One of the most widely used methods for solving average cost MDP problems is the value iteration method. This method, however, is often computationally impractical and restricted in size of solvable MDP problems. We propose acceleration operators that improve the performance of the value iteration for average reward MDP models. These operators are based on two important properties of Markovian operator: contraction mapping and monotonicity. It is well known that the classical relative value iteration methods for average cost criteria MDP do not involve the max-norm contraction or monotonicity property. To overcome this difficulty we propose to combine acceleration operators with variants of value iteration for stochastic shortest path problems associated average reward problems.
We consider the application of the type-I Anderson acceleration to solving general non-smooth fixed-point problems. By interleaving with safe-guarding steps, and employing a Powell-type regularization and a re-start checking for strong linear independence of the updates, we propose the first globally convergent variant of Anderson acceleration assuming only that the fixed-point iteration is non-expansive. We show by extensive numerical experiments that many first order algorithms can be improved, especially in their terminal convergence, with the proposed algorithm. Our proposed method of acce…
Despite the broad use of fixed-point iterations throughout applied mathematics, the optimal convergence rate of general fixed-point problems with nonexpansive nonlinear operators has not been established. This work presents an acceleration mechanism for fixed-point iterations with nonexpansive operators, contractive operators, and nonexpansive operators satisfying a Hölder-type growth condition. We then provide matching complexity lower bounds to establish the exact optimality of the acceleration mechanisms in the nonexpansive and contractive setups. Finally, we provide experiments with CT ima…
Anderson acceleration (AA) as an efficient technique for speeding up the convergence of fixed-point iterations may be designed for accelerating an optimization method. We propose a novel optimization algorithm by adapting Anderson acceleration to the energy adaptive gradient method (AEGD) [arXiv:2010.05109]. The feasibility of our algorithm is examined in light of convergence results for AEGD, though it is not a fixed-point iteration. We also quantify the accelerated convergence rate of AA for gradient descent by a factor of the gain at each implementation of the Anderson mixing. Our experimen…
We study the general approach to accelerating the convergence of the most widely used solution method of Markov decision processes with the total expected discounted reward. Inspired by the monotone behavior of the contraction mappings in the feasible set of the linear programming problem equivalent to the MDP, we establish a class of operators that can be used in combination with a contraction mapping operator in the standard value iteration algorithm and its variants. We then propose two such operators, which can be easily implemented as part of the value iteration algorithm and its variants…
We propose generic acceleration schemes for a wide class of optimization and iterative schemes based on relaxation and inertia. In particular, we introduce methods that automatically tunes the acceleration coefficients online, and establish their convergence. This is made possible by considering the class of fixed-points iterations over averaged operators which encompass gradient methods, ADMM, primal dual algorithms, an so on.
We introduce in this paper an optimal first-order method that allows an easy and cheap evaluation of the local Lipschitz constant of the objective's gradient. This constant must ideally be chosen at every iteration as small as possible, while serving in an indispensable upper bound for the value of the objective function. In the previously existing variants of optimal first-order methods, this upper bound inequality was constructed from points computed during the current iteration. It was thus not possible to select the optimal value for this Lipschitz constant at the beginning of the iteratio…
Gaussian process policy iteration with additive Schwarz acceleration for forward and inverse HJB and mean field game problems
Department of Computing and Mathematical Sciences, California Institute of Technology, CA, USA（计算与数学科学系，加州理工学院，CA，美国）；Department of Mathematics and Risk Management Institute, National University of Singapore, Singapore（数学与风险管理研究所，新加坡国立大学，新加坡）
Primal-Dual Hybrid Gradient (PDHG) and Alternating Direction Method of Multipliers (ADMM) are two widely-used first-order optimization methods. They reduce a difficult problem to simple subproblems, so they are easy to implement and have many applications. As first-order methods, however, they are sensitive to problem conditions and can struggle to reach the desired accuracy. To improve their performance, researchers have proposed techniques such as diagonal preconditioning and inexact subproblems. This paper realizes additional speedup about one order of magnitude. Specifically, we choose non…
The alternating direction multiplier method (ADMM) is widely used in computer graphics for solving optimization problems that can be nonsmooth and nonconvex. It converges quickly to an approximate solution, but can take a long time to converge to a solution of high-accuracy. Previously, Anderson acceleration has been applied to ADMM, by treating it as a fixed-point iteration for the concatenation of the dual variables and a subset of the primal variables. In this paper, we note that the equivalence between ADMM and Douglas-Rachford splitting reveals that ADMM is in fact a fixed-point iteration…
Anderson acceleration (AA) is widely used for accelerating the convergence of nonlinear fixed-point methods $x_{k+1}=q(x_{k})$, $x_k \in \mathbb{R}^n$, but little is known about how to quantify the convergence acceleration provided by AA. As a roadway towards gaining more understanding of convergence acceleration by AA, we study AA($m$), i.e., Anderson acceleration with finite window size $m$, applied to the case of linear fixed-point iterations $x_{k+1}=M x_{k}+b$. We write AA($m$) as a Krylov method with polynomial residual update formulas, and derive recurrence relations for the AA($m$) pol…
This note provides upper bounds on the number of operations required to compute by value iterations a nearly optimal policy for an infinite-horizon discounted Markov decision process with a finite number of states and actions. For a given discount factor, magnitude of the reward function, and desired closeness to optimality, these upper bounds are strongly polynomial in the number of state-action pairs, and one of the provided upper bounds has the property that it is a non-decreasing function of the value of the discount factor.
