灵活预算在Adwords中的威力
The Power of Flexible Budgets in Adwords
AI总结：
针对Adwords问题的灵活预算模型，我们证明了经典算法无法受益于灵活性，但设计了一种竞争比渐近最优为$1-e^{-\delta}$的算法，并刻画了高流量情形下的精确最优竞争比。
AI中文摘要：
搜索广告平台在高流量日通常会超出广告商的平均每日预算进行支出，只要当月总支出保持在月度预算之内。受此实践启发，我们研究了Adwords问题（Mehta等人，2007年）的$D$天泛化版本，其中每个广告商$i$有一个名义（平均）每日预算$B_i$和一个总时间跨度（月度）预算$DB_i$。给定一个灵活性参数$\delta$，平台在任意单日对广告商$i$的支出最多为$\delta B_i$，并受限于$DB_i$的时间跨度支出限制。我们通过与非灵活的离线最优方案（该方案每天对广告商$i$的支出最多为$B_i$）进行基准比较，量化了$\delta$-灵活预算的威力。我们表明，任何程度的灵活性都无法帮助Mehta等人（2007年）经典算法的直接推广。相比之下，对于每个固定的$\delta$，我们设计了一种算法，其竞争比在$D\to\infty$时收敛于$1-e^{-\delta}$，并且我们证明这是渐近最优的。也许令人惊讶的是，这匹配了一个更宽松设置中的最优竞争比，在该设置中，算法每天获得新的支出限制$\delta B_i$，并且在整个时间跨度内最多可支出$\delta D B_i$。在此过程中，我们刻画了在高流量实例上每一对$(D,\delta)$的精确最优竞争比，其中离线基准每天耗尽每个广告商的预算。
英文摘要：
Search advertising platforms routinely spend beyond an advertiser's average daily budget on high-traffic days, so long as total spending over the month stays within the monthly budget. Motivated by this practice, we study a $D$-day generalization of the Adwords problem (Mehta et al. 2007), where each advertiser $i$ has a nominal (average) daily budget $B_i$ and a total horizon (monthly) budget $DB_i$. Given a flexibility parameter $δ$, the platform may spend at most $δB_i$ on advertiser $i$ on any single day, subject to the horizon spending limit of $DB_i$. We quantify the power of $δ$-flexible budgets by benchmarking against the inflexible offline optimum, which may spend at most $B_i$ on advertiser $i$ on each day. We show that no amount of flexibility helps direct generalizations of the classical algorithm of Mehta et al. (2007). By contrast, for every fixed $δ$, we design an algorithm whose competitive ratio converges to $1-e^{-δ}$ as $D\to\infty$, and we show that this is asymptotically optimal. Perhaps surprisingly, this matches the optimal competitive ratio in a more permissive setting where the algorithm receives a fresh spending limit of $δB_i$ each day and may spend up to $δD B_i$ over the horizon. Along the way, we characterize the exact optimal competitive ratio for every pair $(D,δ)$ on high-traffic instances, where the offline benchmark exhausts every advertiser's budget on every day.
