  We present a non-asymptotic lower bound on the eigenspectrum of the design
matrix generated by any linear bandit algorithm with sub-linear regret when the
action set has well-behaved curvature. Specifically, we show that the minimum
eigenvalue of the expected design matrix grows as $\Omega(\sqrt{n})$ whenever
the expected cumulative regret of the algorithm is $O(\sqrt{n})$, where $n$ is
the learning horizon, and the action-space has a constant Hessian around the
optimal arm. This shows that such action-spaces force a polynomial lower bound
rather than a logarithmic lower bound, as shown by \cite{lattimore2017end}, in
discrete (i.e., well-separated) action spaces. Furthermore, while the previous
result is shown to hold only in the asymptotic regime (as $n \to \infty$), our
result for these "locally rich" action spaces is any-time. Additionally, under
a mild technical assumption, we obtain a similar lower bound on the minimum
eigen value holding with high probability.
  We apply our result to two practical scenarios -- \emph{model selection} and
\emph{clustering} in linear bandits. For model selection, we show that an
epoch-based linear bandit algorithm adapts to the true model complexity at a
rate exponential in the number of epochs, by virtue of our novel spectral
bound. For clustering, we consider a multi agent framework where we show, by
leveraging the spectral result, that no forced exploration is necessary -- the
agents can run a linear bandit algorithm and estimate their underlying
parameters at once, and hence incur a low regret.
