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Semi-Stochastic Gradient Descent Methods Jakub Konečný (joint work with Peter Richtárik) University of Edinburgh

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Introduction

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Large scale problem setting Problems are often structured Frequently arising in machine learning Structure – sum of functions is BIG

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Examples Linear regression (least squares) Logistic regression (classification)

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Assumptions Lipschitz continuity of derivative of Strong convexity of

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Gradient Descent (GD) Update rule Fast convergence rate Alternatively, for accuracy we need iterations Complexity of single iteration – (measured in gradient evaluations)

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Stochastic Gradient Descent (SGD) Update rule Why it works Slow convergence Complexity of single iteration – (measured in gradient evaluations) a step-size parameter

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Goal GD SGD Fast convergence gradient evaluations in each iteration Slow convergence Complexity of iteration independent of Combine in a single algorithm

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Semi-Stochastic Gradient Descent S2GD

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Intuition The gradient does not change drastically We could reuse the information from “old” gradient

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Modifying “old” gradient Imagine someone gives us a “good” point and Gradient at point, near, can be expressed as Approximation of the gradient Already computed gradientGradient change We can try to estimate

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The S2GD Algorithm Simplification; size of the inner loop is random, following a geometric rule

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Theorem

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Convergence rate How to set the parameters ? Can be made arbitrarily small, by decreasing For any fixed, can be made arbitrarily small by increasing

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Setting the parameters The accuracy is achieved by setting Total complexity (in gradient evaluations) # of epochs full gradient evaluation cheap iterations # of epochs stepsize # of iterations Fix target accuracy

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Complexity S2GD complexity GD complexity iterations complexity of a single iteration Total

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Related Methods SAG – Stochastic Average Gradient (Mark Schmidt, Nicolas Le Roux, Francis Bach, 2013) Refresh single stochastic gradient in each iteration Need to store gradients. Similar convergence rate Cumbersome analysis MISO - Minimization by Incremental Surrogate Optimization (Julien Mairal, 2014) Similar to SAG, slightly worse performance Elegant analysis

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Related Methods SVRG – Stochastic Variance Reduced Gradient (Rie Johnson, Tong Zhang, 2013) Arises as a special case in S2GD Prox-SVRG (Tong Zhang, Lin Xiao, 2014) Extended to proximal setting EMGD – Epoch Mixed Gradient Descent (Lijun Zhang, Mehrdad Mahdavi, Rong Jin, 2013) Handles simple constraints, Worse convergence rate

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Experiment Example problem, with

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