Mastering the Game of Go Without Human Knowledge

Home , Aja Huang, AlphaGo, AlphaGo Zero, Computer Go, DeepMind, Demis Hassabis

Mastering the Game of Go without Human Knowledge

David Silver*, Julian Schrittwieser*, Karen Simonyan*, Ioannis Antonoglou, Aja Huang, Arthur Guez, Thomas Hubert, Lucas Baker, Matthew Lai, Adrian Bolton, Yutian Chen, Timothy Lillicrap, Fan Hui, Laurent Sifre, George van den Driessche, Thore Graepel, Demis Hassabis.

DeepMind, 5 New Street Square, London EC4A 3TW.

*These authors contributed equally to this work.

A long-standing goal of artificial intelligence is an algorithm that learns, tabula rasa, superhuman proficiency in challenging domains. Recently, AlphaGo became the first program to defeat a world champion in the game of Go. The tree search in AlphaGo evaluated positions and selected moves using deep neural networks. These neural networks were trained by supervised learning from human expert moves, and by reinforcement learning from self- play. Here, we introduce an algorithm based solely on reinforcement learning, without human data, guidance, or domain knowledge beyond game rules. AlphaGo becomes its own teacher: a neural network is trained to predict AlphaGo’s own move selections and also the winner of AlphaGo’s games. This neural network improves the strength of tree search, resulting in higher quality move selection and stronger self-play in the next iteration. Starting tabula rasa, our new program AlphaGo Zero achieved superhuman performance, winning 100-0 against the previously published, champion-defeating AlphaGo.

Much progress towards artiﬁcial intelligence has been made using supervised learning systems that are trained to replicate the decisions of human experts 1–4. However, expert data is often expensive, unreliable, or simply unavailable. Even when reliable data is available it may impose a ceiling on the performance of systems trained in this manner 5. In contrast, reinforcement learning systems are trained from their own experience, in principle allowing them to exceed human capabilities, and to operate in domains where human expertise is lacking. Recently, there has been rapid progress towards this goal, using deep neural networks trained by reinforcement learning. These systems have outperformed humans in computer games such as Atari 6, 7 and 3D virtual en- vironments 8–10. However, the most challenging domains in terms of human intellect – such as the

1 game of Go, widely viewed as a grand challenge for artiﬁcial intelligence 11 – require precise and sophisticated lookahead in vast search spaces. Fully general methods have not previously achieved human-level performance in these domains.

AlphaGo was the ﬁrst program to achieve superhuman performance in Go. The published version 12, which we refer to as AlphaGo Fan, defeated the European champion Fan Hui in October 2015. AlphaGo Fan utilised two deep neural networks: a policy network that outputs move probabilities, and a value network that outputs a position evaluation. The policy network was trained initially by supervised learning to accurately predict human expert moves, and was subsequently reﬁned by policy-gradient reinforcement learning. The value network was trained to predict the winner of games played by the policy network against itself. Once trained, these networks were combined with a Monte-Carlo Tree Search (MCTS) 13–15 to provide a lookahead search, using the policy network to narrow down the search to high-probability moves, and using the value network (in conjunction with Monte-Carlo rollouts using a fast rollout policy) to evaluate positions in the tree. A subsequent version, which we refer to as AlphaGo Lee, used a similar approach (see Methods), and defeated Lee Sedol, the winner of 18 international titles, in March 2016.

Our program, AlphaGo Zero, differs from AlphaGo Fan and AlphaGo Lee 12 in several im- portant aspects. First and foremost, it is trained solely by self-play reinforcement learning, starting from random play, without any supervision or use of human data. Second, it only uses the black and white stones from the board as input features. Third, it uses a single neural network, rather than separate policy and value networks. Finally, it uses a simpler tree search that relies upon this single neural network to evaluate positions and sample moves, without performing any Monte- Carlo rollouts. To achieve these results, we introduce a new reinforcement learning algorithm that incorporates lookahead search inside the training loop, resulting in rapid improvement and precise and stable learning. Further technical differences in the search algorithm, training procedure and network architecture are described in Methods.

2 1 Reinforcement Learning in AlphaGo Zero

Our new method uses a deep neural network fθ with parameters θ. This neural network takes as an input the raw board representation s of the position and its history, and outputs both move probabilities and a value, (p, v) = fθ(s). The vector of move probabilities p represents the probability of selecting each move (including pass), pa = P r(a|s). The value v is a scalar evaluation, estimating the probability of the current player winning from position s. This neural network combines the roles of both policy network and value network 12 into a single architecture. The neural network consists of many residual blocks 4 of convolutional layers 16, 17 with batch normalisation 18 and rectiﬁer non-linearities 19 (see Methods).

The neural network in AlphaGo Zero is trained from games of self-play by a novel reinforcement learning algorithm. In each position s, an MCTS search is executed, guided by the neural

network fθ. The MCTS search outputs probabilities π of playing each move. These search probabilities usually select much stronger moves than the raw move probabilities p of the neural network

20, 21 fθ(s); MCTS may therefore be viewed as a powerful policy improvement operator . Self-play with search – using the improved MCTS-based policy to select each move, then using the game winner z as a sample of the value – may be viewed as a powerful policy evaluation operator. The main idea of our reinforcement learning algorithm is to use these search operators repeatedly in a policy iteration procedure 22, 23: the neural network’s parameters are updated to make the move

probabilities and value (p, v) = fθ(s) more closely match the improved search probabilities and self-play winner (ππ, z); these new parameters are used in the next iteration of self-play to make the search even stronger. Figure 1 illustrates the self-play training pipeline.

The Monte-Carlo tree search uses the neural network fθ to guide its simulations (see Figure 2). Each edge (s, a) in the search tree stores a prior probability P (s, a), a visit count N(s, a), and an action-value Q(s, a). Each simulation starts from the root state and iteratively selects moves that maximise an upper conﬁdence bound Q(s, a) + U(s, a), where U(s, a) ∝ P (s, a)/(1 + N(s, a)) 12, 24, until a leaf node s0 is encountered. This leaf position is expanded and evaluated just

3 Figure 1: Self-play reinforcement learning in AlphaGo Zero. a The program plays a game s1, ..., sT against itself.

In each position st, a Monte-Carlo tree search (MCTS) αθ is executed (see Figure 2) using the latest neural network fθ. Moves are selected according to the search probabilities computed by the MCTS, at ∼ πt. The terminal position sT is scored according to the rules of the game to compute the game winner z. b Neural network training in AlphaGo

Zero. The neural network takes the raw board position st as its input, passes it through many convolutional layers with parameters θ, and outputs both a vector pt, representing a probability distribution over moves, and a scalar value vt, representing the probability of the current player winning in position st. The neural network parameters θ are updated so as to maximise the similarity of the policy vector pt to the search probabilities πt , and to minimise the error between the predicted winner vt and the game winner z (see Equation 1). The new parameters are used in the next iteration of self-play a.

4 Figure 2: Monte-Carlo tree search in AlphaGo Zero. a Each simulation traverses the tree by selecting the edge with maximum action-value Q, plus an upper conﬁdence bound U that depends on a stored prior probability P and visit count N for that edge (which is incremented once traversed). b The leaf node is expanded and the associated position s is evaluated by the neural network (P (s, ·),V (s)) = fθ(s); the vector of P values are stored in the outgoing edges from s. c Action-values Q are updated to track the mean of all evaluations V in the subtree below that action. d Once the search is complete, search probabilities π are returned, proportional to N 1/τ , where N is the visit count of each move from the root state and τ is a parameter controlling temperature.

0 0 0 once by the network to generate both prior probabilities and evaluation, (P (s , ·),V (s )) = fθ(s ). Each edge (s, a) traversed in the simulation is updated to increment its visit count N(s, a), and to P 0 update its action-value to the mean evaluation over these simulations, Q(s, a) = 1/N(s, a) s0|s,a→s0 V (s ), where s, a → s0 indicates that a simulation eventually reached s0 after taking move a from position s.

MCTS may be viewed as a self-play algorithm that, given neural network parameters θ and a root position s, computes a vector of search probabilities recommending moves to play, π =

1/τ αθ(s), proportional to the exponentiated visit count for each move, πa ∝ N(s, a) , where τ is a temperature parameter.

The neural network is trained by a self-play reinforcement learning algorithm that uses

MCTS to play each move. First, the neural network is initialised to random weights θ0. At each subsequent iteration i ≥ 1, games of self-play are generated (Figure 1a). At each time-step t, an MCTS search πt = αθi−1 (st) is executed using the previous iteration of neural network fθi−1 , and a move is played by sampling the search probabilities πt. A game terminates at step T when

5 both players pass, when the search value drops below a resignation threshold, or when the game

exceeds a maximum length; the game is then scored to give a ﬁnal reward of rT ∈ {−1, +1} (see

Methods for details). The data for each time-step t is stored as (st,ππt, zt) where zt = ±rT is the game winner from the perspective of the current player at step t. In parallel (Figure 1b), new network parameters θi are trained from data (s,πππ, z) sampled uniformly among all time-steps of the last iteration(s) of self-play. The neural network (p, v) = fθi (s) is adjusted to minimise the error between the predicted value v and the self-play winner z, and to maximise the similarity of the neural network move probabilities p to the search probabilities π. Speciﬁcally, the parameters θ are adjusted by gradient descent on a loss function l that sums over mean-squared error and cross-entropy losses respectively,

2 > 2 (p, v) = fθ(s), l = (z − v) − π log p + c||θ|| (1) where c is a parameter controlling the level of L2 weight regularisation (to prevent overﬁtting).

2 Empirical Analysis of AlphaGo Zero Training

We applied our reinforcement learning pipeline to train our program AlphaGo Zero. Training started from completely random behaviour and continued without human intervention for approximately 3 days.

Over the course of training, 4.9 million games of self-play were generated, using 1,600 simulations for each MCTS, which corresponds to approximately 0.4s thinking time per move. Param- eters were updated from 700,000 mini-batches of 2,048 positions. The neural network contained 20 residual blocks (see Methods for further details).

Figure 3a shows the performance of AlphaGo Zero during self-play reinforcement learning, as a function of training time, on an Elo scale 25. Learning progressed smoothly throughout training, and did not suffer from the oscillations or catastrophic forgetting suggested in prior literature

6 Figure 3: Empirical evaluation of AlphaGo Zero. a Performance of self-play reinforcement learning. The plot

shows the performance of each MCTS player αθi from each iteration i of reinforcement learning in AlphaGo Zero. Elo ratings were computed from evaluation games between different players, using 0.4 seconds of thinking time per move (see Methods). For comparison, a similar player trained by supervised learning from human data, using the KGS data-set, is also shown. b Prediction accuracy on human professional moves. The plot shows the accuracy of the neural network fθi , at each iteration of self-play i, in predicting human professional moves from the GoKifu data-set. The accuracy measures the percentage of positions in which the neural network assigns the highest probability to the human move. The accuracy of a neural network trained by supervised learning is also shown. c Mean-squared error

(MSE) on human professional game outcomes. The plot shows the MSE of the neural network fθi , at each iteration of self-play i, in predicting the outcome of human professional games from the GoKifu data-set. The MSE is between 1 the actual outcome z ∈ {−1, +1} and the neural network value v, scaled by a factor of 4 to the range [0, 1]. The MSE of a neural network trained by supervised learning is also shown.

