1,008 research outputs found
Greedy Algorithms for Cone Constrained Optimization with Convergence Guarantees
Greedy optimization methods such as Matching Pursuit (MP) and Frank-Wolfe
(FW) algorithms regained popularity in recent years due to their simplicity,
effectiveness and theoretical guarantees. MP and FW address optimization over
the linear span and the convex hull of a set of atoms, respectively. In this
paper, we consider the intermediate case of optimization over the convex cone,
parametrized as the conic hull of a generic atom set, leading to the first
principled definitions of non-negative MP algorithms for which we give explicit
convergence rates and demonstrate excellent empirical performance. In
particular, we derive sublinear () convergence on general
smooth and convex objectives, and linear convergence () on
strongly convex objectives, in both cases for general sets of atoms.
Furthermore, we establish a clear correspondence of our algorithms to known
algorithms from the MP and FW literature. Our novel algorithms and analyses
target general atom sets and general objective functions, and hence are
directly applicable to a large variety of learning settings.Comment: NIPS 201
Recovery and convergence rate of the Frank-Wolfe Algorithm for the m-EXACT-SPARSE Problem
We study the properties of the Frank-Wolfe algorithm to solve the
m-EXACT-SPARSE reconstruction problem, where a signal y must be expressed as a
sparse linear combination of a predefined set of atoms, called dictionary. We
prove that when the signal is sparse enough with respect to the coherence of
the dictionary, then the iterative process implemented by the Frank-Wolfe
algorithm only recruits atoms from the support of the signal, that is the
smallest set of atoms from the dictionary that allows for a perfect
reconstruction of y. We also prove that under this same condition, there exists
an iteration beyond which the algorithm converges exponentially
On Matching Pursuit and Coordinate Descent
Two popular examples of first-order optimization methods over linear spaces are coordinate descent and matching pursuit algorithms, with their randomized variants. While the former targets the optimization by moving along coordinates, the latter considers a generalized notion of directions. Exploiting the connection between the two algorithms, we present a unified analysis of both, providing affine invariant sublinear O(1/t) rates on smooth objectives and linear convergence on strongly convex objectives. As a byproduct of our affine invariant analysis of matching pursuit, our rates for steepest coordinate descent are the tightest known. Furthermore, we show the first accelerated convergence rate O(1/t^2) for matching pursuit and steepest coordinate descent on convex objectives
Linearly Convergent Frank-Wolfe with Backtracking Line-Search
Structured constraints in Machine Learning have recently brought the
Frank-Wolfe (FW) family of algorithms back in the spotlight. While the
classical FW algorithm has poor local convergence properties, the Away-steps
and Pairwise FW variants have emerged as improved variants with faster
convergence. However, these improved variants suffer from two practical
limitations: they require at each iteration to solve a 1-dimensional
minimization problem to set the step-size and also require the Frank-Wolfe
linear subproblems to be solved exactly. In this paper, we propose variants of
Away-steps and Pairwise FW that lift both restrictions simultaneously. The
proposed methods set the step-size based on a sufficient decrease condition,
and do not require prior knowledge of the objective. Furthermore, they inherit
all the favorable convergence properties of the exact line-search version,
including linear convergence for strongly convex functions over polytopes.
Benchmarks on different machine learning problems illustrate large performance
gains of the proposed variants
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