Bryan T. Kelly

Antoine Didisheim, Bryan Kelly, Semyon Malamud

We introduce a methodology for designing and training deep neural networks (DNN) that we call "Deep Regression Ensembles" (DRE). It bridges the gap between DNN and two-layer neural networks trained with random feature regression. Each layer of DRE has two components, randomly drawn input weights and output weights trained myopically (as if the final output layer) using linear ridge regression. Within a layer, each neuron uses a different subset of inputs and a different ridge penalty, constituting an ensemble of random feature ridge regressions. Our experiments show that a single DRE architecture is at par with or exceeds state-of-the-art DNN in many data sets. Yet, because DRE neural weights are either known in closed-form or randomly drawn, its computational cost is orders of magnitude smaller than DNN.

Teng Andrea Xu, Bryan Kelly, Semyon Malamud

The recent discovery of the equivalence between infinitely wide neural networks (NNs) in the lazy training regime and Neural Tangent Kernels (NTKs) (Jacot et al., 2018) has revived interest in kernel methods. However, conventional wisdom suggests kernel methods are unsuitable for large samples due to their computational complexity and memory requirements. We introduce a novel random feature regression algorithm that allows us (when necessary) to scale to virtually infinite numbers of random features. We illustrate the performance of our method on the CIFAR-10 dataset.

Zhimin Chen, Bryan Kelly, Semyon Malamud

Machine learning (ML) methods are highly flexible, but their ability to approximate the true data-generating process is fundamentally constrained by finite samples. We characterize a universal lower bound, the Limits-to-Learning Gap (LLG), quantifying the unavoidable discrepancy between a model's empirical fit and the population benchmark. Recovering the true population $R^2$, therefore, requires correcting observed predictive performance by this bound. Using a broad set of variables, including excess returns, yields, credit spreads, and valuation ratios, we find that the implied LLGs are large. This indicates that standard ML approaches can substantially understate true predictability in financial data. We also derive LLG-based refinements to the classic Hansen and Jagannathan (1991) bounds, analyze implications for parameter learning in general-equilibrium settings, and show that the LLG provides a natural mechanism for generating excess volatility.