Lipschitz bandits admit near-optimal regret $\widetilde O_d(T^{(d+1)/(d+2)})$ with little memory under full adaptivity, or with few batches under unrestricted memory. We characterize the minimax expected pseudo-regret over $T$ rounds with $W$ bits of memory and at most $B$ batches in dimension $d$. For every memory budget $W$, we prove a lower bound of order $T^{\frac{d+2}{d+3}} (1+(B-1)W)^{-\frac{1}{d(d+3)}}$. When $W\gtrsim_d\log(eT)$, algorithms with fixed batch boundaries attain, up to logarithmic factors, the larger of this bound and the optimal $B$-batch regret with unrestricted memory. The analysis separates fine-scale comparisons from the regional information needed to allocate their samples at low regret. Attaining $\widetilde O_d(T^{(d+1)/(d+2)})$ regret requires both $B=Ω_d(\log\log T)$ and $(B-1)W=\widetildeΩ_d(T^{d/(d+2)})$. With one bit of memory, minimax regret is $\widetildeΘ_d(TB^{-1/(d+2)})$ under adaptive batch boundaries while $Θ_d(T)$ under fixed boundaries.

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