We consider the setting of online convex optimization (OCO) with \textit{exp-concave} losses. The best regret bound known for this setting is $O(n\log{}T)$, where $n$ is the dimension and $T$ is the number of prediction rounds (treating all other quantities as constants and assuming $T$ is sufficiently large), and is attainable via the well-known Online Newton Step algorithm (ONS). However, ONS requires on each iteration to compute a projection (according to some matrix-induced norm) onto the feasible convex set, which is often computationally prohibitive in high-dimensional settings and when the feasible set admits a non-trivial structure. In this work we consider projection-free online algorithms for exp-concave and smooth losses, where by projection-free we refer to algorithms that rely only on the availability of a linear optimization oracle (LOO) for the feasible set, which in many applications of interest admits much more efficient implementations than a projection oracle. We present an LOO-based ONS-style algorithm, which using overall $O(T)$ calls to a LOO, guarantees in worst case regret bounded by $\widetilde{O}(n^{2/3}T^{2/3})$ (ignoring all quantities except for $n,T$). However, our algorithm is most interesting in an important and plausible low-dimensional data scenario: if the gradients (approximately) span a subspace of dimension at most $\rho$, $\rho << n$, the regret bound improves to $\widetilde{O}(\rho^{2/3}T^{2/3})$, and by applying standard deterministic sketching techniques, both the space and average additional per-iteration runtime requirements are only $O(\rho{}n)$ (instead of $O(n^2)$). This improves upon recently proposed LOO-based algorithms for OCO which, while having the same state-of-the-art dependence on the horizon $T$, suffer from regret/oracle complexity that scales with $\sqrt{n}$ or worse.
We consider the forecast aggregation problem in repeated settings, where the forecasts are done on a binary event. At each period multiple experts provide forecasts about an event. The goal of the aggregator is to aggregate those forecasts into a subjective accurate forecast. We assume that experts are Bayesian; namely they share a common prior, each expert is exposed to some evidence, and each expert applies Bayes rule to deduce his forecast. The aggregator is ignorant with respect to the information structure (i.e., distribution over evidence) according to which experts make their prediction. The aggregator observes the experts' forecasts only. At the end of each period the actual state is realized. We focus on the question whether the aggregator can learn to aggregate optimally the forecasts of the experts, where the optimal aggregation is the Bayesian aggregation that takes into account all the information (evidence) in the system. We consider the class of partial evidence information structures, where each expert is exposed to a different subset of conditionally independent signals. Our main results are positive; We show that optimal aggregation can be learned in polynomial time in a quite wide range of instances of the partial evidence environments. We provide a tight characterization of the instances where learning is possible and impossible.
We revisit the use of Stochastic Gradient Descent (SGD) for solving convex optimization problems that serve as highly popular convex relaxations for many important low-rank matrix recovery problems such as \textit{matrix completion}, \textit{phase retrieval}, and more. The computational limitation of applying SGD to solving these relaxations in large-scale is the need to compute a potentially high-rank singular value decomposition (SVD) on each iteration in order to enforce the low-rank-promoting constraint. We begin by considering a simple and natural sufficient condition so that these relaxations indeed admit low-rank solutions. This condition is also necessary for a certain notion of low-rank-robustness to hold. Our main result shows that under this condition which involves the eigenvalues of the gradient vector at optimal points, SGD with mini-batches, when initialized with a "warm-start" point, produces iterates that are low-rank with high probability, and hence only a low-rank SVD computation is required on each iteration. This suggests that SGD may indeed be practically applicable to solving large-scale convex relaxations of low-rank matrix recovery problems. Our theoretical results are accompanied with supporting preliminary empirical evidence. As a side benefit, our analysis is quite simple and short.
Computing the leading eigenvector of a symmetric real matrix is a fundamental primitive of numerical linear algebra with numerous applications. We consider a natural online extension of the leading eigenvector problem: a sequence of matrices is presented and the goal is to predict for each matrix a unit vector, with the overall goal of competing with the leading eigenvector of the cumulative matrix. Existing regret-minimization algorithms for this problem either require to compute an eigen decompostion every iteration, or suffer from a large dependency of the regret bound on the dimension. In both cases the algorithms are not practical for large scale applications.
In this paper we present new algorithms that avoid both issues. On one hand they do not require any expensive matrix decompositions and on the other, they guarantee regret rates with a mild dependence on the dimension at most. In contrast to previous algorithms, our algorithms also admit implementations that enable to leverage sparsity in the data to further reduce computation. We extend our results to also handle non-symmetric matrices.
We study the stochastic optimization of canonical correlation analysis (CCA), whose objective is nonconvex and does not decouple over training samples. Although several stochastic gradient based optimization algorithms have been recently proposed to solve this problem, no global convergence guarantee was provided by any of them. Inspired by the alternating least squares/power iterations formulation of CCA, and the shift-and-invert preconditioning method for PCA, we propose two globally convergent meta-algorithms for CCA, both of which transform the original problem into sequences of least squares problems that need only be solved approximately. We instantiate the meta-algorithms with state-of-the-art SGD methods and obtain time complexities that significantly improve upon that of previous work. Experimental results demonstrate their superior performance.
