Modular neural networks outperform nonmodular neural networks on tasks ranging from visual question answering to robotics. These performance improvements are thought to be due to modular networks' superior ability to model the compositional and combinatorial structure of real-world problems. However, a theoretical explanation of how modularity improves generalizability, and how to leverage task modularity while training networks remains elusive. Using recent theoretical progress in explaining neural network generalization, we investigate how the amount of training data required to generalize on a task varies with the intrinsic dimensionality of a task's input. We show theoretically that when applied to modularly structured tasks, while nonmodular networks require an exponential number of samples with task dimensionality, modular networks' sample complexity is independent of task dimensionality: modular networks can generalize in high dimensions. We then develop a novel learning rule for modular networks to exploit this advantage and empirically show the improved generalization of the rule, both in- and out-of-distribution, on high-dimensional, modular tasks.

In order to navigate safely and reliably in off-road and unstructured environments, robots must detect anomalies that are out-of-distribution (OOD) with respect to the training data. We present an analysis-by-synthesis approach for pixel-wise anomaly detection without making any assumptions about the nature of OOD data. Given an input image, we use a generative diffusion model to synthesize an edited image that removes anomalies while keeping the remaining image unchanged. Then, we formulate anomaly detection as analyzing which image segments were modified by the diffusion model. We propose a novel inference approach for guided diffusion by analyzing the ideal guidance gradient and deriving a principled approximation that bootstraps the diffusion model to predict guidance gradients. Our editing technique is purely test-time that can be integrated into existing workflows without the need for retraining or fine-tuning. Finally, we use a combination of vision-language foundation models to compare pixels in a learned feature space and detect semantically meaningful edits, enabling accurate anomaly detection for off-road navigation.