Beyond Non-Degeneracy: Revisiting Certainty Equivalent Heuristic for Online Linear Programming

TL;DR

This paper analyzes the Certainty Equivalence (CE) heuristic for online linear programming with general (non-discrete) request distributions. It proves that CE achieves uniformly near-optimal hindsight regret under mild distributional assumptions, with rates that interpolate between $O((\log T)^2)$ (when $\beta=0$) and $O(T^{1/2 - 1/(2(1+\beta))})$ (when $\beta>0$), without relying on fluid non-degeneracy or second-order growth. A key methodological advance is an empirical-process concentration analysis of the dual offline problems, enabling robust regret bounds via peeling arguments that do not rely on martingale structures. The results clarify that dual uniqueness—not non-degeneracy—is the critical determinant of CE performance in both discrete and non-discrete settings, resolving a long-standing paradox and highlighting CE’s applicability to a broad class of online decision problems with continuous reward distributions.

Abstract

The Certainty Equivalent heuristic (CE) is a widely-used algorithm for various dynamic resource allocation problems in OR and OM. Despite its popularity, existing theoretical guarantees of CE are limited to settings satisfying restrictive fluid regularity conditions, particularly, the non-degeneracy conditions, under the widely held belief that the violation of such conditions leads to performance deterioration and necessitates algorithmic innovation beyond CE. In this work, we conduct a refined performance analysis of CE within the general framework of online linear programming. We show that CE achieves uniformly near-optimal regret (up to a polylogarithmic factor in $T$) under only mild assumptions on the underlying distribution, without relying on any fluid regularity conditions. Our result implies that, contrary to prior belief, CE effectively beats the curse of degeneracy for a wide range of problem instances with continuous conditional reward distributions, highlighting the distinction of the problem's structure between discrete and non-discrete settings. Our explicit regret bound interpolates between the mild $(\log T)^2$ regime and the worst-case $\sqrt{T}$ regime with a parameter $β$ quantifying the minimal rate of probability accumulation of the conditional reward distributions, generalizing prior findings in the multisecretary setting. To achieve these results, we develop novel algorithmic analytical techniques. Drawing tools from the empirical processes theory, we establish strong concentration analysis of the solutions to random linear programs, leading to improved regret analysis under significantly relaxed assumptions. These techniques may find potential applications in broader online decision-making contexts.

Figures (3)

• Figure 1: Function $V^{\rm fluid}$ and dual function $f_{d}$ in degenerate and non-degenerate settings.
• Figure 2: Function $V^{\rm fluid}$ and dual function $f_{d}$ for two different parameters in a non-discrete setting.
• Figure 3: Function $V^{\rm fluid}$ and dual function $f_{d}$ for multisecretary instances with gap.

Theorems & Definitions (51)

• Theorem 3.1: Achievable regret of CE
• Corollary 3.1
• Proposition 3.2: Fundamental Regret Lower Bound
• Lemma 4.1
• Corollary 4.2
• Lemma 4.3
• Lemma 4.4
• Lemma 4.5
• Proposition 4.6
• Lemma 5.1: Regret Decomposition of CE