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Lijun Ding
Publication date
2018
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Mathematical Statistics
and research's language is
English
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We introduce a few variants on Frank-Wolfe style algorithms suitable for large scale optimization. We show how to modify the standard Frank-Wolfe algorithm using stochastic gradients, approximate subproblem solutions, and sketched decision variables in order to scale to enormous problems while preserving (up to constants) the optimal convergence rate $mathcal{O}(frac{1}{k})$.
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We study projection-free methods for constrained Riemannian optimization. In particular, we propose the Riemannian Frank-Wolfe (RFW) method. We analyze non-asymptotic convergence rates of RFW to an optimum for (geodesically) convex problems, and to a critical point for nonconvex objectives. We also present a practical setting under which RFW can attain a linear convergence rate. As a concrete example, we specialize Rfw to the manifold of positive definite matrices and apply it to two tasks: (i) computing the matrix geometric mean (Riemannian centroid); and (ii) computing the Bures-Wasserstein barycenter. Both tasks involve geodesically convex interval constraints, for which we show that the Riemannian linear oracle required by RFW admits a closed-form solution; this result may be of independent interest. We further specialize RFW to the special orthogonal group and show that here too, the Riemannian linear oracle can be solved in closed form. Here, we describe an application to the synchronization of data matrices (Procrustes problem). We complement our theoretical results with an empirical comparison of Rfw against state-of-the-art Riemannian optimization methods and observe that RFW performs competitively on the task of computing Riemannian centroids.
Projection-free optimization via different variants of the Frank-Wolfe (FW), a.k.a. Conditional Gradient method has become one of the cornerstones in optimization for machine learning since in many cases the linear minimization oracle is much cheaper to implement than projections and some sparsity needs to be preserved. In a number of applications, e.g. Poisson inverse problems or quantum state tomography, the loss is given by a self-concordant (SC) function having unbounded curvature, implying absence of theoretical guarantees for the existing FW methods. We use the theory of SC functions to provide a new adaptive step size for FW methods and prove global convergence rate O(1/k) after k iterations. If the problem admits a stronger local linear minimization oracle, we construct a novel FW method with linear convergence rate for SC functions.
This paper proposes a new variant of Frank-Wolfe (FW), called $k$FW. Standard FW suffers from slow convergence: iterates often zig-zag as update directions oscillate around extreme points of the constraint set. The new variant, $k$FW, overcomes this problem by using two stronger subproblem oracles in each iteration. The first is a $k$ linear optimization oracle ($k$LOO) that computes the $k$ best update directions (rather than just one). The second is a $k$ direction search ($k$DS) that minimizes the objective over a constraint set represented by the $k$ best update directions and the previous iterate. When the problem solution admits a sparse representation, both oracles are easy to compute, and $k$FW converges quickly for smooth convex objectives and several interesting constraint sets: $k$FW achieves finite $frac{4L_f^3D^4}{gammadelta^2}$ convergence on polytopes and group norm balls, and linear convergence on spectrahedra and nuclear norm balls. Numerical experiments validate the effectiveness of $k$FW and demonstrate an order-of-magnitude speedup over existing approaches.
In this paper, we propose approximate Frank-Wolfe (FW) algorithms to solve convex optimization problems over graph-structured support sets where the textit{linear minimization oracle} (LMO) cannot be efficiently obtained in general. We first demonstrate that two popular approximation assumptions (textit{additive} and textit{multiplicative gap errors)}, are not valid for our problem, in that no cheap gap-approximate LMO oracle exists in general. Instead, a new textit{approximate dual maximization oracle} (DMO) is proposed, which approximates the inner product rather than the gap. When the objective is $L$-smooth, we prove that the standard FW method using a $delta$-approximate DMO converges as $mathcal{O}(L / delta t + (1-delta)(delta^{-1} + delta^{-2}))$ in general, and as $mathcal{O}(L/(delta^2(t+2)))$ over a $delta$-relaxation of the constraint set. Additionally, when the objective is $mu$-strongly convex and the solution is unique, a variant of FW converges to $mathcal{O}(L^2log(t)/(mu delta^6 t^2))$ with the same per-iteration complexity. Our empirical results suggest that even these improved bounds are pessimistic, with significant improvement in recovering real-world images with graph-structured sparsity.
The stochastic Frank-Wolfe method has recently attracted much general interest in the context of optimization for statistical and machine learning due to its ability to work with a more general feasible region. However, there has been a complexity gap in the guaranteed convergence rate for stochastic Frank-Wolfe compared to its deterministic counterpart. In this work, we present a new generalized stochastic Frank-Wolfe method which closes this gap for the class of structured optimization problems encountered in statistical and machine learning characterized by empirical loss minimization with a certain type of ``linear prediction property (formally defined in the paper), which is typically present loss minimization problems in practice. Our method also introduces the notion of a ``substitute gradient that is a not-necessarily-unbiased sample of the gradient. We show that our new method is equivalent to a particular randomized coordinate mirror descent algorithm applied to the dual problem, which in turn provides a new interpretation of randomized dual coordinate descent in the primal space. Also, in the special case of a strongly convex regularizer our generalized stochastic Frank-Wolfe method (as well as the randomized dual coordinate descent method) exhibits linear convergence. Furthermore, we present computational experiments that indicate that our method outperforms other stochastic Frank-Wolfe methods consistent with the theory developed herein.
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