Alice X. Zheng

Akshay Bansal, Ian George, Soumik Ghosh et al.

In many quantum tasks, there is an unknown quantum object that one wishes to learn. An online strategy for this task involves adaptively refining a hypothesis to reproduce such an object or its measurement statistics. A common evaluation metric for such a strategy is its regret, or roughly the accumulated errors in hypothesis statistics. We prove a sublinear regret bound for learning over general subsets of positive semidefinite matrices via the regularized-follow-the-leader algorithm and apply it to various settings where one wishes to learn quantum objects. For concrete applications, we present a sublinear regret bound for learning quantum states, effects, channels, interactive measurements, strategies, co-strategies, and the collection of inner products of pure states. Our bound applies to many other quantum objects with compact, convex representations. In proving our regret bound, we establish various matrix analysis results useful in quantum information theory. This includes a generalization of Pinsker's inequality for arbitrary positive semidefinite operators with possibly different traces, which may be of independent interest and applicable to more general classes of divergences.

Zhixiang Eddie Xu, Matt J. Kusner, Kilian Q. Weinberger et al.

As machine learning transitions increasingly towards real world applications controlling the test-time cost of algorithms becomes more and more crucial. Recent work, such as the Greedy Miser and Speedboost, incorporate test-time budget constraints into the training procedure and learn classifiers that provably stay within budget (in expectation). However, so far, these algorithms are limited to the supervised learning scenario where sufficient amounts of labeled data are available. In this paper we investigate the common scenario where labeled data is scarce but unlabeled data is available in abundance. We propose an algorithm that leverages the unlabeled data (through Laplace smoothing) and learns classifiers with budget constraints. Our model, based on gradient boosted regression trees (GBRT), is, to our knowledge, the first algorithm for semi-supervised budgeted learning.

Zhixiang Eddie Xu, Gao Huang, Kilian Q. Weinberger et al.

A feature selection algorithm should ideally satisfy four conditions: reliably extract relevant features; be able to identify non-linear feature interactions; scale linearly with the number of features and dimensions; allow the incorporation of known sparsity structure. In this work we propose a novel feature selection algorithm, Gradient Boosted Feature Selection (GBFS), which satisfies all four of these requirements. The algorithm is flexible, scalable, and surprisingly straight-forward to implement as it is based on a modification of Gradient Boosted Trees. We evaluate GBFS on several real world data sets and show that it matches or out-performs other state of the art feature selection algorithms. Yet it scales to larger data set sizes and naturally allows for domain-specific side information.

Alice X. Zheng, Irina Rish, Alina Beygelzimer

We consider the problem of diagnosing faults in a system represented by a Bayesian network, where diagnosis corresponds to recovering the most likely state of unobserved nodes given the outcomes of tests (observed nodes). Finding an optimal subset of tests in this setting is intractable in general. We show that it is difficult even to compute the next most-informative test using greedy test selection, as it involves several entropy terms whose exact computation is intractable. We propose an approximate approach that utilizes the loopy belief propagation infrastructure to simultaneously compute approximations of marginal and conditional entropies on multiple subsets of nodes. We apply our method to fault diagnosis in computer networks, and show the algorithm to be very effective on realistic Internet-like topologies. We also provide theoretical justification for the greedy test selection approach, along with some performance guarantees.