{"id":168900,"date":"2015-11-01T00:00:00","date_gmt":"2015-11-01T00:00:00","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/msr-research-item\/an-accelerated-randomized-proximal-coordinate-gradient-method-and-its-application-to-regularized-empirical-risk-minimization\/"},"modified":"2018-10-16T21:35:09","modified_gmt":"2018-10-17T04:35:09","slug":"an-accelerated-randomized-proximal-coordinate-gradient-method-and-its-application-to-regularized-empirical-risk-minimization","status":"publish","type":"msr-research-item","link":"https:\/\/www.microsoft.com\/en-us\/research\/publication\/an-accelerated-randomized-proximal-coordinate-gradient-method-and-its-application-to-regularized-empirical-risk-minimization\/","title":{"rendered":"An Accelerated Randomized Proximal Coordinate Gradient Method and its Application to Regularized Empirical Risk Minimization"},"content":{"rendered":"<div class=\"asset-content\">\n<p>We consider the problem of minimizing the sum of two convex functions: one is smooth and given by a gradient oracle, and the other is separable over blocks of coordinates and has a simple known structure over each block. We develop an accelerated randomized proximal coordinate gradient (APCG) method for minimizing such convex composite functions. For strongly convex functions, our method achieves faster linear convergence rates than existing randomized proximal coordinate gradient methods. Without strong convexity, our method enjoys accelerated sublinear convergence rates. We show how to apply the APCG method to solve the regularized empirical risk minimization (ERM) problem and devise efficient implementations that avoid full-dimensional vector operations. For ill-conditioned ERM problems, our method obtains improved convergence rates than the state-of-the-art stochastic dual coordinate ascent method.<\/p>\n<\/div>\n<p><!-- .asset-content --><\/p>\n","protected":false},"excerpt":{"rendered":"<p>We consider the problem of minimizing the sum of two convex functions: one is smooth and given by a gradient oracle, and the other is separable over blocks of coordinates and has a simple known structure over each block. We develop an accelerated randomized proximal coordinate gradient (APCG) method for minimizing such convex composite functions. 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Examples abound, such as training neural networks with stochastic gradient descent, segmenting images with submodular optimization, or efficiently searching a game tree with bandit algorithms. We aim to advance the mathematical foundations of both discrete and continuous optimization and to leverage these advances to develop new algorithms with a broad set of AI applications. 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