Regret Lower Bound and Optimal Algorithm for High-Dimensional Contextual Linear Bandit
This work addresses the challenge of regret optimization in high-dimensional bandits for machine learning practitioners, offering a theoretically optimal and adaptive solution.
The paper tackles the high-dimensional contextual linear bandit problem by proving a new minimax regret lower bound that unifies existing results based on a margin parameter, and proposes a computationally efficient UCB-based algorithm that matches this lower bound without requiring forced sampling.
In this paper, we consider the multi-armed bandit problem with high-dimensional features. First, we prove a minimax lower bound, $\mathcal{O}\big((\log d)^{\frac{α+1}{2}}T^{\frac{1-α}{2}}+\log T\big)$, for the cumulative regret, in terms of horizon $T$, dimension $d$ and a margin parameter $α\in[0,1]$, which controls the separation between the optimal and the sub-optimal arms. This new lower bound unifies existing regret bound results that have different dependencies on T due to the use of different values of margin parameter $α$ explicitly implied by their assumptions. Second, we propose a simple and computationally efficient algorithm inspired by the general Upper Confidence Bound (UCB) strategy that achieves a regret upper bound matching the lower bound. The proposed algorithm uses a properly centered $\ell_1$-ball as the confidence set in contrast to the commonly used ellipsoid confidence set. In addition, the algorithm does not require any forced sampling step and is thereby adaptive to the practically unknown margin parameter. Simulations and a real data analysis are conducted to compare the proposed method with existing ones in the literature.