Breaking the Log Barrier: a Novel Universal Restart Strategy for Faster Las Vegas Algorithms

Let $\mathcal{A}$ 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 $\mathcal{A}$ and whose expected running time is $O(\ell^\star_p\log\ell^\star_p)$, where $\ell^\star_p=\Theta\left(\inf_{q\in (0,1]}Q_p(q)/q\right)$ 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 $\mathcal{A}$ and whose expected running time is $O\big(\inf_{q\in (0,1]}\frac{Q_p(q)}{q}\,\psi\big(\log Q_p(q),\,\log (1/q)\big)\big)$ where $\psi(a,b)=1+\min\left\{a+b,a\log^2 a,\,b\log^2 b\right\}$. This quantity is, up to a multiplicative factor, better than: 1) the universal restart strategy of [LSZ93], 2) any $q$-quantile of $p$ for $q\in(0,1]$, 3) the original algorithm, and 4) any quantity of the form $\phi^{-1}(\mathbb{E}[\phi(T)])$ for a large class of concave functions $\phi$. The latter extends the recent restart strategy of [Zam22] achieving $O\left(e^{\mathbb{E}[\ln(T)]}\right)$, and can be thought of as algorithmic reverse Jensen's inequalities. Finally, we show that the behavior of $\frac{t\phi''(t)}{\phi'(t)}$ at infinity controls the existence of reverse Jensen's inequalities by providing a necessary and a sufficient condition for these inequalities to hold.