Algebraic geometry is essential to learning theory. In hierarchical learning machines such as layered neural networks and gaussian mixtures, the asymptotic normality does not hold, since Fisher information matricesare singular. In this paper, the rigorous asymptotic form of the stochastic complexity is clarified based on resolution of singularities and two different problems are studied.

Hidetoshi Nishimori Department of Physics, Tokyo Institute of Technology, Oh-Okayama, Meguro-ku, Tokyo 152-8551, Japan nishi@stat.phys.titech.ac.jp Abstract We introduce stagewise processing in error-correcting codes and image restoration, by extracting information from the former stage and using it selectively to improve the performance of the latter one. Both mean-field analysis using the cavity method and simulations showthat it has the advantage of being robust against uncertainties in hyperparameter estimation. 1 Introduction In error-correcting codes [1] and image restoration [2], the choice of the so-called hyperparameters is an important factor in determining their performances. In error correction, they determine the statistical significance given to the paritychecking termsand the received bits. Similarly in image restoration, they determine the statistical weights given to the prior knowledge and the received data. It was shown, by the use of inequalities, that the choice of the hyperparameters is optimal whenthere is a match between the source and model priors [3].

Several regular programs will return is March 15, 2002.

In this paper, we present algorithms that perform gradient ascent of the average reward in a partially observable Markov decision process (POMDP). These algorithms are based on GPOMDP, an algorithm introduced in a companion paper (Baxter & Bartlett, this volume), which computes biased estimates of the performance gradient in POMDPs. The algorithm's chief advantages are that it uses only one free parameter beta, which has a natural interpretation in terms of bias-variance trade-off, it requires no knowledge of the underlying state, and it can be applied to infinite state, control and observation spaces. We show how the gradient estimates produced by GPOMDP can be used to perform gradient ascent, both with a traditional stochastic-gradient algorithm, and with an algorithm based on conjugate-gradients that utilizes gradient information to bracket maxima in line searches. Experimental results are presented illustrating both the theoretical results of (Baxter & Bartlett, this volume) on a toy problem, and practical aspects of the algorithms on a number of more realistic problems.

Gradient-based approaches to direct policy search in reinforcement learning have received much recent attention as a means to solve problems of partial observability and to avoid some of the problems associated with policy degradation in value-function methods. In this paper we introduce GPOMDP, a simulation-based algorithm for generating a biased estimate of the gradient of the average reward in Partially Observable Markov Decision Processes POMDPs controlled by parameterized stochastic policies. A similar algorithm was proposed by (Kimura et al. 1995). The algorithm's chief advantages are that it requires storage of only twice the number of policy parameters, uses one free beta (which has a natural interpretation in terms of bias-variance trade-off), and requires no knowledge of the underlying state. We prove convergence of GPOMDP, and show how the correct choice of the parameter beta is related to the mixing time of the controlled POMDP. We briefly describe extensions of GPOMDP to controlled Markov chains, continuous state, observation and control spaces, multiple-agents, higher-order derivatives, and a version for training stochastic policies with internal states. In a companion paper (Baxter et al., this volume) we show how the gradient estimates generated by GPOMDP can be used in both a traditional stochastic gradient algorithm and a conjugate-gradient procedure to find local optima of the average reward.

For more details, see Nau et al. 's preconditions can include logical inferences, 's preconditions two methods for traveling from one location can include Horn-clause inferencing, numeric to another: (1) traveling by airplane and (2) computations, and calls to external programs. 's expressive power can be used to create a totally ordered list of subtasks. Suppose domain representations for complex application that all these subtasks are primitive except for domains. For example, the Horn 4. if t is primitive (i.e., there is an operator for t) then clauses can include calls to attached procedures 5. nondeterministically choose an operator o for t We believe the primary 14. endif's higher level of expressivity made it possible to formulate highly expressive domain algorithms in's data structures to make them faster; for example, we found that a simple change to the data structure We intend to make more optimizations in the near future. (Aha and Breslow 1997).

Domain-independent planning is a hard combinatorial problem. Taking into account plan quality makes the task even more difficult. This article introduces Planning by Rewriting (PbR), a new paradigm for efficient high-quality domain-independent planning. PbR exploits declarative plan-rewriting rules and efficient local search techniques to transform an easy-to-generate, but possibly suboptimal, initial plan into a high-quality plan. In addition to addressing the issues of planning efficiency and plan quality, this framework offers a new anytime planning algorithm. We have implemented this planner and applied it to several existing domains. The experimental results show that the PbR approach provides significant savings in planning effort while generating high-quality plans.

This paper presents an implemented system for recognizing the occurrence of events described by simple spatial-motion verbs in short image sequences. The semantics of these verbs is specified with event-logic expressions that describe changes in the state of force-dynamic relations between the participants of the event. An efficient finite representation is introduced for the infinite sets of intervals that occur when describing liquid and semi-liquid events. Additionally, an efficient procedure using this representation is presented for inferring occurrences of compound events, described with event-logic expressions, from occurrences of primitive events. Using force dynamics and event logic to specify the lexical semantics of events allows the system to be more robust than prior systems based on motion profile.

The chief aim of this paper is to propose mean-field approximations for a broad class of Belief networks, of which sigmoid and noisy-or networks can be seen as special cases. The approximations are based on a powerful mean-field theory suggested by Plefka. We show that Saul, Jaakkola and Jordan' s approach is the first order approximation in Plefka's approach, via a variational derivation. The application of Plefka's theory to belief networks is not computationally tractable. To tackle this problem we propose new approximations based on Taylor series. Small scale experiments show that the proposed schemes are attractive.

This paper presents GRT, a domain-independent heuristic planning system for STRIPS worlds. GRT solves problems in two phases. In the pre-processing phase, it estimates the distance between each fact and the goals of the problem, in a backward direction. Then, in the search phase, these estimates are used in order to further estimate the distance between each intermediate state and the goals, guiding so the search process in a forward direction and on a best-first basis. The paper presents the benefits from the adoption of opposite directions between the preprocessing and the search phases, discusses some difficulties that arise in the pre-processing phase and introduces techniques to cope with them. Moreover, it presents several methods of improving the efficiency of the heuristic, by enriching the representation and by reducing the size of the problem. Finally, a method of overcoming local optimal states, based on domain axioms, is proposed. According to it, difficult problems are decomposed into easier sub-problems that have to be solved sequentially. The performance results from various domains, including those of the recent planning competitions, show that GRT is among the fastest planners.