<p>A general framework with a series of different methods that can improve the estimate of convex function (or functional) values using only noisy observations of the true input is studied. Technically, our methods catch the bias introduced by the convexity and remove this bias from a baseline estimate. Comprehensive theoretical analyses are conducted to show that the proposed methods can strictly reduce the expected estimate error under mild conditions. When applied, the methods require no specific knowledge about the problem except the convexity and the evaluation of the function. Therefore, they can serve as off-the-shelf tools to obtain good estimates for a wide range of problems, including optimization problems with random objective functions or constraints, and functionals of probability distributions such as the entropy and the Wasserstein distance. Numerical experiments on a wide variety of problems show that our methods can significantly improve the quality of the estimate compared with the baseline method.</p>