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AlphaGo: Mastering the game of Go with deep neural networks and tree search

June 2024

tl;dr: MCTS trees with neural nets bets human in the game of Go.

Overall impression

Value iteration and policy iterations are systematic, iterative method that solves MDP problems. Yet even with the improved policy iteration, it still have to perform time-consuming operation to update the value of EVERY state. A standard 19x19 Go board has roughly 2e170 possible states. This vast amount of states will be intractable to solve with a vanilla value iteration or policy iteration technique.

AlphaGo and its successors use a Monte Carlo tree search algorithm to find its moves guided by a value network and a policy network, trained on from human and computer play. Both the value network and policy network takes in the current state of the board and produces a singular state value function V(s) of the board position and the state-action value function Q(s, a) of all possible moves given the current board position. Neural networks are used to reduce the effective depth and breadth of the search tree: evaluating positions using a value network, and sampling actions using a policy network.

Every leaf node (an unexplored board position) in the MCTS is evaluated in two very different ways: by the value network; and second, by the outcome of a random rollout played out using the fast rollout policy. Note that a single evaluation of the value network also approached the accuracy of Monte Carlo rollouts using the RL policy network, but using 15,000 times less computation. This is very similar to a fast-slow system design, intuition vs reasoning, system 1 vs system 2 by Nobel laureate Daniel Kahneman (We can see similar design in more recent work such as DriveVLM).

Key ideas

  • MCTS: policy estimation, focuses on decision-making from the current state. It has 4 steps process of selection-expansion-simulation-backprop.
    • Selection: Follow the most promising path based on previous simulations until you reach a leaf node (a position that hasn’t been fully explored yet).
    • Expansion: add one or more child nodes to represent the possible next moves.
    • Simulation: From the new node, play out a random game until the end (this is called a “rollout”).
    • Backpropagation: Go back through the nodes on the path you took and update their values based on the result of the game. If you won, increase the value; if you lost, decrease it.
  • MCTS guided by value network and policy network.
    • Value network reduce the search depth by summarizing values of sub-trees, so we can avoid going deep for good estimations. Policy network to prune search space. Balanced breadth and width.
    • MCTS used both value network and reward from rollout.
    • Policy network reduce the breadth of the search tree by identifying sensible moves, so we can avoid non-sensible moves.
    • Value network V evaluates winning rate from a state (棋面).
    • Trained with state-outcome pairs. Trained with much more self-play data to reduce overfit.
    • Policy network evaluates action distribution
    • Value network is more like instinct (heuristic), value network provides policy gradient to update policy network. Tesla learned collision network, and heuristic network for hybrid A-star.
  • With autonomous driving
    • World model
    • AlphaGo tells us how to extract very good policy with a good world model (simulation)
    • Autonomous driving still needs a very good simulation to be able to leverage alphaGo algorithm. —> It this a dead end, vs FSD v12?
    • Tesla AI day 2021 and 2022 are heavily affected by AlphaGo. FSDv12 is a totally different ballgame though.
    • Go is limited and noiseless.
  • Policy networks
    • P_s trained with SL, more like e2e
    • P_p trained with SL, shallower for fast rollout in MCTS
    • P_r trained with RL to play with P_s

Technical details

  • Summary of technical details, such as important training details, or bugs of previous benchmarks.

Notes

  • Value iteration is to MCTS as Dijkstra's algorithm is to (hybrid) A-star: both value iteration and Dijkstra's systematically consider all possibilities (covering the entire search space) without heuristics, while MCTS and A* use heuristics to focus on the most promising options, making them more efficient for large and complex problems.
  • Question: PN is essentially already e2e, why need VN and MCTS?
    • My guess: Small scale SL generate PN not strong enough, so need RL and MCTS to boost performance.
    • E2E demonstrates that with enough data, e2e SL can generate strong enough PN itself.
    • Maybe later MCTS will come back again to generate superhuman behavior for driving.