Statistical Complexity and Optimal Algorithms for Non-linear Ridge Bandits
This work addresses the challenge of non-linear bandits for researchers in sequential decision-making, providing optimal algorithms for a specific function class.
The paper tackles the problem of sequential decision-making with non-linear mean outcomes, identifying a distinct 'burn-in period' with fixed cost and showing that a two-stage algorithm is statistically optimal for ridge functions, while classical methods like UCB are suboptimal.
We consider the sequential decision-making problem where the mean outcome is a non-linear function of the chosen action. Compared with the linear model, two curious phenomena arise in non-linear models: first, in addition to the "learning phase" with a standard parametric rate for estimation or regret, there is an "burn-in period" with a fixed cost determined by the non-linear function; second, achieving the smallest burn-in cost requires new exploration algorithms. For a special family of non-linear functions named ridge functions in the literature, we derive upper and lower bounds on the optimal burn-in cost, and in addition, on the entire learning trajectory during the burn-in period via differential equations. In particular, a two-stage algorithm that first finds a good initial action and then treats the problem as locally linear is statistically optimal. In contrast, several classical algorithms, such as UCB and algorithms relying on regression oracles, are provably suboptimal.