Quantum algorithms for hedging and the Sparsitron

A paradigmatic algorithm for online learning is the Freund/Schapire Hedge algorithm with multiplicative weight updates. For multiple time steps, the algorithm constructs an allocation into different strategies or experts for which it is guaranteed that a certain regret is never much greater than the minimally achievable, even in an adversarial situation. This work presents quantum algorithms for such online learning in an oracular setting. For $T$ time steps and $N$ strategies, we exhibit run times of about $O \left ({\rm poly} (T) \sqrt{N} \right)$ for passively estimating the hedging losses and for actively betting according to quantum sampling. In addition, we discuss a quantum analogue of a machine learning algorithm, the Sparsitron, which is based on the Hedge algorithm. The quantum algorithm inherits the provable learning guarantees from the classical algorithm and exhibits polynomial speedups. The speedups shown here may find relevance in both finance, for example for estimating hedging losses, and machine learning, for example for learning a generalized linear model or an Ising model.