7 26–28. Surprisingly, AlphaGo Zero outperformed AlphaGo Lee after just 36 hours; for comparison, AlphaGo Lee was trained over several months. After 72 hours, we evaluated AlphaGo Zero against the exact version of AlphaGo Lee that defeated Lee Sedol, under the 2 hour time controls and match conditions as were used in the man-machine match in Seoul (see Methods). AlphaGo Zero used a single machine with 4 Tensor Processing Units (TPUs) 29, while AlphaGo Lee was distributed over many machines and used 48 TPUs. AlphaGo Zero defeated AlphaGo Lee by 100 games to 0 (see Extended Data Figure 5 and Supplementary Information).

To assess the merits of self-play reinforcement learning, compared to learning from human data, we trained a second neural network (using the same architecture) to predict expert moves in the KGS data-set; this achieved state-of-the-art prediction accuracy compared to prior work 12, 30–33 (see Extended Data Table 1 and 2 respectively). Supervised learning achieved better initial performance, and was better at predicting the outcome of human professional games (Figure 3). Notably, although supervised learning achieved higher move prediction accuracy, the self-learned player performed much better overall, defeating the human-trained player within the ﬁrst 24 hours of training. This suggests that AlphaGo Zero may be learning a strategy that is qualitatively different to human play.

To separate the contributions of architecture and algorithm, we compared the performance of the neural network architecture in AlphaGo Zero with the previous neural network architecture used in AlphaGo Lee (see Figure 4). Four neural networks were created, using either separate policy and value networks, as in AlphaGo Lee, or combined policy and value networks, as in AlphaGo Zero; and using either the convolutional network architecture from AlphaGo Lee or the residual network architecture from AlphaGo Zero. Each network was trained to minimise the same loss function (Equation 1) using a ﬁxed data-set of self-play games generated by AlphaGo Zero after 72 hours of self-play training. Using a residual network was more accurate, achieved lower error, and improved performance in AlphaGo by over 600 Elo. Combining policy and value together into a single network slightly reduced the move prediction accuracy, but reduced the value error and boosted playing performance in AlphaGo by around another 600 Elo. This is partly due to

8 Figure 4: Comparison of neural network architectures in AlphaGo Zero and AlphaGo Lee. Comparison of neural network architectures using either separate (“sep”) or combined policy and value networks (“dual”), and using either convolutional (“conv”) or residual networks (“res”). The combinations “dual-res” and “sep-conv” correspond to the neural network architectures used in AlphaGo Zero and AlphaGo Lee respectively. Each network was trained on a ﬁxed data-set generated by a previous run of AlphaGo Zero. a Each trained network was combined with AlphaGo Zero’s search to obtain a different player. Elo ratings were computed from evaluation games between these different players, using 5 seconds of thinking time per move. b Prediction accuracy on human professional moves (from the GoKifu data-set) for each network architecture. c Mean-squared error on human professional game outcomes (from the GoKifu data-set) for each network architecture.

9 improved computational efﬁciency, but more importantly the dual objective regularises the network to a common representation that supports multiple use cases.

3 Knowledge Learned by AlphaGo Zero

AlphaGo Zero discovered a remarkable level of Go knowledge during its self-play training process. This included fundamental elements of human Go knowledge, and also non-standard strategies beyond the scope of traditional Go knowledge.

Figure 5 shows a timeline indicating when professional joseki (corner sequences) were discovered (Figure 5a, Extended Data Figure 1); ultimately AlphaGo Zero preferred new joseki vari- ants that were previously unknown (Figure 5b, Extended Data Figure 2). Figure 5c and the Sup- plementary Information show several fast self-play games played at different stages of training. Tournament length games played at regular intervals throughout training are shown in Extended Data Figure 3 and Supplementary Information. AlphaGo Zero rapidly progressed from entirely random moves towards a sophisticated understanding of Go concepts including fuseki (opening), tesuji (tactics), life-and-death, ko (repeated board situations), yose (endgame), capturing races, sente (initiative), shape, influence and territory, all discovered from first principles. Surprisingly, shicho (“ladder” capture sequences that may span the whole board) – one of the first elements of Go knowledge learned by humans – were only understood by AlphaGo Zero much later in training.

4 Final Performance of AlphaGo Zero

We subsequently applied our reinforcement learning pipeline to a second instance of AlphaGo Zero using a larger neural network and over a longer duration. Training again started from completely random behaviour and continued for approximately 40 days.

Over the course of training, 29 million games of self-play were generated. Parameters were updated from 3.1 million mini-batches of 2,048 positions each. The neural network contained

10 Figure 5: Go knowledge learned by AlphaGo Zero. a Five human joseki (common corner sequences) discovered during AlphaGo Zero training. The associated timestamps indicate the first time each sequence occured (taking account of rotation and reflection) during self-play training. Extended Data Figure 1 provides the frequency of occurence over training for each sequence. b Five joseki favoured at different stages of self-play training. Each displayed corner sequence was played with the greatest frequency, among all corner sequences, during an iteration of self-play training. The timestamp of that iteration is indicated on the timeline. At 10 hours a weak corner move was preferred. At 47 hours the 3-3 invasion was most frequently played. This joseki is also common in human professional play; however AlphaGo Zero later discovered and preferred a new variation. Extended Data Figure 2 provides the frequency of occurence over time for all five sequences and the new variation. c The first 80 moves of three self-play games that were played at different stages of training, using 1,600 simulations (around 0.4s) per search. At 3 hours, the game focuses greedily on capturing stones, much like a human beginner. At 19 hours, the game exhibits the fundamentals of life-and-death, influence and territory. At 70 hours, the game is beautifully balanced, involving multiple battles and a complicated ko fight, eventually resolving into a half-point win for white. See Supplementary Information for the full games. 40 residual blocks. The learning curve is shown in Figure 6a. Games played at regular intervals throughout training are shown in Extended Data Figure 4 and Supplementary Information.

We evaluated the fully trained AlphaGo Zero using an internal tournament against AlphaGo Fan, AlphaGo Lee, and several previous Go programs. We also played games against the strongest existing program, AlphaGo Master – a program based on the algorithm and architecture presented in this paper but utilising human data and features (see Methods) – which defeated the strongest human professional players 60–0 in online games 34 in January 2017. In our evaluation, all programs were allowed 5 seconds of thinking time per move; AlphaGo Zero and AlphaGo Master each played on a single machine with 4 TPUs; AlphaGo Fan and AlphaGo Lee were distributed over 176 GPUs and 48 TPUs respectively. We also included a player based solely on the raw neural network of AlphaGo Zero; this player simply selected the move with maximum probability.

Figure 6b shows the performance of each program on an Elo scale. The raw neural network, without using any lookahead, achieved an Elo rating of 3,055. AlphaGo Zero achieved a rating of 5,185, compared to 4,858 for AlphaGo Master, 3,739 for AlphaGo Lee and 3,144 for AlphaGo Fan.

Finally, we evaluated AlphaGo Zero head to head against AlphaGo Master in a 100 game match with 2 hour time controls. AlphaGo Zero won by 89 games to 11 (see Extended Data Figure 6) and Supplementary Information.

5 Conclusion

Our results comprehensively demonstrate that a pure reinforcement learning approach is fully fea- sible, even in the most challenging of domains: it is possible to train to superhuman level, without human examples or guidance, given no knowledge of the domain beyond basic rules. Further- more, a pure reinforcement learning approach requires just a few more hours to train, and achieves much better asymptotic performance, compared to training on human expert data. Using this ap-

12 Figure 6: Performance of AlphaGo Zero. a Learning curve for AlphaGo Zero using larger 40 block residual network

over 40 days. The plot shows the performance of each player αθi from each iteration i of our reinforcement learning algorithm. Elo ratings were computed from evaluation games between different players, using 0.4 seconds per search (see Methods). b Final performance of AlphaGo Zero. AlphaGo Zero was trained for 40 days using a 40 residual block neural network. The plot shows the results of a tournament between: AlphaGo Zero, AlphaGo Master (defeated top human professionals 60-0 in online games), AlphaGo Lee (defeated Lee Sedol), AlphaGo Fan (defeated Fan Hui), as well as previous Go programs Crazy Stone, Pachi and GnuGo. Each program was given 5 seconds of thinking time per move. AlphaGo Zero and AlphaGo Master played on a single machine on the Google Cloud; AlphaGo Fan and AlphaGo Lee were distributed over many machines. The raw neural network from AlphaGo Zero is also included, which directly selects the move a with maximum probability pa, without using MCTS. Programs were evaluated on an Elo scale 25: a 200 point gap corresponds to a 75% probability of winning.

13 proach, AlphaGo Zero defeated the strongest previous versions of AlphaGo, which were trained from human data using handcrafted features, by a large margin.

Humankind has accumulated Go knowledge from millions of games played over thousands of years, collectively distilled into patterns, proverbs and books. In the space of a few days, starting tabula rasa, AlphaGo Zero was able to rediscover much of this Go knowledge, as well as novel strategies that provide new insights into the oldest of games.

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17 Supplementary Information

Supplementary Information is available in the online version of the paper.

Acknowledgements

We thank A. Cain for work on the visuals; A. Barreto, G. Ostrovski, T. Ewalds, T. Schaul, J. Oh and N. Heess for reviewing the paper; and the rest of the DeepMind team for their support.

Author Contributions

D.S., J.S., K.S., I.A., A.G., L.S. and T.H. designed and implemented the reinforcement learning algorithm in AlphaGo Zero. A.H., J.S., M.L., D.S. designed and implemented the search in AlphaGo Zero. L.B., J.S., A.H, F.H., T.H., Y.C, D.S. designed and implemented the evaluation framework for AlphaGo Zero. D.S., A.B., F.H., A.G., T.L., T.G., L.S., G.v.d.D., D.H. managed and advised on the project. D.S., T.G., A.G. wrote the paper.

Author Information

Reprints and permissions information is available at www.nature.com/reprints. The authors de- clare no competing ﬁnancial interests. Readers are welcome to comment on the online version of the paper. Correspondence and requests for materials should be addressed to D.S. (davidsil- [email protected]).

18 Methods

Reinforcement learning Policy iteration 20, 21 is a classic algorithm that generates a sequence of improving policies, by alternating between policy evaluation – estimating the value function of the current policy – and policy improvement – using the current value function to generate a better policy. A simple approach to policy evaluation is to estimate the value function from the outcomes of sampled trajectories 35, 36. A simple approach to policy improvement is to select actions greedily with respect to the value function 20. In large state spaces, approximations are necessary to evaluate each policy and to represent its improvement 22, 23.

Classiﬁcation-based reinforcement learning 37 improves the policy using a simple Monte- Carlo search. Many rollouts are executed for each action; the action with the maximum mean value provides a positive training example, while all other actions provide negative training examples; a policy is then trained to classify actions as positive or negative, and used in subsequent rollouts. This may be viewed as a precursor to the policy component of AlphaGo Zero’s training algorithm when τ → 0.

A more recent instantiation, classiﬁcation-based modiﬁed policy iteration (CBMPI), also performs policy evaluation by regressing a value function towards truncated rollout values, similar to the value component of AlphaGo Zero; this achieved state-of-the-art results in the game of Tetris 38. However, this prior work was limited to simple rollouts and linear function approximation using handcrafted features.

The AlphaGo Zero self-play algorithm can similarly be understood as an approximate policy iteration scheme in which MCTS is used for both policy improvement and policy evaluation. Policy improvement starts with a neural network policy, executes an MCTS based on that policy’s recommendations, and then projects the (much stronger) search policy back into the function space of the neural network. Policy evaluation is applied to the (much stronger) search policy: the outcomes of self-play games are also projected back into the function space of the neural network. These projection steps are achieved by training the neural network parameters to match the search

19 probabilities and self-play game outcome respectively.

Guo et al. 7 also project the output of MCTS into a neural network, either by regressing a value network towards the search value, or by classifying the action selected by MCTS. This approach was used to train a neural network for playing Atari games; however, the MCTS was ﬁxed — there was no policy iteration — and did not make any use of the trained networks.

