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+# Some Interesting Properties of 1-step K-sample Lookahead Operator
+
+1-step K-sample lookahead operator (which we will simply call lookahead operator from now on) in reinforcement learning is a sample-based policy improvement operator that works by first sampling $K$ action candidates from the base policy and then proceeding with the action with largest q-value:
+
+$$L_{K, \pi}(a \mid s) = E_{a_1, \ldots, a_K \sim \pi(a \mid s)}\Bbb{1}\left[a = \arg \max_{a' \in \\{a_1, \ldots, a_K\\}} Q(s, a')\right].$$
+
+It can be thought of as an imperfect approximation of the optimal max-q operator. It is very useful when dealing with large action spaces, where the exact maximum of the q-function over all actions cannot be computed. In this note, we will list various properties of this operator that might be useful when using it in practice.
+
+### Lookahead operator is indeed a policy improvement operator
+
+Follows from the policy improvement theorem:
+
+$$E_{a \sim L_{K, \pi}(a \mid s)} Q(s, a) = E_{a_1, \ldots, a_K \sim \pi(a \mid s)} \max_{a' \in \\{a_1, \ldots, a_K\\}} Q(s, a') \geq E_{a \sim \pi(a \mid s)} Q(s, a) = V_{\pi}.$$
+
+
+### Repeated application of lookahead increases the effective number of candidates ###
+
+Suppose we define a new policy
+
+$$\pi^{\ast}(a \mid s) = L_{K, \pi}(a \mid s).$$
+
+What is then $L_{K, \pi^{\ast}}(a \mid s)$, i.e. the result of chaining two applications of the lookahead operator? Here is one way to think about it: one invocation of $\pi^{\ast}$ is effectively samping $K$ action candidates and returning the best of them. Lookahead operator $L_{K, \pi^{\ast}}(a \mid s)$ invokes $\pi^{\ast}$ $K$ times, so it effectively samples $K$ groups of $K$ candidates each, selects the best candidate in each group and then selects the best action across all $K$ groups. Since the same q-function is used to select both inside groups and across groups, this procedure is equivalent to simply doing lookahead with $K^2$ candidates.
+
+Stacking another lookahead on top will be equivalent to doing a single lookahead with $K^3$ candidates and so on. In other words, reaching the effective power of max-q improvement operator in an action space with $N$ actions requires just $O(log_K N)$ lookahead compositions, which is a very reasonable number even for very large action spaces.
+
+
+### Lookahead-based Reinforcement Learning
+
+The above property suggests a straightforward RL algorithm:
+
+1. Evaluate the current policy $\pi$ to get $q_{\pi}(s, a)$,
+    * e.g. by sampling some trajectories from $\pi$ and then training a model to predict return-to-go.
+2. Distill $L_{K, \pi}(a \mid s)$ into a new policy $\pi^{\ast}$,
+    * e.g. by sampling some trajectories from $L$ and then applying supervised training on this data.
+3. Set $\pi$ to $\pi^{\ast}$ and go to the first step unless done.
+
+This algorithm
+* Converges to the optimal policy in MDPs and does it fast even in large action spaces (assuming that Q can be approximated well).
+* Does not require to explicitly maximize the Q-function over all actions, as this is covered by the repeated application of lookahead.
+* Is very robust, as it only uses supervised learning as a learning subroutine. No sketchy non-stationary loss functions!
+
+#### Repeated Policy Evaluation is Not Necessary
+
+Perhaps the most surprising property of the above algorithm is that re-evaluating the policy each time it was updated is not necessary: while $Q_{\pi}$ is not equal to $Q_{\pi^{\ast}}$ (which is expected: $\pi^{\ast}$ is an improvement over $\pi$), these q-functions rank actions identically due to constraints imposed on $\pi^{\ast}$ by the fact that it's not an arbitrary policy, but a lookahead policy made using $\pi$ and $Q_{\pi}$. In practice, however, it would still make sense to re-train $Q$ occasionally because as $\pi$ evolves, optimization emphasis shifts to the regions of state space where the original approximation of $Q$ was likely poor.
+
+Let us now prove that $Q_{\pi}$ and $Q_{\pi^{\ast}}$ rank actions identically.