In many online learning problems we are interested in predicting local information about some universe of items. For example, we may want to know whether two items are in the same cluster rather than computing an assignment of items to clusters; we may want to know which of two teams will win a game rather than computing a ranking of teams. Although finding the optimal clustering or ranking is typically intractable, it may be possible to predict the relationships between items as well as if you could solve the global optimization problem exactly. Formally, we consider an online learning problem in which a learner repeatedly guesses a pair of labels (l(x), l(y)) and receives an adversarial payoff depending on those labels. The learner's goal is to receive a payoff nearly as good as the best fixed labeling of the items. We show that a simple algorithm based on semidefinite programming can obtain asymptotically optimal regret in the case where the number of possible labels is O(1), resolving an open problem posed by Hazan, Kale, and Shalev-Schwartz. Our main technical contribution is a novel use and analysis of the log determinant regularizer, exploiting the observation that log det(A + I) upper bounds the entropy of any distribution with covariance matrix A.
Our techniques differ from those of @cite_7 primarily by our choice of regularizer. Like us, they work with a semidefinite relaxation for the space of labelings (though they do not describe their approach in these terms) and solve an appropriately regularized problem. Their regularizer is the von Neumann entropy, and this leads to their regret bound of @math . We are able to improve this bound by using the log determinant. Moreover, because the log determinant is a natural analog of entropy in the setting of semidefinite programming relaxations for constraint satisfaction, we are able to give a conceptually simple analysis.
@cite_7
2140843581 2963274201 2097476744 2146130798
We study convex relaxations of the image labeling problem on a continuous domain with regularizers based on metric interaction potentials. The generic framework ensures existence of minimizers and covers a wide range of relaxations of the original combinatorial problem. We focus on two specific relaxations that differ in flexibility and simplicity—one can be used to tightly relax any metric interaction potential, while the other covers only Euclidean metrics but requires less computational effort. For solving the nonsmooth discretized problem, we propose a globally convergent Douglas-Rachford scheme and show that a sequence of dual iterates can be recovered in order to provide a posteriori optimality bounds. In a quantitative comparison to two other first-order methods, the approach shows competitive performance on synthetic and real-world images. By combining the method with an improved rounding technique for nonstandard potentials, we were able to routinely recover integral solutions within @math - @math of the global optimum for the combinatorial image labeling problem. This paper resolves one of the longest standing basic problems in the streaming computational model. Namely, optimal construction of quantile sketches. An e approximate quantile sketch receives a stream of items x1,·,xn and allows one to approximate the rank of any query item up to additive error e n with probability at least 1-δ.The rank of a query x is the number of stream items such that xi ≤ x. The minimal sketch size required for this task is trivially at least 1 e.Felber and Ostrovsky obtain a O((1 e)log(1 e)) space sketch for a fixed δ.Without restrictions on the nature of the stream or the ratio between e and n, no better upper or lower bounds were known to date. This paper obtains an O((1 e)log log (1 δ)) space sketch and a matching lower bound. This resolves the open problem and proves a qualitative gap between randomized and deterministic quantile sketching for which an Ω((1 e)log(1 e)) lower bound is known. One of our contributions is a novel representation and modification of the widely used merge-and-reduce construction. This modification allows for an analysis which is both tight and extremely simple. The same technique was reported, in private communications, to be useful for improving other sketching objectives and geometric coreset constructions. This paper proves best known guarantees for exact reconstruction of a sparse signal f from few non-adaptive universal linear measurements. We consider Fourier measurements (random sample of frequencies of f) and random Gaussian measurements. The method for reconstruction that has recently gained momentum in the sparse approximation theory is to relax this highly non-convex problem to a convex problem, and then solve it as a linear program. What are best guarantees for the reconstruction problem to be equivalent to its convex relaxation is an open question. Recent work shows that the number of measurements k(r,n) needed to exactly reconstruct any r-sparse signal f of length n from its linear measurements with convex relaxation is usually O(r poly log (n)). However, known guarantees involve huge constants, in spite of very good performance of the algorithms in practice. In attempt to reconcile theory with practice, we prove the first guarantees for universal measurements (i.e. which work for all sparse functions) with reasonable constants. For Gaussian measurements, k(r,n) lsim 11.7 r [1.5 + log(n r)], which is optimal up to constants. For Fourier measurements, we prove the best known bound k(r, n) = O(r log(n) middot log2(r) log(r log n)), which is optimal within the log log n and log3 r factors. Our arguments are based on the technique of geometric functional analysis and probability in Banach spaces. We use convex relaxation techniques to provide a sequence of regularized low-rank solutions for large-scale matrix completion problems. Using the nuclear norm as a regularizer, we provide a simple and very efficient convex algorithm for minimizing the reconstruction error subject to a bound on the nuclear norm. Our algorithm SOFT-IMPUTE iteratively replaces the missing elements with those obtained from a soft-thresholded SVD. With warm starts this allows us to efficiently compute an entire regularization path of solutions on a grid of values of the regularization parameter. The computationally intensive part of our algorithm is in computing a low-rank SVD of a dense matrix. Exploiting the problem structure, we show that the task can be performed with a complexity of order linear in the matrix dimensions. Our semidefinite-programming algorithm is readily scalable to large matrices; for example SOFT-IMPUTE takes a few hours to compute low-rank approximations of a 106 X 106 incomplete matrix with 107 observed entries, and fits a rank-95 approximation to the full Netflix training set in 3.3 hours. Our methods achieve good training and test errors and exhibit superior timings when compared to other competitive state-of-the-art techniques.