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We derive oracle inequalities for the problems of isotonic and convexregression using the combination of $Q$-aggregation procedure and sparsitypattern aggregation. This improves upon the previous results including theoracle inequalities for the constrained least squares estimator. One of theimprovements is that our oracle inequalities are sharp, i.e., with leadingconstant 1. It allows us to obtain bounds for the minimax regret thusaccounting for model misspecification, which was not possible based on theprevious results. Another improvement is that we obtain oracle inequalitiesboth with high probability and in expectation.

Sharp oracle bounds for monotone and convex regression through aggregation