[hal-00736203, v1] Empirical risk minimization is optimal for the convex aggregation problem

Let F be a finite model of cardinality M and denote by conv(F ) its convex hull. The problem of convex aggregation is to construct a procedure having a risk as close as possible to the minimal risk over conv(F ). Consider the bounded regression model with respect to the squared risk denoted by R(·). If f̂ n denotes the empirical risk minimization procedure over conv(F ) then we prove that for any x > 0, with probability greater than 1− 4 exp(−x), R(f̂ n ) ≤ min f∈conv(F ) R(f) + c0 max ( ψ n (M), x n ) where c0 > 0 is an absolute constant and ψ (C) n (M) is the optimal rate of convex aggregation defined in [37] by ψ (C) n (M) = M/n when M ≤ √ n and ψ (C) n (M) = √ log ( eM/ √ n ) /n when M > √ n.