Modern Deep Reinforcement Learning (RL) algorithms require estimates of the maximal Q-value, which are difficult to compute in continuous domains with an infinite number of possible actions. In this work, we introduce a new update rule for online and offline RL which directly models the maximal value using Extreme Value Theory (EVT), drawing inspiration from economics. By doing so, we avoid computing Q-values using out-of-distribution actions which is often a substantial source of error. Our key insight is to introduce an objective that directly estimates the optimal soft-value functions (LogSumExp) in the maximum entropy RL setting without needing to sample from a policy. Using EVT, we derive our \emph{Extreme Q-Learning} framework and consequently online and, for the first time, offline MaxEnt Q-learning algorithms, that do not explicitly require access to a policy or its entropy. Our method obtains consistently strong performance in the D4RL benchmark, outperforming prior works by \emph{10+ points} on the challenging Franka Kitchen tasks while offering moderate improvements over SAC and TD3 on online DM Control tasks. Visualizations and code can be found on our website at https://div99.github.io/XQL/.
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