Black-box Acceleration of Las Vegas Algorithms and Algorithmic Reverse Jensen's Inequalities
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Let 𝒜 be a Las Vegas algorithm, i.e. an algorithm whose running time T is a random variable drawn according to a certain probability distribution p. In 1993, Luby, Sinclair and Zuckerman [LSZ93] proved that a simple universal restart strategy can, for any probability distribution p, provide an algorithm executing 𝒜 and whose expected running time is O(ℓ^⋆_plogℓ^⋆_p), where ℓ^⋆_p=Θ(inf_q∈ (0,1]Q_p(q)/q) is the minimum expected running time achievable with full prior knowledge of the probability distribution p, and Q_p(q) is the q-quantile of p. Moreover, the authors showed that the logarithmic term could not be removed for universal restart strategies and was, in a certain sense, optimal. In this work, we show that, quite surprisingly, the logarithmic term can be replaced by a smaller quantity, thus reducing the expected running time in practical settings of interest. More precisely, we propose a novel restart strategy that executes 𝒜 and whose expected running time is O(inf_q∈ (0,1]Q_p(q)/q ψ(log Q_p(q), log (1/q))) where ψ(a,b)=1+min{a+b,alog^2 a, blog^2 b}. This quantity is, up to a multiplicative factor, better than: 1) the universal restart strategy of [LSZ93], 2) any q-quantile of p for q∈(0,1], 3) the original algorithm, and 4) any quantity of the form ϕ^-1(𝔼[ϕ(T)]) for a large class of concave functions ϕ. The latter extends the recent restart strategy of [Zam22] achieving O(e^𝔼[ln(T)]), and can be thought of as algorithmic reverse Jensen's inequalities. Finally, we show that the behavior of tϕ”(t)/ϕ'(t) at infinity controls the existence of reverse Jensen's inequalities by providing a necessary and a sufficient condition for these inequalities to hold.
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