Optimal Regret Algorithm for Pseudo-1d Bandit Convex Optimization - Saha, Aadirupa; Natarajan, Nagarajan; Netrapalli, Praneeth; Jain, Prateek (2021 International Conference on Machine Learning, 2021)
We study online learning with bandit feedback (i.e. learner has access to only zeroth-order oracle) where cost/reward functions admit a "pseudo-1d" structure, i.e. where the output of is one-dimensional. At each round, the learner observes context , plays prediction (e.g. ) for some and observes loss where is a convex Lipschitz-continuous function. The goal is to minimize the standard regret metric. This pseudo-1d bandit convex optimization problem () arises frequently in domains such as online decision-making or parameter-tuning in large systems. For this problem, we first show a lower bound of for the regret of any algorithm, where is the number of rounds. We propose a new algorithm that combines randomized online gradient descent with a kernelized exponential weights method to exploit the pseudo-1d structure effectively, guaranteeing the [optimal] regret bound mentioned above, up to additional logarithmic factors. In contrast, applying state-of-the-art online convex optimization methods leads to regret, that is significantly suboptimal in .