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We prove that the hit-and-run random walk is rapidly mixing for an arbitrary logconcave distribution starting from any point in the support. This extends the work of [26], where this was shown for an important special case, and settles the main conjecture formulated there. From this result, we derive asymptotically faster algorithms in the general oracle model for sampling, rounding, integration and maximization of logconcave functions, improving or generalizing the main results of [24, 25, 1] and [16] respectively. The algorithms for integration and optimization both use sampling and are surprisingly similar.

Training a learning algorithm is a costly task. A major goal of active learning is to reduce this cost. In this paper we introduce a new algorithm, KQBC, which is capable of actively learning large scale problems by using selective sampling. The algorithm overcomes the costly sampling step of the well known Query By Committee (QBC) algorithm by projecting onto a low dimensional space. KQBC also enables the use of kernels, providing a simple way of extending QBC to the non-linear scenario. Sampling the low dimension space is done using the hit and run random walk. We demonstrate the success of this novel algorithm by applying it to both artificial and a real world problems. 1

Abstract. The use of simulation for high-dimensional intractable computations has revolutionized applied mathematics. Designing, improving and understanding the new tools leads to (and leans on) fascinating mathematics, from representation theory through micro-local analysis. 1.

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We draw on the observation that the amount of heat diffusing outside of a heated body in a short period of time is proportional to its surface area, to design a simple algorithm for approximating the surface area of a convex body given by a membership oracle. Our method has a complexity of O ∗ (n 4), where n is the dimension, compared to O ∗ (n 8.5) for the previous best algorithm. We show that our complexity cannot be improved given the current state-of-the-art in volume estimation. 1

Let K be a polytope in Rn defined by m linear inequalities. We give a new Markov Chain algorithm to draw a nearly uniform sample from K. The underlying Markov Chain is the first to have a mixing time that is strongly polynomial when started from a “central ” point x0. If s is the and ɛ is an upper bound on the desired total variation distance from the uniform, it is sufficient to take O ( mn ( n log(sm) + log 1 ɛ steps of the random walk. We use this result to design an affine interior point algorithm that does a single random walk to solve linear programs approximately. More precisely, suppose Q = {z ∣ ∣Bz ≤ 1} contains a point z such that cT z ≥ d and r: = supz∈Q ‖Bz ‖ + 1, where B is an m × n matrix. Then, after τ = O ( mn ( n ln ( ))) mr 1 ɛ + ln δ steps, the random walk is at a point xτ for which cT xτ ≥ d(1 − ɛ) with probability greater than 1 − δ. The fact that this algorithm has a run-time that is provably polynomial is notable since the analogous deterministic affine algorithm analyzed by Dikin has no known polynomial guarantees. supremum over all chords pq passing through x0 of |p−x0| |q−x0| 1

Abstract. We consider the problem of determining whether a given set S in R n is approximately convex, i.e., if there is a convex set K ∈ R n such that the volume of their symmetric difference is at most ǫ vol(S) for some given ǫ. When the set is presented only by a membership oracle and a random oracle, we show that the problem can be solved with high probability using poly(n)(c/ǫ) n oracle calls and computation time. We complement this result with an exponential lower bound for the natural algorithm that tests convexity along “random ” lines. We conjecture that a simple 2-dimensional version of this algorithm has polynomial complexity. 1

Star-shaped bodies are an important nonconvex generalization of convex bodies (e.g., linear programming with violations). Here we present an efficient algorithm for sampling a given star-shaped body. The complexity of the algorithm grows polynomially in the dimension and inverse polynomially in the fraction of the volume taken up by the kernel of the star-shaped body. The analysis is based on a new isoperimetric inequality. Our main technical contribution is a tool for proving such inequalities when the domain is not convex. As a consequence, we obtain a polynomial algorithm for computing the volume of such a set as well. In contrast, linear optimization over star-shaped sets is NP-hard.

In the thesis, we study ergodicity of adaptive Markov Chain Monte Carlo methods (MCMC) based on two conditions (Diminishing Adaptation and Containment which together imply ergodicity), explain the advantages of adaptive MCMC, and apply the theoretical result for some applications. First we give some examples to explain several facts: 1. Diminishing Adaptation alone may destroy ergodicity; 2. Containment is not necessary for ergodicity; 3. under some additional condition, Containment is necessary for ergodicity. Since Diminishing Adaptation is relatively easy to check and Containment is abstract, we focus on the sufficient conditions of Containment. In order to study Containment, we consider the quantitative bounds of the distance between samplers and targets in total variation norm. From early results, the quantitative bounds are connected with nested drift conditions for polynomial rates of convergence. For ergodicity of adaptive MCMC, assuming that all samplers simultaneously satisfy nested polynomial drift conditions, we find that either when the number of nested