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UBC Theses and Dissertations

Greed is good : greedy optimization methods for large-scale structured problems Nutini, Julie


This work looks at large-scale machine learning, with a particular focus on greedy methods. A recent trend caused by big datasets is to use optimization methods that have a cheap iteration cost. In this category are (block) coordinate descent and Kaczmarz methods, as the updates of these methods only rely on a reduced subspace of the problem at each iteration. Prior to our work, the literature cast greedy variations of these methods as computationally expensive with comparable convergence rates to randomized versions. In this dissertation, we show that greed is good. Specifically, we show that greedy coordinate descent and Kaczmarz methods have efficient implementations and can be faster than their randomized counterparts for certain common problem structures in machine learning. We show linear convergence for greedy (block) coordinate descent methods under a revived relaxation of strong convexity from 1963, which we call the Polyak-Lojasiewicz (PL) inequality. Of the proposed relaxations of strong convexity in the recent literature, we show that the PL inequality is the weakest condition that still ensures a global minimum. Further, we highlight the exploitable flexibility in block coordinate descent methods, not only in the different types of selection rules possible, but also in the types of updates we can use. We show that using second-order or exact updates with greedy block coordinate descent methods can lead to superlinear or finite convergence (respectively) for popular machine learning problems. Finally, we introduce the notion of “active-set complexity”, which we define as the number of iterations required before an algorithm is guaranteed to reach the optimal active manifold, and show explicit bounds for two common problem instances when using the proximal gradient or the proximal coordinate descent method.

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