This work was conducted during Chaobing Song's visit to Professor Yi Ma's group at UC Berkeley.

This work was conducted during Chaobing Song's visit to Professor Yi Ma's group at UC Berkeley.

To do so, we use an infinite-dimensional version of Girsanov's theorem to condition

Mean field approximation and propagation of chaos for mSGLD . . . . . . . . . . 4 S3 T echnical results 4 S4 Quantitative propagation of chaos 8 S4.1 Existence of strong solutions to the particle SDE . . . . . . . . . . . . . . . . . . If F = R, then we simply note C( E). S2.1 Presentation of the modified SGLD and its continuous counterpart The proof is postponed to Section S4.4 Consider now the mean-field SDE starting from a random variable W The proof is postponed to Section S4.4 Then, there exists L 0 such that the following hold. In what follows, we bound separately the two terms in the right-hand side.

This work was conducted during Chaobing Song's visit to Professor Yi Ma's group at UC Berkeley.

The modern theory of stochastic control typically assumes complete knowledge of the underlying system dynamics. While significant theoretical advancements have been made in this area, see Øksendal and Sulem 2019; Fleming and Soner 2006, the practical application of stochastic control often faces challenges when the system model is uncertain or unknown. In recent years, Reinforcement learning (RL) has emerged as a promising approach to address this issue, enabling agents to learn optimal control policies through trial-and-error interactions with the environment. However, RL's success often hinges on the availability of vast amounts of data, and the learned control policies can be difficult to interpret, especially when deep learning techniques are employed, see Sutton 2018. To bridge the gap between fully model-based and model-free approaches, research has increasingly focused on model-based reinforcement learning.

To investigate the feasibility of strongly solving Minishogi (Gogo Shogi), it is necessary to know the number of its reachable positions from the initial position. However, there currently remains a significant gap between the lower and upper bounds of the value, since checking the legality of a Minishogi position is difficult. In this paper, the authors estimate the number of reachable positions by generating candidate positions using uniform random sampling and measuring the proportion of those reachable by a series of legal moves from the initial position. The experimental results reveal that the number of reachable Minishogi positions is approximately $2.38\times 10^{18}$.