A broad class of boosting algorithms can be interpreted as performing coordinate-wise gradient descent to minimize some potential function of the margins of a data set. This class includes AdaBoost, LogitBoost, and other widely used and well-studied boosters. In this paper we show that for a broad class of convex potential functions, any such boosting algorithm is highly susceptible to random classification noise. We do this by showing that for any such booster and any nonzero random classification noise rate R, there is a simple data set of examples which is efficiently learnable by such a booster if there is no noise, but which cannot be learned to accuracy better than 1/2 if there is random classification noise at rate R. This negative result is in contrast with known branching program based boosters which do not fall into the convex potential function framework and which can provably learn to high accuracy in the presence of random classification noise.