Acquisition of Chess Knowledge in AlphaZero

1.1 Our approach

• The investigation uses three complementary approaches: linear concept probes, behavioral comparisons during training, and direct analysis of network layers and activations.Behavioral analysis tracks prior move probabilities without Monte Carlo tree search on curated positions and compares them with top-level human play; direct analysis uses NMF and activation–input covariance.Source: §1.1 · p. 2

1.2 Summary of results

• Many human chess concepts are reliably reconstructed from AlphaZero’s internal representations, often more accurately than from the input, although reconstruction remains incomplete.Sparse linear probes cover concepts from passed pawns and mobility to full position evaluation, revealing computed information while distinguishing AlphaZero’s knowledge from the probes.Source: §1.2 · p. 2
• Concepts emerge consistently across training runs and network layers, with many appearing early alongside rapid, dramatic changes in move selection.What-when-where plots track concept regression accuracy across training time and network depth.Source: §1.2 · pp. 2–3
• AlphaZero’s value function shifts from primarily material-based evaluation early in training toward greater reliance on king safety and mobility later.Concept-weight evolution is used as a surrogate for the system’s changing high-level playing style.Source: §1.2 · p. 3
• AlphaZero’s development shares similarities with human chess history but does not recapitulate it, while self-play policies show striking similarities across training runs.The paper also provides a qualitative assessment of how playstyle changes over training.Source: §1.2 · p. 3
• Unsupervised analyses provide preliminary views of AlphaZero’s representations by decomposing activations into factors and measuring covariance between activations and inputs.NMF identifies factors related to move selection, while covariance analysis examines relationships between early-layer activations and input features.Source: §1.2 · p. 3

2.1 Interpretability

• The study relies on post-hoc interpretability because AlphaZero is a fixed, high-performing system, primarily using concept-based explanations alongside direct analysis of network activations.Interpretability distinguishes inherently interpretable models from post-hoc explanations of already trained models; the paper’s approach is post-hoc and concept-focused.Source: §2.1 · pp. 3–4
• Concept-based methods probe human-understandable domain concepts, while mechanistic and network-dissection approaches investigate learned representations and interpretable hidden units.Chess concepts provide a natural starting point because many have simplified, computable forms in modern chess engines.Source: §2.1 · pp. 3–4
• Post-hoc explanations may lack fidelity: the paper’s probing measures proximate correlation between human concepts and representations, not causation or alignment with the model’s reasoning.Causal understanding remains difficult and may require generative models conditioned on concept values to simulate interventions.Source: §2.1 · p. 4
• Prior explainability work in reinforcement learning includes saliency maps, recurrent-state models, trajectory analysis, representation learning, symbolic methods, and concept-based analysis, including contemporaneous work on AlphaZero trained on Hex.These approaches address transparency or post-hoc understanding, but specialized policy-design methods are less applicable to pre-existing systems.Source: §2.1 · p. 4

2.2 Chess and AI research

• Chess is a model system for AI research because it combines extensive human knowledge, verifiable positions, and intuition with calculation, while neural engines such as AlphaZero learn evaluation functions rather than relying solely on hardcoded rules.This makes strong chess networks potential objects of study for understanding learned game knowledge and AI behavior.Source: §2.2 · pp. 4–5
• Chess-explanation research has used hierarchical explanations, focused feature saliency, human-attention analysis, natural-language methods, and concept-based systems such as DecodeChess.Unlike approaches aimed at explaining MCTS or search-tree computations, this work studies AlphaZero’s value assessment and candidate moves to capture intuitive play.Source: §2.2 · p. 5

3 AlphaZero: Network structure and training

• AlphaZero combines a neural network’s policy and value predictions with Monte Carlo tree search to evaluate positions and select moves.The value head predicts expected game outcome v, while the policy head predicts next-move probabilities p; both guide MCTS.Source: §3 · p. 5

3.2 AlphaZero training iterations

• Training uses self-play positions and MCTS move probabilities, updating parameters by stochastic gradient descent while periodically refreshing networks used for self-play.The setup runs for 1,000,000 steps, samples from the previous 1 million positions, refreshes checkpoints every 1,000 steps, and uses 800 MCTS simulations per move.Source: §3 · p. 5, §3.2 · p. 8

