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  • Publication
    Competing With Strategies
    (2013-01-01) Han, Wei; Rakhlin, Alexander; Sridharan, Karthik
    We study the problem of online learning with a notion of regret defined with respect to a set of strategies. We develop tools for analyzing the minimax rates and for deriving regret-minimization algorithms in this scenario. While the standard methods for minimizing the usual notion of regret fail, through our analysis we demonstrate existence of regret-minimization methods that compete with such sets of strategies as: autoregressive algorithms, strategies based on statistical models, regularized least squares, and follow the regularized leader strategies. In several cases we also derive efficient learning algorithms
  • Publication
    Learning DNF From Random Walks
    (2003-10-11) Bshouty, Nader; Mossel, Elchanan; O'Donnell, Ryan; Servedio, Rocco A
    We consider a model of learning Boolean functions from examples generated by a uniform random walk on {0, 1}n. We give a polynomial time algorithm for learning decision trees and DNF formulas in this model. This is the first efficient algorithm for learning these classes in a natural passive learning model where the learner has no influence over the choice of examples used for learning.
  • Publication
    An Information Inequality for the Bayes Risk Under Truncated Squared Error Loss
    (1993) Brown, Lawrence D
    A bound is given for the Bayes risk of an estimator under truncated squared error loss. The bound derives from an information inequality for the risk under this loss. It is then used to provide new proofs for some classical results of asymptotic theory.
  • Publication
    Cover Trees for Nearest Neighbor
    (2006-01-01) Beygelzimer, Alina; Kakade, Sham M; Langford, John
    We present a tree data structure for fast nearest neighbor operations in general n-point metric spaces (where the data set consists of n points). The data structure requires O(n) space regardless of the metric's structure yet maintains all performance properties of a navigating net (Krauthgamer & Lee, 2004b). If the point set has a bounded expansion constant c, which is a measure of the intrinsic dimensionality, as defined in (Karger & Ruhl, 2002), the cover tree data structure can be constructed in O (c6n log n) time. Furthermore, nearest neighbor queries require time only logarithmic in n, in particular O (c12 log n) time. Our experimental results show speedups over the brute force search varying between one and several orders of magnitude on natural machine learning datasets.
  • Publication
    Faster Ridge Regression via the Subsampled Randomized Hadamard Transform
    (2013-01-01) Lu, Yichao; Dhillon, Paramveer S.; Foster, Dean P; Ungar, Lyle H
    We propose a fast algorithm for ridge regression when the number of features is much larger than the number of observations (p≫n). The standard way to solve ridge regression in this setting works in the dual space and gives a running time of O(n2p). Our algorithm Subsampled Randomized Hadamard Transform - Dual Ridge Regression (SRHT-DRR) runs in time O(np log(n)) and works by preconditioning the design matrix by a Randomized Walsh-Hadamard Transform with a subsequent subsampling of features. We provide risk bounds for our SRHT-DRR algorithm in the fixed design setting and show experimental results on synthetic and real datasets.
  • Publication
    Exploration in Metric State Spaces
    (2003-01-01) Kakade, Sham M; Kearns, Michael J; Langford, John
    We present metric- E3 a provably near-optimal algorithm for reinforcement learning in Markov decision processes in which there is a natural metric on the state space that allows the construction of accurate local models. The algorithm is a generalization of the E3 algorithm of Kearns and Singh, and assumes a black box for approximate planning. Unlike the original E3 , metric-E3 finds a near optimal policy in an amount of time that does not directly depend on the size of the state space, but instead depends on the covering number of the state space. Informally, the covering number is the number of neighborhoods required for accurate local modeling.
  • Publication
    Reaching Consensus on Social Networks
    (2009-01-01) Mossel, Elchanan; Schoenebeck, Grant
    Research in sociology studies the effectiveness of social networks in achieving computational tasks. Typically the agents who are supposed to achieve a task are unaware of the underlying social network except their immediate friends. They have limited memory, communication, and coordination. These limitations result in computational obstacles in achieving otherwise trivial computational problems. One of the simplest problems studied in the social sciences involves reaching a consensus among players between two alternatives which are otherwise indistinguishable. In this paper we formalize the computational model of social networks. We then analyze the consensus problem as well as the problem of reaching a consensus which is identical to the majority of the original signals. In both models we seek to minimize the time it takes players to reach a consensus.
  • Publication
    Competing in the Dark: An Efficient Algorithm for Bandit Linear Optimization
    (2009-01-01) Abernethy, Jacob D; Hazan, Elad; Rakhlin, Alexander
    We introduce an efficient algorithm for the problem of online linear optimization in the bandit setting which achieves the optimal O*(√T)regret. The setting is a natural generalization of the nonstochastic multiarmed bandit problem, and the existence of an efficient optimal algorithm has been posed as an open problem in a number of recent papers. We show how the difficulties encountered by previous approaches are overcome by the use of a self-concordant potential function. Our approach presents a novel connection between online learning and interior point methods.
  • Publication
    An Alternative Prior Process for Nonparametric Bayesian Clustering
    (2010-01-01) Wallach, Hanna M; Jensen, Shane T; Dicker, Lee; Heller, Katherine A
    Prior distributions play a crucial role in Bayesian approaches to clustering. Two commonly-used prior distributions are the Dirichlet and Pitman-Yor processes. In this paper, we investigate the predictive prob- abilities that underlie these processes, and the implicit "rich-get-richer" characteristic of the resulting partitions. We explore an alternative prior for nonparametric Bayesian clustering-the uniform process-for applications where the "rich-get-richer" property is undesirable. We also explore the cost of this process: partitions are no longer exchangeable with respect to the ordering of variables. We present new asymptotic and simulation-based results for the clustering characteristics of the uniform process and compare these with known results for the Dirichlet and Pitman-Yor processes. We compare performance on a real document clustering task, demonstrating the practical advantage of the uniform process despite its lack of exchangeability over orderings.
  • Publication
    Optimization, Learning, and Games With Predictable Sequences
    (2013-01-01) Rakhlin, Alexander; Sridharan, Karthik
    We provide several applications of Optimistic Mirror Descent, an online learning algorithm based on the idea of predictable sequences. First, we recover the Mirror Prox algorithm for offline optimization, prove an extension to Holder-smooth functions, and apply the results to saddle-point type problems. Next, we prove that a version of Optimistic Mirror Descent (which has a close relation to the Exponential Weights algorithm) can be used by two strongly-uncoupled players in a finite zero-sum matrix game to converge to the minimax equilibrium at the rate of O((log T)/T). This addresses a question of Daskalakis et al 2011. Further, we consider a partial information version of the problem. We then apply the results to convex programming and exhibit a simple algorithm for the approximate Max Flow problem.