Sequential decision making problems, such as structured prediction, robotic control, and game playing, require a combination of planning policies and generalisation of those plans. In this paper, we present Expert Iteration (ExIt), a novel reinforcement learning algorithm which decomposes the problem into separate planning and generalisation tasks. Planning new policies is performed by tree search, while a deep neural network generalises those plans. Subsequently, tree search is improved by using the neural network policy to guide search, increasing the strength of new plans. In contrast, standard deep Reinforcement Learning algorithms rely on a neural network not only to generalise plans, but to discover them too. We show that ExIt outperforms REINFORCE for training a neural network to play the board game Hex, and our final tree search agent, trained tabula rasa, defeats MoHex, the previous state-of-the-art Hex player.
AlphaGo Zero (AG0) @cite_2 , presents an independently developed version of ExIt, Our original version, with only policy networks, was published before AG0 was published, but after its submission. Our value networks were developed before AG0 was published, and published after @cite_2 and showed that it achieves state-of-the-art performance in Go. We include a detailed comparison of these closely related works in the appendix.
@cite_2
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During the development of AlphaGo, its many hyper-parameters were tuned with Bayesian optimization multiple times. This automatic tuning process resulted in substantial improvements in playing strength. For example, prior to the match with Lee Sedol, we tuned the latest AlphaGo agent and this improved its win-rate from 50 to 66.5 in self-play games. This tuned version was deployed in the final match. Of course, since we tuned AlphaGo many times during its development cycle, the compounded contribution was even higher than this percentage. It is our hope that this brief case study will be of interest to Go fans, and also provide Bayesian optimization practitioners with some insights and inspiration. The game of Go has long been viewed as the most challenging of classic games for artificial intelligence owing to its enormous search space and the difficulty of evaluating board positions and moves. Here we introduce a new approach to computer Go that uses ‘value networks’ to evaluate board positions and ‘policy networks’ to select moves. These deep neural networks are trained by a novel combination of supervised learning from human expert games, and reinforcement learning from games of self-play. Without any lookahead search, the neural networks play Go at the level of stateof-the-art Monte Carlo tree search programs that simulate thousands of random games of self-play. We also introduce a new search algorithm that combines Monte Carlo simulation with value and policy networks. Using this search algorithm, our program AlphaGo achieved a 99.8 winning rate against other Go programs, and defeated the human European Go champion by 5 games to 0. This is the first time that a computer program has defeated a human professional player in the full-sized game of Go, a feat previously thought to be at least a decade away. Policy gradient reinforcement learning has been applied to two-player alternate-turn zero-sum games, e.g., in AlphaGo, self-play REINFORCE was used to improve the neural net model after supervised learning. In this paper, we emphasize that two-player zero-sum games with alternating turns, which have been previously formulated as Alternating Markov Games (AMGs), are different from standard MDP because of their two-agent nature. We exploit the difference in associated Bellman equations, which leads to different policy iteration algorithms. As policy gradient method is a kind of generalized policy iteration, we show how these differences in policy iteration are reflected in policy gradient for AMGs. We formulate an adversarial policy gradient and discuss potential possibilities for developing better policy gradient methods other than self-play REINFORCE. The core idea is to estimate the minimum rather than the mean for the “critic”. Experimental results on the game of Hex show the modified Monte Carlo policy gradient methods are able to learn better pure neural net policies than the REINFORCE variants. To apply learned neural weights to multiple board sizes Hex, we describe a board-size independent neural net architecture. We show that when combined with search, using a single neural net model, the resulting program consistently beats MoHex 2.0, the state-of-the-art computer Hex player, on board sizes from 9×9 to 13×13. In citepapa, Papadimitriou formalized the notion of routing stability in BGP as the following coalitional game theoretic problem: Given a network with a multicommodity flow satisfying node capacity and demand constraints, the payoff of a node is the total flow originated or terminated at it. A payoff allocation is in the core if and only if there is no subset of nodes that can increase their payoff by seceding from the network. We answer one of the open problems in citepapa by proving that for any network, the core is non-empty in both the transferable (where the nodes can compensate each other with side payments) and the non-transferable case. In the transferable case we show that such an allocation can be computed in polynomial time. We also generalize this result to the case where a strictly concave utility function is associated with each commodity.