Self-play reinforcement learning in games Our approach is most directly applicable to zero-sum games of perfect information. We follow the formalism of alternating Markov games described in previous work 12, noting that algorithms based on value or policy iteration extend naturally to this setting 39.

Self-play reinforcement learning has previously been applied to the game of Go. NeuroGo 40, 41 used a neural network to represent a value function, using a sophisticated architecture based on Go knowledge regarding connectivity, territory and eyes. This neural network was trained by temporal-difference learning 42 to predict territory in games of self-play, building on prior work 43. A related approach, RLGO 44, represented the value function instead by a linear combination of features, exhaustively enumerating all 3 × 3 patterns of stones; it was trained by temporal- difference learning to predict the winner in games of self-play. Both NeuroGo and RLGO achieved a weak amateur level of play.

Monte-Carlo tree search (MCTS) may also be viewed as a form of self-play reinforcement learning 45. The nodes of the search tree contain the value function for the positions encountered during search; these values are updated to predict the winner of simulated games of self-play. MCTS programs have previously achieved strong amateur level in Go 46, 47, but used substantial domain expertise: a fast rollout policy, based on handcrafted features 48 13, that evaluates positions by running simulations until the end of the game; and a tree policy, also based on handcrafted features, that selects moves within the search tree 47.

Self-play reinforcement learning approaches have achieved high levels of performance in other games: chess 49–51, checkers 52, backgammon 53, othello 54, Scrabble 55 and most recently

20 poker 56. In all of these examples, a value function was trained by regression 54–56 or temporal- difference learning 49–53 from training data generated by self-play. The trained value function was used as an evaluation function in an alpha-beta search 49–54, a simple Monte-Carlo search 55, 57, or counterfactual regret minimisation 56. However, these methods utilised handcrafted input features 49–53, 56 or handcrafted feature templates 54, 55. In addition, the learning process used supervised learning to initialise weights 58, hand-selected weights for piece values 49, 51, 52, handcrafted restric- tions on the action space 56, or used pre-existing computer programs as training opponents 49, 50 or to generate game records 51.

Many of the most successful and widely used reinforcement learning methods were ﬁrst introduced in the context of zero-sum games: temporal-difference learning was ﬁrst introduced for a checkers-playing program 59, while MCTS was introduced for the game of Go 13. However, very similar algorithms have subsequently proven highly effective in video games 6–8, 10, robotics 60, industrial control 61–63, and online recommendation systems 64, 65.

AlphaGo versions We compare three distinct versions of AlphaGo:

1. AlphaGo Fan is the previously published program 12 that played against Fan Hui in October 2015. This program was distributed over many machines using 176 GPUs.

2. AlphaGo Lee is the program that defeated Lee Sedol 4–1 in March, 2016. It was previously unpublished but is similar in most regards to AlphaGo Fan 12. However, we highlight several key differences to facilitate a fair comparison. First, the value network was trained from the outcomes of fast games of self-play by AlphaGo, rather than games of self-play by the policy network; this procedure was iterated several times – an initial step towards the tabula rasa algorithm presented in this paper. Second, the policy and value networks were larger than those described in the original paper – using 12 convolutional layers of 256 planes respectively – and were trained for more iterations. This player was also distributed over many machines using 48 TPUs, rather than GPUs, enabling it to evaluate neural networks faster during search.

21 3. AlphaGo Master is the program that defeated top human players by 60–0 in January, 2017 34. It was previously unpublished but uses the same neural network architecture, reinforcement learning algorithm, and MCTS algorithm as described in this paper. However, it uses the same handcrafted features and rollouts as AlphaGo Lee 12 and training was initialised by supervised learning from human data.

4. AlphaGo Zero is the program described in this paper. It learns from self-play reinforcement learning, starting from random initial weights, without using rollouts, with no human supervision, and using only the raw board history as input features. It uses just a single machine in the Google Cloud with 4 TPUs (AlphaGo Zero could also be distributed but we chose to use the simplest possible search algorithm).

Domain Knowledge Our primary contribution is to demonstrate that superhuman performance can be achieved without human domain knowledge. To clarify this contribution, we enumerate the domain knowledge that AlphaGo Zero uses, explicitly or implicitly, either in its training procedure or its Monte-Carlo tree search; these are the items of knowledge that would need to be replaced for AlphaGo Zero to learn a different (alternating Markov) game.

1. AlphaGo Zero is provided with perfect knowledge of the game rules. These are used during MCTS, to simulate the positions resulting from a sequence of moves, and to score any simulations that reach a terminal state. Games terminate when both players pass, or after 19 · 19 · 2 = 722 moves. In addition, the player is provided with the set of legal moves in each position.

2. AlphaGo Zero uses Tromp-Taylor scoring 66 during MCTS simulations and self-play training. This is because human scores (Chinese, Japanese or Korean rules) are not well-deﬁned if the game terminates before territorial boundaries are resolved. However, all tournament and evaluation games were scored using Chinese rules.

3. The input features describing the position are structured as a 19 × 19 image; i.e. the neural network architecture is matched to the grid-structure of the board.

22 4. The rules of Go are invariant under rotation and reflection; this knowledge has been utilised in AlphaGo Zero both by augmenting the data set during training to include rotations and reflections of each position, and to sample random rotations or reflections of the position during MCTS (see Search Algorithm). Aside from komi, the rules of Go are also invariant to colour transposition; this knowledge is exploited by representing the board from the perspective of the current player (see Neural network architecture)

AlphaGo Zero does not use any form of domain knowledge beyond the points listed above. It only uses its deep neural network to evaluate leaf nodes and to select moves (see section below). It does not use any rollout policy or tree policy, and the MCTS is not augmented by any other heuristics or domain-speciﬁc rules. No legal moves are excluded – even those ﬁlling in the player’s own eyes (a standard heuristic used in all previous programs 67).

The algorithm was started with random initial parameters for the neural network. The neural network architecture (see Neural Network Architecture) is based on the current state of the art in image recognition 4, 18, and hyperparameters for training were chosen accordingly (see Self- Play Training Pipeline). MCTS search parameters were selected by Gaussian process optimisation 68, so as to optimise self-play performance of AlphaGo Zero using a neural network trained in a preliminary run. For the larger run (40 block, 40 days), MCTS search parameters were re- optimised using the neural network trained in the smaller run (20 block, 3 days). The training algorithm was executed autonomously without human intervention.

Self-Play Training Pipeline AlphaGo Zero’s self-play training pipeline consists of three main

components, all executed asynchronously in parallel. Neural network parameters θi are continually

optimised from recent self-play data; AlphaGo Zero players αθi are continually evaluated; and the

best performing player so far, αθ∗ , is used to generate new self-play data.

Optimisation Each neural network fθi is optimised on the Google Cloud using TensorFlow, with 64 GPU workers and 19 CPU parameter servers. The batch-size is 32 per worker, for a total mini-batch size of 2,048. Each mini-batch of data is sampled uniformly at random from

23 all positions from the most recent 500,000 games of self-play. Neural network parameters are optimised by stochastic gradient descent with momentum and learning rate annealing, using the loss in Equation 1. The learning rate is annealed according to the standard schedule in Extended Data Table 3. The momentum parameter is set to 0.9. The cross-entropy and mean-squared error losses are weighted equally (this is reasonable because rewards are unit scaled, r ∈ {−1, +1}) and the L2 regularisation parameter is set to c = 10−4. The optimisation process produces a new checkpoint every 1,000 training steps. This checkpoint is evaluated by the evaluator and it may be used for generating the next batch of self-play games, as we explain next.

Evaluator To ensure we always generate the best quality data, we evaluate each new neural

network checkpoint against the current best network fθ∗ before using it for data generation. The neural network fθi is evaluated by the performance of an MCTS search αθi that uses fθi to evaluate leaf positions and prior probabilities (see Search Algorithm). Each evaluation consists of 400 games, using an MCTS with 1,600 simulations to select each move, using an inﬁnitesimal temperature τ → 0 (i.e. we deterministically select the move with maximum visit count, to give the strongest possible play). If the new player wins by a margin of > 55% (to avoid selecting on noise

alone) then it becomes the best player αθ∗ , and is subsequently used for self-play generation, and also becomes the baseline for subsequent comparisons.

Self-Play The best current player αθ∗ , as selected by the evaluator, is used to generate data.

In each iteration, αθ∗ plays 25,000 games of self-play, using 1,600 simulations of MCTS to select each move (this requires approximately 0.4s per search). For the ﬁrst 30 moves of each game, the temperature is set to τ = 1; this selects moves proportionally to their visit count in MCTS, and ensures a diverse set of positions are encountered. For the remainder of the game, an inﬁnitesimal temperature is used, τ → 0. Additional exploration is achieved by adding Dirichlet noise to the

prior probabilities in the root node s0, speciﬁcally P (s, a) = (1 − )pa + ηa, where η ∼ Dir(0.03) and = 0.25; this noise ensures that all moves may be tried, but the search may still overrule bad moves. In order to save computation, clearly lost games are resigned. The resignation threshold vresign is selected automatically to keep the fraction of false positives (games that could have been

24 won if AlphaGo had not resigned) below 5%. To measure false positives, we disable resignation in 10% of self-play games and play until termination.

Supervised Learning For comparison, we also trained neural network parameters θSL by supervised learning. The neural network architecture was identical to AlphaGo Zero. Mini-batches of

data (s,πππ, z) were sampled at random from the KGS data-set, setting πa = 1 for the human expert move a. Parameters were optimised by stochastic gradient descent with momentum and learning rate annealing, using the same loss as in Equation 1, but weighting the mean-squared error component by a factor of 0.01. The learning rate was annealed according to the standard schedule in Extended Data Table 3. The momentum parameter was set to 0.9, and the L2 regularisation parameter was set to c = 10−4.

By using a combined policy and value network architecture, and by using a low weight on the value component, it was possible to avoid overﬁtting to the values (a problem described in prior work 12). After 72 hours the move prediction accuracy exceeded the state of the art reported in previous work 12, 30–33, reaching 60.4% on the KGS test set; the value prediction error was also substantially better than previously reported 12. The validation set was composed of professional games from GoKifu. Accuracies and mean squared errors are reported in Extended Data Table 1 and Extended Data Table 2 respectively.

Search Algorithm AlphaGo Zero uses a much simpler variant of the asynchronous policy and value MCTS algorithm (APV-MCTS) used in AlphaGo Fan and AlphaGo Lee.

Each node s in the search tree contains edges (s, a) for all legal actions a ∈ A(s). Each edge stores a set of statistics,

{N(s, a),W (s, a),Q(s, a),P (s, a)},

where N(s, a) is the visit count, W (s, a) is the total action-value, Q(s, a) is the mean action-value, and P (s, a) is the prior probability of selecting that edge. Multiple simulations are executed in parallel on separate search threads. The algorithm proceeds by iterating over three phases (a–c in Figure 2), and then selects a move to play (d in Figure 2).

25 Select (Figure 2a). The selection phase is almost identical to AlphaGo Fan 12; we recapitulate here for completeness. The ﬁrst in-tree phase of each simulation begins at the root node of the search tree, s0, and ﬁnishes when the simulation reaches a leaf node sL at time-step L. At each of these time-steps, t < L, an action is selected according to the statistics in the search tree,

24 at = argmax Q(st, a) + U(st, a) , using a variant of the PUCT algorithm , a

pP N(s, b) U(s, a) = c P (s, a) b puct 1 + N(s, a) where cpuct is a constant determining the level of exploration; this search control strategy initially prefers actions with high prior probability and low visit count, but asympotically prefers actions with high action-value.

