Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Strategic Linear Contextual Bandits

Thomas Kleine Buening, Aadirupa Saha, Christos Dimitrakakis, Haifeng Xu

TL;DR

This work introduces a strategic variant of linear contextual bandits where arms can misreport privately observed contexts to gain more pulls, framing the problem as online learning with mechanism design. It presents two mechanisms—Greedy Grim Trigger Mechanism (GGTM) when $θ^*$ is known and Optimistic Grim Trigger Mechanism (OptGTM) when $θ^*$ is unknown—that incentivize truthfulness and bound regret under Nash equilibria. GGTM achieves ${\tilde{O}}(K^2 \sqrt{KT})$ strategic regret, while OptGTM attains ${\tilde{O}}(d\sqrt{KT})$ under truthful play and at most ${\tilde{O}}(dK^2 \sqrt{KT})$ in any NE, highlighting a trade-off between incentive design and regret minimization. The paper further demonstrates, via experiments, that OptGTM robustly limits manipulation and outperforms incentive-unaware methods like LinUCB when arms strategically adapt. Overall, the work advances understanding at the intersection of online learning and mechanism design and offers practical insights for safeguarding recommender systems against strategic manipulation.

Abstract

Motivated by the phenomenon of strategic agents gaming a recommender system to maximize the number of times they are recommended to users, we study a strategic variant of the linear contextual bandit problem, where the arms can strategically misreport privately observed contexts to the learner. We treat the algorithm design problem as one of mechanism design under uncertainty and propose the Optimistic Grim Trigger Mechanism (OptGTM) that incentivizes the agents (i.e., arms) to report their contexts truthfully while simultaneously minimizing regret. We also show that failing to account for the strategic nature of the agents results in linear regret. However, a trade-off between mechanism design and regret minimization appears to be unavoidable. More broadly, this work aims to provide insight into the intersection of online learning and mechanism design.

Strategic Linear Contextual Bandits

TL;DR

This work introduces a strategic variant of linear contextual bandits where arms can misreport privately observed contexts to gain more pulls, framing the problem as online learning with mechanism design. It presents two mechanisms—Greedy Grim Trigger Mechanism (GGTM) when is known and Optimistic Grim Trigger Mechanism (OptGTM) when is unknown—that incentivize truthfulness and bound regret under Nash equilibria. GGTM achieves strategic regret, while OptGTM attains under truthful play and at most in any NE, highlighting a trade-off between incentive design and regret minimization. The paper further demonstrates, via experiments, that OptGTM robustly limits manipulation and outperforms incentive-unaware methods like LinUCB when arms strategically adapt. Overall, the work advances understanding at the intersection of online learning and mechanism design and offers practical insights for safeguarding recommender systems against strategic manipulation.

Abstract

Motivated by the phenomenon of strategic agents gaming a recommender system to maximize the number of times they are recommended to users, we study a strategic variant of the linear contextual bandit problem, where the arms can strategically misreport privately observed contexts to the learner. We treat the algorithm design problem as one of mechanism design under uncertainty and propose the Optimistic Grim Trigger Mechanism (OptGTM) that incentivizes the agents (i.e., arms) to report their contexts truthfully while simultaneously minimizing regret. We also show that failing to account for the strategic nature of the agents results in linear regret. However, a trade-off between mechanism design and regret minimization appears to be unavoidable. More broadly, this work aims to provide insight into the intersection of online learning and mechanism design.
Paper Structure (48 sections, 22 theorems, 124 equations, 3 figures, 4 algorithms)

This paper contains 48 sections, 22 theorems, 124 equations, 3 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Linear Contextual Bandits.
  4. Strategic Multi-Armed Bandits.
  5. Modeling Incentives in Recommender Systems.
  6. Strategic Learning.
  7. Strategic Linear Contextual Bandits
  8. Strategic Arms and Nash Equilibrium
  9. Strategic Regret
  10. Regularity Assumptions.
  11. The Necessity of Mechanism Design
  12. Warm-Up: $\theta^*$ is Known in Advance
  13. The Greedy Grim Trigger Mechanism
  14. Grim Trigger Condition.
  15. Selection Rule.
  16. ...and 33 more sections

Key Result

Proposition 3.3

On any non-trivial problem instance, the incentive-unaware greedy algorithm that in round $t$ plays ${i_t = \mathop{\mathrm{argmax}}\limits_{i \in [K]} \langle \theta^*, x_{t,i} \rangle}$ (with ties broken uniformly) suffers linear regret $\Omega(T)$ when the arms act according to any Nash equilibri

Figures (3)

  • Figure 1: Comparison of the strategic regret of $\textnormal{OptGTM}$ and LinUCB. The strategic arms adapt their strategies gradually over the course of 20 epochs. $\textnormal{OptGTM}$ performs similarly across all epochs, whereas LinUCB performs increasingly worse as the arms adapt to the algorithm (Figure \ref{['fig:epoch_regret']}). Figure \ref{['fig:before_regret']} and \ref{['fig:after_regret']} provide a closer look at the regret of the algorithms across the $T$ rounds in the initial epoch, where the arms are truthful, and the final epoch after the arms have adapted to the algorithms.
  • Figure 2: Total context manipulation ${\sum_{t,i} \lVert x_{t,i}^* - x_{t,i} \rVert_2}$.
  • Figure 3: Utility of the arms for each of the 10 runs.

Theorems & Definitions (51)

  • Definition 3.1: Nash Equilibrium
  • Definition 3.2: $\varepsilon$-Nash Equilibrium
  • Proposition 3.3
  • proof : Proof Sketch
  • Theorem 4.1
  • proof : Proof Sketch.
  • Theorem 4.2
  • proof : Proof Sketch.
  • Remark 4.3
  • Theorem 5.1
  • ...and 41 more