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On the Peril of (Even a Little) Nonstationarity in Satisficing Regret Minimization

Yixuan Zhang, Ruihao Zhu, Qiaomin Xie

Abstract

Motivated by the principle of satisficing in decision-making, we study satisficing regret guarantees for nonstationary $K$-armed bandits. We show that in the general realizable, piecewise-stationary setting with $L$ stationary segments, the optimal regret is $Θ(L\log T)$ as long as $L\geq 2$. This stands in sharp contrast to the case of $L=1$ (i.e., the stationary setting), where a $T$-independent $Θ(1)$ satisficing regret is achievable under realizability. In other words, the optimal regret has to scale with $T$ even if just a little nonstationarity presents. A key ingredient in our analysis is a novel Fano-based framework tailored to nonstationary bandits via a \emph{post-interaction reference} construction. This framework strictly extends the classical Fano method for passive estimation as well as recent interactive Fano techniques for stationary bandits. As a complement, we also discuss a special regime in which constant satisficing regret is again possible.

On the Peril of (Even a Little) Nonstationarity in Satisficing Regret Minimization

Abstract

Motivated by the principle of satisficing in decision-making, we study satisficing regret guarantees for nonstationary -armed bandits. We show that in the general realizable, piecewise-stationary setting with stationary segments, the optimal regret is as long as . This stands in sharp contrast to the case of (i.e., the stationary setting), where a -independent satisficing regret is achievable under realizability. In other words, the optimal regret has to scale with even if just a little nonstationarity presents. A key ingredient in our analysis is a novel Fano-based framework tailored to nonstationary bandits via a \emph{post-interaction reference} construction. This framework strictly extends the classical Fano method for passive estimation as well as recent interactive Fano techniques for stationary bandits. As a complement, we also discuss a special regime in which constant satisficing regret is again possible.
Paper Structure (38 sections, 6 theorems, 50 equations, 1 figure, 2 algorithms)

This paper contains 38 sections, 6 theorems, 50 equations, 1 figure, 2 algorithms.

Table of Contents

  1. Introduction
  2. Our Contributions
  3. Related Work
  4. Nonstationary bandits.
  5. Stationary satisficing regret.
  6. Threshold-based pure exploration and other satisficing notions.
  7. Notation
  8. Problem Formulation
  9. Nonstationary Bandits
  10. Satisficing Regret.
  11. Satisficing nonstationary bandits.
  12. A Fano-Based Framework for Nonstationary Bandit Problems
  13. Conditional Fano.
  14. Classical Fano Reduction: Small Regret Implies Correct Identification
  15. Post-Interaction Reference and Two Complementary Approaches
  16. ...and 23 more sections

Key Result

Theorem 1

Suppose $L\ge 3$ and $\Delta^2 T \ge L$. Then $\inf_{\pi \in \Pi}\sup_{\nu \in \mathcal{E}_{L,\Delta}}R_T^S(\pi;\nu)\gtrsim L\log (\Delta^2T/L)/\Delta.$

Figures (1)

  • Figure 1: Schematic illustration: for $L=3$ one can create a separated local window to enable a clean identification reduction; for $L=2$ the analogous separation fails (motivating the information-budget approach).

Theorems & Definitions (6)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Lemma 1
  • Lemma 2