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.

翻译：令 $\mathcal{A}$ 为一个拉斯维加斯算法，即其运行时间 $T$ 为依特定概率分布 $p$ 抽取的随机变量。1993年，Luby、Sinclair与Zuckerman [LSZ93] 证明：对于任意概率分布 $p$，存在一种简单通用重启策略，使得执行 $\mathcal{A}$ 的算法的期望运行时间为 $O(\ell^\star_p\log\ell^\star_p)$，其中 $\ell^\star_p=\Theta\left(\inf_{q\in (0,1]}Q_p(q)/q\right)$ 为完全已知分布 $p$ 时能达到的最小期望运行时间，而 $Q_p(q)$ 为 $p$ 的 $q$ 分位数。此外，作者指出该对数项无法从通用重启策略中移除，且在某种意义上具有最优性。本文中，我们出人意料地证明：该对数项可被更小的量替代，从而在相关实际场景中降低期望运行时间。具体而言，我们提出一种新型重启策略，其执行 $\mathcal{A}$ 的期望运行时间为 $O\big(\inf_{q\in (0,1]}\frac{Q_p(q)}{q}\,\psi\big(\log Q_p(q),\,\log (1/q)\big)\big)$，其中 $\psi(a,b)=1+\min\left\{a+b,a\log^2 a,\,b\log^2 b\right\}$。该量在乘法因子意义下优于：1) [LSZ93] 的通用重启策略；2) 任意 $q\in(0,1]$ 对应的 $p$ 的 $q$ 分位数；3) 原始算法；以及 4) 对于一大类凹函数 $\phi$ 满足 $\phi^{-1}(\mathbb{E}[\phi(T)])$ 的任意量。后者推广了 [Zam22] 中近期提出的、实现 $O\left(e^{\mathbb{E}[\ln(T)]}\right)$ 的重启策略，并可视为算法逆向詹森不等式。最后，我们通过给出此类不等式成立的充分必要条件，证明 $\frac{t\phi''(t)}{\phi'(t)}$ 在无穷远处的行为控制着逆向詹森不等式的存在性。