  We study the optimal batch-regret tradeoff for batch linear contextual
bandits. For any batch number $M$, number of actions $K$, time horizon $T$, and
dimension $d$, we provide an algorithm and prove its regret guarantee, which,
due to technical reasons, features a two-phase expression as the time horizon
$T$ grows. We also prove a lower bound theorem that surprisingly shows the
optimality of our two-phase regret upper bound (up to logarithmic factors) in
the \emph{full range} of the problem parameters, therefore establishing the
exact batch-regret tradeoff.
  Compared to the recent work \citep{ruan2020linear} which showed that $M =
O(\log \log T)$ batches suffice to achieve the asymptotically minimax-optimal
regret without the batch constraints, our algorithm is simpler and easier for
practical implementation. Furthermore, our algorithm achieves the optimal
regret for all $T \geq d$, while \citep{ruan2020linear} requires that $T$
greater than an unrealistically large polynomial of $d$.
  Along our analysis, we also prove a new matrix concentration inequality with
dependence on their dynamic upper bounds, which, to the best of our knowledge,
is the first of its kind in literature and maybe of independent interest.
