A Note on Zeroth-Order Optimization on the Simplex

We construct a zeroth-order gradient estimator for a smooth function defined on the probability simplex. The proposed estimator queries the simplex only. We prove that projected gradient descent and the exponential weights algorithm, when run with this estimator instead of exact gradients, converge at a $\mathcal{O}(T^{-1\mathrm{/}4})$ rate.