An Efficient Algorithm for Minimizing Ordered Norms in Fractional Load Balancing

2025-11-14

AI Summary

This paper studies the optimization problem of minimizing the ordered norm of the load vector in fractional load balancing. To overcome the bottleneck of conventional methods—which rely heavily on linear optimization oracle calls—we propose a stochastic online algorithm based on smoothed approximation and regularized Follow-the-Regularized-Leader (FTRL). Our method incorporates dynamic cost budgeting, non-uniform updates, and rejection sampling, and employs martingale analysis with adaptive step-size control. We present the first efficient approximation algorithm applicable to *arbitrary* ordered norms, achieving a $(1+varepsilon)$-approximate solution with high probability. The algorithm requires only $Oig((n+d)(varepsilon^{-2} + log log d) log(n+d)ig)$ oracle calls—significantly improving upon prior complexity bounds. Our approach combines theoretical rigor with broad applicability across diverse norm structures in load-balancing settings.

Technology Category

• Search and Optimization: Algorithm Configuration
• Machine Learning: Online Learning & Bandits
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments

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, $mathcal{O}((n+d) (varepsilon^{-2}+loglog d)log (n+d))$ 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.

Problem

• Minimizing ordered norms in fractional load balancing problems
• Developing efficient approximation algorithms for convex optimization
• Addressing limitations of existing techniques for arbitrary norms

Innovation

• Randomized algorithm using resource price mechanism
• Dynamic cost budgets for non-uniform customer updates
• Smooth approximations of ordered norms with stability