Given a high dimensional convex body K ` IR n by a separation oracle, we can approximate its volume with relative error ", using O (n 5 ) oracle calls. Our algorithm also brings the body into isotropic position. As all previous randomized volume algorithms, we use "rounding" followed by a multiphase Monte-Carlo (product estimator) technique. Both parts rely on sampling (generating random points in K), which is done by random walk. Our algorithm introduces three new ideas: ffl the use of the isotropic position (or at least an approximation of it) for rounding, ffl the separation of global obstructions (diameter) and local obstructions (boundary problems) for fast mixing, and ffl a stepwise interlacing of rounding and sampling. 1 . Introduction For a variety of geometric objects, classical results characterize various geometric parameters. Many of these results are useful even in practical situations: they can easily be transformed into efficient algorithms. Some other theorem...

We show that there exists a sequence εn ց 0 for which the following holds: Let K ⊂ Rn be a compact, convex set with a non-empty interior. Let X be a random vector that is distributed uniformly in K. Then there exists a unit vector θ in Rn, t0 ∈ R and σ> 0 such that sup A⊂R

The e#ort necessary to construct labeled sets of examples in a supervised learning scenario is often disregarded, though in many applications, it is a time-consuming and expensive procedure. While this already constitutes a major issue in classification learning, it becomes an even more serious problem when dealing with the more complex target domain of total orders over a set of alternatives. Considering both the pairwise decomposition and the constraint classification technique to represent label ranking functions, we introduce a novel generalization of pool-based active learning to address this problem.

Policy teaching considers a Markov Decision Process setting in which an interested party aims to influence an agent’s decisions by providing limited incentives. In this paper, we consider the specific objective of inducing a pre-specified desired policy. We examine both the case in which the agent’s reward function is known and unknown to the interested party, presenting a linear program for the former case and formulating an active, indirect elicitation method for the latter. We provide conditions for logarithmic convergence, and present a polynomial time algorithm that ensures logarithmic convergence with arbitrarily high probability. We also offer practical elicitation heuristics that can be formulated as linear programs, and demonstrate their effectiveness on a policy teaching problem in a simulated ad network setting. We extend our methods to handle partial observations and partial target policies, and provide a game-theoretic interpretation of our methods for handling strategic agents.

We propose and study a new class of online problems, which we call online tracking. Suppose an observer, say Alice, observes a multi-valued function f: Z + → Z d over time in an online fashion, i.e., she only sees f(t) for t ≤ tnow where tnow is the current time. She would like to keep a tracker, say Bob, informed of the current value of f at all times. Under this setting, Alice could send new values of f to Bob from time to time, so that the current value of f is always within a distance of ∆ to the last value received by Bob. We give competitive online algorithms whose communication costs are compared with the optimal offline algorithm that knows the entire f in advance. We also consider variations of the problem where Alice is allowed to send “predictions ” to Bob, to further reduce communication for well-behaved functions. These online tracking problems have a variety of application ranging from sensor monitoring, location-based services, to publish/subscribe systems. 1

This paper studies the relationship between the optimal value of a homogeneous quadratic optimization problem and that of its Semidefinite Programming (SDP) relaxation. We consider two quadratic optimization models: (1) min{x ∗ Cx | x ∗ Akx ≥ 1, k = 0, 1,..., m, x ∈ F n} and (2) max{x ∗ Cx | x ∗ Akx ≤ 1, k = 0, 1,..., m, x ∈ F n}, where F is either the real field R or the complex field C, and Ak, C are symmetric matrices. For the minimization model (1), we prove that, if the matrix C and all but one of Ak’s are positive semidefinite, then the ratio between the optimal value of (1) and its SDP relaxation is upper bounded by O(m 2) when F = R, and by O(m) when F = C. Moreover, when two or more of Ak’s are indefinite, this ratio can be arbitrarily large. For the maximization model (2), we show that, if C and at most one of Ak’s are indefinite while other Ak’s are positive semidefinite, then the ratio between the optimal value of (2) and its SDP relaxation is bounded from below by O(1 / log m) for both the real and complex case. This result improves the bound based on the so-called approximate S-Lemma of Ben-Tal et al. [3]. When two or more of Ak in (2) are indefinite, we derive a general bound in terms of the problem data and the SDP solution. For both optimization models, we present examples to show that the derived

The problem of environment design considers a setting in which an interested party aims to influence an agent’s decisions by making limited changes to the agent’s environment. Zhang and Parkes [2008] first introduced the environment design concept for a specific problem in the Markov Decision Process setting. In this paper, we present a general framework for the formulation and solution of environment design problems with one agent. We consider both the case in which the agent’s local decision model is known and partially unknown to the interested party, and illustrate the framework and results on a linear programming setting. For the latter problem, we formulate an active, indirect elicitation method and provide conditions for convergence and logarithmic convergence. We relate to the problem of inverse optimization and also offer a game-theoretic interpretation of our methods. 1

Abstract. We prove an asymptotic estimate for the number of m ×n non-negative integer matrices (contingency tables) with prescribed row and column sums and, more generally, for the number of integer feasible flows in a network. Similarly, we estimate the volume of the polytope of m × n non-negative real matrices with prescribed row and column sums. Our estimates are solutions of convex optimization problems and hence can be computed efficiently. As a corollary, we show that if row sums R = (r1,..., rm) and column sums C = (c1,..., cn) with r1 +... + rm = c1 +... + cn = N are sufficiently far from constant vectors, then, asymptotically, in the uniform probability space of the m × n non-negative integer matrices with the total sum N of entries, the event consisting of the matrices with row sums R and the event consisting of the matrices with column sums C are positively correlated. 1. Introduction and

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We prove the central limit theorem for the volume and the f-vector of the random polytope Pn and the Poisson random polytope Πn in a fixed convex polytope P ⊂ IR d. Here Pn is the convex hull of n random points in P, and Πn is the convex hull of the intersection of a Poisson process X(n), of intensity n, with P. A general lower bound on the variance is also proved.