3.1 AlphaZero neural network

• The network receives an oriented 8×8×119 representation encoding the current position and seven preceding plies, with pieces, repetition, turn, castling rights, and move counters.Using a history of eight plies marginally improves performance over representing only the current position, and follows.Source: §3.1 · pp. 5–6, Figure 1 · p. 7

3.1.1 Layers

• The 20-layer ResNet torso transforms inputs through residual convolutional blocks with ReLU and activation clipping, then feeds separate value and policy heads.Each torso layer has 8×8×256 activations; the policy head produces an 8×8×73 move tensor, while the value head computes v.Source: §3.1.1 · p. 6

4 Encoding of human conceptual knowledge

• The study uses sparse linear probing to test which human chess concepts are represented across AlphaZero layers and training steps.Probe performance on held-out data measures how much concept-relevant information activations carry.Source: §4 · p. 8, §4.2 · p. 9

4.1 Concepts

• Concepts are pre-specified functions of the input position, ranging from binary bishop-pair detection to integer or continuous measures such as pawn differences, mobility, and engine scores.They encode domain-specific chess knowledge rather than arbitrary properties of network states.Source: §4.1 · p. 8
• The concept set includes Stockfish 8 evaluation sub-functions for material, pawn structure, mobility, king safety, threats, and related features, with phase-adjusted total values used as high-level concepts.These concepts are drawn from Stockfish’s public evaluation API, enabling comparison with observations in.Source: §4.1 · p. 8
• In addition to Stockfish concepts, the authors implement 116 custom concepts covering forks, pins, contested files, and diverse pawn-structure features.These custom concepts target more specific, lower-level chess features.Source: §4.1 · p. 9
• Activations and concept values come from 105 randomly selected ChessBase games, deduplicated by FEN and split into training, validation, and test sets.Continuous concepts use 105 unique training positions plus 3×10^4 validation and test positions, while binary concepts are balanced and range from 50,363 to 105 training examples.Source: §4.1 · p. 9

4.2 Probing concept learning with sparse linear regression

• For each layer, training step, and concept, linear probes predict continuous targets and logistic probes predict binary targets from activations, with sparse regularization selected by validation cross-validation.Each block has 8×8×256 = 16,384 activation dimensions, motivating regularization and held-out testing.Source: §4.2 · p. 9
• Input-regression and random-concept controls distinguish concept-specific network representations from separability or dimensionality effects.The input control assesses what network computation adds, while the random target is sampled from N(0,1).Source: §4.2 · p. 9
• Probe results should not be interpreted as newly created Shannon information, because the data processing inequality permits networks to lose but not create input information.The authors instead frame layers as making existing information more available to a computationally bounded probe, following.Source: §4.2 · pp. 9–10

4.3.1 Scoring concept regression

• Concept presence is measured by regression accuracy at each layer and checkpoint, using r2 for continuous concepts and normalized accuracy above random guessing for binary concepts.Scores are plotted over layer and training-step axes; fixed human-play positions isolate representation changes from shifts in AlphaZero’s self-play distribution.Source: §4.3.1 · p. 10

4.3.2 The evolution of human concepts in AlphaZero

• For the Stockfish 8 total score, regression accuracy is low early, begins increasing substantially after 16,000 training steps, and exceeds r2 > 0.75 after 10 layers at 64,000 steps and beyond.Accuracy plateaus at 128,000 training steps.Source: §4.3.2 · p. 12
• Threat concepts are progressively computed across layers, with mate-threat representations indicating that AlphaZero models opponent moves and their consequences, consistent with at least single-move lookahead.Specific threat concepts are more accurately regressed than the general threats concept, potentially because distributed representations are penalized by sparse probes.Source: §4.3.2 · p. 12
• For threats, piece-count difference, and imbalance, regression accuracy can peak early and drop later, suggesting intermediate computations are discarded or become nonlinear rather than remaining linearly available.Some pawn-related concepts remain poorly decodable, whereas others are trivially predictable; sparse probing and input-level availability help explain these extremes.Source: §4.3.2 · pp. 12–13

