Probabilistic Invariant Learning with Randomized Linear Classifiers
This work addresses the challenge of balancing invariance and resource efficiency in machine learning models, offering a novel approach for tasks involving sets, graphs, and spherical data.
The paper tackles the problem of designing models that are both expressive and invariant to known transformations, which often requires high computational or memory resources, by introducing Randomized Linear Classifiers (RLCs) that achieve probabilistic invariance and universality with less resources, as demonstrated empirically on tasks where deterministic invariant neural networks struggle.
Designing models that are both expressive and preserve known invariances of tasks is an increasingly hard problem. Existing solutions tradeoff invariance for computational or memory resources. In this work, we show how to leverage randomness and design models that are both expressive and invariant but use less resources. Inspired by randomized algorithms, our key insight is that accepting probabilistic notions of universal approximation and invariance can reduce our resource requirements. More specifically, we propose a class of binary classification models called Randomized Linear Classifiers (RLCs). We give parameter and sample size conditions in which RLCs can, with high probability, approximate any (smooth) function while preserving invariance to compact group transformations. Leveraging this result, we design three RLCs that are provably probabilistic invariant for classification tasks over sets, graphs, and spherical data. We show how these models can achieve probabilistic invariance and universality using less resources than (deterministic) neural networks and their invariant counterparts. Finally, we empirically demonstrate the benefits of this new class of models on invariant tasks where deterministic invariant neural networks are known to struggle.