This work proposes an algorithm improving the dimensionality dependence for gradient-free optimisation over cross-polytopes, which has many applications such as adversarial attacks, explainable AI and sparse regression. For bandit convex optimisation with two-point feedback over cross-polytopes, the state-of-the-art algorithms have a dimensionality dependence of $\mathcal{O}(\sqrt{d\log d})$, while the known lower bound is of the form $\Omega(\sqrt{d(\log d)^{-1}})$. We propose a mirror descent algorithm equipped with a symmetric version of the negative $\frac{1}{2}$-Tsallis entropy. Combined with an $\ell_1$-ellipsoidal smoothing-based gradient estimator, the proposed algorithm guarantees a dimensionality dependence on $\mathcal{O}(\sqrt{d})$, which improves the state-of-the-art algorithms by a factor of $\sqrt{\log d}$. The idea can be further applied to optimising non-smooth and non-convex functions. We propose an algorithm with a convergence depending on $\mathcal{O}(d)$, which is the best-known dimensionality dependence.

Abstract This paper proposes a new family of algorithms for the online optimisation of composite objectives. The algorithms can be interpreted as the combination of the exponentiated gradient and p-norm algorithm. Combined with algorithmic ideas of adaptivity and optimism, the proposed algorithms achieve a sequence-dependent regret upper bound, matching the best-known bounds for sparse target decision variables. Furthermore, the algorithms have efficient implementations for popular composite objectives and constraints and can be converted to stochastic optimisation algorithms with the optimal accelerated rate for smooth objectives.