4.4 Systematic semantic differences in Stockfish score outliers

• These residual patterns can reflect systematic differences of opinion between AlphaZero’s fixed-depth MCTS and Stockfish’s dynamically deepening exchange search, rather than mere regression noise.AlphaZero must encode some look-ahead information that Stockfish can incorporate through search code, including partially simulated exchange sequences.Source: §4.4 · pp. 13–14
• The most extreme Stockfish-score residuals consistently identify positions where White can take Black’s queen, and these outliers cluster in AlphaZero’s activation space.All 12 outliers with regression more favorable to White than Stockfish involved Black’s queen being capturable, often midway through a queen exchange.Source: §4.4 · pp. 13–14

4.5 Challenges and limitations for concept probing

• Sparse linear probes may underestimate spatially represented concepts because sparsity cannot combine distributed board information, motivating more sophisticated probing architectures.Candidate approaches include data-dependent sparsity, information bottlenecks, minimum-description-length probing, and Bayesian probing.Source: §4.5 · p. 14
• Human-defined ground truth is inherently contestable because Stockfish implementations and chess grandmasters may assess positions differently.The study relies on Stockfish 8’s public API and independently implemented low-level features as its ground truth.Source: §4.5 · p. 14
• Regression accuracy can rise gradually with depth as convolutional layers propagate spatial information, so low early accuracy does not necessarily mean a concept is absent.Long-distance queen captures become predictable later than nearby captures, and threat-related concepts show similar gradual increases.Source: §4.5 · p. 15
• High probe scores establish linear decodability, not that AlphaZero genuinely represents or uses the human concept, because confounding and entanglement remain unresolved.Input or internal interventions may help, but internal interventions can push the network off-distribution and produce incoherent downstream computation.Source: §4.5 · p. 15

4.6 Relating human concepts to AlphaZero’s value function

• Material begins influencing AlphaZero’s value predictions at 16,000 steps, while king safety, threats, and mobility appear from 32,000 steps; space emerges late and imbalance contributes negligibly.Piece values converge toward commonly accepted values after 128,000 training steps.Source: §4.6 · p. 16

5 Progression through AlphaZero and human history

• AlphaZero narrows plausible first moves from uniform exploration, whereas human play expands from an early dominance of 1. e4 into a more balanced opening distribution.AlphaZero’s first-move entropy decreases through training, while recorded human first-move entropy increases over five centuries.Source: §5 · p. 16, §5.2 · p. 17

5.1 Five centuries of data

• The analysis uses a subsample of ChessBase Mega Database, including all 135 available recorded games from 1400–1800 and roughly 10,000 games per decade from the twentieth century onward.Sampling before 1970 required games of at least 20 moves, while later games came from players with the highest ELO ratings; period-quality differences were treated as part of human chess progression.Source: §5.1 · pp. 16–17

5.2 First move progression

• 1.87 bits of information were encoded by top-level human first moves at the end of the twentieth century, compared with 0.33 bits between 1400 and 1800; AlphaZero’s prior fell from 4.32 to 2.76–3.02 bits after 1 million steps.The three reported AlphaZero seeds produced 2.76, 2.81, and 3.02 bits after 1 million training steps, with nonzero mass remaining on every move.Source: §5.2 · p. 17

5.3 The Ruy Lopez

• In the Ruy Lopez, AlphaZero converged across seeds on either 3... Nf6 or 3...a6, but rapidly developed a Berlin preference despite humans taking longer to appreciate its benefits.The two lines are theoretically similar, while the Berlin defense was historically viewed as slightly worse for Black before becoming a fashionable modern equalizing system.Source: §5.3 · p. 18

5.4 Remarks

• This comparison is narrow, covering the expansion or narrowing of initial opening options rather than human middle-game themes or endgame understanding.The authors note that estimating human preferences for those later phases requires a different approach.Source: §5.4 · p. 18

