This paper analyzes the behavior of solving Life-and-Death (L&D) problems in the game of Go using current state-of-the-art computer Go solvers with two techniques: the Relevance-Zone Based Search (RZS) and the relevance-zone pattern table. We examined the solutions derived by relevance-zone based solvers on seven L&D problems from the renowned book “Life and Death Dictionary” written by Cho Chikun, a Go grandmaster, and found several interesting results. First, for each problem, the solvers identify a relevance-zone that highlights the critical areas for solving. Second, the solvers discover a series of patterns, including some that are rare. Finally, the solvers even find different answers compared to the given solutions for two problems. We also identified two issues with the solver: (a) it misjudges values of rare patterns, and (b) it tends to prioritize living directly rather than maximizing territory, which differs from the behavior of human Go players. We suggest possible approaches to address these issues in future work.

Strength estimation and adjustment are crucial in designing human-AI interactions, particularly in games where AI surpasses human players. This paper introduces a novel strength system, including a *strength estimator* (SE) and an SE-based Monte Carlo tree search, denoted as *SE-MCTS*, which predicts strengths from games and offers different playing strengths with human styles. The strength estimator calculates strength scores and predicts ranks from games without direct human interaction. SE-MCTS utilizes the strength scores in a Monte Carlo tree search to adjust playing strength and style. We first conduct experiments in Go, a challenging board game with a wide range of ranks. Our strength estimator significantly achieves over 80% accuracy in predicting ranks by observing 15 games only, whereas the previous method reached 49% accuracy for 100 games. For strength adjustment, SE-MCTS successfully adjusts to designated ranks while achieving a 51.33% accuracy in aligning to human actions, outperforming a previous state-of-the-art, with only 42.56% accuracy. To demonstrate the generality of our strength system, we further apply SE and SE-MCTS to chess and obtain consistent results. These results show a promising approach to strength estimation and adjustment, enhancing human-AI interactions in games. Our code is available at https://rlg.iis.sinica.edu.tw/papers/strength-estimator.

Planning with options -- a sequence of primitive actions -- has been shown effective in reinforcement learning within complex environments. Previous studies have focused on planning with predefined options or learned options through expert demonstration data. Inspired by MuZero, which learns superhuman heuristics without any human knowledge, we propose a novel approach, named *OptionZero*. OptionZero incorporates an *option network* into MuZero, providing autonomous discovery of options through self-play games. Furthermore, we modify the dynamics network to provide environment transitions when using options, allowing searching deeper under the same simulation constraints. Empirical experiments conducted in 26 Atari games demonstrate that OptionZero outperforms MuZero, achieving a 131.58% improvement in mean human-normalized score. Our behavior analysis shows that OptionZero not only learns options but also acquires strategic skills tailored to different game characteristics. Our findings show promising directions for discovering and using options in planning. Our code is available at https://rlg.iis.sinica.edu.tw/papers/optionzero.

Although AlphaZero has achieved superhuman performance in board games, recent studies reveal its limitations in handling scenarios requiring a comprehensive understanding of the entire board, such as recognizing long-sequence patterns in Go. To address this challenge, we propose ResTNet, a network that interleaves residual and Transformer blocks to bridge local and global knowledge. ResTNet improves playing strength across multiple board games, increasing win rate from 54.6% to 60.8% in 9x9 Go, 53.6% to 60.9% in 19x19 Go, and 50.4% to 58.0% in 19x19 Hex. In addition, ResTNet effectively processes global information and tackles two long-sequence patterns in 19x19 Go, including circular pattern and ladder pattern. It reduces the mean square error for circular pattern recognition from 2.58 to 1.07 and lowers the attack probability against an adversary program from 70.44% to 23.91%. ResTNet also improves ladder pattern recognition accuracy from 59.15% to 80.01%. By visualizing attention maps, we demonstrate that ResTNet captures critical game concepts in both Go and Hex, offering insights into AlphaZero's decision-making process. Overall, ResTNet shows a promising approach to integrating local and global knowledge, paving the way for more effective AlphaZero-based algorithms in board games. Our code is available at https://rlg.iis.sinica.edu.tw/papers/restnet.

Human-like agents have long been one of the goals in pursuing artificial intelligence. Although reinforcement learning (RL) has achieved superhuman performance in many domains, relatively little attention has been focused on designing human-like RL agents. As a result, many reward-driven RL agents often exhibit unnatural behaviors compared to humans, raising concerns for both interpretability and trustworthiness. To achieve human-like behavior in RL, this paper first formulates human-likeness as trajectory optimization, where the objective is to find an action sequence that closely aligns with human behavior while also maximizing rewards, and adapts the classic receding-horizon control to human-like learning as a tractable and efficient implementation. To achieve this, we introduce Macro Action Quantization (MAQ), a human-like RL framework that distills human demonstrations into macro actions via Vector-Quantized VAE. Experiments on D4RL Adroit benchmarks show that MAQ significantly improves human-likeness, increasing trajectory similarity scores, and achieving the highest human-likeness rankings among all RL agents in the human evaluation study. Our results also demonstrate that MAQ can be easily integrated into various off-the-shelf RL algorithms, opening a promising direction for learning human-like RL agents. Our code is available at https://rlg.iis.sinica.edu.tw/papers/MAQ.

