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We give near-optimal algorithms for computing an ellipsoidal rounding of a convex polytope whose vertices are given in a stream. The approximation factor is linear in the dimension (as in John's theorem) and only loses an excess logarithmic factor in the aspect ratio of the polytope. Our algorithms are nearly optimal in two senses: first, their runtimes nearly match those of the most efficient known algorithms for the offline version of the problem. Second, their approximation factors nearly match a lower bound we show against a natural class of geometric streaming algorithms. In contrast to existing works in the streaming setting that compute ellipsoidal roundings only for centrally symmetric convex polytopes, our algorithms apply to general convex polytopes. We also show how to use our algorithms to construct coresets from a stream of points that approximately preserve both the ellipsoidal rounding and the convex hull of the original set of points. 

We introduce and study the problem of dueling optimization with a monotone adversary, a generalization of (noiseless) dueling convex optimization. The goal is to design an online algorithm to find a minimizer x* for a function f:X→R, for X \subseteq R^d. In each round, the algorithm submits a pair of guesses x1 and x2, and the adversary responds with any point in the space that is at least as good as both guesses. The cost of each query is the suboptimality of the worst of the two guesses; i.e., max(f(x1) − f(x*),f(x2) − f(x*)). The goal is to minimize the number of iterations required to find an ε-optimal point and to minimize the total cost (regret) of the guesses over many rounds. Our main result is an efficient randomized algorithm for several natural choices of the function f and set X that incurs cost O(d) and iteration complexity O(d log(1/ε)^2). Moreover, our dependence on d is asymptotically optimal, as we show examples in which any randomized algorithm for this problem must incur Ω(d) cost and iteration complexity. 

We give efficient deterministic one-pass streaming algorithms for finding an ellipsoidal approximation of a symmetric convex polytope. The algorithms are near-optimal in that their approximation factors differ from that of the optimal offline solution only by a factor sub-logarithmic in the aspect ratio of the polytope