6 Rapid increase of basic knowledge

• Material and mobility concepts are largely learned between 10k and 30k training steps, preceding an inflection where basic opening play yields tangible advantages.Stockfish’s material evaluation sub-function strongly indicates AlphaZero’s value assessment, with mobility progressively incorporated during the same period.Source: §6 · p. 18

6.1 Discovery of standard openings

• Recognizable opening theory develops between 25k and 60k steps: 1. d4 and 1. e4 are rapidly adopted, while strong responses after 1. e4 gain 80% of prior mass.The four moves 1...c5, c6, e5 or e6 initially account for 20% of prior mass and reach 80% after 25k steps.Source: §6.1 · p. 18
• After basic openings emerge, AlphaZero progressively refines opening theory through repeated self-play updates over hundreds of thousands of training steps.This sequence follows rapid discovery in the 25k–60k window and subsequent long-term refinement.Source: §6.1 · pp. 19–20
• The timing of opening discovery depends on training configuration, including the rate at which self-play networks are refreshed, and was not tested under weaker settings.The onset is therefore not attributed to depletion of the initial random training buffer.Source: §6.1 · p. 19

6.3 Are tactical skills acquired before positional skills?

• At 64k versus 32k steps, AlphaZero’s estimated ELO gains were 365 points in tactical openings and 343 points in positional openings.The corresponding tactical and positional win rates were 82.3% and 76.6%, respectively.Source: §6.3 · p. 21
• At 128k versus 64k steps, estimated ELO gains were 152 points in tactical openings and 123 points in positional openings, providing tentative evidence for earlier tactical acquisition.The tactical advantage persisted across both checkpoint comparisons, but the authors describe the explanation as circumstantial and call for further study.Source: §6.3 · pp. 20–21

7.1.1 Methodology

• NMF discovers concept-agnostic activation structure by approximating each board-square representation with non-negative combinations of fewer global factors.The method reshapes 8×8×256 activations into square-by-channel matrices and approximates them as ˆzl ≈ΩF, using 36 factors per block across 720 block/factor pairs.Source: §7.1 · p. 21, §7.1.1 · pp. 21–22

7.1.2 Results

• NMF interpretations are provisional because many later-layer factors remain unexplained and require quantitative validation, ideally by intervening on inputs.Uninterpretable factors may reflect an unsuitable factor count, broken spatial correspondence, or complexity requiring reference to earlier layers.Source: §7.1.1 · p. 22, §7.1.2 · p. 22
• Unsupervised analyses reveal interpretable computations for threats, potential moves, opponent exchanges, and move selection across AlphaZero’s layers.NMF factors show move computations developing over multiple blocks, opponent-move anticipation early in the network, and factors associated with potential exchanges and move quality.Source: §7.1.2 · p. 22, Figure 8 · p. 23

7.2 Covariance between inputs and activations

• Activation–input covariance shows that some spatially aligned activations propagate local piece features through all layers, whereas others respond to evolving global patterns.These analyses use the fully trained network and estimate covariances over 1.4 million randomly selected positions from grandmaster games.Source: §7.2 · pp. 23–24

7.2.1 Feature detectors

• First-layer activations detect localized 3×3 piece patterns around a square, while other channels detect long-range diagonal, horizontal, or vertical move types.Some apparent global correlations arise because opening-specific piece configurations are correlated with other board locations in the sampled positions.Source: §7.2.1 · pp. 24–25

7.2.2 Persisted feature detectors

• A detector for opposing central pawns persists from layer 1 through layer 20, indicating that useful local information can be propagated and reused throughout the network.The activation-input covariance remains largely similar across layers for activation i = (5,4,10).Source: §7.2.2 · p. 25, §7.2.4 · p. 27

7.2.3 Evolving feature detectors

• Another detector evolves from a central light-squared bishop to an opposing pawn on e4 and finally a knight on c3 attacking e4.This progression illustrates how lower-layer localized patterns can become increasingly global as activations move upward through the network.Source: §7.2.3 · pp. 25–26, §7.2.4 · p. 28

