SGD Provably Prioritizes a Shortcut Spurious Feature in the XOR Model
For machine learning practitioners and theorists, this paper offers a rigorous theoretical understanding of how neural networks over-rely on spurious correlations, which is a known problem but previously lacked such a detailed mechanistic explanation.
This work provides the first end-to-end theoretical characterization of spurious feature learning in two-layer ReLU networks trained with SGD on the XOR model, showing that SGD learns the spurious feature first and exponentially fast, with the spurious component inhibiting signal learning. The analysis reveals phase transitions and shows that with maximally strong spurious correlation, the spurious feature dominates even at the sample complexity threshold where XOR would be learned in isolation.
Neural networks are known to be susceptible to over-reliance on spurious correlations. However, the precise mechanism by which models exploit shortcut features is not fully understood, and algorithms to mitigate this behavior rely on as yet unjustified assumptions about the learned representations. In this work, we provide the first end-to-end theoretical characterization of spurious feature learning for two-layer ReLU neural networks trained by online minibatch SGD on the logistic loss. We consider data drawn from the high-dimensional Boolean hypercube with a quadratic signal function (namely XOR) and a linear spurious correlation. We show that SGD learns the spurious feature first, and exponentially fast. Moreover, the optimization dynamics couple the spurious and signal features, with a stronger spurious component inhibiting signal feature learning. Our analysis reveals precise phase transitions in the learning dynamics. In the first phase, alignment between the signs of the spurious feature and second-layer weight drives rapid growth of the spurious feature. In the second phase, large majority group margin slows learning and the signal feature remains suppressed. When the spurious correlation is maximally strong, we show theoretically that the spurious feature dominates even at the sample complexity threshold where XOR would be learned in isolation (i.e., if the spurious feature was absent). In contrast, when the correlation strength is constant, we provide preliminary empirical evidence that the model can eventually learn the XOR signal, although the spurious feature is not forgotten.