1,954 research outputs found
Let's Make Block Coordinate Descent Go Fast: Faster Greedy Rules, Message-Passing, Active-Set Complexity, and Superlinear Convergence
Block coordinate descent (BCD) methods are widely-used for large-scale
numerical optimization because of their cheap iteration costs, low memory
requirements, amenability to parallelization, and ability to exploit problem
structure. Three main algorithmic choices influence the performance of BCD
methods: the block partitioning strategy, the block selection rule, and the
block update rule. In this paper we explore all three of these building blocks
and propose variations for each that can lead to significantly faster BCD
methods. We (i) propose new greedy block-selection strategies that guarantee
more progress per iteration than the Gauss-Southwell rule; (ii) explore
practical issues like how to implement the new rules when using "variable"
blocks; (iii) explore the use of message-passing to compute matrix or Newton
updates efficiently on huge blocks for problems with a sparse dependency
between variables; and (iv) consider optimal active manifold identification,
which leads to bounds on the "active set complexity" of BCD methods and leads
to superlinear convergence for certain problems with sparse solutions (and in
some cases finite termination at an optimal solution). We support all of our
findings with numerical results for the classic machine learning problems of
least squares, logistic regression, multi-class logistic regression, label
propagation, and L1-regularization
An Efficient Primal-Dual Prox Method for Non-Smooth Optimization
We study the non-smooth optimization problems in machine learning, where both
the loss function and the regularizer are non-smooth functions. Previous
studies on efficient empirical loss minimization assume either a smooth loss
function or a strongly convex regularizer, making them unsuitable for
non-smooth optimization. We develop a simple yet efficient method for a family
of non-smooth optimization problems where the dual form of the loss function is
bilinear in primal and dual variables. We cast a non-smooth optimization
problem into a minimax optimization problem, and develop a primal dual prox
method that solves the minimax optimization problem at a rate of
{assuming that the proximal step can be efficiently solved}, significantly
faster than a standard subgradient descent method that has an
convergence rate. Our empirical study verifies the efficiency of the proposed
method for various non-smooth optimization problems that arise ubiquitously in
machine learning by comparing it to the state-of-the-art first order methods
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