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：设 $\mathcal{A}$ 为拉斯维加斯算法，其运行时间 $T$ 为服从概率分布 $p$ 的随机变量。1993 年，Luby、Sinclair 与 Zuckerman [LSZ93] 证明了：针对任意概率分布 $p$，存在一种简单通用重启策略，使得执行 $\mathcal{A}$ 的算法之期望运行时间为 $O(\ell^\star_p\log\ell^\star_p)$，其中 $\ell^\star_p=\Theta\left(\inf_{q\in (0,1]}Q_p(q)/q\right)$ 为完全已知概率分布 $p$ 的先验信息时可达到的最小期望运行时间，且 $Q_p(q)$ 表示 $p$ 的 $q$ 分位数。此外，作者指出通用重启策略无法消除对数项，且该对数项在特定意义上已是最优。在本工作中，我们出人意料地发现：对数项可被更小的量替代，从而在实际场景中缩减期望运行时间。具体而言，我们提出一种执行 $\mathcal{A}$ 的新重启策略，其期望运行时间为 $O\big(\inf_{q\in (0,1]}\frac{Q_p(q)}{q}\,\psi\big(\log Q_p(q),\,\log (1/q)\big)\big)$，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) 原始算法本身；4) 对一类广泛凹函数 $\phi$ 成立的形如 $\phi^{-1}(\mathbb{E}[\phi(T)])$ 的任何量。最后一项推广了 [Zam22] 中实现 $O\left(e^{\mathbb{E}[\ln(T)]}\right)$ 的重启策略，可视为算法形式的反向延森不等式。我们进一步通过给出这些不等式成立的充要条件，证明了 $\frac{t\phi''(t)}{\phi'(t)}$ 在无穷远处的行为决定了反向延森不等式的存在性。