Provable learning of Noisy-or Networks

TL;DR

This paper introduces a provably efficient tensor decomposition method for learning noisy-or networks, a type of nonlinear latent variable model, addressing challenges posed by correlated noise and systematic errors.

Contribution

It develops a novel tensor decomposition approach for noisy-or networks, handling systematic errors and extending provable learning to nonlinear latent variable models.

Findings

Successfully applied tensor decomposition to noisy-or networks

Analyzed tensor methods in the presence of correlated noise

Provided theoretical guarantees for learning in nonlinear models

Abstract

Many machine learning applications use latent variable models to explain structure in data, whereby visible variables (= coordinates of the given datapoint) are explained as a probabilistic function of some hidden variables. Finding parameters with the maximum likelihood is NP-hard even in very simple settings. In recent years, provably efficient algorithms were nevertheless developed for models with linear structures: topic models, mixture models, hidden markov models, etc. These algorithms use matrix or tensor decomposition, and make some reasonable assumptions about the parameters of the underlying model. But matrix or tensor decomposition seems of little use when the latent variable model has nonlinearities. The current paper shows how to make progress: tensor decomposition is applied for learning the single-layer {\em noisy or} network, which is a textbook example of a Bayes net, and used for example in the classic QMR-DT software for diagnosing which disease(s) a patient may have by observing the symptoms he/she exhibits. The technical novelty here, which should be useful in other settings in future, is analysis of tensor decomposition in presence of systematic error (i.e., where the noise/error is correlated with the signal, and doesn't decrease as number of samples goes to infinity). This requires rethinking all steps of tensor decomposition methods from the ground up. For simplicity our analysis is stated assuming that the network parameters were chosen from a probability distribution but the method seems more generally applicable.