Expand and evaluate (Figure 2b). The leaf node sL is added to a queue for neural network evaluation, (di(p), v) = fθ(di(sL)), where di is a dihedral reﬂection or rotation selected uniformly at random from i ∈ [1..8].

Positions in the queue are evaluated by the neural network using a mini-batch size of 8; the search thread is locked until evaluation completes. The leaf node is expanded and each edge (sL, a) is initialised to {N(sL, a) = 0,W (sL, a) = 0,Q(sL, a) = 0,P (sL, a) = pa}; the value v is then backed up.

Backup (Figure 2c). The edge statistics are updated in a backward pass through each step t ≤ L. The visit counts are incremented, N(st, at) = N(st, at)+1, and the action-value is updated to the mean value, W (s , a ) = W (s , a ) + v, Q(s , a ) = W (st,at) . We use virtual loss to ensure t t t t t t N(st,at) each thread evaluates different nodes 69.

Play (Figure 2d). At the end of the search AlphaGo Zero selects a move a to play in the root 1/τ P 1/τ position s0, proportional to its exponentiated visit count, π(a|s0) = N(s0, a) / b N(s0, b) , where τ is a temperature parameter that controls the level of exploration. The search tree is reused at subsequent time-steps: the child node corresponding to the played action becomes the new root node; the subtree below this child is retained along with all its statistics, while the remainder of

26 the tree is discarded. AlphaGo Zero resigns if its root value and best child value are lower than a threshold value vresign.

Compared to the MCTS in AlphaGo Fan and AlphaGo Lee, the principal differences are that AlphaGo Zero does not use any rollouts; it uses a single neural network instead of separate policy and value networks; leaf nodes are always expanded, rather than using dynamic expansion; each search thread simply waits for the neural network evaluation, rather than performing evaluation and backup asynchronously; and there is no tree policy. A transposition table was also used in the large (40 block, 40 day) instance of AlphaGo Zero.

Neural Network Architecture The input to the neural network is a 19 × 19 × 17 image stack comprising 17 binary feature planes. 8 feature planes Xt consist of binary values indicating the i presence of the current player’s stones (Xt = 1 if intersection i contains a stone of the player’s colour at time-step t; 0 if the intersection is empty, contains an opponent stone, or if t < 0). A further 8 feature planes, Yt, represent the corresponding features for the opponent’s stones. The ﬁnal feature plane, C, represents the colour to play, and has a constant value of either 1 if black is to play or 0 if white is to play. These planes are concatenated together to give input features st = [Xt,Yt,Xt−1,Yt−1, ..., Xt−7,Yt−7,C]. History features Xt,Yt are necessary because Go is not fully observable solely from the current stones, as repetitions are forbidden; similarly, the colour feature C is necessary because the komi is not observable.

The input features st are processed by a residual tower that consists of a single convolutional block followed by either 19 or 39 residual blocks 4.

The convolutional block applies the following modules:

1. A convolution of 256 ﬁlters of kernel size 3 × 3 with stride 1

2. Batch normalisation 18

3. A rectiﬁer non-linearity

27 Each residual block applies the following modules sequentially to its input:

1. A convolution of 256 ﬁlters of kernel size 3 × 3 with stride 1

2. Batch normalisation

3. A rectiﬁer non-linearity

4. A convolution of 256 ﬁlters of kernel size 3 × 3 with stride 1

5. Batch normalisation

6. A skip connection that adds the input to the block

7. A rectiﬁer non-linearity

The output of the residual tower is passed into two separate “heads” for computing the policy and value respectively. The policy head applies the following modules:

1. A convolution of 2 ﬁlters of kernel size 1 × 1 with stride 1

2. Batch normalisation

3. A rectiﬁer non-linearity

4. A fully connected linear layer that outputs a vector of size 192 + 1 = 362 corresponding to logit probabilities for all intersections and the pass move

The value head applies the following modules:

1. A convolution of 1 ﬁlter of kernel size 1 × 1 with stride 1

2. Batch normalisation

3. A rectiﬁer non-linearity

28 4. A fully connected linear layer to a hidden layer of size 256

5. A rectiﬁer non-linearity

6. A fully connected linear layer to a scalar

7.A tanh non-linearity outputting a scalar in the range [−1, 1]

The overall network depth, in the 20 or 40 block network, is 39 or 79 parameterised layers respectively for the residual tower, plus an additional 2 layers for the policy head and 3 layers for the value head.

We note that a different variant of residual networks was simultaneously applied to computer Go 33 and achieved amateur dan-level performance; however this was restricted to a single-headed policy network trained solely by supervised learning.

Neural Network Architecture Comparison Figure 4 shows the results of a comparison between network architectures. Speciﬁcally, we compared four different neural networks:

1. dual-res: The network contains a 20-block residual tower, as described above, followed by both a policy head and a value head. This is the architecture used in AlphaGo Zero.

2. sep-res: The network contains two 20-block residual towers. The ﬁrst tower is followed by a policy head and the second tower is followed by a value head.

3. dual-conv: The network contains a non-residual tower of 12 convolutional blocks, followed by both a policy head and a value head.

4. sep-conv: The network contains two non-residual towers of 12 convolutional blocks. The ﬁrst tower is followed by a policy head and the second tower is followed by a value head. This is the architecture used in AlphaGo Lee.

Each network was trained on a ﬁxed data-set containing the ﬁnal 2 million games of self- play data generated by a previous run of AlphaGo Zero, using stochastic gradient descent with

29 the annealing rate, momentum, and regularisation hyperparameters described for the supervised learning experiment; however, cross-entropy and mean-squared error components were weighted equally, since more data was available.

Evaluation We evaluated the relative strength of AlphaGo Zero (Figure 3a and 6) by measuring the Elo rating of each player. We estimate the probability that player a will defeat player b by a logistic function p(a defeats b) = 1 , and estimate the ratings e(·) by Bayesian 1+exp (celo(e(b)−e(a)) 25 logistic regression, computed by the BayesElo program using the standard constant celo = 1/400.

Elo ratings were computed from the results of a 5 second per move tournament between AlphaGo Zero, AlphaGo Master, AlphaGo Lee, and AlphaGo Fan. The raw neural network from AlphaGo Zero was also included in the tournament. The Elo ratings of AlphaGo Fan, Crazy Stone, Pachi and GnuGo were anchored to the tournament values from prior work 12, and correspond to the players reported in that work. The results of the matches of AlphaGo Fan against Fan Hui and AlphaGo Lee against Lee Sedol were also included to ground the scale to human references, as otherwise the Elo ratings of AlphaGo are unrealistically high due to self-play bias.

The Elo ratings in Figure 3a, 4a and 6a were computed from the results of evaluation games between each iteration of player αθi during self-play training. Further evaluations were also performed against baseline players with Elo ratings anchored to the previously published values 12.

We measured the head-to-head performance of AlphaGo Zero against AlphaGo Lee, and the 40 block instance of AlphaGo Zero against AlphaGo Master, using the same player and match conditions as were used against Lee Sedol in Seoul, 2016. Each player received 2 hours of thinking time plus 3 byoyomi periods of 60 seconds per move. All games were scored using Chinese rules with a komi of 7.5 points.

Data Availability The datasets used for validation and testing are the GoKifu dataset (available from http://gokifu.com/) and the KGS dataset (available from https://u-go.net/gamerecords/).

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34 KGS train KGS test GoKifu validation

Supervised learning (20 block) 62.0 60.4 54.3 Supervised learning (12 layer 12) 59.1 55.9 - Reinforcement learning (20 block) - - 49.0 Reinforcement learning (40 block) - - 51.3

Extended Data Table 1: Move prediction accuracy. Percentage accuracies of move prediction for neural networks trained by reinforcement learning (i.e. AlphaGo Zero) or supervised learning respectively. For supervised learning, the network was trained for 3 days on KGS data (amateur games); comparative results are also shown from Silver et al 12. For reinforcement learning, the 20 block network was trained for 3 days and the 40 block network was trained for 40 days. Networks were also evaluated on a validation set based on professional games from the GoKifu data set.

KGS train KGS test GoKifu validation

Supervised learning (20 block) 0.177 0.185 0.207 Supervised learning (12 layer 12) 0.19 0.37 - Reinforcement learning (20 block) - - 0.177 Reinforcement learning (40 block) - - 0.180

Extended Data Table 2: Game outcome prediction error. Mean squared error on game outcome predictions for neural networks trained by reinforcement learning (i.e. AlphaGo Zero) or supervised learning respectively. For supervised learning, the network was trained for 3 days on KGS data (amateur games); comparative results are also shown from Silver et al 12. For reinforcement learning, the 20 block network was trained for 3 days and the 40 block network was trained for 40 days. Networks were also evaluated on a validation set based on professional games from the GoKifu data set. Thousands of steps Reinforcement learning Supervised learning

0–200 10−2 10−1 200–400 10−2 10−2 400–600 10−3 10−3 600–700 10−4 10−4 700–800 10−4 10−5 >800 10−4 -

Extended Data Table 3: Learning rate schedule. Learning rate used during reinforcement learning and supervised learning experiments, measured in thousands of steps (mini-batch updates). 5-3 point press Small avalanche 4.00e-03 4.00e-04

3.50e-03 3.50e-04 10 6 4 2 3 5 8 3.00e-03 3.00e-04 5 3 7 2 4 1 9 2.50e-03 2.50e-04 1 6 11 2.00e-03 2.00e-04

1.50e-03 1.50e-04 Frequency Frequency

1.00e-03 1.00e-04

5.00e-04 5.00e-05

0.00e+00 0.00e+00 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70 Hours Hours

Attach and draw back Knight's move pincer 2.50e-04 2.50e-05

2.00e-04 8 4 3 5 2.00e-05 3 6 6 2 1 1 4 1.50e-04 1.50e-05

7 2 1.00e-04 1.00e-05 Frequency Frequency

5 5.00e-05 5.00e-06

0.00e+00 0.00e+00 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70 Hours Hours

Pincer 3-3 point 4.00e-04

3.50e-04 12 6 4 10 3.00e-04 7 1 5 8 2.50e-04 9 2.00e-04 2 11 1.50e-04 Frequency 1.00e-04 3 5.00e-05

0.00e+00

0 10 20 30 40 50 60 70 Hours

Extended Data Figure 1: Frequency of occurence over time during training, for each joseki from Figure 5a (corner sequences common in professional play that were discovered by AlphaGo Zero). The corresponding joseki are reproduced in this ﬁgure as insets. 1-1 point Outside attachment 4.50e-04 1 6.00e-04

4.00e-04 5.00e-04 3.50e-04 5 7

3.00e-04 4.00e-04 9 3 1 4

2.50e-04 8 2 6 3.00e-04 2.00e-04

Frequency 1.50e-04 2 Frequency 2.00e-04

1.00e-04 4 3 1.00e-04 5.00e-05 10

0.00e+00 0.00e+00 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70 Hours Hours

Knight's move approach One-space jump 6.00e-04 6.00e-03 6 5.00e-04 5.00e-03 7 10 8 8 2 3

4.00e-04 5 1 4.00e-03 4 1 3 9 6 7 3.00e-04 3.00e-03 4 2 5 Frequency 2.00e-04 Frequency 2.00e-03

1.00e-04 1.00e-03

0.00e+00 0.00e+00 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70 Hours Hours

3-3 invasion 3-3 point knight's move 3.50e-02 12 2.50e-03 10 7 6 11 13 17 8 11 15 13 10 3.00e-02 8 5 4 2 2.00e-03 19 5 6 4 2 9 14 2.50e-02 9 1 3 18 7 1 3 12 1.50e-03 2.00e-02

