# How does policy network learn in AlphaZero?

I'm currently trying to understand how AlphaZero works. There is one thing with the training of the AlphaZero's policy head that confuses me. Basically, in AlphaGo Zero's paper (where the major part of AlphaZero algorithm is explained) a combined neural network is used to estimate both, the value of the position and a good policy. More precisely, the loss function used is:

$$L = (z-v) - \pi^t \log(\textbf{p}) + c \Vert \theta \Vert$$

where $$z$$ is the outcome of the game, $$v$$ is the value estimated by the neural network, $$\pi$$ is the policy calculated by the MCTS and $$\textbf{p}$$ is the policy predicted by the neural network.

I would like to focus on the policy head loss. Basically, we are trying to minimize the difference between the policy calculated by the MCTS and the policy predicted by the neural network. That makes sense when the player has won the game, but it doesn't (at least from my point of view) when the player has lost it. You would be teaching your neural network a policy that has lost. Maybe the loss was unavoidable, but if it wasn't that's definitely not the policy we want to learn.

I have programmed a slightly simplified version that works well with Tic Tac Toe. But for Connect 4 some problems related to this arise. Basically, it learns a bad policy. At the beginning of the training, the values estimated for each board are quite random, and that makes the policy shift to a random (and wrong) direction. At the same time, that makes the value function to be wrong (because we are losing games that we could have easily won), worsening even more the policy.

I suppose that with enough training this problem disappears. The correct value and policy should backpropagate from the leaf nodes of our simulation. Even if the neural network policy gives a probability of 0 to the optimal action, thanks to the Dirichlet noise added to the probabilities the MCTS can find that optimal action and learn it.

However, several things confuse me:

1. In AlphaGo's paper, they take into account whether if the outcome of the game has been a win or a loss when training the policy network with reinforcement learning. More precisely, the optimization made is

$$\Delta p \propto \frac{\delta \log{p(a_t|s_t)}}{\delta p} z_t$$

where $$z_t = 1$$ if we have won or $$z_t = -1$$ if we have lost. So DeepMind's take into account if the action was good or not and change the direction to optimize.

2. I haven't found anywhere in AlphaGo Zero's paper that we are training just with the examples where the player has won, so they might be using all the data gathered, including also the examples where the player has lost. As far as I know, they don't mention anything related to this problem.

3. $$\pi$$ (the policy provided by the MCTS) is calculated as the exponentiated visit count of each action

$$\pi(a|s_0) = \frac{N(s_0,a)^{1/\tau}}{\sum_b N(s_0,b)^{1/\tau}}$$

where $$\tau$$ is a parameter that controls the temperature''. DeepMind's team sets $$\tau = 1$$ during the 30 first movements to ensure exploration. After that, they set it to $$\tau \approx 0$$ to ensure that the action that is considered the best one (and thus has been simulated more times) is the one played. However, that means $$\pi$$ is something like

$$[0,0,\dots,0,1,0, \dots, 0]$$

making the policy changes a bit agressive and especially harmful if the movement is not the good one (making it even harder to recover from a bad action).

Am I missing something? Is this the intended way of working of the algorithm? Is there any way to overcome the learning of bad policies?