基于奇异值分解的井字棋评价函数信息压缩及性能评估
Information Compression and Performance Evaluation of Tic-Tac-Toe's Evaluation Function Using Singular Value Decomposition
摘要：
利用奇异值分解（SVD）方法对井字棋的评价函数进行了近似，并研究了近似精度对胜率的影响。我们首先准备了井字棋的完美评价函数，并通过将评价函数视为九阶张量来进行低秩近似。我们发现，我们可以减少70%的评价函数的信息量，而不会显着降低性能。近似准确度和获胜率之间存在强相关性，但并不完全成正比。我们还研究了评价函数的分解方法如何影响性能。我们考虑了两种分解方法：将评价函数作为矩阵的简单SVD和高阶SVD的Tucker分解（HOSVD）。在相同的压缩比下，采用HOSVD得到的近似评价函数的策略比SVD得到的策略具有更高的获胜率。这些结果表明，SVD可以有效地压缩棋盘游戏策略，并存在一个最佳的压缩方法，取决于游戏。
英文摘要：
We approximated the evaluation function for the game Tic-Tac-Toe by singular value decomposition (SVD) and investigated the effect of approximation accuracy on winning rate. We first prepared the perfect evaluation function of Tic-Tac-Toe and performed low-rank approximation by considering the evaluation function as a ninth-order tensor. We found that we can reduce the amount of information of the evaluation function by 70% without significantly degrading the performance. Approximation accuracy and winning rate were strongly correlated but not perfectly proportional. We also investigated how the decomposition method of the evaluation function affects the performance. We considered two decomposition methods: simple SVD regarding the evaluation function as a matrix and the Tucker decomposition by higher-order SVD (HOSVD). At the same compression ratio, the strategy with the approximated evaluation function obtained by HOSVD exhibited a significantly higher winning rate than that obtained by SVD. These results suggest that SVD can effectively compress board game strategies and an optimal compression method that depends on the game exists.
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