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In this paper we initiate a study of non parametric contextual bandits undershape constraints on the mean reward function. Specifically, we study a settingwhere the context is one dimensional, and the mean reward function is isotonicwith respect to this context. We propose a policy for this problem and showthat it attains minimax rate optimal regret. Moreover, we show that the samepolicy enjoys automatic adaptation; that is, for subclasses of the parameterspace where the true mean reward functions are also piecewise constant with $k$pieces, this policy remains minimax rate optimal simultaneously for all $k \geq1.$ Automatic adaptation phenomena are well-known for shape constrainedproblems in the offline setting;  %The phenomenon of automatic adaptation of shape constrained methods is knownto occur in offline problems;  we show that such phenomena carry over to the online setting.  The main technical ingredient underlying our policy is a procedure to deriveconfidence bands for an underlying isotonic function using the isotonicquantile estimator. The confidence band we propose is valid under heavy tailednoise, and its average width goes to $0$ at an adaptively optimal rate. Weconsider this to be an independent contribution to the isotonic regressionliterature.

Regret Minimization in Isotonic, Heavy-Tailed Contextual Bandits via Adaptive Confidence Bands