We consider the problem of efficiently computing the maximum likelihood estimator in Generalized Linear Models (GLMs) when the number of observations is much larger than the number of coefficients ( @math ). In this regime, optimization algorithms can immensely benefit from approximate second order information. We propose an alternative way of constructing the curvature information by formulating it as an estimation problem and applying a Stein-type lemma, which allows further improvements through sub-sampling and eigenvalue thresholding. Our algorithm enjoys fast convergence rates, resembling that of second order methods, with modest per-iteration cost. We provide its convergence analysis for the general case where the rows of the design matrix are samples from a sub-gaussian distribution. We show that the convergence has two phases, a quadratic phase followed by a linear phase. Finally, we empirically demonstrate that our algorithm achieves the highest performance compared to various algorithms on several datasets.
There are many composite variants of the aforementioned methods, that mostly combine two or more techniques. Well-known composite algorithms are the combinations of sub-sampling and Quasi-Newton @cite_6 @cite_26 , SGD and GD @cite_36 , NG and NM @cite_28 , NG and low-rank approximation @cite_39 , sub-sampling and eigenvalue thresholding @cite_21 .
@cite_26 @cite_36 @cite_28 @cite_21 @cite_6 @cite_39
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A general, inexact, efficient proximal quasi-Newton algorithm for composite optimization problems has been proposed by Scheinberg and Tang (Math Program 160:495–529, 2016) and a sublinear global convergence rate has been established. In this paper, we analyze the global convergence rate of this method, in the both exact and inexact settings, in the case when the objective function is strongly convex. We also investigate a practical variant of this method by establishing a simple stopping criterion for the subproblem optimization. Furthermore, we consider an accelerated variant, based on FISTA of Beck and Teboulle (SIAM 2:183–202, 2009), to the proximal quasi-Newton algorithm. (SIAM 22:1042–1064, 2012) considered a similar accelerated method, where the convergence rate analysis relies on very strong impractical assumptions on Hessian estimates. We present a modified analysis while relaxing these assumptions and perform a numerical comparison of the accelerated proximal quasi-Newton algorithm and the regular one. Our analysis and computational results show that acceleration may not bring any benefit in the quasi-Newton setting. We consider the problem of minimizing a sum of @math functions over a convex parameter set @math where @math . In this regime, algorithms which utilize sub-sampling techniques are known to be effective. In this paper, we use sub-sampling techniques together with low-rank approximation to design a new randomized batch algorithm which possesses comparable convergence rate to Newton's method, yet has much smaller per-iteration cost. The proposed algorithm is robust in terms of starting point and step size, and enjoys a composite convergence rate, namely, quadratic convergence at start and linear convergence when the iterate is close to the minimizer. We develop its theoretical analysis which also allows us to select near-optimal algorithm parameters. Our theoretical results can be used to obtain convergence rates of previously proposed sub-sampling based algorithms as well. We demonstrate how our results apply to well-known machine learning problems. Lastly, we evaluate the performance of our algorithm on several datasets under various scenarios. We introduce a framework for quasi-Newton forward--backward splitting algorithms (proximal quasi-Newton methods) with a metric induced by diagonal @math rank- @math symmetric positive definite matrices. This special type of metric allows for a highly efficient evaluation of the proximal mapping. The key to this efficiency is a general proximal calculus in the new metric. By using duality, formulas are derived that relate the proximal mapping in a rank- @math modified metric to the original metric. We also describe efficient implementations of the proximity calculation for a large class of functions; the implementations exploit the piece-wise linear nature of the dual problem. Then, we apply these results to acceleration of composite convex minimization problems, which leads to elegant quasi-Newton methods for which we prove convergence. The algorithm is tested on several numerical examples and compared to a comprehensive list of alternatives in the literature. Our quasi-Newton splitting algorithm with the prescribed metric compares favorably against state-of-the-art. The algorithm has extensive applications including signal processing, sparse recovery, machine learning and classification to name a few. We present a unified framework for the design and convergence analysis of a class of algorithms based on approximate solution of proximal point subproblems. Our development further enhances the constructive approximation approach of the recently proposed hybrid projection–proximal and extragradient–proximal methods. Specifically, we introduce an even more flexible error tolerance criterion, as well as provide a unified view of these two algorithms. Our general method possesses global convergence and local (super)linear rate of convergence under standard assumptions, while using a constructive approximation criterion suitable for a number of specific implementations. For example, we show that close to a regular solution of a monotone system of semismooth equations, two Newton iterations are sufficient to solve the proximal subproblem within the required error tolerance. Such systems of equations arise naturally when reformulating the nonlinear complementarity problem. *Research of the first author is suppo...