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Since TD-trained networks with a raw input encoding were able toachieve parity with Neurogammon, it was hoped that by addingNeurogammon's hand-designed features to the raw encoding, the TD netsmight then be able to surpass Neurogammon. This was indeed foundto be the case: the TD nets with the additional features, whichform the basis of version 1.0 and subsequent versions of TD-Gammon,have greatly surpassed Neurogammon and all other previous computerprograms. Among the indicators contributing to this assessment(for more details, see the ) arenumerous tests of TD-Gammon in play against several world-classhuman grandmasters, including Bill Robertie and Paul Magriel, bothnoted authors and highly respected former World Champions.

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TD-Gammon was designed as a way to explore the capability of multilayerneural networks trained by TD() to learn complex nonlinearfunctions. It was also designed to provide a detailed comparison of theTD learning approach with the alternative approach of supervisedtraining on a corpus of expert-labeled exemplars. The latter methodologywas used a few years ago in the development of Neurogammon, the author'sprevious neural-network backgammon program. Neurogammon was trained bybackpropagation on a data base of recorded expert games. Its inputrepresentation included both the raw board information (number ofcheckers at each location), as well as a few hand-crafted "features"that encoded important expert concepts. Neurogammon achieved a strongintermediate level of play, which enabled it to win in convincing stylethe backgammon championship at the 1989 International Computer Olympiad[].Thus by comparing TD-Gammon with Neurogammon, one can get a senseof the potential of TD learning relative to the more establishedapproach of supervised learning.

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A third key ingredient has been found by a close examination of theearly phases of the learning process. As stated previously, duringthe first few thousand training games, the network learns a numberof elementary concepts, such as bearing off as many checkers aspossible, hitting the opponent, playing safe (i.e., not leavingexposed blots that can be hit by the opponent) and building newpoints. It turns out that these early elementary concepts can allbe expressed by an evaluation function that is linear in the rawinput variables. Thus what appears to be happening in the TDlearning process is that the neural network first extracts thelinear component of the evaluation function, while nonlinearconcepts emerge later in learning. (This is also frequentlyseen in backpropagation: in many applications, when training amultilayer net on a complex task, the network first extracts thelinearly separable part of the problem.)

If TD-Gammon has been an exciting new development in the world ofbackgammon, it has been even more exciting for the fields of neuralnetworks and machine learning. By combining the TD approach totemporal credit assignment with the MLP architecture for nonlinearfunction approximation, rather surprising results have beenobtained, to say the least. The TD self-play approach has greatlysurpassed the alternative approach of supervised training on expertexamples, and has achieved a level of play well beyond what onecould have expected, based on prior theoretical and empirical workin reinforcement learning. Hence there is now considerable interestwithin the machine learning community in trying to extract theprinciples underlying the success of TD-Gammon's self-teachingprocess. This could form the basis for further theoretical progressin the understanding of TD methods, and it could also provide someindication as to other classes of applications where TD learningmight also be successful. While a complete understanding of thelearning process is still far away, some important insights havebeen obtained, and are described in more detail here.

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A number of researchers are currently investigating applications ofTD() to other games such as chess and Go. SebastianThrun has obtained encouraging preliminary results with a TD-chesslearning system that learns by playing against a publicly availablechess program, Gnuchess (Thrun, personal communication).Schraudolph, et al. have also obtained encouraging early resultsusing TD() to learn to play Go [].Finally, Jean-FrancoisIsabelle has obtained good results applying the TD self-learningprocedure to Othello []. The best network reportedin that studywas able to defeat convincingly an "intermediate-advanced"conventional Othello program.

During training, the neural network itself is used to select movesfor both sides. At each time step during the course of a game, theneural network scores every possible legal move. (We interpret thenetwork's score as an estimate of expected outcome, or "equity" ofthe position. This is a natural interpretation which is exact incases where TD() has been proven to converge.) The movethat is selected is then the move with maximum expected outcome forthe side making the move. In other words, the neural network islearning from the results of playing against itself. This self-playtraining paradigm is used even at the start of learning, when thenetwork's weights are random, and hence its initial strategy is arandom strategy. Initially, this methodology would appear unlikelyto produce any sensible learning, because random strategy isexceedingly bad, and because the games end up taking an incrediblylong time: with random play on both sides, games often last severalhundred or even several thousand time steps. In contrast, in normalhuman play games usually last on the order of 50-60 time steps.

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Temporal Difference Learning and TD-Gammon

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Programming a computer to play high-level backgammon has been found tobe a rather difficult undertaking. In certain simplified endgamesituations, it is possible to design a program that plays perfectlyvia table look-up. However, such an approach is not feasible for thefull game, due to the enormous number of possible states (estimatedat over 10 to the power of 20). Furthermore, the brute-forcemethodology of deep searches, which has worked so well in games suchas chess, checkers and Othello, is not feasible due to the highbranching ratio resulting from the probabilistic dice rolls. At eachply there are 21 dice combinations possible, with an average of about20 legal moves per dice combination, resulting in a branching ratioof several hundred per ply. This is much larger than in checkers andchess (typical branching ratios quoted for these games are 8-10 forcheckers and 30-40 for chess), and too large to reach significantdepth even on the fastest available supercomputers.

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