Estimating the Probability of Sampling a Trained Neural Network at Random

• URL: http://arxiv.org/abs/2501.18812v2
• Date: Tue, 08 Apr 2025 00:36:01 GMT
• Title: Estimating the Probability of Sampling a Trained Neural Network at Random
• Authors: Adam Scherlis, Nora Belrose,
• Abstract summary: We analyze an algorithm for estimating the size of a localized neighborhood in a neural network.<n>We show that this lower bound can be improved with an importance-sampling method.<n>We find that overfit, badly-generalizing neighborhoods are smaller, indicating a more complex learned behavior.
• Score: 1.9358739203360091
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We present and analyze an algorithm for estimating the size, under a Gaussian or uniform measure, of a localized neighborhood in neural network parameter space with behavior similar to an ``anchor'' point. We refer to this as the "local volume" of the anchor. We adapt an existing basin-volume estimator, which is very fast but in many cases only provides a lower bound. We show that this lower bound can be improved with an importance-sampling method using gradient information that is already provided by popular optimizers. The negative logarithm of local volume can also be interpreted as a measure of the anchor network's information content. As expected for a measure of complexity, this quantity increases during language model training. We find that overfit, badly-generalizing neighborhoods are smaller, indicating a more complex learned behavior. This smaller volume can also be interpreted in an MDL sense as suboptimal compression. Our results are consistent with a picture of generalization we call the "volume hypothesis": that neural net training produces good generalization primarily because the architecture gives simple functions more volume in parameter space, and the optimizer samples from the low-loss manifold in a volume-sensitive way. We believe that fast local-volume estimators are a promising practical metric of network complexity and architectural inductive bias for interpretability purposes.

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