In the experts problem, on each of T days, an agent needs to follow the advice of one of n experts. After each day, the loss associated with each expert’s advice is revealed. A fundamental result in learning theory says that the agent can achieve vanishing regret, i.e. their cumulative loss is within o(T) of the cumulative loss of the best-in-hindsight expert. Can the agent perform well without sufficient space to remember all the experts? We extend a nascent line of research on this question in two directions: (1) We give a new algorithm against the oblivious adversary, improving over the memory- regret tradeoff obtained by [Peng-Zhang’23], and nearly matching the lower bound of [Srinivas-Woodruff-Xu-Zhou'22]. (2) We also consider an adaptive adversary who can observe past experts chosen by the agent. In this setting we give both a new algorithm and a novel lower bound, proving that roughly \sqrt{n} memory is both necessary and sufficient for obtaining o(T) regret. Based on joint work with Aviad Rubinstein

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