Rethinking Parameter Counting in Deep Models: Effective Dimensionality Revisited
This work provides a foundational insight into generalization in deep learning, addressing a core theoretical problem for researchers and practitioners.
The paper tackles the puzzle of why neural networks generalize well despite having more parameters than data points, and explains phenomena like double descent by introducing effective dimensionality as a measure of parameter space complexity determined by data, showing it outperforms other generalization measures.
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.