The Frank-Wolfe method (a.k.a. conditional gradient algorithm) for smooth optimization has regained much interest in recent years in the context of large scale optimization and machine learning. A key advantage of the method is that it avoids projections - the computational bottleneck in many applications - replacing it by a linear optimization step. Despite this advantage, the known convergence rates of the FW method fall behind standard first order methods for most settings of interest. It is an active line of research to derive faster linear optimization-based algorithms for various settings of convex optimization.
In this paper we consider the special case of optimization over strongly convex sets, for which we prove that the vanila FW method converges at a rate of 1/t2. This gives a quadratic improvement in convergence rate compared to the general case, in which convergence is of the order 1/t, and known to be tight. We show that various balls induced by lp norms, Schatten norms and group norms are strongly convex on one hand and on the other hand, linear optimization over these sets is straightforward and admits a closed-form solution. We further show how several previous fastrate results for the FW method follow easily from our analysis.
Low-rank and nonsmooth matrix optimization problems capture many fundamental tasks in statistics and machine learning. While significant progress has been made in recent years in developing efficient methods for \textit{smooth} low-rank optimization problems that avoid maintaining high-rank matrices and computing expensive high-rank SVDs, advances for nonsmooth problems have been slow paced. In this paper we consider standard convex relaxations for such problems. Mainly, we prove that under a \textit{strict complementarity} condition and under the relatively mild assumption that the nonsmooth objective can be written as a maximum of smooth functions, approximated variants of two popular \textit{mirror-prox} methods: the Euclidean \textit{extragradient method} and mirror-prox with \textit{matrix exponentiated gradient updates}, when initialized with a "warm-start", converge to an optimal solution with rate $O(1/t)$, while requiring only two \textit{low-rank} SVDs per iteration. Moreover, for the extragradient method we also consider relaxed versions of strict complementarity which yield a trade-off between the rank of the SVDs required and the radius of the ball in which we need to initialize the method. We support our theoretical results with empirical experiments on several nonsmooth low-rank matrix recovery tasks, demonstrating both the plausibility of the strict complementarity assumption, and the efficient convergence of our proposed low-rank mirror-prox variants.
We revisit Blackwell's celebrated approachability problem which considers a repeated vector-valued game between a player and an adversary. Motivated by settings in which the action set of the player or adversary (or both) is difficult to optimize over, for instance when it corresponds to the set of all possible solutions to some NP-Hard optimization problem, we ask what can the player guarantee \textit{efficiently}, when only having access to these sets via approximation algorithms with ratios $\alpha_{\mX} \geq 1$ and $ 1 \geq \alpha_{\mY} > 0$, respectively. Assuming the player has monotone preferences, in the sense that he does not prefer a vector-valued loss $\ell_1$ over $\ell_2$ if $\ell_2 \leq \ell_1$, we establish that given a Blackwell instance with an approachable target set $S$, the downward closure of the appropriately-scaled set $\alpha_{\mX}\alpha_{\mY}^{-1}S$ is \textit{efficiently} approachable with optimal rate. In case only the player's or adversary's set is equipped with an approximation algorithm, we give simpler and more efficient algorithms.
Convex optimization over the spectrahedron, that is, the set of all real n × n positive semidefinite matrices with unit trace, has important applications in machine learning, signal processing, and statistics, mainly as a convex relaxation for optimization problems with low-rank matrices. It is also one of the most prominent examples in the theory of first order methods for convex optimization in which non-Euclidean methods can be significantly preferable to their Euclidean counterparts. In particular, the desirable choice is the matrix exponentiated gradient (MEG) method, which is based on the Bregman distance induced by the (negative) von Neumann entropy. Unfortunately, implementing MEG requires a full singular value decomposition (SVD) computation on each iteration, which is not scalable to high-dimensional problems. In this work, we propose efficient implementations of MEG, with both deterministic and stochastic gradients, which are tailored for optimization with low-rank matrices, and only use a single low-rank SVD computation on each iteration. We also provide efficiently computable certificates for the correct convergence of our methods. Mainly, we prove that, under a strict complementarity condition, the suggested methods converge from a warm-start initialization with similar rates to their full SVD–based counterparts. Finally, we bring empirical experiments that both support our theoretical findings and demonstrate the practical appeal of our methods. Funding: This work was supported by the Israel Science Foundation [Grant 1108/18].
We consider variants of the classical Frank-Wolfe algorithm for constrained smooth convex minimization, that instead of access to the standard oracle for minimizing a linear function over the feasible set, have access to an oracle that can find an extreme point of the feasible set that is closest in Euclidean distance to a given vector. We first show that for many feasible sets of interest, such an oracle can be implemented with the same complexity as the standard linear optimization oracle. We then show that with such an oracle we can design new Frank-Wolfe variants which enjoy significantly improved complexity bounds in case the set of optimal solutions lies in the convex hull of a subset of extreme points with small diameter (e.g., a low-dimensional face of a polytope). In particular, for many $0\text{--}1$ polytopes, under quadratic growth and strict complementarity conditions, we obtain the first linearly convergent variant with rate that depends only on the dimension of the optimal face and not on the ambient dimension.
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