The framework of Integral Quadratic Constraints (IQC) introduced by (2014) reduces the computation of upper bounds on the convergence rate of several optimization algorithms to semi-definite programming (SDP). In particular, this technique was applied to Nesterov's accelerated method (NAM). For quadratic functions, this SDP was explicitly solved leading to a new bound on the convergence rate of NAM, and for arbitrary strongly convex functions it was shown numerically that IQC can improve bounds from Nesterov (2004). Unfortunately, an explicit analytic solution to the SDP was not provided. In this paper, we provide such an analytical solution, obtaining a new general and explicit upper bound on the convergence rate of NAM, which we further optimize over its parameters. To the best of our knowledge, this is the best, and explicit, upper bound on the convergence rate of NAM for strongly convex functions.
Several recent works have revisited NAM and computed bounds on its convergence rate based on different techniques. In addition to @cite_6 , the following works are relevant. @cite_14 views NAM as a linear coupling between GD and Mirror Descent, and, for @math , re-derives the previously known bound @math @cite_12 , with the choice @math and @math , which is different from Nesterov's bound. This rate is not of the type that we consider in this paper.
@cite_14 @cite_12 @cite_6
2075660001 2963487351 2769394111 2807821938
In this paper we analyze the randomized block-coordinate descent (RBCD) methods proposed in Nesterov (SIAM J Optim 22(2):341---362, 2012), Richtarik and Takaă? (Math Program 144(1---2):1---38, 2014) for minimizing the sum of a smooth convex function and a block-separable convex function, and derive improved bounds on their convergence rates. In particular, we extend Nesterov's technique developed in Nesterov (SIAM J Optim 22(2):341---362, 2012) for analyzing the RBCD method for minimizing a smooth convex function over a block-separable closed convex set to the aforementioned more general problem and obtain a sharper expected-value type of convergence rate than the one implied in Richtarik and Takaă? (Math Program 144(1---2):1---38, 2014). As a result, we also obtain a better high-probability type of iteration complexity. In addition, for unconstrained smooth convex minimization, we develop a new technique called randomized estimate sequence to analyze the accelerated RBCD method proposed by Nesterov (SIAM J Optim 22(2):341---362, 2012) and establish a sharper expected-value type of convergence rate than the one given in Nesterov (SIAM J Optim 22(2):341---362, 2012). Nesterov's accelerated gradient descent (AGD), an instance of the general family of momentum methods,'' provably achieves faster convergence rate than gradient descent (GD) in the convex setting. While these methods are widely used in modern nonconvex applications, including training of deep neural networks, whether they are provably superior to GD in the nonconvex setting remains open. This paper studies a simple variant of Nesterov's AGD, and shows that it escapes saddle points and finds a second-order stationary point in ( O (1 ^ 7 4 ) ) iterations, matching the best known convergence rate, which is faster than the ( O (1 ^ 2 ) ) iterations required by GD. To the best of our knowledge, this is the first direct acceleration (single-loop) algorithm that is provably faster than GD in general nonconvex setting---all previous nonconvex accelerated algorithms rely on more complex mechanisms such as nested loops and proximal terms. Our analysis is based on two key ideas: (1) the use of a simple Hamiltonian function, inspired by a continuous-time perspective, which AGD monotonically decreases on each step even for nonconvex functions, and (2) a novel framework called improve or localize, which is useful for tracking the long-term behavior of gradient-based optimization algorithms. We believe that these techniques may deepen our understanding of both acceleration algorithms and nonconvex optimization. Nesterov's accelerated gradient descent (AGD), an instance of the general family of "momentum methods", provably achieves faster convergence rate than gradient descent (GD) in the convex setting. However, whether these methods are superior to GD in the nonconvex setting remains open. This paper studies a simple variant of AGD, and shows that it escapes saddle points and finds a second-order stationary point in @math iterations, faster than the @math iterations required by GD. To the best of our knowledge, this is the first Hessian-free algorithm to find a second-order stationary point faster than GD, and also the first single-loop algorithm with a faster rate than GD even in the setting of finding a first-order stationary point. Our analysis is based on two key ideas: (1) the use of a simple Hamiltonian function, inspired by a continuous-time perspective, which AGD monotonically decreases per step even for nonconvex functions, and (2) a novel framework called improve or localize, which is useful for tracking the long-term behavior of gradient-based optimization algorithms. We believe that these techniques may deepen our understanding of both acceleration algorithms and nonconvex optimization. In this paper, we investigate the non-asymptotic stationary convergence behavior of Stochastic Mirror Descent (SMD) for nonconvex optimization. We focus on a general class of nonconvex nonsmooth stochastic optimization problems, in which the objective can be decomposed into a relatively weakly convex function (possibly non-Lipschitz) and a simple non-smooth convex regularizer. We prove that SMD, without the use of mini-batch, is guaranteed to converge to a stationary point in a convergence rate of @math . The efficiency estimate matches with existing results for stochastic subgradient method, but is evaluated under a stronger stationarity measure. Our convergence analysis applies to both the original SMD and its proximal version, as well as the deterministic variants, for solving relatively weakly convex problems.