This article presents MiniZero, a zero-knowledge learning framework that supports four state-of-the-art algorithms, including AlphaZero, MuZero, Gumbel AlphaZero, and Gumbel MuZero. While these algorithms have demonstrated super-human performance in many games, it remains unclear which among them is most suitable or efficient for specific tasks. Through MiniZero, we systematically evaluate the performance of each algorithm in the two board games, 9 × 9Goand 8 × 8Othello, as well as 57 Atari games. For the two board games, using more simulations generally results in higher performance. However, the choice between AlphaZero and MuZero may differ based on game properties. For Atari games, both MuZero and Gumbel MuZero are worth considering. Since each game has unique characteristics, different algorithms and simulations yield varying results. In addition, we introduce an approach, called progressive simulation, which progressively increases the simulation budget during training to allocate computation more efficiently. Our empirical results demonstrate that progressive simulation achieves significantly superior performance in the two board games. By making our framework and trained models publicly available, this article contributes a benchmark for future research on zero-knowledge learning algorithms, assisting researchers in algorithm selection and comparison against these zero-knowledge learning baselines. The code and data are available online.

This paper describes a Relevance-Zone pattern table (RZT) that can be used to replace a traditional transposition table. An RZT stores exact game values for patterns that are discovered during a Relevance-Zone-Based Search (RZS), which is the current state-of-the-art in solving life-and-death (L&D) problems in Go. Positions that share the same pattern can reuse the same exact game value in the RZT. The pattern matching scheme for RZTs is implemented using a radix tree, taking into consideration patterns with different shapes. To improve the efficiency of table lookups, we designed a heuristic that prevents redundant lookups. The heuristic can safely skip previously queried patterns for a given position, reducing the overhead to 10% of the original cost. We also analyze the time complexity of the RZT both theoretically and empirically. Experiments show the overhead of traversing the radix tree in practice during lookup remain flat logarithmically in relation to the number of entries stored in the table. Experiments also show that the use of an RZT instead of a traditional transposition table significantly reduces the number of searched nodes on two data sets of$7 \times 7$and$19 \times 19$L&D Go problems.

Game solving is a similar, yet more difficult task than mastering a game. Solving a game typically means to find the game-theoretic value (outcome given optimal play), and optionally a full strategy to follow in order to achieve that outcome. The AlphaZero algorithm has demonstrated super-human level play, and its powerful policy and value predictions have also served as heuristics in game solving. However, to solve a game and obtain a full strategy, a winning response must be found for all possible moves by the losing player. This includes very poor lines of play from the losing side, for which the AlphaZero self-play process will not encounter. AlphaZero-based heuristics can be highly inaccurate when evaluating these out-of-distribution positions, which occur throughout the entire search. To address this issue, this paper investigates applying online fine-tuning while searching and proposes two methods to learn tailor-designed heuristics for game solving. Our experiments show that using online fine-tuning can solve a series of challenging 7x7 Killall-Go problems, using only 23.54\% of computation time compared to the baseline without online fine-tuning. Results suggest that the savings scale with problem size. Our method can further be extended to any tree search algorithm for problem solving. Our code is available at https://rlg.iis.sinica.edu.tw/papers/neurips2023-online-fine-tuning-solver.

Goal-achieving problems are puzzles that set up a specific situation with a clear objective. An example that is well-studied is the category of life-and-death (L&D) problems for Go, which helps players hone their skill of identifying region safety. Many previous methods like lambda search try null moves first, then derive so-called relevance zones (RZs), outside of which the opponent does not need to search. This paper first proposes a novel RZ-based approach, called the RZ-Based Search (RZS), to solving L&D problems for Go. RZS tries moves before determining whether they are null moves post-hoc. This means we do not need to rely on null move heuristics, resulting in a more elegant algorithm, so that it can also be seamlessly incorporated into AlphaZero's super-human level play in our solver. To repurpose AlphaZero for solving, we also propose a new training method called Faster to Life (FTL), which modifies AlphaZero to entice it to win more quickly. We use RZS and FTL to solve L&D problems on Go, namely solving 68 among 106 problems from a professional L&D book while a previous state-of-the-art program TSUMEGO-EXPLORER solves 11 only. Finally, we discuss that the approach is generic in the sense that RZS is applicable to solving many other goal-achieving problems for board games.

