Tree Search Algorithms: Beyond Monte Carlo Tree Search


Academic supervisors

IECL, LORIA, INRIA Center of Université de Lorraine.



In recent years, AlphaZero’s success in mastering complex games such
as chess and Go has demonstrated the power of reinforcement learning
coupled with tree search algorithms (such as MCTS) and powerful
function approximators (deep neural networks). However, even with
these remarkable achievements, intriguing theoretical questions remain
unexplored. This thesis concerns the study of the theoretical
foundations of game algorithms, focusing specifically on deterministic
two-player games. The main objectives include improving existing tree
search algorithms, the development of more sophisticated probabilistic
models of deterministic game, the corresponding inference algorithms,
and the development of benchmarks.

Phd topic

The first avenue of exploration is to advance the tree search
literature, in particular those concerning
MIN-MAX trees adapted to two-player games. Building on
established algorithms such as $A^*$ [14],
AlphaBeta [7] and MTD(f) [13], The aim of this
research is to propose new search algorithms. Potential improvements
include optimized pruning strategies and innovative approaches to the
exploration of the search space such as the use of optimistic approach
through regularity assumptions (DOO/SOO [10,11,2],
SEQUOOL [1]). The theoretical analysis will focus on the
mathematical properties of the proposed algorithms, establishing their
strengths and limitations.

The second line of research focuses on identifying and analyzing
probabilistic models of deterministic games, which are less trivial
than Pearl’s model [12]. By developing more sophisticated
models, the aim is to provide a richer representation of the
complexity inherent in deterministic games. This includes
incorporating elements such as dependency through common tree
ancestors [3,6,4] to better reflect real-world
scenarios. Theoretical analysis will examine inference algorithms to
update model parameters based on observations acquired during the
research and their modeling quality for a varied set of
deterministic games

On the basis of these probabilistic models, a third objective is to to
study the corresponding average complexity of several
state-of-the-art algorithms, to explore alternative strategies for
decision decision making in deterministic games. The theory acquired
will contribute to a better understanding of how probabilistic models
can guide decision-making processes in the context of games (at
present, most of these algorithms are compared on the Pearl model and
are essentially indifferentiable… being all optimal in this

A final dimension of this research, which cuts across all the previous
dimensions, involves the constitution of a benchmark database
for functions and games
(in a spirit similar to what is done for
function approximation [9,5]). Carefully selected problems
will provide a comprehensive evaluation of the proposed methods. For
games, the focus will be on identifying AlphaZero’s potential
weaknesses, and propose algorithms that may exploit these

Finally, depending on the progress made on the above dimensions,
future perspectives of the work may include the case of problems with
partial information, or the implementation of a dynamic programming
scheme similar to AlphaZero.


P. L. Bartlett, V. Gabillon, and M. Valko.

A simple parameter-free and adaptive approach to optimization under a
minimal local smoothness assumption.

In A. Garivier and S. Kale, editors, Algorithmic Learning
Theory, ALT 2019, 22-24 March 2019, Chicago, Illinois, USA
, volume 98 of
Proceedings of Machine Learning Research, pages 184–206. PMLR, 2019.

L. Busoniu, R. Munos, and E. Páll.

An analysis of optimistic, best-first search for minimax sequential
decision making.

In Adaptive Dynamic Programming and Reinforcement Learning
(ADPRL), 2014 IEEE Symposium on
, page 1–8. IEEE, IEEE, 2014.

L. Devroye and O. Kamoun.

Random minimax game trees.

In D. Aldous and R. Pemantle, editors, Random Discrete
, pages 55–80, New York, NY, 1996. Springer New York.

J. Grosse, C. Zhang, and P. Hennig.

Probabilistic dag search.

In C. de Campos and M. H. Maathuis, editors, Proceedings of the
Thirty-Seventh Conference on Uncertainty in Artificial Intelligence
, volume
161 of Proceedings of Machine Learning Research, pages 1424–1433.
PMLR, 27–30 Jul 2021.

N. Hansen, D. Brockhoff, O. Mersmann, T. Tusar, D. Tusar, O. A. ElHara, P. R.
Sampaio, A. Atamna, K. Varelas, U. Batu, D. M. Nguyen, F. Matzner, and
A. Auger.

COmparing Continuous Optimizers: numbbo/COCO on Github, Mar. 2019.

P. Hennig, D. Stern, and T. Graepel.

Coherent inference on optimal play in game trees.

In Y. W. Teh and M. Titterington, editors, Proceedings of the
Thirteenth International Conference on Artificial Intelligence and
, volume 9 of Proceedings of Machine Learning Research,
pages 326–333, Chia Laguna Resort, Sardinia, Italy, 13–15 May 2010. PMLR.

D. E. Knuth and R. W. Moore.

An analysis of alpha-beta pruning.

Artificial Intelligence, 6(4):293–326, 1975.

M. Lanctot, E. Lockhart, J.-B. Lespiau, V. Zambaldi, S. Upadhyay,
J. Pérolat, S. Srinivasan, F. Timbers, K. Tuyls, S. Omidshafiei,
D. Hennes, D. Morrill, P. Muller, T. Ewalds, R. Faulkner, J. Kramár,
B. D. Vylder, B. Saeta, J. Bradbury, D. Ding, S. Borgeaud, M. Lai,
J. Schrittwieser, T. Anthony, E. Hughes, I. Danihelka, and J. Ryan-Davis.

OpenSpiel: A framework for reinforcement learning in games.

CoRR, abs/1908.09453, 2019.

W. Li, H. Li, J. Honorio, and Q. Song.

Pyxab – a python library for $\mathcal{X}$-armed bandit and online
blackbox optimization algorithms, 2023.

R. Munos.

Optimistic optimization of a deterministic function without the
knowledge of its smoothness.

In J. Shawe-Taylor, R. Zemel, P. Bartlett, F. Pereira, and
K. Weinberger, editors, Advances in Neural Information Processing
, volume 24. Curran Associates, Inc., 2011.

R. Munos.

From bandits to monte-carlo tree search: The optimistic principle
applied to optimization and planning.

Foundations and Trends® in Machine Learning, 7(1):1–129,

J. Pearl.

Asymptotic properties of minimax trees and game-searching procedures.

Artificial Intelligence, 14(2):113–138, 1980.

A. Plaat, J. Schaeffer, W. Pijls, and A. de Bruin.

Best-first fixed-depth minimax algorithms.

Artificial Intelligence, 87(1):255–293, 1996.

S. Russell and P. Norvig.

Artificial Intelligence: A Modern Approach.

Prentice Hall, 3 edition, 2010.

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