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Register a new userConditional Gradient Algorithms for Rank-One Matrix Approximations with a Sparsity Constraint
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Added by
Ronny Luss
Publication date
2011
fields
Informatics Engineering
and research's language is
English
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The sparsity constrained rank-one matrix approximation problem is a difficult mathematical optimization problem which arises in a wide array of useful applications in engineering, machine learning and statistics, and the design of algorithms for this problem has attracted intensive research activities. We introduce an algorithmic framework, called ConGradU, that unifies a variety of seemingly different algorithms that have been derived from disparate approaches, and allows for deriving new schemes. Building on the old and well-known conditional gradient algorithm, ConGradU is a simplified version with unit step size and yields a generic algorithm which either is given by an analytic formula or requires a very low computational complexity. Mathematical properties are systematically developed and numerical experiments are given.
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This article derives lower bounds on the convergence rate of continuous-time gradient-based optimization algorithms. The algorithms are subjected to a time-normalization constraint that avoids a reparametrization of time in order to make the discussion of continuous-time convergence rates meaningful. We reduce the multi-dimensional problem to a single dimension, recover well-known lower bounds from the discrete-time setting, and provide insight into why these lower bounds occur. We present algorithms that achieve the proposed lower bounds, even when the function class under consideration includes certain nonconvex functions.
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The conditional gradient method (CGM) is widely used in large-scale sparse convex optimization, having a low per iteration computational cost for structured sparse regularizers and a greedy approach to collecting nonzeros. We explore the sparsity acquiring properties of a general penalized CGM (P-CGM) for convex regularizers and a reweighted penalized CGM (RP-CGM) for nonconvex regularizers, replacing the usual convex constraints with gauge-inspired penalties. This generalization does not increase the per-iteration complexity noticeably. Without assuming bounded iterates or using line search, we show $O(1/t)$ convergence of the gap of each subproblem, which measures distance to a stationary point. We couple this with a screening rule which is safe in the convex case, converging to the true support at a rate $O(1/(delta^2))$ where $delta geq 0$ measures how close the problem is to degeneracy. In the nonconvex case the screening rule converges to the true support in a finite number of iterations, but is not necessarily safe in the intermediate iterates. In our experiments, we verify the consistency of the method and adjust the aggressiveness of the screening rule by tuning the concavity of the regularizer.
The low-rank matrix completion problem can be solved by Riemannian optimization on a fixed-rank manifold. However, a drawback of the known approaches is that the rank parameter has to be fixed a priori. In this paper, we consider the optimization problem on the set of bounded-rank matrices. We propose a Riemannian rank-adaptive method, which consists of fixed-rank optimization, rank increase step and rank reduction step. We explore its performance applied to the low-rank matrix completion problem. Numerical experiments on synthetic and real-world datasets illustrate that the proposed rank-adaptive method compares favorably with state-of-the-art algorithms. In addition, it shows that one can incorporate each aspect of this rank-adaptive framework separately into existing algorithms for the purpose of improving performance.
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