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Register a new userIncremental Majorization-Minimization Optimization with Application to Large-Scale Machine Learning
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Added by
Julien Mairal
Publication date
2014
fields
Informatics Engineering
and research's language is
English
Authors
Julien Mairal
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Majorization-minimization algorithms consist of successively minimizing a sequence of upper bounds of the objective function. These upper bounds are tight at the current estimate, and each iteration monotonically drives the objective function downhill. Such a simple principle is widely applicable and has been very popular in various scientific fields, especially in signal processing and statistics. In this paper, we propose an incremental majorization-minimization scheme for minimizing a large sum of continuous functions, a problem of utmost importance in machine learning. We present convergence guarantees for non-convex and convex optimization when the upper bounds approximate the objective up to a smooth error; we call such upper bounds first-order surrogate functions. More precisely, we study asymptotic stationary point guarantees for non-convex problems, and for convex ones, we provide convergence rates for the expected objective function value. We apply our scheme to composite optimization and obtain a new incremental proximal gradient algorithm with linear convergence rate for strongly convex functions. In our experiments, we show that our method is competitive with the state of the art for solving machine learning problems such as logistic regression when the number of training samples is large enough, and we demonstrate its usefulness for sparse estimation with non-convex penalties.
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Majorization-minimization algorithms consist of iteratively minimizing a majorizing surrogate of an objective function. Because of its simplicity and its wide applicability, this principle has been very popular in statistics and in signal processing. In this paper, we intend to make this principle scalable. We introduce a stochastic majorization-minimization scheme which is able to deal with large-scale or possibly infinite data sets. When applied to convex optimization problems under suitable assumptions, we show that it achieves an expected convergence rate of $O(1/sqrt{n})$ after $n$ iterations, and of $O(1/n)$ for strongly convex functions. Equally important, our scheme almost surely converges to stationary points for a large class of non-convex problems. We develop several efficient algorithms based on our framework. First, we propose a new stochastic proximal gradient method, which experimentally matches state-of-the-art solvers for large-scale $ell_1$-logistic regression. Second, we develop an online DC programming algorithm for non-convex sparse estimation. Finally, we demonstrate the effectiveness of our approach for solving large-scale structured matrix factorization problems.
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.
We propose and analyze algorithms for distributionally robust optimization of convex losses with conditional value at risk (CVaR) and $chi^2$ divergence uncertainty sets. We prove that our algorithms require a number of gradient evaluations independent of training set size and number of parameters, making them suitable for large-scale applications. For $chi^2$ uncertainty sets these are the first such guarantees in the literature, and for CVaR our guarantees scale linearly in the uncertainty level rather than quadratically as in previous work. We also provide lower bounds proving the worst-case optimality of our algorithms for CVaR and a penalized version of the $chi^2$ problem. Our primary technical contributions are novel bounds on the bias of batch robust risk estimation and the variance of a multilevel Monte Carlo gradient estimator due to [Blanchet & Glynn, 2015]. Experiments on MNIST and ImageNet confirm the theoretical scaling of our algorithms, which are 9--36 times more efficient than full-batch methods.
Quantum computers promise to enhance machine learning for practical applications. Quantum machine learning for real-world data has to handle extensive amounts of high-dimensional data. However, conventional methods for measuring quantum kernels are impractical for large datasets as they scale with the square of the dataset size. Here, we measure quantum kernels using randomized measurements to gain a quadratic speedup in computation time and quickly process large datasets. Further, we efficiently encode high-dimensional data into quantum computers with the number of features scaling linearly with the circuit depth. The encoding is characterized by the quantum Fisher information metric and is related to the radial basis function kernel. We demonstrate the advantages of our methods by classifying images with the IBM quantum computer. To achieve further speedups we distribute the quantum computational tasks between different quantum computers. Our approach is exceptionally robust to noise via a complementary error mitigation scheme. Using currently available quantum computers, the MNIST database can be processed within 220 hours instead of 10 years which opens up industrial applications of quantum machine learning.
In this paper, we consider a class of nonsmooth nonconvex optimization problems whose objective is the sum of a block relative smooth function and a proper and lower semicontinuous block separable function. Although the analysis of block proximal gradient (BPG) methods for the class of block $L$-smooth functions have been successfully extended to Bregman BPG methods that deal with the class of block relative smooth functions, accelerated Bregman BPG methods are scarce and challenging to design. Taking our inspiration from Nesterov-type acceleration and the majorization-minimization scheme, we propose a block alternating Bregman Majorization-Minimization framework with Extrapolation (BMME). We prove subsequential convergence of BMME to a first-order stationary point under mild assumptions, and study its global convergence under stronger conditions. We illustrate the effectiveness of BMME on the penalized orthogonal nonnegative matrix factorization problem.
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