T. A. Marsland

A comparison of minimax tree search algorithms

Abstract Although theoretic performance measures of most game-searching algorithms exist, for various reasons their practicality is limited. This paper examines and extends the existing search methods, and reports on empirical performance studies on trees with useful size and ordering properties. Emphasis is placed on trees that are strongly ordered, i.e., similar to those produced by many current game-playing programs.

Parallel Game-Tree Search

The design issues affecting a parallel implementation of the alpha-beta search algorithm are discussed with emphasis on a tree decomposition scheme that is intended for use on well ordered trees. In particular, the principal variation splitting method has been implemented, and experimental results are presented which show how such refinements as progressive deepening, narrow window searching, and the use of memory tables affect the performance of multiprocessor based chess playing programs. When dealing with parallel processing systems, communication delays are perhaps the greatest source of lost time. Therefore, an implementation of our tree decomposition based algorithm is presented, one that operates with a modest amount of message passing within a network of processors. Since our system has low search overhead, the principal basis for comparison is the communication overhead, which in turn is shown to have two components.

Global snapshots for distributed debugging

The widespread adoption of distributed computing has accentuated the need for an effective set of support tools to facilitate debugging and monitoring. In providing such support, one fundamental problem is that of constructing a global snapshot or global state of a distributed computation. This paper examines global snapshot algorithms from a distributed debugging perspective, and proposes an abstract framework based on global snapshots, which is defined to form a consistent state of the entire system. It is shown that by using a property preserving algorithm, this framework can be superimposed on the underlying computation, but not interfere with it.<<ETX>>

Low overhead alternatives to SSS

Abstract Of the many minimax algorithms, sss∗ is noteworthy because it usually searches the smallest game trees. Its success can be attributed to the accumulation and use of information acquired while traversing the tree. The main disadvantages of sss∗ are its high storage needs and management costs. This paper describes a class of methods, based on the popular alpha-beta algorithm, that acquire and use information to guide a tree search. They retain a given search direction and yet are as good as sss∗ , even while searching random trees. Further, although some of these new algorithms also require substantial storage, they are more flexible and can be programmed to use only the space available, at the cost of some degradation in performance.

Multi-cut ab-pruning in game-tree search

The efficiency of the αβ-algorithm as a minimax search procedure can be attributed to its effective pruning at the so-called cut-nodes; ideally only one move is examined there to establish the minimax value. This paper explores the benefits of investing additional search effort at cut-nodes by also expanding some of the remaining moves. Our results show a strong correlation between the number of promising move alternatives at cut-nodes and a new principal variation emerging. Furthermore, a new forward-pruning method is introduced that uses this additional information to ignore potentially futile subtrees. We also provide experimental results with the new pruning method in the domain of chess.

Tuning evaluation functions by maximizing concordance

Abstract Heuristic search effectiveness depends directly upon the quality of heuristic evaluations of states in a search space. Given the large amount of research effort devoted to computer chess throughout the past half-century, insufficient attention has been paid to the issue of determining if a proposed change to an evaluation function is beneficial. We argue that the mapping of an evaluation function from chess positions to heuristic values is of ordinal, but not interval scale. We identify a robust metric suitable for assessing the quality of an evaluation function, and present a novel method for computing this metric efficiently. Finally, we apply an empirical gradient-ascent procedure, also of our design, over this metric to optimize feature weights for the evaluation function of a computer-chess program. Our experiments demonstrate that evaluation function weights tuned in this manner give equivalent performance to hand-tuned weights.

Learning extension parameters in game-tree search

The strength of a program for playing an adversary game like chess or checkers is greatly influenced by how selectively it explores the various branches of the game tree. Typically, some branch paths are discontinued early while others are explored more deeply. Finding the best set of parameters to control these extensions is a difficult, time-consuming, and tedious task. In this paper we describe a method for automatically tuning search-extension parameters in adversary search. Based on the new method, two learning variants are introduced: one for offline learning and the other for online learning. The two approaches are compared and experimental results provided in the domain of chess.

Risk management in game-tree pruning

Abstract In the half century since minimax was first suggested as a strategy for adversary game search, various search algorithms have been developed. The standard approach has been to use improvements to the Alpha–Beta ( α – β ) algorithm. Some of the more powerful improvements examine continuations beyond the nominal search depth if they are of special interest, while others terminate the search early. The latter case is referred to as forward pruning. In this paper we discuss some important aspects of forward pruning, especially regarding risk-management, and propose ways of making risk-assessment. Finally, we introduce two new pruning methods based on some of the principles discussed here, and present experimental results from application of the methods in an established chess program.

Accuracy and savings in depth-limited capture search

Capture search, an expensive part of any chess program, is conducted at every leaf node of the approximating game tree. Often an exhaustive capture search is not feasible, and yet limiting the search depth compromises the result. Our experiments confirm that for chess a deeper search results in less error, and show that a shallow search does not provide significant savings. It is therefore better to do an arbitrary depth capture search. If a limit is used for search termination, an odd depth is preferable.

EVALUATION FUNCTION TUNING VIA ORDINAL CORRELATION

Heuristic search effectiveness depends directly upon the quality of heuristic evaluations of states in the search space. We show why ordinal correlation is relevant to heuristic search, present a metric for assessing the quality of a static evaluation function, and apply it to learn feature weights for a computer chess program.