1.50e-02 20 1.00e-03 Frequency

Frequency 1.00e-02 5.00e-04 5.00e-03 16 at 9

0.00e+00 0.00e+00 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70 Hours Hours

Extended Data Figure 2: Frequency of occurence over time during training, for each joseki from Figure 5b, (corner sequences that AlphaGo Zero favoured for at least one iteration), and one additional variation. The corresponding joseki are reproduced in this ﬁgure as insets. Game 1, B: AG Zero, W: AG Zero, Result: B+R Game 2, B: AG Zero, W: AG Zero, Result: B+R Game 3, B: AG Zero, W: AG Zero, Result: W+R Game 4, B: AG Zero, W: AG Zero, Result: B+R 2 90 63 13 65 51 77 93 23 48 88 98 28 10044 42 48 82 81 60 58 30 7 76 59 89 70 94 50 22 55 76 98 43 23 22 18 16 80 79 8 99 72 70 69 21 61 56 57 4 3 33 58 45 47 26 24 18 65 67 90 96 86 92 17 94 99 87 86 21 24 15 14 20 52 78 49 45 77 97 98 74 71 68 87 59 31 6 2 1 35 44 21 25 17 20 66 59 56 83 30 8 94 95 25 26 7 64 58 67 37 80 79 75 73 86 88 91 26 25 5 34 46 23 68 19 93 3 5 81 12 52 9 96 27 28 11 2 9 41 84 55 29 27 22 32 69 79 41 4 25 77 51 29 30 5 13 83 81 78 66 64 62 54 89 90 39 36 28 99 39 14 10 38 50 59 31 32 56 12 85 82 67 65 63 53 52 49 43 41 40 48 50 92 95 98 61 32 33 44 57 1 97 33 34 6 93 92 94 51 50 48 42 44 76 49 72 88 91 96 84 54 46 16 91 37 35 10 8 46 38 96 10095 11 77 74 43 73 10097 100 80 40 31 6 39 36 38 75 78 82 68 29 87 69 99 60 75 64 24 40 54 60 4 47 91 87 85 83 79 60 67 26 66 61 46 47 15 13 86 62 80 81 73 42 85 11 18 15 53 17 24 45 12 11 41 71 21 84 62 78 7 83 61 16 23 10 5 6 14 30 36 34 40 39 20 45 27 35 95 53 57 68 75 69 19 65 10 20 18 14 12 15 4 1 32 29 16 35 56 52 54 22 37 97 55 66 72 76 84 3 62 63 93 19 17 13 9 2 3 33 31 57 27 53 55 42 38 51 47 72 70 58 49 73 71 70 74 1 85 88 89 92 8 7 9 19 28 64 63 71 43 89 36 82 37 34 74 90

Game 5, B: AG Zero, W: AG Zero, Result: B+R Game 6, B: AG Zero, W: AG Zero, Result: W+R Game 7, B: AG Zero, W: AG Zero, Result: B+R Game 8, B: AG Zero, W: AG Zero, Result: B+R 38 15 21 71 30 73 17 25 39 36 35 28 37 39 48 16 14 7 10 13 80 76 20 13 70 26 24 25 16 13 29 28 40 30 96 27 95 23 21 22 85 16 30 25 26 24 20 33 47 3 5 6 11 94 50 17 9 77 12 3 11 22 1 71 12 3 84 41 24 2 26 97 80 79 86 32 31 27 1 23 96 4 2 12 79 45 20 8 74 75 14 10 19 74 23 69 14 15 78 47 46 88 87 93 95 55 54 21 64 62 65 66 65 19 58 48 97 95 18 34 91 97 49 53 72 27 66 67 70 67 18 64 77 42 9810099 59 58 56 46 57 69 68 75 68 81 83 92 93 61 60 72 73 76 100 55 60 57 80 31 63 62 90 78 71 97 94 98 87 61 72 63 62 82 93 29 65 64 89 93 92 82 90 91 10098 73 74 76 59 45 56 99 98 67 66 95 81 78 86 61 63 99 79 44 90 100 69 68 29 79 77 60 80 59 44 48 51 89 85 86 92 83 88 70 27 28 56 55 57 40 52 51 47 38 35 49 88 87 72 96 91 94 44 41 87 23 26 9 5 54 49 34 58 50 53 36 37 92 91 71 83 53 90 63 45 42 43 33 25 24 7 2 6 36 28 33 18 4 37 39 34 1 75 20 8 4 10 94 4 2 10 19 74 82 59 44 41 17 77 15 40 1 31 30 32 18 29 8 96 88 84 32 31 35 17 15 16 45 32 33 11 7 5 6 21 3 5 6 9 69 52 62 66 40 65 49 43 76 51 57 22 39 35 34 19 99 89 83 42 41 43 38 46 50 43 52 53 9 22 14 12 7 8 11 58 67 68 60 42 47 46 50 54 55 82 81 84 85 37 36 38 52 85 54 13 70 61 64 48 56 86 51 73 at 61 75 at 67 78 at 58 81 at 67 84 at 58 89 at 67 Game 9, B: AG Zero, W: AG Zero, Result: W+R Game 10, B: AG Zero, W: AG Zero, Result: W+R Game 11, B: AG Zero, W: AG Zero, Result: W+R Game 12, B: AG Zero, W: AG Zero, Result: W+R 79 15 59 70 78 77 66 64 65 61 62 18 16 14 9 10 13 45 58 57 61 69 68 76 20 71 60 55 5 46 17 5 7 8 11 40 98 4 1 94 3 43 42 44 17 28 54 53 84 3 80 73 72 74 58 59 47 4 6 2 12 97 38 7 93 89 97 3 5 29 2 55 87 82 57 81 75 67 63 52 54 39 19 6 87 83 84 95 95 89 78 71 67 56 51 52 85 83 88 49 48 5 82 92 96 32 87 88 48 83 70 60 18 5610086 43 50 53 99100 77 76 96 92 98 90 85 84 99 51 79 78 88 97 91 86 50 45 29 8 22 80 11 81 36 82 93 99 49 46 83 98 37 13 66 64 27 62 44 41 39 96 28 91 12 10 65 63 42 40 94 82 80 74 41 42 68 75 66 90 31 47 14 33 57 55 64 81 71 70 72 70 74 73 65 67 13 9 94 21 69 68 16 11 12 93 36 26 23 37 63 62 6510056 75 73 43 30 43 69 72 71 79 34 22 20 19 10 9 13 37 32 24 25 99 52 66 93 91 89 86 88 45 44 67 26 24 44 47 40 38 85 14 7610077 40 25 81 15 8 1 33 26 2 22 31 22 1 54 53 92 58 60 90 87 68 3 46 69 52 23 4 28 48 46 41 39 31 30 20 18 16 2 7 1 11 15 30 38 75 33 4 23 80 17 6 7 96 38 34 24 25 23 21 28 35 20 21 51 50 48 94 49 59 61 27 95 78 76 47 25 27 55 45 42 49 36 21 19 15 32 86 16 6 8 10 9 14 74 73 35 26 24 19 18 97 91 90 30 27 29 32 31 34 35 84 79 85 51 53 29 50 57 56 37 34 33 35 17 12 72 41 39 36 54 64 63 62 60 59 58 61 89 at 3 92 at 82 95 at 3 98 at 82 77 at 46

Game 13, B: AG Zero, W: AG Zero, Result: W+R Game 14, B: AG Zero, W: AG Zero, Result: W+R Game 15, B: AG Zero, W: AG Zero, Result: W+R Game 16, B: AG Zero, W: AG Zero, Result: W+R 58 62 67 57 27 29 36 34 57 87 57 56 58 97 88 91 56 55 44 45 59 16 15 18 20 26 25 23 24 38 32 33 47 48 13 92 14 46 1 66 20 17 19 11 58 39 53 1 6 7 14 28 2 5 35 30 1 49 4 59 60 46 18 4 41 14 15 52 22 8 5 4 56 31 65 64 5010053 54 62 35 63 43 42 16 37 21 10 11 22 47 37 51 61 99 98 5 13 79 84 38 36 61 12 9 19 13 54 61 67 55 51 21 40 78 97100 17 44 46 40 66 71 73 74 77 64 66 76 45 79 85 29 70 72 75 80 88 99 50 75 78 84 28 68 69 76 81 93 90 65 38 90 75 55 89 43 63 77 68 63 76 90 52 42 50 49 96 82 92 87 36 74 65 91 95 96 97 86 65 53 62 73 74 95 43 41 44 48 47 94 91 95 99 98 34 35 64 51 50 73 95 17 77 91 98 51 64 41 42 85 30 96 45 81 24 89 37 76 48 49 57 94 12 9 21 94 69 67 60 14 72 69 70 40 82 24 26 31 60 39 23 79 80 46 47 10 11 100 74 84 82 80 85 87 68 22 71 20 39 36 33 54 23 22 25 86 78 77 42 45 52 53 72 92 97 96 8 3 75 92 90 83 81 79 78 48 88 4 18 7 3 11 21 19 37 31 26 34 2 32 3 27 6 10 83 30 34 2 43 28 2 30 82 24 40 41 54 56 26 3 98100 1 6 7 71 70 49 13 52 28 15 6 8 10 9 23 32 33 9 7 8 12 5 32 27 29 81 62 58 25 55 68 86 93 44 83 16 15 18 19 72 93 39 59 16 27 29 25 12 17 35 38 33 31 60 59 89 63 66 67 84 85 20 73 94 61 87 70 71 69 88 80 at 68 83 at 63 86 at 68 89 at 63 93 at 68 99 at 44

Game 17, B: AG Zero, W: AG Zero, Result: W+R Game 18, B: AG Zero, W: AG Zero, Result: B+R Game 19, B: AG Zero, W: AG Zero, Result: B+R Game 20, B: AG Zero, W: AG Zero, Result: W+R

25 27 79 80 67 66 70 17 15 13 22 1 26 65 62 58 86 27100 9 14 11 12 10 11 65 59 57 49 47 53 19 3 11 21 21 20 24 28 69 61 56 4 54 57 3 99 98 96 88 97 25 16 2 12 3 60 63 44 58 50 4 48 51 9 18 20 23 37 5 1 23 74 72 68 59 53 55 51 85 86 84 87 14 15 64 68 56 52 7 6 36 3 97 82 71 75 52 88 85 81 10 16 13 41 66 61 45 43 54 8 10 14 56 96 81 36 73 60 42 12 64 62 61 79 40 71 69 67 55 84 78 76 77 41 43 63 87 72 68 71 42 62 27 97 41 35 83 76 56 58 59 98 94 46 92 94 25 24 30 40 17 83 34 29 75 66 65 57 54 45 99 95 9310091 82 29 13 26 98 90 84 91 63 64 55 39 36 60 81 83 84 28 34 92 42 58 22 46 37 89 73 44 37 40 31 38 29 85 24 70 86 15 33 90 39 87 89 96 54 52 46 57 95 100 93 74 52 69 42 41 29 30 27 77 75 25 80 93 31 35 91 95 86 88 53 51 45 49 13 15 40 10 8 93 8 94 95 53 80 70 32 28 21 24 9 76 74 99 12 32 85 17 14 45 39 38 32 6 7 78 51 23 22 5 37 36 18 20 22 79 78 100 66 76 80 79 48 2 16 94 33 31 30 18 9 4 7 26 50 49 1 20 28 26 30 32 34 17 19 21 96 8 72 1 4 68 62 63 81 72 47 43 49 50 2 38 47 50 44 92 99 11 5 6 5 90 46 92 19 18 2 23 31 33 35 97 90 6 7 73 60 67 64 61 48 71 70 55 44 16 98 19 82 89 91 35 33 34 48 39 38 88 87 59 69 65 77 82 73 74 75 47 89 78 43 at 40 67 at 54 77 at 69 83 at 48

