Online Local Learning via Semidefinite Programming

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.Comment: 10 page

Label optimal regret bounds for online local learning

We resolve an open question from (Christiano, 2014b) posed in COLT'14 regarding the optimal dependency of the regret achievable for online local learning on the size of the label set. In this framework the algorithm is shown a pair of items at each step, chosen from a set of $n$ items. The learner then predicts a label for each item, from a label set of size $L$ and receives a real valued payoff. This is a natural framework which captures many interesting scenarios such as collaborative filtering, online gambling, and online max cut among others. (Christiano, 2014a) designed an efficient online learning algorithm for this problem achieving a regret of $O(\sqrt{nL^3T})$, where $T$ is the number of rounds. Information theoretically, one can achieve a regret of $O(\sqrt{n \log L T})$. One of the main open questions left in this framework concerns closing the above gap. In this work, we provide a complete answer to the question above via two main results. We show, via a tighter analysis, that the semi-definite programming based algorithm of (Christiano, 2014a), in fact achieves a regret of $O(\sqrt{nLT})$. Second, we show a matching computational lower bound. Namely, we show that a polynomial time algorithm for online local learning with lower regret would imply a polynomial time algorithm for the planted clique problem which is widely believed to be hard. We prove a similar hardness result under a related conjecture concerning planted dense subgraphs that we put forth. Unlike planted clique, the planted dense subgraph problem does not have any known quasi-polynomial time algorithms. Computational lower bounds for online learning are relatively rare, and we hope that the ideas developed in this work will lead to lower bounds for other online learning scenarios as well.Comment: 13 pages; Changes from previous version: small changes to proofs of Theorems 1 & 2, a small rewrite of introduction as well (this version is the same as camera-ready copy in COLT '15

Robust Cardiac Motion Estimation using Ultrafast Ultrasound Data: A Low-Rank-Topology-Preserving Approach

Cardiac motion estimation is an important diagnostic tool to detect heart diseases and it has been explored with modalities such as MRI and conventional ultrasound (US) sequences. US cardiac motion estimation still presents challenges because of the complex motion patterns and the presence of noise. In this work, we propose a novel approach to estimate the cardiac motion using ultrafast ultrasound data. -- Our solution is based on a variational formulation characterized by the L2-regularized class. The displacement is represented by a lattice of b-splines and we ensure robustness by applying a maximum likelihood type estimator. While this is an important part of our solution, the main highlight of this paper is to combine a low-rank data representation with topology preservation. Low-rank data representation (achieved by finding the k-dominant singular values of a Casorati Matrix arranged from the data sequence) speeds up the global solution and achieves noise reduction. On the other hand, topology preservation (achieved by monitoring the Jacobian determinant) allows to radically rule out distortions while carefully controlling the size of allowed expansions and contractions. Our variational approach is carried out on a realistic dataset as well as on a simulated one. We demonstrate how our proposed variational solution deals with complex deformations through careful numerical experiments. While maintaining the accuracy of the solution, the low-rank preprocessing is shown to speed up the convergence of the variational problem. Beyond cardiac motion estimation, our approach is promising for the analysis of other organs that experience motion.Comment: 15 pages, 10 figures, Physics in Medicine and Biology, 201