Time-Inconsistent MDPs with Entropy Regularization: Equilibrium Existence and Policy Iteration
Zhongtai Securities Institute for Financial Studies, Shandong University（山东大学中泰证券金融研究院）；Shandong Province Key Laboratory of Financial Risk（山东省金融风险重点实验室）
A detailed comparison is made between four different iterative procedures: Picard, Ishikawa, Mann and Picard-Krasnoselskii, within the framework of the BLUES function method and the variational iteration method. The resulting modified methods are subsequently applied to a nonlinear reaction-diffusion-advection differential equation to generate approximations to the known exact solution. The differences between the BLUES function method and the variational iteration method are illustrated by studying the approximants and the error between the obtained approximants and the exact solution.
Markov decision processes (MDPs) are used to model stochastic systems in many applications. Several efficient algorithms to compute optimal policies have been studied in the literature, including value iteration (VI) and policy iteration. However, these do not scale well especially when the discount factor for the infinite horizon discounted reward, $λ$, gets close to $1$. In particular, the running time scales as $O \left( 1/(1-λ) \right)$ for these algorithms. In this paper, our goal is to design new algorithms that scale better than previous approaches when $λ$ approaches $1$. Our main cont…
We consider nonlinear convergence acceleration methods for fixed-point iteration $x_{k+1}=q(x_k)$, including Anderson acceleration (AA), nonlinear GMRES (NGMRES), and Nesterov-type acceleration (corresponding to AA with window size one). We focus on fixed-point methods that converge asymptotically linearly with convergence factor $ρ<1$ and that solve an underlying fully smooth and non-convex optimization problem. It is often observed that AA and NGMRES substantially improve the asymptotic convergence behavior of the fixed-point iteration, but this improvement has not been quantified theoretica…
King Abdullah University of Science and Technology（阿卜杜拉国王科技大学）
We propose an accelerated version of stochastic variance reduced coordinate descent -- ASVRCD. As other variance reduced coordinate descent methods such as SEGA or SVRCD, our method can deal with problems that include a non-separable and non-smooth regularizer, while accessing a random block of partial derivatives in each iteration only. However, ASVRCD incorporates Nesterov's momentum, which offers favorable iteration complexity guarantees over both SEGA and SVRCD. As a by-product of our theory, we show that a variant of Allen-Zhu (2017) is a specific case of ASVRCD, recovering the optimal or…
This paper describes and establishes the iteration-complexity of a doubly accelerated inexact proximal point (D-AIPP) method for solving the nonconvex composite minimization problem whose objective function is of the form $f+h$ where $f$ is a (possibly nonconvex) differentiable function whose gradient is Lipschitz continuous and $h$ is a closed convex function with bounded domain. D-AIPP performs two types of iterations, namely, inner and outer ones. Its outer iterations correspond to the ones of the accelerated inexact proximal point scheme. Its inner iterations are the ones performed by an a…
We study the asymptotic convergence of AA($m$), i.e., Anderson acceleration with window size $m$ for accelerating fixed-point methods $x_{k+1}=q(x_{k})$, $x_k \in R^n$. Convergence acceleration by AA($m$) has been widely observed but is not well understood. We consider the case where the fixed-point iteration function $q(x)$ is differentiable and the convergence of the fixed-point method itself is root-linear. We identify numerically several conspicuous properties of AA($m$) convergence: First, AA($m$) sequences $\{x_k\}$ converge root-linearly but the root-linear convergence factor depends st…
The state-of-art seismic imaging techniques treat inversion tasks such as FWI and LSRTM as PDE-constrained optimization problems. Due to the large-scale nature, gradient-based optimization algorithms are preferred in practice to update the model iteratively. Higher-order methods converge in fewer iterations but often require higher computational costs, more line search steps, and bigger memory storage. A balance among these aspects has to be considered. We propose using Anderson acceleration (AA), a popular strategy to speed up the convergence of fixed-point iterations, to accelerate the steep…
Nesterov's well-known scheme for accelerating gradient descent in convex optimization problems is adapted to accelerating stationary iterative solvers for linear systems. Compared with classical Krylov subspace acceleration methods, the proposed scheme requires more iterations, but it is trivial to implement and retains essentially the same computational cost as the unaccelerated method. An explicit formula for a fixed optimal parameter is derived in the case where the stationary iteration matrix has only real eigenvalues, based only on the smallest and largest eigenvalues. The fixed parameter…
