Stochastic Caching via Subset Entropy
A classic approach to beyond worst-case algorithm design is to impose stochastic assumptions on the input. However, a limiting feature of stochastic analyses is that, by the min-max principle, performance on worst-case distributions mirrors that of randomized algorithms on worst-case inputs. In other words, the same shortcoming of worst-case analysis -- its inability to distinguish "easy" and "hard" instances -- reappears as an inability to distinguish "easy" and "hard" distributions. This raises a natural question: Can we characterize "easy" input distributions with useful beyond worst-case bounds? A canonical example is the stochastic caching problem (Aho et al. 1971). When the page requests are drawn i.i.d. from the uniform distribution, the best achievable competitive ratio is $O(\log k)$, matching the performance of the best randomized algorithm on worst-case instances (Fiat et al. 1991). However, when the input distribution has less entropy, intuition suggests that we should be able to do better by exploiting the information provided by the distribution. We formalize this by defining a new information-theoretic parameter called subset entropy which we use to give a fine-grained characterization of the competitive ratio of stochastic caching, including a new analysis for the well-known LRU algorithm on stochastic inputs. While our technical results are for the caching problem, we believe the broader principle -- parameterizing algorithmic performance by an entropy measure of the input -- is of independent interest and might apply to other online/stochastic optimization problems. Indeed, for problems such as (comparison-based) sorting, online matching, load balancing, etc., the hardest stochastic instances involve high-entropy distributions. We hope our work is a step toward a broader theory of fine-grained algorithmic performance for this class of problems.