arXiv subjects

M. Opper

Publications and source records attributed to M. Opper.

5 recordsLinked to original sources

arXiv2009

Optimal control as a graphical model inference problem

We reformulate a class of non-linear stochastic optimal control problems introduced by Todorov (2007) as a Kullback-Leibler (KL) minimization problem. As a result, the optimal control computation reduces to an inference computation and approximate inference methods can be applied to efficiently compute approximate optimal controls. We show how this KL control theory contains the path integral control method as a special case. We provide an example of a block stacking task and a multi-agent cooperative game where we demonstrate how approximate inference can be successfully applied to instances that are too complex for exact computation. We discuss the relation of the KL control approach to other inference approaches to control.

math.OC↗

arXiv2002

Statistical Mechanics of Learning: A Variational Approach for Real Data

Using a variational technique, we generalize the statistical physics approach of learning from random examples to make it applicable to real data. We demonstrate the validity and relevance of our method by computing approximate estimators for generalization errors that are based on training data alone.

cond-mat.dis-nn↗

arXiv2001

Retarded Learning: Rigorous Results from Statistical Mechanics

We study learning of probability distributions characterized by an unknown symmetry direction. Based on an entropic performance measure and the variational method of statistical mechanics we develop exact upper and lower bounds on the scaled critical number of examples below which learning of the direction is impossible. The asymptotic tightness of the bounds suggests an asymptotically optimal method for learning nonsmooth distributions.

cond-mat.dis-nn↗

arXiv1999

Bayes-optimal performance in a discrete space

We study a simple model of unsupervised learning where the single symmetry breaking vector has binary components $\pm 1$. We calculate exactly the Bayes-optimal performance of an estimator which is required to lie in the same discrete space. We also show that, except for very special cases, such an estimator cannot be obtained by minimization of a class of variationally optimal potentials.

cond-mat.dis-nn↗

arXiv1995

Storage of correlated patterns in a perceptron

We calculate the storage capacity of a perceptron for correlated gaussian patterns. We find that the storage capacity $α_c$ can be less than 2 if similar patterns are mapped onto different outputs and vice versa. As long as the patterns are in general position we obtain, in contrast to previous works, that $α_c \geq 1$ in agreement with Cover's theorem. Numerical simulations confirm the results.

cond-mat↗