42,352 research outputs found

    Connectivity of Direct Products of Graphs

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    Let κ(G)\kappa(G) be the connectivity of GG and G×HG\times H the direct product of GG and HH. We prove that for any graphs GG and KnK_n with n≥3n\ge 3, κ(G×Kn)=min{nκ(G),(n−1)δ(G)}\kappa(G\times K_n)=min\{n\kappa(G),(n-1)\delta(G)\}, which was conjectured by Guji and Vumar.Comment: 5 pages, accepted by Ars Com

    Learning to Control in Metric Space with Optimal Regret

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    We study online reinforcement learning for finite-horizon deterministic control systems with {\it arbitrary} state and action spaces. Suppose that the transition dynamics and reward function is unknown, but the state and action space is endowed with a metric that characterizes the proximity between different states and actions. We provide a surprisingly simple upper-confidence reinforcement learning algorithm that uses a function approximation oracle to estimate optimistic Q functions from experiences. We show that the regret of the algorithm after KK episodes is O(HL(KH)d−1d)O(HL(KH)^{\frac{d-1}{d}}) where LL is a smoothness parameter, and dd is the doubling dimension of the state-action space with respect to the given metric. We also establish a near-matching regret lower bound. The proposed method can be adapted to work for more structured transition systems, including the finite-state case and the case where value functions are linear combinations of features, where the method also achieve the optimal regret
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