Online Learning with Unknown Constraints
This addresses the problem of safe online decision-making for applications where constraints are initially unknown, representing a novel theoretical framework rather than an incremental improvement.
The paper tackles online learning with unknown safety constraints that must be satisfied at every round, developing a meta-algorithm that minimizes regret relative to the best safe action while maintaining safety with high probability. The algorithm achieves √T regret for linear constraints using a transformation that balances exploration and constraint satisfaction.
We consider the problem of online learning where the sequence of actions played by the learner must adhere to an unknown safety constraint at every round. The goal is to minimize regret with respect to the best safe action in hindsight while simultaneously satisfying the safety constraint with high probability on each round. We provide a general meta-algorithm that leverages an online regression oracle to estimate the unknown safety constraint, and converts the predictions of an online learning oracle to predictions that adhere to the unknown safety constraint. On the theoretical side, our algorithm's regret can be bounded by the regret of the online regression and online learning oracles, the eluder dimension of the model class containing the unknown safety constraint, and a novel complexity measure that captures the difficulty of safe learning. We complement our result with an asymptotic lower bound that shows that the aforementioned complexity measure is necessary. When the constraints are linear, we instantiate our result to provide a concrete algorithm with $\sqrt{T}$ regret using a scaling transformation that balances optimistic exploration with pessimistic constraint satisfaction.