  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.
