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Bellman optimality equation

Reinforcement learning · Formula · Not yet linked from a lesson

V∗(s)=max⁡aE ⁣[r+γV∗(s′)∣s,a]V^{*}(s) = \max_{a} \mathbb{E}\!\left[r + \gamma V^{*}(s') \mid s, a\right]

The same recursion with a maximum where the policy used to be. It describes the best achievable value without reference to any particular policy, and the greedy choice over it is an optimal policy.

Symbols

V∗V^{*}
The optimal value function.
max⁡a\max_{a}
Taking the best available action rather than the usual one.
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