Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userA Unified Analysis of First-Order Methods for Smooth Games via Integral Quadratic Constraints
194
0
0.0
(
0
)
Added by
Guodong Zhang
Publication date
2020
fields
Informatics Engineering
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
The theory of integral quadratic constraints (IQCs) allows the certification of exponential convergence of interconnected systems containing nonlinear or uncertain elements. In this work, we adapt the IQC theory to study first-order methods for smooth and strongly-monotone games and show how to design tailored quadratic constraints to get tight upper bounds of convergence rates. Using this framework, we recover the existing bound for the gradient method~(GD), derive sharper bounds for the proximal point method~(PPM) and optimistic gradient method~(OG), and provide emph{for the first time} a global convergence rate for the negative momentum method~(NM) with an iteration complexity $mathcal{O}(kappa^{1.5})$, which matches its known lower bound. In addition, for time-varying systems, we prove that the gradient method with optimal step size achieves the fastest provable worst-case convergence rate with quadratic Lyapunov functions. Finally, we further extend our analysis to stochastic games and study the impact of multiplicative noise on different algorithms. We show that it is impossible for an algorithm with one step of memory to achieve acceleration if it only queries the gradient once per batch (in contrast with the stochastic strongly-convex optimization setting, where such acceleration has been demonstrated). However, we exhibit an algorithm which achieves acceleration with two gradient queries per batch.
rate research
Read More
We present a unified convergence analysis for first order convex optimization methods using the concept of strong Lyapunov conditions. Combining this with suitable time scaling factors, we are able to handle both convex and strong convex cases, and establish contraction properties of Lyapunov functions for many existing ordinary differential equation models. Then we derive prevailing first order optimization algorithms, such as proximal gradient methods, heavy ball methods (also known as momentum methods), Nesterov accelerated gradient methods, and accelerated proximal gradient methods from numerical discretizations of corresponding dynamical systems. We also apply strong Lyapunov conditions to the discrete level and provide a more systematical analysis framework. Another contribution is a novel second order dynamical system called Hessian-driven Nesterov accelerated gradient flow which can be used to design and analyze accelerated first order methods for smooth and non-smooth convex optimizations.
In this paper, we provide a unified convergence analysis for a class of shuffling-type gradient methods for solving a well-known finite-sum minimization problem commonly used in machine learning. This algorithm covers various variants such as randomized reshuffling, single shuffling, and cyclic/incremental gradient schemes. We consider two different settings: strongly convex and non-convex problems. Our main contribution consists of new non-asymptotic and asymptotic convergence rates for a general class of shuffling-type gradient methods to solve both non-convex and strongly convex problems. While our rate in the non-convex problem is new (i.e. not known yet under standard assumptions), the rate on the strongly convex case matches (up to a constant) the best-known results. However, unlike existing works in this direction, we only use standard assumptions such as smoothness and strong convexity. Finally, we empirically illustrate the effect of learning rates via a non-convex logistic regression and neural network examples.
We provide improved convergence rates for various emph{non-smooth} optimization problems via higher-order accelerated methods. In the case of $ell_infty$ regression, we achieves an $O(epsilon^{-4/5})$ iteration complexity, breaking the $O(epsilon^{-1})$ barrier so far present for previous methods. We arrive at a similar rate for the problem of $ell_1$-SVM, going beyond what is attainable by first-order methods with prox-oracle access for non-smooth non-strongly convex problems. We further show how to achieve even faster rates by introducing higher-order regularization. Our results rely on recent advances in near-optimal accelerated methods for higher-order smooth convex optimization. In particular, we extend Nesterovs smoothing technique to show that the standard softmax approximation is not only smooth in the usual sense, but also emph{higher-order} smooth. With this observation in hand, we provide the first example of higher-order acceleration techniques yielding faster rates for emph{non-smooth} optimization, to the best of our knowledge.
We consider minimization of indefinite quadratics with either trust-region (norm) constraints or cubic regularization. Despite the nonconvexity of these problems we prove that, under mild assumptions, gradient descent converges to their global solutions, and give a non-asymptotic rate of convergence for the cubic variant. We also consider Krylov subspace solutions and establish sharp convergence guarantees to the solutions of both trust-region and cubic-regularized problems. Our rates mirror the behavior of these methods on convex quadratics and eigenvector problems, highlighting their scalability. When we use Krylov subspace solutions to approximate the cubic-regularized Newton step, our results recover the strongest known convergence guarantees to approximate second-order stationary points of general smooth nonconvex functions.
In recent years, the success of deep learning has inspired many researchers to study the optimization of general smooth non-convex functions. However, recent works have established pessimistic worst-case complexities for this class functions, which is in stark contrast with their superior performance in real-world applications (e.g. training deep neural networks). On the other hand, it is found that many popular non-convex optimization problems enjoy certain structured properties which bear some similarities to convexity. In this paper, we study the class of textit{quasar-convex functions} to close the gap between theory and practice. We study the convergence of first order methods in a variety of different settings and under different optimality criterions. We prove complexity upper bounds that are similar to standard results established for convex functions and much better that state-of-the-art convergence rates of non-convex functions. Overall, this paper suggests that textit{quasar-convexity} allows efficient optimization procedures, and we are looking forward to seeing more problems that demonstrate similar properties in practice.
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
