Rethinking Parameter Counting: Effective Dimensionality Revisited

Gregory Benton, Wesley Maddox, Andrew Gordon Wilson

Keywords: effective dimension, hessian, generalization, double descent

Abstract: Neural networks appear to have mysterious generalization properties when using parameter counting as a proxy for complexity. Indeed, neural networks often have many more parameters than there are data points, yet still provide good generalization performance. Moreover, when we measure generalization as a function of parameters, we see double descent behaviour, where the test error decreases, increases, and then again decreases. We show that many of these properties become understandable when viewed through the lens of effective dimensionality, which measures the dimensionality of the parameter space determined by the data. We relate effective dimensionality to posterior contraction in Bayesian deep learning, model selection, width-depth tradeoffs, double descent, and functional diversity in loss surfaces, leading to a richer understanding of the interplay between parameters and functions in deep models. We also show that effective dimensionality compares favourably to alternative norm- and flatness- based generalization measures.

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One-sentence Summary: We show how effective dimensionality can shed light on a number of phenomena in modern deep learning including double descent, width-depth trade-offs, and subspace inference, while providing a straightforward and compelling generalization metric.

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