The boosted Frank-Wolfe algorithm accelerates the classical Frank-Wolfe algorithm by better aligning the update direction with the negative gradient. Its analysis, however, has been limited to deterministic convex problems, with step sizes that require either line search or knowledge of the Lipschitz constant of the gradient. We develop a novel step size strategy that does not depend on the Lipschitz constant of the gradient, which allows us to extend the boosted Frank-Wolfe algorithm to the stochastic setting. We prove that boosting with this step size strategy can be combined with many modern gradient estimators, including SAGA, L-SVRG, SAG, Heavy Ball momentum, and zeroth-order estimators, among others, while retaining the worst-case convergence rates of ordinary stochastic Frank-Wolfe. Our analysis also yields the first convergence rates for boosted Frank-Wolfe on nonconvex and quasar-convex objectives, results which are new even for deterministic problems. Experiments on sparse logistic regression and quantum process tomography show that stochastic boosted Frank-Wolfe achieves faster convergence per gradient oracle call (and on wall-clock) compared to the non-boosted baseline.

Kernel quadrature (KQ) is a kernel-based approach to numerical integration, closely related to Bayesian quadrature (BQ) and probabilistic integration [38, 39, 10]. For sufficiently regular integrands, KQ can exploit spectral structure in a reproducing kernel Hilbert space (RKHS) that is invisible to plain Monte Carlo and thereby converge faster than the usual O(N 1/2) rate in the number of points [3, 28]. Unconstrained kernel-based rules, however, may produce numerically unstable weights, motivating longstanding interest in positively weighted rules [13, 21, 29, 46]. In this paper, positive weights mean nonnegative weights that sum to one, i.e., simplex or convex-combination weights. Whether positive-weight KQ can systematically improve over Monte Carlo is a subtle question. Kernel herding and related constructions suggested fast rates under favorable assumptions [13], but the conditional-gradient viewpoint of Bach et al. [4] clarified that the strongest such assumptions are not generally available in infinite-dimensional RKHSs. Subsequent herding-type analyses in broad RKHS settings have therefore mostly remained at the Monte-Carlo scale, except under additional structure or modified algorithms such as sparse herding variants [31, 44, 43]. Beyond herding, subsampling-based positive KQ methods such as thinning [16, 15] and recombination [21, 24] have obtained rates beyond Monte Carlo, but a general mechanism for such improvement in the simple i.i.d.

We introduce regularized Frank-Wolfe, a general and effective algorithm for inference and learning of dense conditional random fields (CRFs). The algorithm optimizes a nonconvex continuous relaxation of the CRF inference problem using vanilla Frank-Wolfe with approximate updates, which are equivalent to minimizing a regularized energy function. Our proposed method is a generalization of existing algorithms such as mean field or concave-convex procedure. This perspective not only offers a unified analysis of these algorithms, but also allows an easy way of exploring different variants that potentially yield better performance. We illustrate this in our empirical results on standard semantic segmentation datasets, where several instantiations of our regularized Frank-Wolfe outperform mean field inference, both as a standalone component and as an end-to-end trainable layer in a neural network. We also show that dense CRFs, coupled with our new algorithms, produce significant improvements over strong CNN baselines.

We introduce regularized Frank-Wolfe, a general and effective algorithm for inference and learning of dense conditional random fields (CRFs). The algorithm optimizes a nonconvex continuous relaxation of the CRF inference problem using vanilla Frank-Wolfe with approximate updates, which are equivalent to minimizing a regularized energy function. Our proposed method is a generalization of existing algorithms such as mean field or concave-convex procedure. This perspective not only offers a unified analysis of these algorithms, but also allows an easy way of exploring different variants that potentially yield better performance. We illustrate this in our empirical results on standard semantic segmentation datasets, where several instantiations of our regularized Frank-Wolfe outperform mean field inference, both as a standalone component and as an end-to-end trainable layer in a neural network. We also show that dense CRFs, coupled with our new algorithms, produce significant improvements over strong CNN baselines.

In this work, we propose an infinite restricted Boltzmann machine (RBM), whose maximum likelihood estimation (MLE) corresponds to a constrained convex optimization. We consider the Frank-Wolfe algorithm to solve the program, which provides a sparse solution that can be interpreted as inserting a hidden unit at each iteration, so that the optimization process takes the form of a sequence of finite models of increasing complexity. As a side benefit, this can be used to easily and efficiently identify an appropriate number of hidden units during the optimization. The resulting model can also be used as an initialization for typical state-of-the-art RBM training algorithms such as contrastive divergence, leading to models with consistently higher test likelihood than random initialization.

We propose a generalized variant of Frank-Wolfe algorithm for solving a class ofsparse/low-rank optimization problems.

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Convex Optimizationwithout Projection Steps.arXivmath.OC, August 2011.