Do neural networks learn to implement algorithms such as look-ahead or search "in the wild"? Or do they rely purely on collections of simple heuristics? We present evidence of learned look-ahead in the policy network of Leela Chess Zero, the currently strongest neural chess engine. We find that Leela internally represents future optimal moves and that these representations are crucial for its final output in certain board states. Concretely, we exploit the fact that Leela is a transformer that treats every chessboard square like a token in language models, and give three lines of evidence (1) activations on certain squares of future moves are unusually important causally; (2) we find attention heads that move important information "forward and backward in time," e.g., from squares of future moves to squares of earlier ones; and (3) we train a simple probe that can predict the optimal move 2 turns ahead with 92% accuracy (in board states where Leela finds a single best line). These findings are an existence proof of learned look-ahead in neural networks and might be a step towards a better understanding of their capabilities.
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Summary
The author examine if a chess NNs encode future moves in their activations. The results suggest that Leela predicts future self-play moves and that the activations can be manipulated to change the predicted moves.
Strengths
# originality Linear probes for difference concepts are well known in the chess literature, but the activation patching and explicitly looking at future moves as concepts are novel. # quality The analysis looks good, but the lack of consideration of alternative hypothesizes or attempts to disprove the results weakens the results. # clarity The paper is clear and flows well. # significance I think this work is interesting and presents some good ideas for chess AI and the broader XAI community.
Weaknesses
My main concerns with this paper is that it does not test the methods outside of a very limited scope. Even looking a puzzles where Leela gets the next move wrong would be a good first step, i.e. what is the accuracy of your move predictor when the model is wrong, or can you patch to get the correct move. I'm also not that surprised by this result and don't think the main motivation is addressed. The Leela NN is trained to predict the future actions of itself, and those are based on an explicit tree model of the game MCTS. The model internally doing a depth 3 search seems very plausible and I don't think proves the motivating claim of "look-ahead in neural networks", as the heuristics could simply be on the depth 3 tree instead of the depth 1 tree. One options for this analysis would be to look at a chess NN that's not trained on selfplay data (Maia, CrazyAra, etc).
Questions
Could you give a more formal definition of " look-ahead in neural networks" and explain how this work proves it? Does the move probe work on human games or non-puzzle positions? Figure 8, what is the accuracy measuring? There are 64 squares on a chess board, how is random getting above 2% accuracy? What would a falsification look like in this method? The patching for example seems so show that you can disrupt the chosen square, but this assumes the activations are encoding this linearly. If they were non-linear or only partially in the patch how would that changes these results?
Rating
4
Confidence
4
Soundness
2
Presentation
3
Contribution
3
Limitations
Discussed above
Summary
**Update after rebuttal:** To me personally, the authors' response clarifies all open questions and misunderstandings and adds interesting new results. I remain convinced that the work is ready for publication and interesting to the NeurIPS audience, which means my score remains 'Accept'. The paper investigates whether it is possible to reliably identify functional signatures of look-ahead in a Leela Chess Zero (policy) network. This analysis in a complex domain (chess) with a relatively large network is challenging. To do this, the paper uses three interpretability techniques for transformers: activation patching to measure the effect of specific interventions on neuron-activations and the sub-network that they influence, analysis and structured ablation of attention patterns, and training of simple read-out probes to test whether certain information is represented in the internal state or not. Additionally, the paper comes up with a method to automatically collect a large, high-quality dataset of challenging chess situations (that simpler chess engines cannot solve) that have a short and unique solution. This allows reliably identifying candidate-targets to aim the look-ahead investigation at. It also allows automatically constructing highly non-trivial interventions on the board state, which are crucial for meaningful activation patching. The paper finds clear and convincing evidence (from multiple angles and with various control experiments) that at least in these situations, the Leela Chess Zero network performs a one-step look-ahead of its next move (an opponent move is in-between, making it a two-step look-ahead in terms of game steps).
Strengths
* Very well written paper, with clear intro, and explanation of the fairly involved methodology, and great supporting figures. * Very high-quality case-study of interpretability and analysis of a concrete capability/mechanism “in the wild” (meaning in contrast to networks trained on synthetic data to facilitate or simplify the analysis). By its very nature, interpretability work is typically bespoke, and while some general techniques can be developed, I think the field greatly benefits from a body of well executed case studies that provide at least an abstract recipe and approach for others to adapt. * Great experimental work, with convincing evidence, important controls, and multiple investigations aimed at the same question to provide strong evidence. Despite the challenging setting, experiments are conducted with care and rigor, and attention to ruling out some obvious alternatives that could lead to the same findings. * The filtered dataset is a contribution in itself, as well as the particular model used in this paper, which may easily be picked up by follow-up research.