Extended Data Figure 3: AlphaGo Zero (20 block) self-play games. The 3 day training run was subdivided into 20 periods. The best player from each period (as selected by the evaluator) played a single game against itself, with 2 hour time controls. 100 moves are shown for each game; full games are provided in Supplementary Information. Game 1, B: AG Zero, W: AG Zero, Result: B+R Game 2, B: AG Zero, W: AG Zero, Result: W+R Game 3, B: AG Zero, W: AG Zero, Result: B+R Game 4, B: AG Zero, W: AG Zero, Result: W+R 48 61 86 85 89 11 80 97 27 34 49 72 50 73 85 58 57 98 99 11 67 42 86 78 84 75 83 82 81 87 14 7 84 96 52 39 74 14 13 82 17 8 9 7 5 21 93 94 81 51 52 34 82 61 84 80 79 29 76 73 77 45 36 31 15 53 4 15 56 47 18 10 2 6 19 3 5 92 80 96 97 55 64 39 40 4 62 36 63 83 2 85 78 74 75 5 3 82 95 88 81 26 62 61 67 66 60 63 20 95 87 98 66 65 41 71 72 57 90 86 49 59 47 6 29 19 33 10 16 75 74 68 59 79 81 52 91 90 99 76 68 37 33 30 88 94 54 56 57 46 42 37 59 64 69 72 70 100 77 69 70 73 79 89 52 53 58 48 42 28 16 55 65 73 80 49 58 54 88 92 93 63 31 60 45 30 58 8 75 2 56 68 48 96 43 45 74 47 95 98 62 61 43 44 99 35 77 70 5 78 83 38 48 46 65 50 9 79 25 83 41 76 71 53 50 10099 51 64 55 72 20 63 23 94 18 21 69 67 43 51 77 88 89 16 28 69 67 68 70 92 89 62 90 47 45 49 50 93 90 91 14 44 56 29 66 23 21 38 51 65 10012 3 97 44 95 92 85 84 12 9 15 35 27 26 31 25 22 9 12 71 4 24 13 78 71 64 10012 37 53 86 20 10 11 59 28 24 25 27 36 97 96 11 10 15 57 43 1 32 38 35 36 54 29 1 23 87 19 18 8 1 2 23 30 32 34 33 26 1 8 19 54 22 98 42 40 3 39 34 28 25 24 22 17 6 7 22 21 32 4 41 35 91 38 24 7 6 17 91 76 60 46 41 30 27 26 31 33 13 60 37 39 40 16 13 14 46 87 17 44 93 40 55 66 32 18 94 at 85 20 at 13

Game 5, B: AG Zero, W: AG Zero, Result: W+R Game 6, B: AG Zero, W: AG Zero, Result: W+R Game 7, B: AG Zero, W: AG Zero, Result: W+R Game 8, B: AG Zero, W: AG Zero, Result: B+R 76 75 82 79 42 58 57 48 77 76 68 77 66 67 74 49 51 78 57 55 41 60 39 40 9 89 85 40 44 35 46 47 43 7 32 75 43 45 39 44 29 35 33 72 64 65 92 91 85 25 44 48 56 53 54 61 21 20 93 5 7 88 79 77 83 39 2 74 42 45 49 33 4 78 47 2 40 41 46 4 69 70 1 94 86 84 87 45 36 58 50 3 22 91 6 4 8 95 87 80 2 78 81 41 73 42 48 34 71 93 37 43 46 23 92 98 84 82 5 36 38 34 5 32 30 11 72 19 95 47 24 94 90 11 37 53 54 31 36 89 38 28 97 86 51 55 60 92 71 73 100 62 90 100 71 50 52 89 59 63 52 17 19 99 69 65 70 93 12 63 36 75 42 63 64 66 96 16 88 65 64 70 69 18 76 68 67 62 61 94 93 97 19 30 66 67 51 60 28 61 99 53 66 68 90 95 21 20 25 31 29 50 94 55 59 73 35 96 74 72 70 64 60 61 67 5610089 72 99 24 22 23 27 6 90 98 97 56 54 21 20 24 23 17 35 10 24 33 34 55 73 71 69 63 62 65 79 98 18 11 26 91 85 87 10074 53 52 17 18 68 62 33 34 26 28 23 25 32 41 54 56 52 83 3 81 85 17 10 9 28 1 13 83 81 3 84 37 99 76 75 15 16 1 7 19 57 29 4 27 88 12 2 6 96 30 29 1 21 22 57 50 15 3 13 82 80 84 59 91 8 14 12 15 82 80 86 78 79 77 9 14 8 6 22 58 31 30 98 18 11 8 7 5 81 31 26 27 20 37 39 59 51 47 46 14 12 43 92 88 86 87 96 95 38 49 13 10 25 26 97 32 13 10 9 14 16 83 40 38 49 16 45 48 44 27 15 58 80 at 66

Game 9, B: AG Zero, W: AG Zero, Result: W+R Game 10, B: AG Zero, W: AG Zero, Result: W+R Game 11, B: AG Zero, W: AG Zero, Result: B+R Game 12, B: AG Zero, W: AG Zero, Result: W+R 22 18 17 20 90 89 53 73 71 59 87 86 45 63 67 64 65 33 34 31 9 32 92 21 10 11 28 70 100 93 10 11 51 31 72 45 68 65 20 62 17 66 22 11 10 28 5 7 23 22 91 76 87 23 12 3 2 99 94 12 1 52 48 49 67 66 4 2 46 24 25 16 3 12 40 33 6 2 8 30 86 88 4 19 14 15 97 14 15 50 80 75 23 13 36 35 38 90 89 85 80 16 13 26 29 16 13 29 76 58 40 26 42 43 37 14 95 93 84 79 77 81 25 27 83 82 70 28 63 62 61 57 50 47 31 27 41 39 26 65 71 98 73 84 30 69 82 60 30 28 32 29 15 25 27 70 78 78 79 63 64 72 81 64 77 84 44 80 81 97 75 69 66 86 58 99 96 97 83 78 48 49 78 79 39 20 24 38 94 82 83 87 85 67 68 10098 95 39 79 72 74 77 35 59 57 92 86 87 35 32 52 84 71 70 59 61 17 21 37 49 96 69 100 98 80 46 48 27 85 33 19 100 99 76 73 60 75 57 56 52 36 81 77 76 74 60 56 49 45 44 9 88 26 22 23 9 9 18 97 98 68 55 56 34 16 18 36 58 53 59 61 33 75 51 96 52 8 43 42 91 25 24 47 8 36 37 8 21 90 95 96 69 85 58 54 63 41 29 66 99 38 37 1 31 32 95 94 61 4 7 47 3 21 18 46 19 42 2 7 44 7 4 93 94 83 1 57 62 3 47 19 65 50 54 60 64 68 13 1 11 34 35 30 39 89 50 92 62 6 5 17 20 74 43 41 6 5 5 6 15 91 92 82 53 72 40 44 46 43 51 55 67 48 12 10 93 88 90 91 54 53 40 41 55 54 56 34 38 51 89 88 71 73 42 45 14 55 74 24 at 17

Game 13, B: AG Zero, W: AG Zero, Result: B+R Game 14, B: AG Zero, W: AG Zero, Result: W+R Game 15, B: AG Zero, W: AG Zero, Result: W+R Game 16, B: AG Zero, W: AG Zero, Result: W+R 41 41 35 30 39 40 43 49 99 47 40 36 37 65 27 97 96 19 20 37 38 42 7 6 13 14 44 39100 30 29 28 18 74 13 26 39 45 22 21 29 16 15 17 5 92 88 67 10098 21 4 29 31 36 1 8 23 15 4 28 42 2 31 17 2 20 43 27 4 23 28 30 2 25 12 13 84 81 86 87 3 99 25 27 28 32 33 9 21 20 16 40 32 19 44 42 46 67 24 26 14 85 90 89 93 73 23 22 26 10 22 17 18 97 5 5 14 16 38 65 25 11 18 80 91 78 74 72 82 24 34 60 25 19 26 41 33 71 15 35 94 53 27 24 70 73 83 95 79 77 75 87 51 43 59 58 68 69 66 68 29 64 84 86 96 99100 90 77 91 72 95 63 82 83 81 97 93 95 92 89 88 76 71 72 84 87 89 90 93 57 98 91 89 92 75 79 77 69 62 62 50 66 98 91 85 52 70 73 69 85 86 75 88 94 95 93 64 87 90 80 78 76 61 70 76 57 11 18 65 75 71 76 58 94 48 79 78 66 63 74 68 53 94 88 66 33 59 60 55 10 61 63 79 14 72 73 77 59 47 67 61 62 64 65 92 51 51 54 83 52 51 41 71 9 54 53 56 67 2 64 12 13 74 57 56 3 46 7 1 9 55 60 81 96 83 80 48 47 37 3 35 50 58 7 3 11 48 52 99 97 33 1 31 82 20 4 22 31 37 38 44 1 8 58 16 15 17 5 68 70 55 45 44 56 6 8 10 11 54 49 45 46 36 34 52 60 55 6 8 10 9 49 53 10098 96 81 32 30 84 24 19 21 42 23 32 40 34 39 45 7 6 54 69 78 12 82 38 56 57 61 59 12 50 63 85 34 86 50 46 43 49 47 35 36 48 80 at 13 62 at 55

Game 17, B: AG Zero, W: AG Zero, Result: B+R Game 18, B: AG Zero, W: AG Zero, Result: W+R Game 19, B: AG Zero, W: AG Zero, Result: W+R Game 20, B: AG Zero, W: AG Zero, Result: W+R 61 60 59 58 60 10 46 86 10 88 26 71 58 59 61 13 15 49 69 67 41 21 19 20 91 92 26 8 6 45 85 54 6 8 12 9 52 24 22 64 77 56 9 11 35 12 14 16 47 48 3 42 16 15 35 5 40 21 4 10093 94 9 1 7 87 55 7 1 13 53 98 92 25 3 23 91 75 10 2 36 45 51 46 39 7 43 44 18 14 1 22 4 23 24 96 55 54 47 97 91 76 79 80 37 52 50 53 40 63 65 64 68 6 20 17 19 29 34 25 22 95 43 93 85 83 84 54 82 55 62 66 42 8 24 25 28 31 30 99 97 67 99 86 90 38 81 92 57 36 30 27 26 38 56 32 44 88 98 34 63 66 10096 65 93 89 41 18 32 23 31 85 86 62 88 87 23 24 33 44 37 79 84 51 58 43 39 45 87 89 41 42 71 78 83 90 49 61 59 60 84 17 22 10098 82 80 81 64 74 77 70 57 56 95 88 87 99 84 73 72 35 57 40 69 75 76 78 77 94 71 89 100 43 82 89 85 81 51 39 11 18 65 63 67 68 48 11 35 75 90 34 5 19 20 97 90 72 74 79 50 13 62 38 28 66 14 50 52 21 18 20 76 72 73 74 30 38 78 98 99 6 57 75 66 70 68 69 73 80 12 36 37 2 27 12 13 51 53 3 19 15 4 36 49 82 29 28 37 2 42 40 27 26 4 70 33 1 21 53 48 56 7 64 65 91 67 97 2 11 33 16 15 17 5 17 16 14 48 47 80 27 33 31 32 41 46 31 25 28 8 34 32 5 95 49 46 3 52 60 58 63 71 83 10 9 29 50 70 68 69 83 81 79 39 44 29 30 74 72 73 94 47 54 61 59 62 76 92 93 94 77 78 86 45 96 96 95 55 at 46

