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Water-Filling is Universally Minimax Optimal

Siddhartha Banerjee, Ramiro N. Deo-Campo Vuong, Robert Kleinberg

Abstract

Allocation of dynamically-arriving (i.e., online) divisible resources among a set of offline agents is a fundamental problem, with applications to online marketplaces, scheduling, portfolio selection, signal processing, and many other areas. The water-filling algorithm, which allocates an incoming resource to maximize the minimum load of compatible agents, is ubiquitous in many of these applications whenever the underlying objectives prefer more balanced solutions; however, the analysis and guarantees differ across settings. We provide a justification for the widespread use of water-filling by showing that it is a universally minimax optimal policy in a strong sense. Formally, our main result implies that water-filling is minimax optimal for a large class of objectives -- including both Schur-concave maximization and Schur-convex minimization -- under $α$-regret and competitive ratio measures. This optimality holds for every fixed tuple of agents and resource counts. Remarkably, water-filling achieves these guarantees as a myopic policy, remaining entirely agnostic to the objective function, agent count, and resource availability. Our techniques notably depart from the popular primal-dual analysis of online algorithms, and instead develop a novel way to apply the theory of majorization in online settings to achieve universality guarantees.

Water-Filling is Universally Minimax Optimal

Abstract

Allocation of dynamically-arriving (i.e., online) divisible resources among a set of offline agents is a fundamental problem, with applications to online marketplaces, scheduling, portfolio selection, signal processing, and many other areas. The water-filling algorithm, which allocates an incoming resource to maximize the minimum load of compatible agents, is ubiquitous in many of these applications whenever the underlying objectives prefer more balanced solutions; however, the analysis and guarantees differ across settings. We provide a justification for the widespread use of water-filling by showing that it is a universally minimax optimal policy in a strong sense. Formally, our main result implies that water-filling is minimax optimal for a large class of objectives -- including both Schur-concave maximization and Schur-convex minimization -- under -regret and competitive ratio measures. This optimality holds for every fixed tuple of agents and resource counts. Remarkably, water-filling achieves these guarantees as a myopic policy, remaining entirely agnostic to the objective function, agent count, and resource availability. Our techniques notably depart from the popular primal-dual analysis of online algorithms, and instead develop a novel way to apply the theory of majorization in online settings to achieve universality guarantees.

Paper Structure

This paper contains 42 sections, 37 theorems, 73 equations, 5 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. Overview of Results and Techniques
  3. Related Work
  4. Preliminaries
  5. Setting and Problem Statement
  6. The Allocation Game
  7. Objectives and Competitive Performance
  8. Majorization
  9. Schur-Monotone Functions
  10. Majorization Minimal Offline Allocations
  11. Water Filling
  12. Main Results
  13. Minimax Optimality of water-filling
  14. Characterizing Worst-Case Sequences
  15. Water-Filling, Nested Sequences and Majorization
  16. ...and 27 more sections

Key Result

Theorem 1

Fix parameters $n,m\in\mathbb{N}$ and $q\in\mathbb{R}_{> 0}$. Given any (possibly randomized) allocation policy $\mathcal{A}$ and any request sequence $E\in\mathcal{E}_{n,m,q}$, there is a nested sequence $E'\in\mathcal{E}^{\textsc{nest}}_{n,m,q}$ for which $\textsc{WF}(E) \preceq \mathbb{E}[\mathca

Figures (5)

  • Figure 1: Example of a request sequence $E = ((N_t, q_t))_{t\in[5]}\in\mathcal{E}_{4,5,12}$ with $n=4$ offline nodes, $m=5$ online nodes and quantity $q=12$. The table gives the quantities and adjacency matrix between offline (columns) and online (rows) nodes, and details the allocation $\textsc{OPT}(E)$, and intermediate loads for offline nodes over time. The diagram on the right depicts the request sequence, with online nodes arriving from top to bottom.
  • Figure 2: water-filling allocations on the \ref{['fig:opt_unified']} sequence. The columns from left to right show the quantity and neighbors of each online node, allocations made by $\textsc{water-filling}\xspace$, and intermediate load vectors.
  • Figure 3: The table depicts competitive ratios for various Schur-monotone objectives. The columns, from left to right, are the name and definition of the function, the form of objective (maximization/minimization), and the minimax competitive ratio for given $n$. Notably, the fractional matching objective with capacity $c=1$, $\textsc{FM}_n(\mathbf{x}) = \sum_{i\in[m]} \min(1, \mathbf{x}(i))$, defines the $M_k$ sequence: $M_k = \frac{1}{k} \textsc{FM}_k(H\vec{1})$. For details, see \ref{['appsec:CR']}.
  • Figure 4: The table depicts the sequence $E\in\mathcal{E}_{4,5,12}$ from \ref{['fig:opt_unified']}, and the corresponding induced nested sequence $\widehat{E}\in\mathcal{E}^{\textsc{nest}}_{4,5,12}$. Note that both sequences have the same number of offline nodes (columns) and online nodes (rows). For clarity, we highlight $\mu_i$, the latest arriving online neighbor of each offline node in red.
  • Figure 5: We consider the same allocation sequence from \ref{['fig:opt_unified']}, and demonstrate the transformations (pruning, permuting and nestification) carried out by \ref{['alg:nest']} After each step, we update water-filling allocations; one can verify that the former increases in the majorization preorder (i.e., becomes less balanced). We also track deleted edges in the pruning process via boxes, so one can verify that these are all added back in the end (the red boxes track edges which are in the support of $\textsc{OPT}(E)$). Finally we track the original optimal allocation $\textsc{OPT}(E)$, which becomes infeasible after pruning, but feasible in the final sequence.

Theorems & Definitions (72)

  • Definition 1: Nested sequences
  • Definition 2: Majorization
  • Definition 3: Schur-monotone Functions
  • Definition 4: water-filling
  • Theorem 1: is Majorization Minimal
  • Lemma 1: Majorization Maximality of Nested Sequences
  • Lemma 2: Policy Deviation
  • Corollary 1: Deterministic Minimax Optimality of
  • Theorem 2: Deterministic and Randomized Separation
  • Corollary 2: Randomized Minimax Optimality of
  • ...and 62 more