Weaknesses
* By the nature of a case study, the paper’s findings are limited to the particular network and the particular domain. The authors have managed to successfully identify and exploit aspects of the architecture (due to the network architecture, the internal state seems to map well onto a 2D chess board) and the data (filtered puzzle set with unique and short solution trajectories). I think this is crucial for the success of the current study and is highly non-trivial and original work, but naturally this means that the method cannot be straightforwardly applied e.g., to a LLM. I do believe though that the work here is highly inspiring to adapt the techniques to other models and domains. * (minor) Throughout the paper I was wondering why the policy network and not the value network was used. As the appendix says, both networks share the same torso and simply add relatively shallow heads, meaning the findings in the paper also apply to the value network. I think it would be good to mention this and a few more details in the main paper (such as training via supervised learning, and fine-tuning to remove history-dependence). I will list these below in ‘Improvements’. **Verdict:** The paper is a great case-study in interpretability and analysis, and I greatly enjoyed reading it. The question tackled in the paper is challenging, and I personally believe that the paper does a great job at producing a number of convincing pieces of evidence, with well chosen control experiments (such as a random network for the probes, and control-ablations for activation-patching and attention-ablations). While there is a challenge to transfer the method to other settings, the paper helps with this by clearly explaining the reasoning behind the various steps. Ultimately, I appreciate a well executed domain-specific case-study over a sloppy execution in a more general setting. Interpretability research is hard, particularly mechanistic interpretability, and I think that the field will be a composition of general techniques (which are also used in the paper) and exemplary case-studies for others to follow and apply to their research questions and domains. Therefore, I think the current paper may well have significant impact beyond the specific findings, which are limited to a particular network and a particular task. I currently recommend acceptance of the paper: all main claims are critically investigated and supported by evidence, the work is novel and very original, and I think is interesting to a wide part of the NeurIPS audience and may be quite impactful for follow-up work. **Improvements:** 1. I think it would be nice to mention a few more details about Leela in the main paper. In particular that the torso is the same as for the value network (and the value-head is relatively shallow and has no additional attention mechanism, so findings also hold for the value network). Maybe something like L425-428 in the appendix plus a brief description of the training (supervised on a set of high-quality trajectories, plus fine-tuning to remove history-dependency). 2. It would be interesting to see how the findings regarding look-ahead evolve during training. This is completely optional, and beyond the scope of the paper (and the authors may not have access to the original training procedure / checkpoints). But it would be interesting to see whether attention heads and the activation patching results develop gradually or sharply and whether that can be related to a similar increase in performance on the filtered puzzle dataset used in the paper.
Questions
1. Are there situations where the weaker Leela gets the puzzle right, but the stronger Leela does not? Or is the stronger Leela strictly better? 2. Related: how has the weaker Leela been trained? Is it plausible that, e.g. stronger MCTS during training of the weaker Leela means look-ahead is much less important, hence why it does not develop look-ahead (or does look-ahead only develop at sufficiently large and strong networks)? [It is perfectly fine to respond that you do not have the answers for these questions and not spend any more time on this; I am just curious.] 3. The probe accuracy in Fig 8 is relatively high in early layers - how does this fit with the activation patching results that show the largest effect only in medium-to-late layers 6-11? 4. As far as I am aware the precise details about the training process of this variant of Leela have not been published (in an academic venue). While this is beyond the scope of this paper, having these details (or as many as possible), e.g, in the appendix, would greatly help the scientific community to use this version of Leela as something that can be reliably reproduced. This may help boost Leela’s popularity for *scientific* research using openly available chess models. Similarly, the model and dataset used in this paper (with the fine-tuning) are a nice contribution. 5. L225 typo: “are seem”.
Rating
7
Confidence
4
Soundness
4
Presentation
4
Contribution
3
Limitations
Limitations are nicely discussed in Sec 5.
