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We introduce a new online learning framework where, at each trial, thelearner is required to select a subset of actions from a given known actionset. Each action is associated with an energy value, a reward and a cost. Thesum of the energies of the actions selected cannot exceed a given energybudget. The goal is to maximise the cumulative profit, where the profitobtained on a single trial is defined as the difference between the maximumreward among the selected actions and the sum of their costs. Action energyvalues and the budget are known and fixed. All rewards and costs associatedwith each action change over time and are revealed at each trial only after thelearner's selection of actions. Our framework encompasses several onlinelearning problems where the environment changes over time; and the solutiontrades-off between minimising the costs and maximising the maximum reward ofthe selected subset of actions, while being constrained to an action energybudget. The algorithm that we propose is efficient and general in that it maybe specialised to multiple natural online combinatorial problems.

MaxHedge: Maximising a Maximum Online