The success of AlphaZero (AZ) has demonstrated that neural-network-based Go AIs can surpass human performance by a large margin. Given that the state space of Go is extremely large and a human player can play the game from any legal state, we ask whether adversarial states exist for Go AIs that may lead them to play surprisingly wrong actions.In this paper, we first extend the concept of adversarial examples to the game of Go: we generate perturbed states that are ``semantically'' equivalent to the original state by adding meaningless moves to the game, and an adversarial state is a perturbed state leading to an undoubtedly inferior action that is obvious even for Go beginners. However, searching the adversarial state is challenging due to the large, discrete, and non-differentiable search space. To tackle this challenge, we develop the first adversarial attack on Go AIs that can efficiently search for adversarial states by strategically reducing the search space. This method can also be extended to other board games such as NoGo. Experimentally, we show that the actions taken by both Policy-Value neural network (PV-NN) and Monte Carlo tree search (MCTS) can be misled by adding one or two meaningless stones; for example, on 58\% of the AlphaGo Zero self-play games, our method can make the widely used KataGo agent with 50 simulations of MCTS plays a losing action by adding two meaningless stones. We additionally evaluated the adversarial examples found by our algorithm with amateur human Go players, and 90\% of examples indeed lead the Go agent to play an obviously inferior action. Ourcode is available at \url{https://PaperCode.cc/GoAttack}.

Monte Carlo tree search (MCTS) has achieved state-of-the-art results in many domains such as Go and Atari games when combining with deep neural networks (DNNs). When more simulations are executed, MCTS can achieve higher performance but also requires enormous amounts of CPU and GPU resources. However, not all states require a long searching time to identify the best action that the agent can find. For example, in 19x19 Go and NoGo, we found that for more than half of the states, the best action predicted by DNN remains unchanged even after searching 2 minutes. This implies that a significant amount of resources can be saved if we are able to stop the searching earlier when we are confident with the current searching result. In this paper, we propose to achieve this goal by predicting the uncertainty of the current searching status and use the result to decide whether we should stop searching. With our algorithm, called Dynamic Simulation MCTS (DS-MCTS), we can speed up a NoGo agent trained by AlphaZero 2.5 times faster while maintaining a similar winning rate, which is critical for training and conducting experiments. Also, under the same average simulation count, our method can achieve a 61\% winning rate against the original program.

AlphaZero has been very successful in many games. Unfortunately, it still consumes a huge amount of computing resources, the majority of which is spent in self-play. Hyperparameter tuning exacerbates the training cost since each hyperparameter configuration requires its own time to train one run, during which it will generate its own self-play records. As a result, multiple runs are usually needed for different hyperparameter configurations. This paper proposes using population based training (PBT) to help tune hyperparameters dynamically and improve strength during training time. Another significant advantage is that this method requires a single run only, while incurring a small additional time cost, since the time for generating self-play records remains unchanged though the time for optimization is increased following the AlphaZero training algorithm. In our experiments for 9x9 Go, the PBT method is able to achieve a higher win rate for 9x9 Go than the baselines, each with its own hyperparameter configuration and trained individually. For 19x19 Go, with PBT, we are able to obtain improvements in playing strength. Specifically, the PBT agent can obtain up to 74% win rate against ELF OpenGo, an open-source state-of-the-art AlphaZero program using a neural network of a comparable capacity. This is compared to a saturated non-PBT agent, which achieves a win rate of 47% against ELF OpenGo under the same circumstances.

This paper proposes an approach to strength adjustment for MCTS-based game-playing programs. In this approach, we use a softmax policy with a strength index z to choose moves. Most importantly, we filter low quality moves by excluding those that have a lower simulation count than a pre-defined threshold ratio of the maximum simulation count. We perform a theoretical analysis, reaching the result that the adjusted policy is guaranteed to choose moves exceeding a lower bound in strength by using a threshold ratio. The approach is applied to the Go program ELF OpenGo. The experiment results show that z is highly correlated to the empirical strength; namely, given a threshold ratio 0.1, z is linearly related to the Elo rating with regression error 47.95 Elo where −2≤z ≤2. Meanwhile, the covered strength range is about 800 Elo ratings in the interval of z in [−2,2]. With the ease of strength adjustment using z, we present two methods to adjust strength and predict opponents’ strengths dynamically. To our knowledge, this result is state-of-the-art in terms of the range of strengths in Elo rating while maintaining a controllable relationship between the strength and a strength index.

This paper proposes a new approach to a novel value network architecture for the gameGo, called a multilabeled (ML) value network. In the ML value network, different values (win rates) are trained simultaneously for different settings of komi, a compensation given to balance the initiative of playing first. The ML value network has three advantages: 1) it outputs values for different komi; (2) it supports dynamic komi; and (3) it lowers the mean squared error (MSE). This paper also proposes a new dynamic komi method to improve game-playing strength. This paper also performs experiments to demonstrate the merits of the architecture. First, the MSE of the ML value network is generally lower than the value network alone. Second, the program based on the ML value network wins by a rate of 67.6% against the program based on the value network alone. Third, the program with the proposed dynamic komi method significantly improves the playing strength over the baseline that does not use dynamic komi, especially for handicap games. To our knowledge, up to date, no handicap games have been played openly by programs using value networks. This paper provides these programs with a useful approach to playing handicap games.