[Paper Review] Regularized Contextual Bandits

Xavier Fontaine, Quentin Berthet+1 more|arXiv (Cornell University)|Oct 11, 2018

Advanced Bandit Algorithms Research3 citations

TL;DR

This paper proposes a regularized contextual bandit algorithm that splits the context space into bins and solves regularized multi-armed bandit problems independently in each bin. It achieves problem-dependent fast rates and problem-independent intermediate rates via a novel margin condition, improving convergence over classical nonparametric contextual bandits under Hölder-smooth reward functions.

ABSTRACT

We consider the stochastic contextual bandit problem with additional regularization. The motivation comes from problems where the policy of the agent must be close to some baseline policy which is known to perform well on the task. To tackle this problem we use a nonparametric model and propose an algorithm splitting the context space into bins, and solving simultaneously - and independently - regularized multi-armed bandit instances on each bin. We derive slow and fast rates of convergence, depending on the unknown complexity of the problem. We also consider a new relevant margin condition to get problem-independent convergence rates, ending up in intermediate convergence rates interpolating between the aforementioned slow and fast rates.

Motivation & Objective

• To address contextual bandit problems where the agent's policy must remain close to a known baseline policy.
• To develop a nonparametric algorithm that maintains regularization while adapting to unknown problem complexity.
• To derive convergence rates that interpolate between slow and fast rates using a new margin condition.
• To establish theoretical guarantees for regret under Hölder-smooth reward functions.
• To provide a method that balances exploration and regularization without requiring prior knowledge of the problem's complexity.

Proposed method

• The algorithm partitions the d-dimensional context space into B^d bins, treating each bin as an independent regularized multi-armed bandit problem.
• Within each bin, it applies a regularized UCB-style policy to balance exploration and proximity to the baseline policy.
• The regularization term enforces that the learned policy remains close to a pre-specified baseline policy to ensure stability and performance.
• A novel margin condition is introduced to derive intermediate convergence rates between slow and fast rates.
• Theoretical analysis uses nonparametric regression techniques, including regressogram-style binning and Hölder continuity assumptions on reward functions.
• Optimal bin size B is selected as a function of T to minimize regret, balancing bias and variance.

Experimental results

Research questions

• RQ1 Can regularization improve convergence rates in contextual bandits when the policy must remain close to a baseline?
• RQ2 What is the optimal trade-off between exploration and regularization in nonparametric contextual bandits?
• RQ3 How do problem-dependent and problem-independent rates interact under a new margin condition?
• RQ4 Can intermediate convergence rates be derived that interpolate between slow and fast rates?
• RQ5 What is the minimax lower bound for regularized contextual bandits under Hölder-smooth reward functions?

Key findings

• The algorithm achieves a regret bound of O((T / log²T)^(-β(1+α)/(2β+d))) for α ∈ (0,1), which interpolates between slow and fast rates.
• The convergence rate depends on the smoothness β of the reward functions and the dimension d, with improved performance under higher smoothness.
• A new margin condition enables intermediate rates that are faster than slow rates but slower than fast rates, depending on problem structure.
• The regret bound is tighter than classical nonparametric contextual bandits, particularly when the baseline policy is well-chosen.
• A lower bound of Ω(T^(-2β/(2β+d))) is established, matching the minimax rate for nonparametric regression under Hölder smoothness.
• The analysis shows that regularization does not degrade convergence and can improve robustness in high-dimensional or complex contexts.