  Stochastic linear bandits with high-dimensional sparse features are a
practical model for a variety of domains, including personalized medicine and
online advertising. We derive a novel $\Omega(n^{2/3})$ dimension-free minimax
regret lower bound for sparse linear bandits in the data-poor regime where the
horizon is smaller than the ambient dimension and where the feature vectors
admit a well-conditioned exploration distribution. This is complemented by a
nearly matching upper bound for an explore-then-commit algorithm showing that
that $\Theta(n^{2/3})$ is the optimal rate in the data-poor regime. The results
complement existing bounds for the data-rich regime and provide another example
where carefully balancing the trade-off between information and regret is
necessary. Finally, we prove a dimension-free $O(\sqrt{n})$ regret upper bound
under an additional assumption on the magnitude of the signal for relevant
features.
