  We study the logistic bandit, in which rewards are binary with success
probability $\exp(\beta a^\top \theta) / (1 + \exp(\beta a^\top \theta))$ and
actions $a$ and coefficients $\theta$ are within the $d$-dimensional unit ball.
While prior regret bounds for algorithms that address the logistic bandit
exhibit exponential dependence on the slope parameter $\beta$, we establish a
regret bound for Thompson sampling that is independent of $\beta$.
Specifically, we establish that, when the set of feasible actions is identical
to the set of possible coefficient vectors, the Bayesian regret of Thompson
sampling is $\tilde{O}(d\sqrt{T})$. We also establish a $\tilde{O}(\sqrt{d\eta
T}/\lambda)$ bound that applies more broadly, where $\lambda$ is the worst-case
optimal log-odds and $\eta$ is the "fragility dimension," a new statistic we
define to capture the degree to which an optimal action for one model fails to
satisfice for others. We demonstrate that the fragility dimension plays an
essential role by showing that, for any $\epsilon > 0$, no algorithm can
achieve $\mathrm{poly}(d, 1/\lambda)\cdot T^{1-\epsilon}$ regret.
