Projection step In this step, we compute: This moves θ in the direction of steepest descent (∇F) with step size s, and then the function [•] ∆ projects the resulting point onto the simplex; that is, it finds the nearest point that satisfies the constraints (8).
the gradient ∇F(θ k ) is to Schoenemann (2011b), we use an O(n log n) algorithm for the projection step due to Duchi et.