7.2.4 Open questions: distributed encoding

• The covariance method relates individual activation detectors to AlphaZero’s distributed encoding, but connecting micro-level features to macro-level human concepts remains an open direction.Earlier analyses regress recognizable concepts from the distributed activation encoding, whereas this section examines how individual activations correlate with input positions.Source: §7.2.4 · p. 26

Conclusions

• Many human chess concepts can be accurately regressed from AlphaZero’s post-training representations despite self-play training without human games or objectives promoting human-like play or activations.These findings are supported by multiple human-interpretable factors identified in AlphaZero’s activations.Source: §8 · p. 29
• Opening knowledge develops rapidly around the same time that many human concepts become predictable from network activations, suggesting a critical period of rapid knowledge acquisition.The conclusion combines opening-theory analysis with an expert qualitative analysis of AlphaZero’s self-play during training.Source: §8 · p. 29
• The study broadens the range of systems in which human-understandable concepts may be found, including superhuman systems trained entirely through self-play.The authors argue that closer examination of AlphaZero may reveal additional human-understandable concepts.Source: §8 · p. 29
• The authors identify methodological challenges, suggest future research directions, and release AlphaZero activation data online for further investigation.They frame a central next question as whether systems can yield knowledge beyond what humans already understand.Source: §8 · p. 29

A Full concept list

• The study uses 93 concepts from Stockfish 8’s public API, selected from that version because observations compare AlphaZero with Stockfish 8.Source: A Full concept list · p. 34
• Additional concepts were implemented programmatically, including general custom measures and pawn-related measures using mine, opponent, and difference values.The custom concept tables define m as mine, o as opponent, and diff as the difference between corresponding values.Source: A Full concept list · p. 34, Table 3 · p. 35, Table 4 · p. 36

• The extended concept set is presented as a starting point for exploring AlphaZero’s acquired chess knowledge, not as a complete representation of chess knowledge.The authors note that the list cannot capture the full breadth of centuries of chess knowledge or all board patterns.Source: B Regression results for all concepts · p. 36
• Figures 14–18 report regression results for Stockfish concepts from Table 2.Source: B Regression results for all concepts · p. 36, Figure 14 · p. 37, Figure 15 · p. 38
• Figures 19–22 report regression results for custom concepts from Table 3, excluding capture-related concepts.Source: B Regression results for all concepts · p. 36, Figure 19 · p. 42, Figure 20 · p. 43
• Figure 23 reports regression results for custom capture-related concepts from Table 3, while Figures 24–25 cover custom pawn-related concepts from Table 4.Source: B Regression results for all concepts · p. 36, Figure 23 · p. 46, Figure 24 · p. 47

Policy progression

• The policy-network analysis tracks highest-probability human-played move sequences across training iterations for different initial training seeds.Each figure includes a starting position, seed-specific probability distributions, and cumulative probability for sequences outside the displayed set.Source: C AlphaZero policy progression through… · p. 49

Opening-specific sequences

• Figures 26–29 show top-probability sequences from multiple opening positions, including the Italian Game, the Ruy Lopez, the Caro–Kann Defence, and the French Defence.The figures report the top 20, top 8, top 12, and top 20 sequences, respectively, for their specified positions.Source: Figure 26 · p. 50, Figure 27 · p. 51, Figure 28 · p. 52

Start-position policy

• Figures 30 and 31 examine policy probabilities at the chess start position, listing the top 20 move sequences and top 19 moves, respectively.These start-position plots extend the analysis from opening continuations to initial move selection.Source: Figure 30 · p. 54, Figure 31 · p. 55

D Outlier positions in score regression

• Black’s queen can be captured in every negative outlier position shown in Figure 3.These positions are highlighted in red and have true values below zero.Source: D Outlier positions in score regression · p. 56, Figure 32 · p. 56

E Concept regression results for second seed

• Regression plots from seed 1 matched the initial concept-regression plots from seed 3, supporting generalization beyond a single training run.The second-seed analysis covered Stockfish concepts, custom concepts, capture-related concepts, and pawn-related concepts.Source: E Concept regression results for… · p. 57, Figure 33 · p. 58, Figure 38 · p. 63