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 $c_t: [0,1]\to\mathbb{R}_+$ appears at some point $r_t\in M$. In response, an algorithm may change the allocation of the resource, paying movement cost as determined by the metric and service cost $c_t(x_{r_t})$, where $x_{r_t}$ is the fraction of the resource at $r_t$ at the end of time $t$. For example, when the cost functions are $c_t(x)=\alpha x$, this is equivalent to randomized MTS, and when the cost functions are $c_t(x)=\infty\cdot 1_{x<1/k}$, 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.
翻译:我们引入了一个自然的在线分配问题, 将在线优化中几个最基本的问题连接起来。 让 $M 成为美元 。 考虑一个资源可以任意分配到美元。 $t$ 。 每次美元, comvex 单调成本函数 $c_ t: [0, 1\ to\\\\\\\\\\\\\\\R\\\\M$, 在某些点( $美元) 。 作为回应, 一个算法可以改变资源的分配, 支付由美元 (x\\\\\\\ t) 的递增成本, 支付由美元 美元 (x\\\\\\ t) 美元 确定的流动成本, 美元是任意的, 美元是美元 美元 美元 。 当成本函数是美元(x) 时, 当成本(x\\\\\\\\\\\\\\\\\ xxxxxxxxxxxxxxxxxx), 成本功能会变换到 美元, 美元。 我们的递变换到 美元 美元, 美元 美元 货币的货币计算法。