Online metric allocation

We introduce a natural online allocation problem that connects several of the most fundamental problems in online optimization. Let $M$ be an $n$-point metric space. Consider a resource that can be allocated in arbitrary fractions to the points of $M$. At each time $t$, a convex monotone cost function $ct: [0,1]\to\mathbb{R}+$ appears at some point $rt\in M$. In response, an algorithm may change the allocation of the resource, paying movement cost as determined by the metric and service cost $ct(x{rt})$, where $x{rt}$ is the fraction of the resource at $rt$ at the end of time $t$. For example, when the cost functions are $ct(x)=\alpha x$, this is equivalent to randomized MTS, and when the cost functions are $ct(x)=\infty\cdot 1{x<1/k}$, this is equivalent to fractional $k$-server. We give an $O(\log n)$-competitive algorithm for weighted star metrics. Due to the generality of allowed cost functions, classical multiplicative update algorithms do not work for the metric allocation problem. A key idea of our algorithm is to decouple the rate at which a variable is updated from its value, resulting in interesting new dynamics. This can be viewed as running mirror descent with a time-varying regularizer, and we use this perspective to further refine the guarantees of our algorithm. The standard analysis techniques run into multiple complications when the regularizer is time-varying, and we show how to overcome these issues by making various modifications to the default potential function. We also consider the problem when cost functions are allowed to be non-convex. In this case, we give tight bounds of $\Theta(n)$ on tree metrics, which imply deterministic and randomized competitive ratios of $O(n^2)$ and $O(n\log n)$ respectively on arbitrary metrics. Our algorithm is based on an $\ell_2^{2$-regularizer.}