Optimistic Optimisation of Composite Objective with Exponentiated Update

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.