Influence functions and regularity tangents for efficient active learning

In this paper we describe an efficient method for providing a regression model with a sense of curiosity about its data. In the field of machine learning, our framework for representing curiosity is called Active Learning, which means automatically choosing data points for which to query labels in the semisupervised setting. The methods we propose are based on computing a "regularity tangent" vector that can be calculated (with only a constant slow-down) together with the model's parameter vector during training. We then take the inner product of this tangent vector with the gradient vector of the model's loss at a given data point to obtain a measure of the influence of that point on the complexity of the model. There is only a single regularity tangent vector, of the same dimension as the parameter vector. Thus, in the proposed technique, once training is complete, evaluating our "curiosity" about a potential query data point can be done as quickly as calculating the model's loss gradient at that point. The new vector only doubles the amount of storage required by the model. We show that the quantity computed by our technique is an example of an "influence function", and that it measures the expected squared change in model complexity incurred by up-weighting a given data point. We propose a number of ways for using this quantity to choose new training data for a model in the framework of active learning.

Belief propagation generalizes backpropagation

The two most important algorithms in artificial intelligence are backpropagation and belief propagation. In spite of their importance, the connection between them is poorly characterized. We show that when an input to backpropagation is converted into an input to belief propagation so that (loopy) belief propagation can be run on it, then the result of belief propagation encodes the result of backpropagation; thus backpropagation is recovered as a special case of belief propagation. In other words, we prove for apparently the first time that belief propagation generalizes backpropagation. Our analysis is a theoretical contribution, which we motivate with the expectation that it might reconcile our understandings of each of these algorithms, and serve as a guide to engineering researchers seeking to improve the behavior of systems that use one or the other.