We study an online caching problem in which requests can be served by a local cache to avoid retrieval costs from a remote server. The cache can update its state after a batch of requests and store an arbitrarily small fraction of each file. We study no-regret algorithms based on Online Mirror Descent (OMD) strategies. We show that bounds for the regret crucially depend on the diversity of the request process, provided by the diversity ratio R/h, where R is the size of the batch, and h is the maximum multiplicity of a request in a given batch. We characterize the optimality of OMD caching policies w.r.t. regret under different diversity regimes. We also prove that, when the cache must store the entire file, rather than a fraction, OMD strategies can be coupled with a randomized rounding scheme that preserves regret guarantees, even when update costs cannot be neglected. We provide a formal characterization of the rounding problem through optimal transport theory, and moreover we propose a computationally efficient randomized rounding scheme.

翻译：我们研究一类在线缓存问题，其中请求可通过本地缓存服务以避免远程服务器的检索成本。缓存可在批量请求处理后更新状态，并存储每个文件的任意小部分。我们基于在线镜像下降（OMD）策略研究无憾算法。研究表明，遗憾界限关键取决于请求过程的多样性，该多样性由多样性比R/h刻画，其中R为批次大小，h为给定批次中请求的最大重复次数。我们刻画了不同多样性体制下OMD缓存策略关于遗憾的最优性。同时证明，当缓存必须存储整个文件而非部分时，OMD策略可与能保持遗憾保证的随机取整方案相结合，即便更新成本不可忽略。我们通过最优传输理论对取整问题进行了形式化刻画，并进一步提出一种计算高效的随机取整方案。