David Castanon

Ali Siahkamari, Xide Xia, Venkatesh Saligrama et al.

Bregman divergences generalize measures such as the squared Euclidean distance and the KL divergence, and arise throughout many areas of machine learning. In this paper, we focus on the problem of approximating an arbitrary Bregman divergence from supervision, and we provide a well-principled approach to analyzing such approximations. We develop a formulation and algorithm for learning arbitrary Bregman divergences based on approximating their underlying convex generating function via a piecewise linear function. We provide theoretical approximation bounds using our parameterization and show that the generalization error $O_p(m^{-1/2})$ for metric learning using our framework matches the known generalization error in the strictly less general Mahalanobis metric learning setting. We further demonstrate empirically that our method performs well in comparison to existing metric learning methods, particularly for clustering and ranking problems.

Huanyu Ding, David Castanon

The problem of search by multiple agents to find and localize objects arises in many important applications. In this paper, we study a class of multi-agent search problems in which each agent can access only a subset of a discrete search space, with detection performance that depends only on the location. We show that this problem can be reformulated as a minimum cost network optimization problem, and develop a fast specialized algorithm for the solution. We prove that our algorithm is correct, and has worst case computation performance that is faster than general minimum cost flow algorithms. We also address the problem where detection performance depends on both location and agent, which is known to be NP-Hard. We reduce the problem to a submodular maximization problem over a matroid, and provide an approximate algorithm with guaranteed performance. We illustrate the performance of our algorithms with simulations of search problems and compare it with other min-cost flow algorithms.

Kirill Trapeznikov, Venkatesh Saligrama, David Castanon

In many classification systems, sensing modalities have different acquisition costs. It is often {\it unnecessary} to use every modality to classify a majority of examples. We study a multi-stage system in a prediction time cost reduction setting, where the full data is available for training, but for a test example, measurements in a new modality can be acquired at each stage for an additional cost. We seek decision rules to reduce the average measurement acquisition cost. We formulate an empirical risk minimization problem (ERM) for a multi-stage reject classifier, wherein the stage $k$ classifier either classifies a sample using only the measurements acquired so far or rejects it to the next stage where more attributes can be acquired for a cost. To solve the ERM problem, we show that the optimal reject classifier at each stage is a combination of two binary classifiers, one biased towards positive examples and the other biased towards negative examples. We use this parameterization to construct stage-by-stage global surrogate risk, develop an iterative algorithm in the boosting framework and present convergence and generalization results. We test our work on synthetic, medical and explosives detection datasets. Our results demonstrate that substantial cost reduction without a significant sacrifice in accuracy is achievable.