Extended Data Figure 4: AlphaGo Zero (40 block) self-play games. The 40 day training run was subdivided into 20 periods. The best player from each period (as selected by the evaluator) played a single game against itself, with 2 hour time controls. 100 moves are shown for each game; full games are provided in Supplementary Information. Game 1, B: AG Lee, W: AG Zero, Result: W+R Game 2, B: AG Lee, W: AG Zero, Result: W+R Game 3, B: AG Lee, W: AG Zero, Result: W+R Game 4, B: AG Lee, W: AG Zero, Result: W+0.50

30 32 34 96 32 34 36 85 30 28 29 79 83 18 20 30 26 28 29 31 33 35 97 47 51 52 57 28 30 31 33 35 37 84 83 41 45 46 68 61 20 21 77 31 78 67 41 55 58 32 14 16 17 19 21 31 93 27 1 49 48 2 56 62 29 1 51 82 59 43 42 66 2 67 60 69 40 22 1 76 80 66 69 64 73 59 53 52 56 2 29 15 1 59 2 63 50 58 60 53 50 44 65 49 32 27 26 82 43 81 70 71 54 50 81 80 48 50 51 80 81 17 61 70 72 91 93 27 39 24 23 88 62 42 86 68 65 75 61 51 49 97 79 58 75 82 53 73 59 75 55 76 63 64 92 25 89 33 92 85 72 74 57 78 77 73 76 52 49 64 72 67 94 95 74 40 94 74 73 80 71 62 19 63 84 87 91 60 74 36 68 69 71 90 92 87 38 81 75 78 89 77 88 100 18 90 60 54 55 70 54 86 89 93 82 54 53 39 79 98 100 99 98 48 46 56 84 85 10099 91 78 77 79 56 11 52 57 90 87 11 13 14 86 41 65 92 11 88 98 84 83 76 25 58 55 86 23 15 12 16 47 45 13 57 35 43 46 62 16 85 96 95 97 35 34 37 45 44 42 33 40 66 63 64 66 8 21 99 8 36 38 44 8 93 36 34 88 39 67 8 10 14 12 65 40 10 14 15 12 48 47 25 10 37 25 89 90 38 72 68 24 13 20 19 21 9 7 44 4 24 18 16 20 13 9 7 4 17 9 7 4 94 23 22 26 87 28 70 9 7 4 96 99 22 3 15 18 5 6 43 37 19 3 17 22 5 6 26 3 5 6 95 27 3 24 91 61 47 71 11 69 5 6 95 23 39 38 41 42 46 96 10 94 12 98 97 100 45 83 at 58

Game 5, B: AG Lee, W: AG Zero, Result: W+R Game 6, B: AG Lee, W: AG Zero, Result: W+0.50 Game 7, B: AG Lee, W: AG Zero, Result: W+R Game 8, B: AG Lee, W: AG Zero, Result: W+R 66 67 43 65 32 33 83 86 72 74 76 85 25 24 16 18 20 61 58 34 32 35 55 50 51 64 36 37 39 44 24 25 30 31 15 14 88 71 70 73 10 96 15 11 19 23 27 22 20 30 15 14 17 12 59 57 62 40 33 26 28 30 29 37 71 57 56 12 49 3 34 38 13 66 28 26 27 1 16 86 3 75 12 13 26 1 21 28 60 1 19 87 2 38 27 1 31 36 10 46 3 59 41 35 40 42 29 17 18 10087 94 16 14 95 18 69 53 51 33 32 29 63 74 39 84 41 70 58 47 52 88 19 20 98 17 93 54 52 50 34 13 72 73 60 53 54 21 22 99 92 89 80 55 56 96 44 61 48 84 23 97 79 47 46 53 54 85 63 62 99100 45 44 37 52 65 67 91 88 96 89 67 64 9 98 85 9 35 36 66 90 92 93 94 9 65 68 11 87 82 81 33 34 64 95 79 81 93 95 69 10 95 22 51 71 73 21 83 76 45 80 87 92 94 97 96 92 93 97 68 37 48 77 68 69 77 78 89 17 12 5 7 48 57 43 36 5 7 82 68 70 85 8 82 14 98 86 88 7 5 16 23 91 6 56 55 39 6 42 84 47 66 58 63 78 84 10 86 72 83 26 100 22 6 19 94 4 8 47 2 58 40 41 4 8 90 49 83 65 2 55 59 49 41 43 11 74 97 9 7 28 4 2 90 91 43 8 4 21 20 89 80 60 72 50 63 49 45 46 51 44 35 31 91 46 45 60 57 56 62 42 38 3 75 76 99 98 5 6 27 23 42 99 13 11 15 24 90 78 73 59 70 69 74 79 53 52 62 54 38 67 64 61 50 39 40 48 77 7810082 25 24 29 30 32 75 25 18 77 71 75 76 81 61 81 79 80 31

Game 9, B: AG Lee, W: AG Zero, Result: W+R Game 10, B: AG Lee, W: AG Zero, Result: W+R Game 11, B: AG Zero, W: AG Lee, Result: B+R Game 12, B: AG Zero, W: AG Lee, Result: B+1.50 48 30 28 29 91 89 90 42 41 44 45 90 24 22 50 49 24 50 51 53 52 29 27 30 64 34 35 20 21 31 60 72 62 45 88 83 84 24 25 94 37 31 38 21 4 42 41 57 3 22 21 23 25 6 61 63 48 8 7 34 22 1 67 2 85 87 26 1 39 2 36 23 20 43 59 51 39 36 37 4 19 33 31 26 28 62 68 66 18 2 9 32 27 26 57 61 86 96 28 29 40 35 19 48 47 58 56 55 32 65 17 47 10 11 33 24 2310058 56 66 73 30 27 46 49 81 34 32 23 33 44 38 46 74 20 44 67 12 13 25 35 65 68 98 43 47 80 33 76 77 31 32 45 42 73 55 57 92 59 14 15 19 59 64 70 78 79 29 25 26 41 43 87 98 16 75 18 69 76 63 83 50 51 27 6 28 95 86 85 84 58 69 71 74 97 97 30 89 74 82 93 40 56 83 79 78 60 11 13 14 99 85 62 82 92 84 88 73 72 81 76 94 99 54 88 89 91 71 15 12 16 63 89 75 86 8810098 11 91 78 80 83 86 65 75 71 100 90 93 77 75 72 70 37 36 39 44 43 59 61 64 84 87 99 18 77 79 87 96 94 95 76 82 38 40 42 41 55 93 8 93 60 65 67 8 46 63 90 54 66 80 5 97 10 53 54 51 92 78 77 81 95 58 10 72 66 69 73 16 17 85 61 62 60 39 81 17 52 47 50 9 7 4 80 79 96 55 71 70 68 9 7 20 4 34 8 2 45 67 70 52 5 97 1 36 49 38 3 46 48 49 5 6 82 56 12 3 53 57 5 6 19 13 35 7 9 10 12 14 16 68 64 69 53 96 98 99 3 37 40 1 94 95 92 91 52 54 74 15 14 17 18 22 11 13 15 100 21

Game 13, B: AG Zero, W: AG Lee, Result: B+R Game 14, B: AG Zero, W: AG Lee, Result: B+R Game 15, B: AG Zero, W: AG Lee, Result: B+R Game 16, B: AG Zero, W: AG Lee, Result: B+R 59 24 40 41 58 39 57 49 56 61 57 56 49 36 61 57 58 62 63 3 94 20 4 22 23 26 7 8 40 37 50 51 45 47 18 53 55 20 21 5 59 58 9 46 42 43 38 35 34 4 32 67 49 45 39 56 59 3 65 19 21 36 35 9 4 42 60 52 28 46 48 54 2 19 23 2 60 47 1 40 39 33 31 66 53 52 48 44 60 20 21 28 25 11 10 33 43 44 24 22 25 69 44 41 38 37 41 85 69 55 54 24 23 25 64 13 12 35 34 26 17 6 78 68 63 55 53 51 45 48 84 68 22 42 72 53 55 15 14 36 94 29 27 10099 64 62 54 52 50 81 26 71 51 52 27 10016 38 73 74 43 65 42 54 6 88 74 73 97 98 6 82 75 76 67 63 61 64 89 29 30 89 90 96 84 71 82 78 51 47 57 58 62 85 88 41 31 32 91 83 97 74 85 83 70 67 81 80 88 77 50 66 60 59 79 77 84 87 33 34 62 85 84 95 72 70 22 86 77 75 65 87 72 28 19 70 40 79 83 56 78 80 83 91 37 18 79 78 73 69 68 96 90 66 80 68 69 90 92 99 96 43 38 31 76 87 5 71 13 79 8 18 46 93 86 95 98 97 17 93 82 86 31 89 26 94 17 27 99 1 70 5 73 100 45 2 8 46 92 66 65 63 67 80 1 3 35 12 91 76 20 4 23 25 8 2 30 29 5 95 93 1 98 74 75 44 71 47 16 14 12 10 9 7 49 3 77 30 6 64 32 81 61 28 34 11 10 14 92 7 18 15 16 24 95 86 7 9 10 12 14 16 97 94 92 96100 81 82 76 72 50 48 15 13 11 75 98 99 37 33 32 29 30 93 21 19 17 27 90 87 88 89 11 13 15 36 91 39 at 23

Game 17, B: AG Zero, W: AG Lee, Result: B+R Game 18, B: AG Zero, W: AG Lee, Result: B+R Game 19, B: AG Zero, W: AG Lee, Result: B+1.50 Game 20, B: AG Zero, W: AG Lee, Result: B+R 47 45 43 37 99 98100 27 69 62 65 67 46 37 41 32 40 44 18 19 18 31 29 36 39 36 42 24 21 20 25 26 63 59 60 66 68 6 38 36 31 35 29 8 7 13 12 10 25 30 27 28 34 7 8 96 44 16 18 19 13 14 62 28 60 78 3 4 61 21 33 30 27 2 9 15 11 2 41 32 4 33 35 9 2 95 41 43 4 17 21 15 4 59 71 72 58 6 7 64 34 24 23 17 39 10 11 17 14 16 42 43 11 10 93 40 22 20 23 74 17 18 61 69 10 9 11 22 28 20 25 42 12 13 21 9 19 44 26 13 12 97 94 25 15 27 76 19 16 23 29 68 70 80 79 8 26 14 15 20 37 14 26 75 22 24 32 63 66 12 81 83 92 16 39 38 91 77 73 65 30 82 76 75 93 77 38 23 76 29 31 67 64 88 83 82 78 47 45 75 31 53 50 51 52 87 85 84 79 69 22 46 72 92 33 28 39 36 37 45 46 87 89 90 89 88 86 74 70 73 40 35 41 35 34 38 85 74 58 99 90 60 59 71 61 77 79 10088 77 88 47 42 40 44 91 81 57 55100 5 24 5 93 78 83 7 86 87 74 5 49 53 73 71 30 43 33 48 99 93 92 52 56 54 70 71 51 50 58 66 68 89 90 92 95 99 84 85 78 48 50 51 52 86 70 63 34 65 67 69 49 54 10086 94 95 97 84 48 51 50 94 91 73 56 52 48 49 54 65 62 94 91 96 97 6 80 81 85 46 47 87 62 61 59 64 66 68 2 96 98 5 1 3 49 53 80 96 95 72 1 53 1 55 57 67 63 64 8 98 82 3 82 1 54 45 89 6 90 60 32 57 3 72 55 56 98 97 84 81 83 58 55 56 75 76 57 79 80