Thank you for the detailed responses
Thank you for answering my clarifying questions and commenting on my (mostly optional) suggestions for improvement. I am positively surprised to see a full analysis of the value-network too. Overall I am happy with the authors' responses and additional results. I stand by my original score - I think this is a great interpretability case study with interesting results, that is ready for publication, well executed, and interesting to the NeurIPS audience. I have also read the other reviewers' comments and authors' responses and consider all raised issues sufficiently addressed (though I will happily take into account reviewers' future comments in case they disagree). abVe and p7Jw seem to be mainly concerned by the dataset filtering and only showing results on this filtered dataset - while this is understandable criticism at first, I think constructing such a dataset in a reasonable manner is a contribution in itself. The difficulty is that a large neural action/value predictor behaves quite differently from a search-based chess algorithm; in many board states it may be that the neural system does not use look-ahead-search but instead relies on memorization and exploitation of (statistically) similar patterns. It is thus crucial to first identify states where look-ahead-search may be needed. I am also not too worried about the use of the puzzle dataset - previous works (e.g. the 'Grandmaster-level chess without search' that abVe mentioned) have found strong correlation between puzzle performance and actual game-playing strength across a variety of models. The paper claims (and is very clear about this) that convincing evidence of look-ahead-search can be found *in these situations*. It does not claim that this is the main mechanism that the network uses at all times. Without proper filtering, the evidence for this mechanism might quickly "drown" within the (potentially large) number of situations where no look-ahead search is performed. While it would be interesting to know "how often" the network uses its learned search, this question is beyond the scope of the paper, whose goal is to clearly establish that the paper has learned to use look-ahead search at least sometimes (which I consider a highly non-trivial result).
Thank you for your positive feedback about our additional results and taking the time to write detailed and insightful responses to our rebuttal and the overall discussion on our work so far.
Summary
This paper conducts a mechanistic interpretability analysis of Leela and shows evidence that it learns to look ahead in the network.
Strengths
This paper provides a convincing answer to an important question: are networks like Leela's analogues of System 1-style pure intuition, or do they encode some amount of System 2-style calculation? The technique of corrupting the state and using activation patching is clever and effective. It's clear from the experiments that yes, there is some look-ahead being done. The authors are also careful to not overstate their claims: an actual broad search isn't necessarily being done, but at least some 1- or 2-step look-ahead is happening.
Weaknesses
The paper is essentially an existence proof of lookahead, which is a useful contribution, but it would be nice to know something about the the prevalence of lookahead. The dataset is chosen to maximize the probability that lookahead is happening. How often does it happen? I thought the claim that "look-ahead or other sophisticated algorithms should pick up on the importance of this difference, but shallow heuristics should mostly ignore it." required more justification. Some of the tactical patterns picked up by the method are so common that it's conceivable that a "shallow heuristic" is designed for them, rather than look-ahead. The lack of results/explanations regarding why the squares of the 2nd move aren't important weakens the argument somewhat. Why aren't there pawn or king heads, similar to the other piece types? Nitpick: the motivation in the first paragraph about chess being different from other domains because in other domains "models can solve their tasks with a single simple algorithm" also applies to chess.
Questions
How often does look-ahead occur? What is the evidence that look-ahead is occurring as opposed to a shallow heuristic designed to pick up on common tactical patterns? Why aren't the squares of the 2nd move similarly important? Why aren't there pawn or king heads, similar to the other piece types?
Rating
6
Confidence
5
Soundness
4
Presentation
4
Contribution
3
Limitations
Yes
Summary
This paper closely examines, using various forms of activation patching and probes, the inner workings of the state-of-the-art policy network of Leela Chess Zero (for the game of Chess). The authors find several different pieces of evidence that suggest it is likely that the network has learned to carry out some form of look-ahead search, at least in complex states that require this to arrive at the correct solution.
Strengths
I found this to be a very interesting paper, and have little to remark on it. It's always good to see a paper that is actually trying to improve our understanding (and doing a good job at that), rather than just being "here is our new technique and it has bigger numbers". It is well written, using good examples and illustrations.
