  This paper is motivated by recent research in the $d$-dimensional stochastic
linear bandit literature, which has revealed an unsettling discrepancy:
algorithms like Thompson sampling and Greedy demonstrate promising empirical
performance, yet this contrasts with their pessimistic theoretical regret
bounds. The challenge arises from the fact that while these algorithms may
perform poorly in certain problem instances, they generally excel in typical
instances. To address this, we propose a new data-driven technique that tracks
the geometric properties of the uncertainty ellipsoid around the main problem
parameter. This methodology enables us to formulate an instance-dependent
frequentist regret bound, which incorporates the geometric information, for a
broad class of base algorithms, including Greedy, OFUL, and Thompson sampling.
This result allows us to identify and ``course-correct" problem instances in
which the base algorithms perform poorly. The course-corrected algorithms
achieve the minimax optimal regret of order $\tilde{\mathcal{O}}(d\sqrt{T})$
for a $T$-period decision-making scenario, effectively maintaining the
desirable attributes of the base algorithms, including their empirical
efficacy. We present simulation results to validate our findings using
synthetic and real data.
