Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userAcceleration Methods
62
0
0.0
(
0
)
Added by
Damien Scieur
Publication date
2021
fields
Informatics Engineering
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
This monograph covers some recent advances on a range of acceleration techniques frequently used in convex optimization. We first use quadratic optimization problems to introduce two key families of methods, momentum and nested optimization schemes, which coincide in the quadratic case to form the Chebyshev method whose complexity is analyzed using Chebyshev polynomials. We discuss momentum methods in detail, starting with the seminal work of Nesterov (1983) and structure convergence proofs using a few master templates, such as that of emph{optimized gradient methods} which have the key benefit of showing how momentum methods maximize convergence rates. We further cover proximal acceleration techniques, at the heart of the emph{Catalyst} and emph{Accelerated Hybrid Proximal Extragradient} frameworks, using similar algorithmic patterns. Common acceleration techniques directly rely on the knowledge of some regularity parameters of the problem at hand, and we conclude by discussing emph{restart} schemes, a set of simple techniques to reach nearly optimal convergence rates while adapting to unobserved regularity parameters.
rate research
Read More
Structured problems arise in many applications. To solve these problems, it is important to leverage the structure information. This paper focuses on convex problems with a finite-sum compositional structure. Finite-sum problems appear as the sample average approximation of a stochastic optimization problem and also arise in machine learning with a huge amount of training data. One popularly used numerical approach for finite-sum problems is the stochastic gradient method (SGM). However, the additional compositional structure prohibits easy access to unbiased stochastic approximation of the gradient, so directly applying the SGM to a finite-sum compositional optimization problem (COP) is often inefficient. We design new algorithms for solving strongly-convex and also convex two-level finite-sum COPs. Our design incorporates the Katyusha acceleration technique and adopts the mini-batch sampling from both outer-level and inner-level finite-sum. We first analyze the algorithm for strongly-convex finite-sum COPs. Similar to a few existing works, we obtain linear convergence rate in terms of the expected objective error, and from the convergence rate result, we then establish complexity results of the algorithm to produce an $varepsilon$-solution. Our complexity results have the same dependence on the number of component functions as existing works. However, due to the use of Katyusha acceleration, our results have better dependence on the condition number $kappa$ and improve to $kappa^{2.5}$ from the best-known $kappa^3$. Finally, we analyze the algorithm for convex finite-sum COPs, which uses as a subroutine the algorithm for strongly-convex finite-sum COPs. Again, we obtain better complexity results than existing works in terms of the dependence on $varepsilon$, improving to $varepsilon^{-2.5}$ from the best-known $varepsilon^{-3}$.
We study robustness properties of some iterative gradient-based methods for strongly convex functions, as well as for the larger class of functions with sector-bounded gradients, under a relative error model. Proofs of the corresponding convergence rates are based on frequency-domain criteria for the stability of nonlinear systems. Applications are given to inexa
First-order operator splitting methods are ubiquitous among many fields through science and engineering, such as inverse problems, signal/image processing, statistics, data science and machine learning, to name a few. In this paper, we study a geometric property of first-order methods when applying to solve non-smooth optimization problems. With the tool of partial smoothness, we design a framework to analyze the trajectory of the fixed-point sequence generated by first-order methods and show that locally, the fixed-point sequence settles onto a regular trajectory such as a straight line or a spiral. Based on this finding, we discuss the limitation of current widely used inertial acceleration technique, and propose a trajectory following adaptive acceleration algorithm. Global convergence is established for the proposed acceleration scheme based on the perturbation of fixed-point iteration. Locally, we first build connections between the acceleration scheme and the well-studied vector extrapolation technique in the field of numerical analysis, and then discuss local acceleration guarantees of the proposed acceleration scheme. Moreover, our result provides a geometric interpretation of these vector extrapolation techniques. Numerical experiments on various first-order methods are provided to demonstrate the advantage of the proposed adaptive acceleration scheme.
The optimized gradient method (OGM) provides a factor-$sqrt{2}$ speedup upon Nesterovs celebrated accelerated gradient method in the convex (but non-strongly convex) setup. However, this improved acceleration mechanism has not been well understood; prior analyses of OGM relied on a computer-assisted proof methodology, so the proofs were opaque for humans despite being verifiable and correct. In this work, we present a new analysis of OGM based on a Lyapunov function and linear coupling. These analyses are developed and presented without the assistance of computers and are understandable by humans. Furthermore, we generalize OGMs acceleration mechanism and obtain a factor-$sqrt{2}$ speedup in other setups: acceleration with a simpler rational stepsize, the strongly convex setup, and the mirror descent setup.
The aim of this paper is to discuss some advanced aspects of image reconstruction in single-pixel cameras, focusing in particular on detectors in the THz regime. We discuss the reconstruction problem from a computational imaging perspective and provide a comparison of the effects of several state-of-the art regularization techniques. Moreover, we focus on some advanced aspects arising in practice with THz cameras, which lead to nonlinear reconstruction problems: the calibration of the beam reminiscent of the Retinex problem in imaging and phase recovery problems. Finally we provide an outlook to future challenges in the area.
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
