An Efficient Algorithm for Minimizing Ordered Norms in Fractional Load Balancing
Daniel Blankenburg, Antonia Ellerbrock, Thomas Kesselheim, Jens Vygen
TL;DR
This work addresses minimizing an ordered norm of a load vector across $d$ resources when each of $n$ customers contributes via a convex choice set $X_c$, with the overall feasible set $X= extstyleigoplus_c X_c$. The authors develop a randomized algorithm that achieves a $(1+oldsymbol extu01)$-approximation using $Oig((n+d)ig(oldsymbol extu01^{-2}+ ext{loglog d}ig) ext{log}(n+d)ig)$ calls to linearminimization oracles, with a per-call cost $O(d ext{log}d)$, and extend to $ au$-approximate oracles to obtain a $( au(1+oldsymbol extu01))$-approximation. The core technique combines a resource-price mechanism inspired by Follow-the-Regularized-Leader with a differentiable norm surrogate $oldsymbol extPsi$ for ordered norms, together with dynamic per-customer budgets and nonuniform sampling analyzed via martingales. A central contribution is the construction of an efficient $( rac{ ext{log}d}{oldsymbol exteta},oldsymbol exteta)$-approximation $oldsymbol extPsi$ of any ordered norm, derived from generalized relative entropy projections onto the dual space $Y$, and shown to have a contraction property that stabilizes the gradient under additive updates. The framework yields efficient solutions for multi-commodity flow and related resource-sharing problems, significantly widening the applicability of efficient ordered-norm minimization beyond the $ ext{l}_{ ext{infty}}$ case.
Abstract
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, $\mathcal{O}((n+d) (\varepsilon^{-2}+\log\log d)\log (n+d))$ 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.