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Extending Wormald's Differential Equation Method to One-sided Bounds

Patrick Bennett, Calum MacRury

TL;DR

This work introduces a one-sided extension of Wormald's differential equation method that upper-bounds the evolution of multiple random processes when only inequality control on expected one-step changes is available, rather than tight equalities. The authors develop a cooperative-system framework and prove a high-probability upper bound for Z_j(i) driven by a system y_j(t) solving y_j' = f_j(t, y_1,..., y_a), with explicit error terms and stopping-time considerations. The method combines a probabilistic Doob-martingale decomposition with Freedman's concentration inequality and a deterministic comparison via a cooperative, Lipschitz drift, augmented by a critical-interval analysis. They also show how to recover a standard two-sided Wormald-type theorem within the same setting, yielding explicit bounds and extending the technique to classical differential-equation results. The approach has practical impact in bounding online algorithm performance and in proving impossibility results in online stochastic optimization problems.

Abstract

In this note, we formulate a "one-sided" version of Wormald's differential equation method. In the standard "two-sided" method, one is given a family of random variables which evolve over time and which satisfy some conditions including a tight estimate of the expected change in each variable over one time step. These estimates for the expected one-step changes suggest that the variables ought to be close to the solution of a certain system of differential equations, and the standard method concludes that this is indeed the case. We give a result for the case where instead of a tight estimate for each variable's expected one-step change, we have only an upper bound. Our proof is very simple, and is flexible enough that if we instead assume tight estimates on the variables, then we recover the conclusion of the standard differential equation method.

Extending Wormald's Differential Equation Method to One-sided Bounds

TL;DR

This work introduces a one-sided extension of Wormald's differential equation method that upper-bounds the evolution of multiple random processes when only inequality control on expected one-step changes is available, rather than tight equalities. The authors develop a cooperative-system framework and prove a high-probability upper bound for Z_j(i) driven by a system y_j(t) solving y_j' = f_j(t, y_1,..., y_a), with explicit error terms and stopping-time considerations. The method combines a probabilistic Doob-martingale decomposition with Freedman's concentration inequality and a deterministic comparison via a cooperative, Lipschitz drift, augmented by a critical-interval analysis. They also show how to recover a standard two-sided Wormald-type theorem within the same setting, yielding explicit bounds and extending the technique to classical differential-equation results. The approach has practical impact in bounding online algorithm performance and in proving impossibility results in online stochastic optimization problems.

Abstract

In this note, we formulate a "one-sided" version of Wormald's differential equation method. In the standard "two-sided" method, one is given a family of random variables which evolve over time and which satisfy some conditions including a tight estimate of the expected change in each variable over one time step. These estimates for the expected one-step changes suggest that the variables ought to be close to the solution of a certain system of differential equations, and the standard method concludes that this is indeed the case. We give a result for the case where instead of a tight estimate for each variable's expected one-step change, we have only an upper bound. Our proof is very simple, and is flexible enough that if we instead assume tight estimates on the variables, then we recover the conclusion of the standard differential equation method.
Paper Structure (5 sections, 7 theorems, 65 equations)

This paper contains 5 sections, 7 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Motivating Applications
  3. Main Theorem
  4. Proving \ref{['thm:de_one_sided']}
  5. Recovering a Version of Wormald's Theorem

Key Result

Theorem 1

Suppose $f:\mathbb{R}^2 \rightarrow \mathbb{R}$ is Lipschitz continuous, and $y=y(t)$ satisfies Suppose $z=z(t)$ is differentiable and satisfies Then $z(t) \le y(t)$ for all $t \ge c$.

Theorems & Definitions (15)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Remark 4: Simplified Parameters
  • Remark 5: Stopping Time Selection
  • Remark 6: Competitive Functions
  • Corollary 7: of \ref{['thm:de_one_sided']}
  • proof : Proof of \ref{['cor:assuming_conditions']}
  • proof : Proof of \ref{['thm:comparisonmulti']}
  • Theorem 8: Freedman's Inequality freedman1975
  • ...and 5 more