Optimal Decision Tree with Noisy Outcomes

Neural Information Processing Systems

A fundamental task in active learning involves performing a sequence of tests to identify an unknown hypothesis that is drawn from a known distribution. This problem, known as optimal decision tree induction, has been widely studied for decades and the asymptotically best-possible approximation algorithm has been devised for it. We study a generalization where certain test outcomes are noisy, even in the more general case when the noise is persistent, i.e., repeating the test on the scenario gives the same noisy output, disallowing simple repetition as a way to gain confidence. We design new approximation algorithms for both the non-adaptive setting, where the test sequence must be fixed a-priori, and the adaptive setting where the test sequence depends on the outcomes of prior tests. Previous work in the area assumed at most a constant number of noisy outcomes per test and per scenario and provided approximation ratios that were problem dependent (such as the minimum probability of a hypothesis).

Directional Decision Lists

arXiv.org Machine Learning

In this paper we introduce a novel family of decision lists consisting of highly interpretable models which can be learned efficiently in a greedy manner. The defining property is that all rules are oriented in the same direction. Particular examples of this family are decision lists with monotonically decreasing (or increasing) probabilities. On simulated data we empirically confirm that the proposed model family is easier to train than general decision lists. We exemplify the practical usability of our approach by identifying problem symptoms in a manufacturing process.

Algebraic tests of general Gaussian latent tree models

Neural Information Processing Systems

We consider general Gaussian latent tree models in which the observed variables are not restricted to be leaves of the tree. Extending related recent work, we give a full semi-algebraic description of the set of covariance matrices of any such model. In other words, we find polynomial constraints that characterize when a matrix is the covariance matrix of a distribution in a given latent tree model. However, leveraging these constraints to test a given such model is often complicated by the number of constraints being large and by singularities of individual polynomials, which may invalidate standard approximations to relevant probability distributions. Illustrating with the star tree, we propose a new testing methodology that circumvents singularity issues by trading off some statistical estimation efficiency and handles cases with many constraints through recent advances on Gaussian approximation for maxima of sums of high-dimensional random vectors.

Maryland's Attorney General Appeals Redistricting Decision

U.S. News

The decision, written by 4th U.S. Circuit Court Judge Paul Niemeyer, says the state's 2011 congressional map removed roughly 66,000 Republican voters from the 6th district and added around 24,000 Democratic voters, "bringing about the single greatest alteration of voter makeup in any district in the Nation following the 2010 census."

Online Decision-Making in General Combinatorial Spaces

Neural Information Processing Systems

We study online combinatorial decision problems, where one must make sequential decisions in some combinatorial space without knowing in advance the cost of decisions on each trial; the goal is to minimize the total regret over some sequence of trials relative to the best fixed decision in hindsight. Such problems have been studied mostly in settings where decisions are represented by Boolean vectors and costs are linear in this representation. Here we study a general setting where costs may be linear in any suitable low-dimensional vector representation of elements of the decision space. We give a general algorithm for such problems that we call low-dimensional online mirror descent (LDOMD); the algorithm generalizes both the Component Hedge algorithm of Koolen et al. (2010), and a recent algorithm of Suehiro et al. (2012). Our study offers a unification and generalization of previous work, and emphasizes the role of the convex polytope arising from the vector representation of the decision space; while Boolean representations lead to 0-1 polytopes, more general vector representations lead to more general polytopes.