  We derive oracle inequalities for the problems of isotonic and convex
regression using the combination of $Q$-aggregation procedure and sparsity
pattern aggregation. This improves upon the previous results including the
oracle inequalities for the constrained least squares estimator. One of the
improvements is that our oracle inequalities are sharp, i.e., with leading
constant 1. It allows us to obtain bounds for the minimax regret thus
accounting for model misspecification, which was not possible based on the
previous results. Another improvement is that we obtain oracle inequalities
both with high probability and in expectation.
