is independent…
In unconstrained maximum a posteriori (MAP) and maximum likelihood estimation, the inverse of minus the merit-function Hessian matrix is an approximation of the estimate covariance matrix. In the Bayesian context of MAP estimation, it is the covariance of a normal approximation of the posterior around the mode; while in maximum likelihood estimation, it an approximation of the inverse Fisher information matrix, to which the covariance of efficient estimators converge. These measures are routinely used in system identification to evaluate the estimate uncertainties and diagnose problems such as…
Zeroth-order methods have become important tools for solving problems where we have access only to function evaluations. However, the zeroth-order methods only using gradient approximations are $n$ times slower than classical first-order methods for solving n-dimensional problems. To accelerate the convergence rate, this paper proposes the zeroth order randomized subspace Newton (ZO-RSN) method, which estimates projections of the gradient and Hessian by random sketching and finite differences. This allows us to compute the Newton step in a lower dimensional subspace, with small computational c…
A novel gradient stepsize is derived at the motivation of equipping the Barzilai-Borwein (BB) method with two dimensional quadratic termination property. A remarkable feature of the novel stepsize is that its computation only depends on the BB stepsizes in previous iterations and does not require any exact line search or the Hessian, and hence it can easily be extended for nonlinear optimization. By adaptively taking long BB steps and some short steps associated with the new stepsize, we develop an efficient gradient method for quadratic optimization and general unconstrained optimization and …
We present an extensible software framework, hIPPYlib, for solution of large-scale deterministic and Bayesian inverse problems governed by partial differential equations (PDEs) with infinite-dimensional parameter fields (which are high-dimensional after discretization). hIPPYlib overcomes the prohibitive nature of Bayesian inversion for this class of problems by implementing state-of-the-art scalable algorithms for PDE-based inverse problems that exploit the structure of the underlying operators, notably the Hessian of the log-posterior. The key property of the algorithms implemented in hIPPYl…
The limited memory BFGS method (L-BFGS) of Liu and Nocedal (1989) is often considered to be the method of choice for continuous optimization when first- and/or second- order information is available. However, the use of L-BFGS can be complicated in a black-box scenario where gradient information is not available and therefore should be numerically estimated. The accuracy of this estimation, obtained by finite difference methods, is often problem-dependent that may lead to premature convergence of the algorithm. In this paper, we demonstrate an alternative to L-BFGS, the limited memory Covarian…
University of Freiburg（弗莱堡大学）；Ruhr-Universität Bochum（波鸿鲁尔大学）；Vorarlberg University of Applied Sciences（福拉尔贝格应用科学大学）
The Covariance Matrix Adaptation Evolution Strategy (CMA-ES) is a popular method to deal with nonconvex and/or stochastic optimization problems when the gradient information is not available. Being based on the CMA-ES, the recently proposed Matrix Adaptation Evolution Strategy (MA-ES) provides a rather surprising result that the covariance matrix and all associated operations (e.g., potentially unstable eigendecomposition) can be replaced in the CMA-ES by a updated transformation matrix without any loss of performance. In order to further simplify MA-ES and reduce its $\mathcal{O}\big(n^2\big)…
University of Texas at Austin（德克萨斯大学奥斯汀分校）；University of California, Santa Cruz（加州大学圣克鲁兹分校）
In this work, we analyze the properties of the solution to the covariance steering problem for discrete time Gaussian linear systems with a squared Wasserstein distance terminal cost. In our previous work, we have shown that by utilizing the state feedback control policy parametrization, this stochastic optimal control problem can be associated with a difference of convex functions program. Here, we revisit the same covariance control problem but this time we focus on the analysis of the problem. Specifically, we establish the existence of solutions to the optimization problem and derive the f…
In this paper we investigate the convergence properties of a variant of the Covariance Matrix Adaptation Evolution Strategy (CMA-ES). Our study is based on the recent theoretical foundation that the pure rank-mu update CMA-ES performs the natural gradient descent on the parameter space of Gaussian distributions. We derive a novel variant of the natural gradient method where the parameters of the Gaussian distribution are updated along the natural gradient to improve a newly defined function on the parameter space. We study this algorithm on composites of a monotone function with a convex quadr…
Czech Technical University（捷克技术大学）；Charles University in Prague（布拉格查理大学）；Czech Academy of Sciences（捷克科学院）
Surrogate modeling has become a valuable technique for black-box optimization tasks with expensive evaluation of the objective function. In this paper, we investigate the relationship between the predictive accuracy of surrogate models and features of the black-box function landscape. We also study properties of features for landscape analysis in the context of different transformations and ways of selecting the input data. We perform the landscape analysis of a large set of data generated using runs of a surrogate-assisted version of the Covariance Matrix Adaptation Evolution Strategy on the …
We consider stochastic zeroth-order optimization over Riemannian submanifolds embedded in Euclidean space, where the task is to solve Riemannian optimization problem with only noisy objective function evaluations. Towards this, our main contribution is to propose estimators of the Riemannian gradient and Hessian from noisy objective function evaluations, based on a Riemannian version of the Gaussian smoothing technique. The proposed estimators overcome the difficulty of the non-linearity of the manifold constraint and the issues that arise in using Euclidean Gaussian smoothing techniques when …
Bayesian inference provides a systematic framework for integration of data with mathematical models to quantify the uncertainty in the solution of the inverse problem. However, the solution of Bayesian inverse problems governed by complex forward models described by partial differential equations (PDEs) remains prohibitive with black-box Markov chain Monte Carlo (MCMC) methods. We present hIPPYlib-MUQ, an extensible and scalable software framework that contains implementations of state-of-the art algorithms aimed to overcome the challenges of high-dimensional, PDE-constrained Bayesian inverse …
Department of Computer Science, ETH Zurich（苏黎世联邦理工学院计算机科学系）；Max Planck Institute for Intelligent Systems（智能系统马克斯·普朗克研究所）；Amazon（亚马逊公司）；Paul G. Allen School of Computer Science and Engineering, University of Washington（华盛顿大学保罗·G·阿伦计算机科学与工程学校）
