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.