University of Alberta（阿尔伯塔大学）；Harbin Institute of Technology（哈尔滨工业大学）；Huawei Technologies Ltd.（华为技术有限公司）
Anderson mixing has been heuristically applied to reinforcement learning (RL) algorithms for accelerating convergence and improving the sampling efficiency of deep RL. Despite its heuristic improvement of convergence, a rigorous mathematical justification for the benefits of Anderson mixing in RL has not yet been put forward. In this paper, we provide deeper insights into a class of acceleration schemes built on Anderson mixing that improve the convergence of deep RL algorithms. Our main results establish a connection between Anderson mixing and quasi-Newton methods and prove that Anderson mix…
CNRS（法国国家科学研究中心）；ENS（巴黎高等师范学院）；Inria Sierra（法国国家信息与自动化研究所Sierra项目组）；King Abdullah University of Science and Technology（阿卜杜拉国王科技大学）；CISPA Helmholtz Center for Information Security（CISPA亥姆霍兹信息安全中心）
We introduce ProxSkip -- a surprisingly simple and provably efficient method for minimizing the sum of a smooth ($f$) and an expensive nonsmooth proximable ($ψ$) function. The canonical approach to solving such problems is via the proximal gradient descent (ProxGD) algorithm, which is based on the evaluation of the gradient of $f$ and the prox operator of $ψ$ in each iteration. In this work we are specifically interested in the regime in which the evaluation of prox is costly relative to the evaluation of the gradient, which is the case in many applications. ProxSkip allows for the expensive p…
We propose a novel and efficient lifting approach for the optimal control of rigid-body systems with contacts to improve the convergence properties of Newton-type methods. To relax the high nonlinearity, we consider the state, acceleration, contact forces, and control input torques, as optimization variables and the inverse dynamics and acceleration constraints on the contact frames as equality constraints. We eliminate the update of the acceleration, contact forces, and their dual variables from the linear equation to be solved in each Newton-type iteration in an efficient manner. As a result…
Fixed-point iterations are at the heart of numerical computing and are often a computational bottleneck in real-time applications that typically need a fast solution of moderate accuracy. We present neural fixed-point acceleration which combines ideas from meta-learning and classical acceleration methods to automatically learn to accelerate fixed-point problems that are drawn from a distribution. We apply our framework to SCS, the state-of-the-art solver for convex cone programming, and design models and loss functions to overcome the challenges of learning over unrolled optimization and accel…
The alternating direction method of multipliers (ADMM) is a popular approach for solving optimization problems that are potentially non-smooth and with hard constraints. It has been applied to various computer graphics applications, including physical simulation, geometry processing, and image processing. However, ADMM can take a long time to converge to a solution of high accuracy. Moreover, many computer graphics tasks involve non-convex optimization, and there is often no convergence guarantee for ADMM on such problems since it was originally designed for convex optimization. In this paper,…
We show that accelerated gradient descent, averaged gradient descent and the heavy-ball method for non-strongly-convex problems may be reformulated as constant parameter second-order difference equation algorithms, where stability of the system is equivalent to convergence at rate O(1/n 2), where n is the number of iterations. We provide a detailed analysis of the eigenvalues of the corresponding linear dynamical system , showing various oscillatory and non-oscillatory behaviors, together with a sharp stability result with explicit constants. We also consider the situation where noisy gradient…
City University of Hong Kong（香港城市大学）；University of New Hampshire（新罕布什尔大学）；Imperial College London（帝国理工学院）
Robust Markov decision processes (MDPs) allow to compute reliable solutions for dynamic decision problems whose evolution is modeled by rewards and partially-known transition probabilities. Unfortunately, accounting for uncertainty in the transition probabilities significantly increases the computational complexity of solving robust MDPs, which severely limits their scalability. This paper describes new efficient algorithms for solving the common class of robust MDPs with s- and sa-rectangular ambiguity sets defined by weighted $L_1$ norms. We propose partial policy iteration, a new, efficient…
Recently, research on accelerated stochastic gradient descent methods (e.g., SVRG) has made exciting progress (e.g., linear convergence for strongly convex problems). However, the best-known methods (e.g., Katyusha) requires at least two auxiliary variables and two momentum parameters. In this paper, we propose a fast stochastic variance reduction gradient (FSVRG) method, in which we design a novel update rule with the Nesterov's momentum and incorporate the technique of growing epoch size. FSVRG has only one auxiliary variable and one momentum weight, and thus it is much simpler and has much …
