The success of neural networks on problems like Go and protein folding, long considered intractable NP-hard problems, is profound. It suggests our formal understanding of computational hardness, which focuses on worst-case scenarios, may be an incomplete model for how to find useful, approximate solutions in practice.

The enormous compute budget for the original AlphaGo was not about finding the most efficient training method, but about proving a method could work at all. Once a breakthrough is made and the path is clear, subsequent efforts can focus on optimization and achieve similar results with far less compute.

Go's search space is larger than the number of atoms in the universe, making exhaustive search impossible. AlphaGo's core breakthrough was using neural networks to intelligently guide its search, evaluating only the most promising moves and making an intractable problem solvable.

AlphaGo's architecture mimicked human cognition by pairing a 'fast thinking' neural network for intuition with a 'slow thinking' search algorithm for explicit planning. This hybrid model, combining pattern recognition with calculation, proved more powerful for tackling complex problems than either approach alone.

Instead of training on the single best action from its search (a one-hot label), AlphaGo's policy network learns to imitate the entire probability distribution of moves from MCTS. This 'soft label' contains far more information, enabling a much more effective and sample-efficient form of knowledge distillation.

Monte Carlo Tree Search (MCTS) acts as a 'policy improvement operator.' After the search finds a better move distribution, the policy network is trained to directly predict this improved distribution. This distills the expensive search process into the network itself, making it stronger over time.

A core legacy of AlphaGo is turning complex search problems into 'games' for AI agents. AlphaTensor reframed the challenge of finding the fastest matrix multiplication algorithm as a game, allowing it to discover a more efficient method than any human had found in over 50 years, proving the approach's power for scientific discovery.

Humans stop analyzing a game when they intuit a winning or losing position. AlphaGo’s value function mimics this by predicting the eventual outcome from any board state. This allows the search to be drastically shortened, as it doesn't need to play out every possibility to the very end.

Unlike typical reinforcement learning which learns from sparse win/loss signals, AlphaGo's method is remarkably stable. It uses MCTS to generate an 'improved' move for every state, turning the problem into a simple supervised learning task of imitating a better version of itself, avoiding high-variance gradients.