Weaknesses
My only "important" criticism is that I object to the use of "existence proof" in both the abstract and the conclusion. I think this is too strong, and would suggest rephrasing to "evidence". I do find the evidence provided in the paper to be compelling, but still do not think it can be definitively described as proof rather than evidence. --- Minor nitpicky comments: - line 86 has a random period in the middle of a sentence (right before the footnote) - I feel like it would be more natural for the paragraph of lines 222-224 to be moved a bit earlier. It's essentially describing a part of the experiment setup, but after some of the results have already been presented and discussed. - Line 225: "are seem"
Questions
1) Probably a difficult question, but curious to hear your thoughts if you have any: is it definitively possible to draw a hard line between "heuristics" and look-ahead? Especially depth-limited look-ahead (like what you seem to be finding evidence for in this paper)? Here's what I'm thinking. If you only have a very basic heuristic (say, only material count), then you need at least 1 ply on search on top of that to find which moves let you capture something. But you could just add a smarter heuristic that counts the values of pieces your pieces can attack in one move, and then you no longer need one ply for this. With the most basic heuristic, you would need at least 2 plies of search to also consider the responses that the opponent can make. But, you could add a heuristic that counts which pieces defend their own friendly pieces, and that could partially account for what the second ply would typically be searching. Is there a limit to this? Could we conceivably create heuristics that detect the 3-ply situations explored in this paper, without search? 2) As a philosophical follow-up to the above: can we ever really distinguish between look-ahead search and heuristics, if the look-ahead search is depth-limited (I suppose any "learned look-ahead" in non-recurrent neural networks would always be depth-limited)?
Rating
8
Confidence
4
Soundness
3
Presentation
4
Contribution
4
Limitations
There is a good discussion of limitations.
Summary
The paper try to discover and analyze the evidence of learned look-ahead in Chess playing network. They take experiments on filtered Chess puzzle dataset and the policy network of Leela chess policy. The claims are mainly analyzed from three perspectives, including Activation patching, attention head analysis and probing experiments.
Strengths
1. The finding of the paper is interesting and inspiring. I like the idea of getting inside the network to understand how the policy network or transformer work to predict the optimal action. 2. The paper is easy to follow and well-written. 3. Experiments on different perspectives to help verify the conclusion.
Weaknesses
1. The analysis seems to strongly related with the specific setting: Chess and transformer -- where each grid can be considered as a token. This can somehow limit the generality of the analysis method. 2. The main issue about the weakness comes from the experiments. Since this is mainly a paper for discovering phenomenon and analysis, with no newly proposed algorithm or showing how this finding can help anything, the standard for NeurIPS paper requires it to have comprehensive experiments to validate the solidness of the finding. I do like this new finding, but the experiment is not enough to make it solid -- for instance, here are several experiments I think necessary for authors to present: 2.1. Present how this phenomenon happens without explicit designing test/evaluation dataset. The authors do a lot of work in section 2.2 on filtering the testing samples to make it more likely to have this phenomenon -- and they use this to claim that the policy network has this phenomenon. This is quite weird since you also somehow optimize the evaluation dataset to help you find the claimed phenomenon. If this is not a general finding or conclusion, it can make the argument pretty weak and may only work on some human created settings -- an extreme case is that I can manually design some test cases that can definitely present this ability but it lacks generality. I agree with the argument that not all boards present this phenomenon -- but at least the authors need to present more things, like in which condition this phenomenon is likely to happen and which will not, and how/why this can happen. Is there any key issue that prevent or support the emergence of this phenomenon? Also the authors filter out all puzzles that Leela Chess may fail. This is really weird and can largely weaken the conclusion since if you have already tested it to solve the puzzle, it is quite normal that the representation of policy network can have something closely related with future optimal action (in the activation patching experiments or probing one). 2.2 Is this conclusion generalizable on other transformer-based Chess policy? For example in "Grandmaster-Level Chess Without Search " they also have a policy trained by supervised learning with no RL or MCTS like the leela chess does. Will this setting affect the emergence of this behavior? 2.3 Does only the policy network have this phenomenon or the same thing can happen at critic network? For example you can test it on leela-chess/stockfish evaluation function. There are also a lot of ablation studies on modules or hyperparameters to make the conclusion solid. So again, solid and comprehensive experiments and ablation studies are necessary for such analysis/discovering paper, while this paper doesn't offer enough evidence.
Questions
1. There are several papers about transformer/Chess/AlphaZero that are also related work to this paper, including: (1) Zahavy, Tom, et al. "Diversifying ai: Towards creative chess with alphazero." arXiv preprint arXiv:2308.09175 (2023). (2) Feng, Xidong, et al. "Chessgpt: Bridging policy learning and language modeling." Advances in Neural Information Processing Systems 36 (2024). (3) Noever, David, Matt Ciolino, and Josh Kalin. "The chess transformer: Mastering play using generative language models." arXiv preprint arXiv:2008.04057 (2020). (4) Stöckl, Andreas. "Watching a language model learning chess." Proceedings of the International Conference on Recent Advances in Natural Language Processing (RANLP 2021). 2021. 2. Can the author explain more on the activation patching. Is it something like: you replace one neuron in one layer from the neuron of the same place in the forward process of corrupted board? Then I am wondering how you choose this corrupted board. Also in figure 3 why 1st/3rd but no 2nd action? 3. Also I 'd suggest authors to include more illustrative examples/figures in the main paper or appendix to help readers (and those not familiar with chess) to understand the phenomenon.
