We study the problem of minimizing an ordered norm of a load vector (indexed by a set of $d$ resources), where a finite number $n$ of customers $c$ contribute to the load of each resource by choosing a solution $x_c$ in a convex set $X_c \subseteq \mathbb{R}^d_{\geq 0}$; so we minimize $||\sum_{c}x_c||$ for some fixed ordered norm $||\cdot||$. We devise a randomized algorithm that computes a $(1+\varepsilon)$-approximate solution to this problem and makes, with high probability, $\mathcal{O}\left((n+d) (\varepsilon^{-2}+\log\log d)\log (n+d)\right)$ calls to oracles that minimize linear functions (with non-negative coefficients) over $X_c$. While this has been known for the $\ell_{\infty}$ norm via the multiplicative weights update method, existing proof techniques do not extend to arbitrary ordered norms. Our algorithm uses a resource price mechanism that is motivated by the follow-the-regularized-leader paradigm, and is expressed by smooth approximations of ordered norms. We need and show that these have non-trivial stability properties, which may be of independent interest. For each customer, we define dynamic cost budgets, which evolve throughout the algorithm, to determine the allowed step sizes. This leads to non-uniform updates and may even reject certain oracle solutions. Using non-uniform sampling together with a martingale argument, we can guarantee sufficient expected progress in each iteration, and thus bound the total number of oracle calls with high probability.

翻译：我们研究在有限个客户（记为 $n$）中，通过每个客户 $c$ 在凸集 $X_c \subseteq \mathbb{R}^d_{\geq 0}$ 中选择解 $x_c$，最小化由 $d$ 个资源索引的负载向量的有序范数问题，即对于给定的有序范数 $||\cdot||$，最小化 $||\sum_{c}x_c||$。我们设计了一种随机化算法，对该问题计算 $(1+\varepsilon)$-近似解，并以高概率调用 $\mathcal{O}\left((n+d) (\varepsilon^{-2}+\log\log d)\log (n+d)\right)$ 次最小化 $X_c$ 上线性函数（系数非负）的预言机。尽管对于 $\ell_{\infty}$ 范数，该结果可通过乘法权重更新方法实现，但现有证明技术无法推广至任意有序范数。我们的算法采用一种受“跟随正则化领导者”范式启发的资源定价机制，并通过有序范数的光滑近似表达。我们需要证明这些近似具有非平凡的稳定性，该性质可能具有独立的研究价值。对于每个客户，我们定义动态成本预算，该预算在算法过程中演化，以确定允许的步长。这会导致非均匀更新，甚至可能拒绝某些预言机解。利用非均匀采样与鞅论证相结合的方法，我们可保证每次迭代中足够的期望进展，从而以高概率限制总预言机调用次数。