Extended Formulations for Online Linear Bandit Optimization

On-line linear optimization on combinatorial action sets (d-dimensional actions) with bandit feedback, is known to have complexity in the order of the dimension of the problem. The exponential weighted strategy achieves the best known regret bound that is of the order of $d^{2}\sqrt{n}$ (where $d$ is the dimension of the problem, $n$ is the time horizon). However, such strategies are provably suboptimal or computationally inefficient. The complexity is attributed to the combinatorial structure of the action set and the dearth of efficient exploration strategies of the set. Mirror descent with entropic regularization function comes close to solving this problem by enforcing a meticulous projection of weights with an inherent boundary condition. Entropic regularization in mirror descent is the only known way of achieving a logarithmic dependence on the dimension. Here, we argue otherwise and recover the original intuition of exponential weighting by borrowing a technique from discrete optimization and approximation algorithms called `extended formulation'. Such formulations appeal to the underlying geometry of the set with a guaranteed logarithmic dependence on the dimension underpinned by an information theoretic entropic analysis.