Rating
4
Confidence
4
Soundness
3
Presentation
2
Contribution
2
Limitations
See weaknesses.
Thank you for your detailed response. I continue to be concerned that this work is not falsifiable as presented. You only look at features in a highly specific dataset (Lichess puzzles that the model is successful on) and never checked to see if the features are absent when not looking at the dataset. This is also a result of your definition which considers only the positive case, i.e., a model that represents all lines equally before discarding the results and deciding independently would also meet your definition. This is the _"Leela might still use look-ahead along some incorrect continuation"_ scenario that you mentioned. In particular I am concerned by the lack of "real' games, the Lichess puzzles dataset only contains positions that meet certain easy to evaluate heuristics. It is impossible to determine if you are observing these heuristics or more robust features in this test. I would be much more confident of a claim of general look-ahead if they showed the effects outside of a cherry-picked set of samples. Explicitly, presenting a hypothesis for this work and explaining how your tests would disprove it would greatly improve things on this front, currently a negative result simply suggests you're not searching hard enough.
Thank you for the follow-up! > I continue to be concerned that this work is not falsifiable as presented. You only look at features in a highly specific dataset (Lichess puzzles that the model is successful on) and never checked to see if the features are absent when not looking at the dataset. Is your concern (1) that we should check that we get *negative* results on inputs where there is no look-ahead, as a sanity check/baseline? Or (2) that we should present *positive* results on a wider distribution? If (1), it's unfortunately unclear how to find inputs where we can rule out that look-ahead is happening. But we do have other types of baselines in each experiment. If (2), as described above, we only claim look-ahead on this specific distribution. So in this case, the question in our mind would be how interesting this claim is, rather than issues with our evidence. Please let us know in case we misunderstood your point! We unfortunately still don't see why you think our work "isn't falsifiable." > a model that represents all lines equally before discarding the results and deciding independently would also meet your definition. This is the "Leela might still use look-ahead along some incorrect continuation" scenario that you mentioned. We don't think so. If a model discarded results from considering future lines, then the representations related to those future lines would not influence the output of the model. So we would not consider this look-ahead under our definition (because it's lacking the "using those representations to decide on the current move" part). We perform interventions to check this in sections 3.1 and 3.2, so for such a model, we also wouldn't get most of the positive evidence of look-ahead we present. The "Leela might still use look-ahead along some incorrect continuation" scenario is different: here, Leela would not be *discarding* the look-ahead along incorrect lines, but instead would be using it to output an (incorrect) move. Please let us know in case we misunderstood your concern. > the Lichess puzzles dataset only contains positions that meet certain easy to evaluate heuristics. It is impossible to determine if you are observing these heuristics or more robust features in this test Could you clarify what you mean by "easy to evaluate heuristics?" In case you mean chess heuristics to easily find the best move (without look-ahead), then (1) as we've described in the paper (section 2.2), we've specifically tried to make the puzzles *not* solvable by simple heuristics, and expect they are much more difficult than typical chess states, and (2) our experiments give evidence of look-ahead without assuming that look-ahead is required to solve those puzzles (see also the beginning of our response to SpQ5). > Explicitly, presenting a hypothesis for this work and explaining how your tests would disprove it would greatly improve things on this front, currently a negative result simply suggests you're not searching hard enough. Could you say more about what kind of hypothesis you are looking for here? A complete hypothesis in the sense of pseudocode for how Leela might play chess this well without look-ahead is clearly infeasible: all known algorithms for this level of chess performance involve explicit look-ahead/search, so if Leela wasn't using look-ahead, it would have to use some method or set of heuristics that haven't been discovered in decades of chess engine development. (This is certainly plausible a priori, but we naturally couldn't specify any concrete such hypothesis.) That is why we've focused on hypotheses to "explain away" our specific results in our response above (see the alternative explanation we considered and falsified for our results in section 3.1). As mentioned, we now intend to include this in the paper to make it explicit. If this kind of hypothesis is not what you are suggesting, it would be very helpful if you could elaborate a bit more. Thank you!
