  We study the Stochastic Multi-armed Bandit problem under bounded arm-memory.
In this setting, the arms arrive in a stream, and the number of arms that can
be stored in the memory at any time, is bounded. The decision-maker can only
pull arms that are present in the memory. We address the problem from the
perspective of two standard objectives: 1) regret minimization, and 2) best-arm
identification. For regret minimization, we settle an important open question
by showing an almost tight hardness. We show {\Omega}(T^{2/3}) cumulative
regret in expectation for arm-memory size of (n-1), where n is the number of
arms. For best-arm identification, we study two algorithms. First, we present
an O(r) arm-memory r-round adaptive streaming algorithm to find an
{\epsilon}-best arm. In r-round adaptive streaming algorithm for best-arm
identification, the arm pulls in each round are decided based on the observed
outcomes in the earlier rounds. The best-arm is the output at the end of r
rounds. The upper bound on the sample complexity of our algorithm matches with
the lower bound for any r-round adaptive streaming algorithm. Secondly, we
present a heuristic to find the {\epsilon}-best arm with optimal sample
complexity, by storing only one extra arm in the memory.
