Hasan Fallahgoul

Hasan Fallahgoul

Recent advances in machine learning have shown promising results for financial prediction using large, over-parameterized models. This paper provides theoretical foundations and empirical validation for understanding when and how these methods achieve predictive success. I examine two key aspects of high-dimensional learning in finance. First, I prove that within-sample standardization in Random Fourier Features implementations fundamentally alters the underlying Gaussian kernel approximation, replacing shift-invariant kernels with training-set dependent alternatives. Second, I establish information-theoretic lower bounds that identify when reliable learning is impossible no matter how sophisticated the estimator. A detailed quantitative calibration of the polynomial lower bound shows that with typical parameter choices, e.g., 12,000 features, 12 monthly observations, and R-square 2-3%, the required sample size to escape the bound exceeds 25-30 years of data--well beyond any rolling-window actually used. Thus, observed out-of-sample success must originate from lower-complexity artefacts rather than from the intended high-dimensional mechanism.

Hasan Fallahgoul

This paper develops a scale-insensitive framework for neural network significance testing, substantially generalizing existing approaches through three key innovations. First, we replace metric entropy calculations with Rademacher complexity bounds, enabling the analysis of neural networks without requiring bounded weights or specific architectural constraints. Second, we weaken the regularity conditions on the target function to require only Sobolev space membership $H^s([-1,1]^d)$ with $s > d/2$, significantly relaxing previous smoothness assumptions while maintaining optimal approximation rates. Third, we introduce a modified sieve space construction based on moment bounds rather than weight constraints, providing a more natural theoretical framework for modern deep learning practices. Our approach achieves these generalizations while preserving optimal convergence rates and establishing valid asymptotic distributions for test statistics. The technical foundation combines localization theory, sharp concentration inequalities, and scale-insensitive complexity measures to handle unbounded weights and general Lipschitz activation functions. This framework better aligns theoretical guarantees with contemporary deep learning practice while maintaining mathematical rigor.