Thank you for the response. I have read the other comments and will discuss the paper with the other reviewers. I apologize for not getting back to you sooner. > Is your concern (1) that we should check that we get negative results on inputs where there is no look-ahead, as a sanity check/baseline? Or (2) that we should present positive results on a wider distribution? I think either result would strengthen your paper significantly, but I was concerned with the former in my comment. The patching method used for detecting features relies on assumptions about the inner workings of NNs that are based on inferences not formal proofs. So you present good evidence that your methods detect something consistently in the dataset of positive cases, but you never check if they also detect something when the features are absent. > If a model discarded results from considering future lines, then the representations related to those future lines would not influence the output of the model. I agree that this is something you test for, but again the test assumes that the influence of squares is linear. So patching one will simply shift the effects by some regular amount. You do not test more complex combinations. I am not asking you to do so, more asking you to check negative cases so that you can better verify the state of your detectors. > "easy to evaluate heuristics?" The heuristics are listed here: https://github.com/ornicar/lichess-puzzler/blob/master/tagger/model.py#L6 . They are all simple rules that can be checked quickly, the human games are just used to generate potential positions. Testing more complex, i.e. human curated puzzles would help here. > kind of hypothesis you are looking for here? There is previous work showing chess NNs learn heuristics [1], some of which include modeling the future e.g. forks. So some definition of look-ahead that explains how it differs from heuristics and what a model would look like with or without it. Currently you only consider models that have this property. [1] https://arxiv.org/abs/2111.09259
Thank you for your follow-up response. We are very grateful for your time, and for the lively discussion about our work! > but you never check if they also detect something when the features are absent. We apologize if we’re not understanding this request, but we are still not sure what you mean by “when the features are absent”. Our claim is that when (1) we are in a tactically “interesting” position (as defined by our smaller model filtering) and (2) there is a unique principle variation (as evaluated by a powerful engine such as Stockfish) and (3) Leela outputs the correct move, our probe can predict the 3rd move with 92% accuracy. Which one of these three conditions being not present would “make these features absent”? And what would you like to see the probes predict in that case? We appreciate your patience here, and are keen to understand how we can make our result stronger. > They are all simple rules that can be checked quickly We want to mention here that the board positions from the lichess website are first fed through Stockfish NNUE at 40 meganodes (as mentioned in another response, this computation took 50 years of CPU time by the lichess team). We reference `lichess-puzzler/generator/generator.py` line 100, where an advantage value is calculated from Stockfish’s solution, and line 169 where the real board position is accepted or rejected based on Stockfish’s evaluations, not the tagging. The tagging is done after the fact, and based on Stockfish’s best move. > There is previous work showing chess NNs learn heuristics [1], some of which include modeling the future e.g. forks. From [1], the authors probe for the presence of `pawn_fork`, `knight_fork`, `bishop_fork`, and `rook_fork` (please let us know if we missed a concept). Without loss of generality we will quote the description of `rook_fork` from [1], “True if a rook is attacking two pieces of higher value (queen, or king) and is not pinned.” This is considering the future for one step, from the POV of the current player. We wanted to test if the network considers moves more steps down the line, specifically, if it considers an opponent response, and then the model’s response after, which we believe [1] does not study. Again, we thank you for the rich discussion and follow-up responses, we really appreciate it!
I acknowledge that I have read the rebuttal, and thank the authors for still taking the time to engage in an interesting discussion when there were also other reviewers' comments present for which it was probably more urgent to respond. I do not plan to further raise my score (which was already very high), but must say I am very puzzled by some of the perceived "weaknesses" described in some other reviews. I will argue against them in the Reviewer-AC discussion if necessary.
Thank you for your rebuttal! I am satisfied with additional explanations and results. But I am still concerned with the dataset filtering -- I appreciate the authors' transparency about this. But if this is a conclusion that only applies for 20k puzzle boards this can largely weaken my surpriseness about the conclusion, especially that the title seems to ignore this limited scope. I will keep my score for now -- but maybe will increase it after my discussion with other reviewers and ACs. Thanks for your engagement again.
Thank you for your response! We appreciate the time to read through our rebuttal. > if this is a conclusion that only applies for 20k puzzle boards We wanted to mention that the lichess puzzles come from real games played on the lichess website. These games were analyzed via Stockfish at 40 meganodes (which "took more than 50 years of CPU time" by the lichess team), and we further filtered these down using a simple filtering method (discarding board positions that a smaller model can solve). Therefore, our dataset is a subset of board positions from real games filtered for our description of hard positions that might require look-ahead to solve. Thank you for engaging with our discussions again, we really appreciate your responses!
Decision
Accept (poster)