AdaGrad does not adapt to Hölder-smoothness for composite objectives

We exhibit a simple deterministic one-dimensional convex composite optimization problem for which AdaGrad scheme does not achieve the classical convergence rate $\mathcal{O}(n^{-(1+\nu)/2})$ associated with H\"older-smooth objectives. The example highlights a basic mismatch between classical AdaGrad accumulation and composite optimality. A main insight is that the gradient of the smooth term may not vanish at the optimum, causing AdaGrad to keep reducing its stepsize excessively and converge more slowly. We also discuss why alternative accumulation mechanisms based on gradient mappings or on successive gradient differences, avoid this pathology.