Extended Data Figure 5: Tournament games between AlphaGo Zero (20 block, 3 day) versus AlphaGo Lee using 2 hour time controls. 100 moves of the ﬁrst 20 games are shown; full games are provided in Supplementary Information. Game 1, B: AG Master, W: AG Zero, Result: W+R Game 2, B: AG Zero, W: AG Master, Result: B+R Game 3, B: AG Master, W: AG Zero, Result: W+R Game 4, B: AG Zero, W: AG Master, Result: B+R 36 68 51 52 12 31 34 61 60 66 63 56 69 50 49 45 48 67 69 59 6 8 10 9 32 30 19 26 24 7 8 64 55 72 70 71 49 53 6 3 98 7 6 44 41 46 55 17 54 61 66 5 71 60 73 7 1 11 33 35100 67 66 28 68 4 23 9 4 59 58 67 47 52 54 2 13 1 8 53 40 3 65 63 64 70 68 72 4 72 65 60 59 63 25 11 10 62 57 65 73 50 11 10 42 47 73 58 37 74 99 98 91 61 62 64 22 20 17 13 12 48 51 5 5 32 9 12 38 39 6 88 86 84 87 95 90 94 29 21 15 14 76 56 54 26 43 35 62 70 85 71 97 96 89 92 16 75 74 53 16 58 90 96 94 90 100 27 77 74 75 78 49 55 51 52 89 91 92 83 76 89 56 13 93 100 67 59 30 72 87 88 93 19 96 92 84 79 75 99 80 97 98 53 18 45 99 92 77 68 70 71 83 84 95 35 36 95 91 81 85 86 36 57 43 44 98 97 85 65 69 73 46 50 74 82 31 29 27 40 84 76 83 64 57 15 47 43 44 60 23 45 47 83 75 28 22 23 41 42 91 80 17 66 14 33 79 42 31 45 29 22 18 24 78 94 39 42 77 30 25 24 26 46 78 39 87 34 80 48 25 61 20 21 22 21 77 3 54 5 37 38 51 15 43 48 2 78 76 35 33 3 19 79 1 38 81 82 2 38 40 28 27 4 19 30 1 23 20 25 87 93 2 8 56 58 55 41 40 16 82 46 52 49 44 14 34 32 18 20 21 37 36 82 86 96 10099 63 41 37 39 62 24 18 17 32 31 26 27 88 16 14 12 10 9 7 57 69 81 50 79 80 89 88 95 86 34 33 28 29 15 13 11 90 93 94 81 at 53 85 at 78 97 at 88

Game 5, B: AG Master, W: AG Zero, Result: W+R Game 6, B: AG Zero, W: AG Master, Result: B+R Game 7, B: AG Master, W: AG Zero, Result: W+R Game 8, B: AG Zero, W: AG Master, Result: B+R

34 36 44 11 13 17 11 13 15 29 28 33 34 31 32 68 43 16 46 29 14 39 42 43 7 9 10 12 14 18 63 18 17 55 7 9 10 12 14 16 27 26 31 32 3 30 5 74 45 47 63 15 26 2 42 8 4 3 2 19 4 54 8 4 84 25 20 23 3 30 33 27 40 31 80 83 27 45 50 49 85 83 81 82 21 22 35 72 73 62 60 58 48 41 37 38 84 79 60 44 85 25 16 14 51 47 48 52 74 71 79 80 24 18 67 64 61 59 57 28 50 32 59 69 41 44 15 97 53 46 72 67 68 73 77 62 90 70 66 65 25 47 48 55 70 100 70 69 55 54 66 65 69 61 76 75 71 69 24 49 53 58 56 86 99 40 19 96 82 95 61 56 52 78 92 13 98 54 39 57 81 24 81 80 93 94 62 70 91 19 93 64 65 35 46 51 74 26 43 79 78 68 58 89 92 94 95 96 95 52 78 73 26 23 16 22 23 27 91 71 83 90 60 93 98 93 88 96 34 94 33 38 66 77 65 67 21 20 92 57 58 63 97 59 95 87 88 45 82 91 99 9710089 87 54 56 6 61 75 22 20 64 41 5 29 10089 98 9 12 6 64 99 56 96 94 86 49 83 85 86 55 51 50 97 63 76 68 71 21 66 39 38 34 28 35 88 85 99 11 10 57 43 42 78 7 1 11 80 90 52 49 22 20 4 18 2 29 36 30 62 67 1 40 37 24 25 31 74 13 84 1 8 98 2 40 50 53 1 44 84 75 6 8 10 9 81 53 23 21 19 17 92 90 28 5 72 37 15 32 3 30 75 72 73 87 7 6 36 35 38 5 51 17 47 46 76 79 12 77 91 87 88 45 82 36 33 77 76 86 59 60 10037 39 41 48 89 42 at 37

Game 9, B: AG Master, W: AG Zero, Result: W+R Game 10, B: AG Zero, W: AG Master, Result: B+R Game 11, B: AG Master, W: AG Zero, Result: B+R Game 12, B: AG Zero, W: AG Master, Result: B+R 62 84 91 84 61 60 55 56 81 80 76 74 75 98 99 29 81 79 78 80 30 29 3 33 86 88 90 85 81 7 6 73 71 83 63 82 46 56 19 79 77 18 30 32 34 44 38 26 40 25 32 31 18 17 21 28 50 20 23 18 75 74 32 28 26 27 89 13 87 1 8 72 68 3 69 65 54 53 57 78 1 31 33 54 43 2 37 39 34 24 4 19 1 31 85 57 19 3 76 31 93 11 10 67 66 62 58 55 61 24 22 23 28 46 41 53 68 33 20 21 51 84 54 53 77 80 79 5 54 63 58 9 12 17 59 60 52 50 48 47 21 20 26 14 42 45 67100 22 17 27 55 56 82 34 65 92 73 72 59 57 88 70 64 85 51 49 25 27 29 30 29 15 27 80 84 90 92 23 22 25 52 16 35 52 77 68 75 74 78 47 83 91 85 93 35 26 30 88 32 37 64 83 69 66 67 70 76 93 91 92 28 36 98 87 100 36 82 53 71 89 90 94 63 86 48 95 96 97 94 86 93 38 94 97 96 69 71 87 47 34 40 49 99 91 92 73 99 45100 16 16 59 74 89 88 82 39 43 16 23 15 25 50 43 98 95 14 15 58 70 94 81 64 89 58 67 14 15 14 24 47 45 51 21 5 42 12 13 57 60 5 77 95 65 9 12 5 33 83 60 70 71 12 13 99 46 44 43 49 20 41 38 40 10 11 56 62 61 75 72 76 11 10 90 45 24 42 44 35 69 72 61 10 11 95 2 42 48 39 40 4 19 39 35 4 44 2 9 50 48 49 73 13 97 1 8 46 2 37 36 68 59 65 4 9 97 96 41 22 18 17 37 36 6 8 7 52 51 3 55 79 96 7 6 41 38 6 66 64 8 7 10098 86 87 78 66 62 63

Game 13, B: AG Master, W: AG Zero, Result: W+R Game 14, B: AG Zero, W: AG Master, Result: W+R Game 15, B: AG Master, W: AG Zero, Result: W+R Game 16, B: AG Zero, W: AG Master, Result: W+R

12 11 13 15 29 28 33 34 63 31 32 66 84 77 37 39 41 52 6 8 10 9 38 48 88 69 7 9 10 12 14 16 27 26 31 32 64 30 27 28 34 24 65 83 18 17 83 74 36 35 38 5 43 15 51 50 7 1 11 40 87 4 52 68 8 2 25 20 23 1 30 59 2 29 33 4 19 75 73 4 42 40 93 3 48 86 84 21 22 60 20 21 82 76 45 72 47 46 89 85 49 51 47 61 24 18 14 26 36 38 67 22 93 92 6 66 94 53 96 50 94 96 58 25 15 35 37 87 91 94 23 79 54 100 49 35 33 90 83 97 53 92 95 54 61 41 89 85 82 64 63 81 95 34 30 31 81 80 44 70 91 93 56 53 52 86 88 80 62 80 61 86 96 85 97 32 13 82 74 73 99 97 19 57 55 95 58 84 19 39 37 72 79 98 67 99 58 51 44 57 78 92 95 94 100 66 65 77 89 98 60 47 59 16 43 81 96 65 36 92 91 66 65 67 17 72 81 76 80 90 50 40 79 60 59 71 88 23 93 58 57 55 71 77 6 71 82 83 86 46 39 46 5 9 12 68 69 98 87 24 16 21 70 69 68 56 60 75 84 87 88 45100 64 54 52 48 44 45 42 97 11 10 77 90 67 70 89 91 21 22 3 18 41 5 27 29 45 54 15 63 62 2 76 75 73 4 40 42 62 56 3 48 55 49 47 73 72 62 99 13 71 1 8 78 56 8 2 90 25 20 23 1 30 19 20 43 25 16 28 42 46 59 14 61 78 74 36 35 38 5 43 63 17 57 49 50 53 3 98 10070 68 69 7 6 55 7 9 10 12 14 18 99 27 26 31 32 22 44 24 26 64 85 78 37 39 41 51 74 75 76 11 13 17 29 28 33 34 79

Game 17, B: AG Master, W: AG Zero, Result: W+R Game 18, B: AG Zero, W: AG Master, Result: B+R Game 19, B: AG Master, W: AG Zero, Result: W+R Game 20, B: AG Zero, W: AG Master, Result: B+R 50 34 31 51 47 46 43 44 88 89 12 62 65 95 78 77 79 56 45 44 30 3 28 7 6 62 34 32 16 20 21 37 36 81 66 6 8 10 9 61 63 76 43 50 48 55 72 7 8 52 6 38 35 22 23 29 13 1 8 35 33 3 19 1 42 84 7 1 11 94 52 53 4 47 60 9 4 2 56 37 36 32 26 33 11 10 30 25 24 26 44 85 43 70 49 71 74 11 10 68 51 52 39 5 27 53 48 9 12 28 22 23 39 40 15 51 46 44 17 13 12 46 5 55 57 19 18 49 45 31 29 27 38 80 97 45 54 58 17 14 67 16 20 21 25 83 80 41 61 81 80 91 96 93 67 75 59 57 68 73 18 24 41 79 82 10063 99 87 85 88 90 92 69 62 98 60 86 13 82 86 70 78 68 92 81 90 99 83 84 89 69 84 77 67 93 18 58 57 87 83 98 41 43 85 42 15 97 61 64 17 14 59 82 55 100 31 29 27 89 88 38 60 86 14 91 54 13 12 64 70 90 92 91 54 5 29 33 25 41 28 22 23 39 40 50 55 15 88 66 86 11 10 66 63 65 93 72100 50 48 51 64 28 23 26 27 18 21 24 30 25 24 26 47 59 56 42 37 99 2 17 95 75 89 87 4 65 9 4 96 56 77 69 94 67 78 2 45 49 31 3 22 32 5 19 20 15 2 35 33 3 48 53 19 91 92 96 63 1 36 98 72 71 73 74 58 60 59 85 7 8 97 95 74 71 68 6 46 47 34 30 35 36 39 16 42 14 49 34 32 72 16 69 58 61 66 20 21 94 62100 96 76 94 53 52 98 76 75 73 79 38 37 81 80 95 74 71 73 70 54 57 64 84 83 82 76 78 77 75 65 40 at 35 99 at 67 40 at 35 79 at 73 87 at 83 90 at 84 93 at 83 97 at 80

Extended Data Figure 6: AlphaGo Zero (40 block, 40 day) versus AlphaGo Master tournament games using 2 hour time controls. 100 moves of the ﬁrst 20 games are shown; full games are provided in Supplementary Information.