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@@ -397,7 +397,7 @@ Another way to test the impact of search on learning and the data distribution w
 
 # B.3 REGULARIZED POLICY UPDATES (MPO) WITH MCTS
 
-Past work has demonstrated that MuZero’s policy targets suffer from degeneracies at low visit counts [27, 20]. To account for this, we modified the policy targets to use an MPO-style update [1] rather than the visit count distribution, similar to [20]: πMPOt+k ∝ πkθ,t · exp qMCTS/τ 
. Here, qMCTS are the Q-values at the root node of the search tree; $\tau = 0 . 1$ is a temperature parameter; and we use $\pi _ { t + k } ^ { \mathrm { M P O } }$ in place of $\pi _ { t + k } ^ { \mathrm { M C T S } }$ in Equation 8. Note that the Q-values for unvisited actions are set to zero; while it is in general a poor estimate for the true Q-function, we found this choice to outperform setting the Q-function for unvisited actions to the value function. This is likely because unvisited actions are unlikely under the prior and perhaps ought not be reinforced unless good estimates are obtained through exploration. However, this choice leads to a biased MPO update; how to unbias it will be a topic of further research. Similarly, we chose the MPO update for its ease of implementation, but it is likely other forms of regularized policy gradient (e.g. TRPO [59], or more generally natural or mirror policy optimization [72, 2]) would result in quantitively similar findings. We also did not tune $\tau$ for different environments; it is likely that properly tuning it could further improve performance of the agent at small search budgets.
+Past work has demonstrated that MuZero’s policy targets suffer from degeneracies at low visit counts [27, 20]. To account for this, we modified the policy targets to use an MPO-style update [1] rather than the visit count distribution, similar to [20]: πMPOt+k ∝ πkθ,t · exp qMCTS/τ 
. Here, qMCTS are the Q-values at the root node of the search tree; $\tau = 0 . 1$ is a temperature parameter; and we use $\pi _ { t + k } ^ { \mathrm { M P O } }$ in place of $\pi _ { t + k } ^ { \mathrm { M C T S } }$ in Equation 8. Note that the Q-values for unvisited actions are set to zero; while it is in general a poor estimate for the true Q-function, we found this choice to outperform setting the Q-function for unvisited actions to the value function. This is likely because unvisited actions are unlikely under the prior and perhaps ought not be reinforced unless good estimates are obtained through exploration. However, this choice leads to a biased MPO update; how to unbias it will be a topic of further research. Similarly, we chose the MPO update for its ease of implementation, but it is likely other forms of regularized policy gradient (e.g. TRPO [59], or more generally natural or mirror policy optimization [72, 2]) would result in quantitively similar findings. We also did not tune $\tau$ for different environments; it is likely that properly tuning it could further improve performance of the agent at small search budgets.
 
 # B.4 BREADTH-FIRST SEARCH