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We consider the sequential decision-making problem where the mean outcome isa non-linear function of the chosen action. Compared with the linear model, twocurious 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-linearfunction; second, achieving the smallest burn-in cost requires new explorationalgorithms. For a special family of non-linear functions named ridge functionsin the literature, we derive upper and lower bounds on the optimal burn-incost, and in addition, on the entire learning trajectory during the burn-inperiod via differential equations. In particular, a two-stage algorithm thatfirst finds a good initial action and then treats the problem as locally linearis statistically optimal. In contrast, several classical algorithms, such asUCB and algorithms relying on regression oracles, are provably suboptimal.

Statistical Complexity and Optimal Algorithms for Non-linear Ridge Bandits