Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userOn Stochastic Moving-Average Estimators for Non-Convex Optimization
301
0
0.0
(
0
)
Added by
Zhishuai Guo
Publication date
2021
fields
Informatics Engineering
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
In this paper, we demonstrate the power of a widely used stochastic estimator based on moving average (SEMA) on a range of stochastic non-convex optimization problems, which only requires {bf a general unbiased stochastic oracle}. We analyze various stochastic methods (existing or newly proposed) based on the {bf variance recursion property} of SEMA for three families of non-convex optimization, namely standard stochastic non-convex minimization, stochastic non-convex strongly-concave min-max optimization, and stochastic bilevel optimization. Our contributions include: (i) for standard stochastic non-convex minimization, we present a simple and intuitive proof of convergence for a family Adam-style methods (including Adam) with an increasing or large momentum parameter for the first-order moment, which gives an alternative yet more natural way to guarantee Adam converge; (ii) for stochastic non-convex strongly-concave min-max optimization, we present a single-loop stochastic gradient descent ascent method based on the moving average estimators and establish its oracle complexity of $O(1/epsilon^4)$ without using a large mini-batch size, addressing a gap in the literature; (iii) for stochastic bilevel optimization, we present a single-loop stochastic method based on the moving average estimators and establish its oracle complexity of $widetilde O(1/epsilon^4)$ without computing the inverse or SVD of the Hessian matrix, improving state-of-the-art results. For all these problems, we also establish a variance diminishing result for the used stochastic gradient estimators.
rate research
Read More
We lower bound the complexity of finding $epsilon$-stationary points (with gradient norm at most $epsilon$) using stochastic first-order methods. In a well-studied model where algorithms access smooth, potentially non-convex functions through queries to an unbiased stochastic gradient oracle with bounded variance, we prove that (in the worst case) any algorithm requires at least $epsilon^{-4}$ queries to find an $epsilon$ stationary point. The lower bound is tight, and establishes that stochastic gradient descent is minimax optimal in this model. In a more restrictive model where the noisy gradient estimates satisfy a mean-squared smoothness property, we prove a lower bound of $epsilon^{-3}$ queries, establishing the optimality of recently proposed variance reduction techniques.
In this paper, we study decentralized online stochastic non-convex optimization over a network of nodes. Integrating a technique called gradient tracking in decentralized stochastic gradient descent, we show that the resulting algorithm, GT-DSGD, enjoys certain desirable characteristics towards minimizing a sum of smooth non-convex functions. In particular, for general smooth non-convex functions, we establish non-asymptotic characterizations of GT-DSGD and derive the conditions under which it achieves network-independent performances that match the centralized minibatch SGD. In contrast, the existing results suggest that GT-DSGD is always network-dependent and is therefore strictly worse than the centralized minibatch SGD. When the global non-convex function additionally satisfies the Polyak-Lojasiewics (PL) condition, we establish the linear convergence of GT-DSGD up to a steady-state error with appropriate constant step-sizes. Moreover, under stochastic approximation step-sizes, we establish, for the first time, the optimal global sublinear convergence rate on almost every sample path, in addition to the asymptotically optimal sublinear rate in expectation. Since strongly convex functions are a special case of the functions satisfying the PL condition, our results are not only immediately applicable but also improve the currently known best convergence rates and their dependence on problem parameters.
In this paper we propose several adaptive gradient methods for stochastic optimization. Unlike AdaGrad-type of methods, our algorithms are based on Armijo-type line search and they simultaneously adapt to the unknown Lipschitz constant of the gradient and variance of the stochastic approximation for the gradient. We consider an accelerated and non-accelerated gradient descent for convex problems and gradient descent for non-convex problems. In the experiments we demonstrate superiority of our methods to existing adaptive methods, e.g. AdaGrad and Adam.
The Frank-Wolfe method and its extensions are well-suited for delivering solutions with desirable structural properties, such as sparsity or low-rank structure. We introduce a new variant of the Frank-Wolfe method that combines Frank-Wolfe steps and steepest descent steps, as well as a novel modification of the Frank-Wolfe gap to measure convergence in the non-convex case. We further extend this method to incorporate in-face directions for preserving structured solutions as well as block coordinate steps, and we demonstrate computational guarantees in terms of the modified Frank-Wolfe gap for all of these variants. We are particularly motivated by the application of this methodology to the training of neural networks with sparse properties, and we apply our block coordinate method to the problem of $ell_1$ regularized neural network training. We present the results of several numerical experiments on both artificial and real datasets demonstrating significant improvements of our method in training sparse neural networks.
We propose a new majorization-minimization (MM) method for non-smooth and non-convex programs, which is general enough to include the existing MM methods. Besides the local majorization condition, we only require that the difference between the directional derivatives of the objective function and its surrogate function vanishes when the number of iterations approaches infinity, which is a very weak condition. So our method can use a surrogate function that directly approximates the non-smooth objective function. In comparison, all the existing MM methods construct the surrogate function by approximating the smooth component of the objective function. We apply our relaxed MM methods to the robust matrix factorization (RMF) problem with different regularizations, where our locally majorant algorithm shows advantages over the state-of-the-art approaches for RMF. This is the first algorithm for RMF ensuring, without extra assumptions, that any limit point of the iterates is a stationary point.
suggested questions
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
