6,342 research outputs found

    Chasing Ghosts: Competing with Stateful Policies

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    We consider sequential decision making in a setting where regret is measured with respect to a set of stateful reference policies, and feedback is limited to observing the rewards of the actions performed (the so called "bandit" setting). If either the reference policies are stateless rather than stateful, or the feedback includes the rewards of all actions (the so called "expert" setting), previous work shows that the optimal regret grows like Θ(T)\Theta(\sqrt{T}) in terms of the number of decision rounds TT. The difficulty in our setting is that the decision maker unavoidably loses track of the internal states of the reference policies, and thus cannot reliably attribute rewards observed in a certain round to any of the reference policies. In fact, in this setting it is impossible for the algorithm to estimate which policy gives the highest (or even approximately highest) total reward. Nevertheless, we design an algorithm that achieves expected regret that is sublinear in TT, of the form O(T/log1/4T)O( T/\log^{1/4}{T}). Our algorithm is based on a certain local repetition lemma that may be of independent interest. We also show that no algorithm can guarantee expected regret better than O(T/log3/2T)O( T/\log^{3/2} T)

    Lower Bounds for Oblivious Near-Neighbor Search

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    We prove an Ω(dlgn/(lglgn)2)\Omega(d \lg n/ (\lg\lg n)^2) lower bound on the dynamic cell-probe complexity of statistically oblivious\mathit{oblivious} approximate-near-neighbor search (ANN\mathsf{ANN}) over the dd-dimensional Hamming cube. For the natural setting of d=Θ(logn)d = \Theta(\log n), our result implies an Ω~(lg2n)\tilde{\Omega}(\lg^2 n) lower bound, which is a quadratic improvement over the highest (non-oblivious) cell-probe lower bound for ANN\mathsf{ANN}. This is the first super-logarithmic unconditional\mathit{unconditional} lower bound for ANN\mathsf{ANN} against general (non black-box) data structures. We also show that any oblivious static\mathit{static} data structure for decomposable search problems (like ANN\mathsf{ANN}) can be obliviously dynamized with O(logn)O(\log n) overhead in update and query time, strengthening a classic result of Bentley and Saxe (Algorithmica, 1980).Comment: 28 page

    Simulating quantum computation by contracting tensor networks

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    The treewidth of a graph is a useful combinatorial measure of how close the graph is to a tree. We prove that a quantum circuit with TT gates whose underlying graph has treewidth dd can be simulated deterministically in TO(1)exp[O(d)]T^{O(1)}\exp[O(d)] time, which, in particular, is polynomial in TT if d=O(logT)d=O(\log T). Among many implications, we show efficient simulations for log-depth circuits whose gates apply to nearby qubits only, a natural constraint satisfied by most physical implementations. We also show that one-way quantum computation of Raussendorf and Briegel (Physical Review Letters, 86:5188--5191, 2001), a universal quantum computation scheme with promising physical implementations, can be efficiently simulated by a randomized algorithm if its quantum resource is derived from a small-treewidth graph.Comment: 7 figure
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