Asymptotically Optimal Regularization in Smooth Parametric Models

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Smoothness, Low Noise and Fast Rates (2010, background): “However, for any fixed d and B, even if d B(2), asymptotically as the number of samples increase, the excess risk of norm-constrained or norm-regularized regression actually behaves as L(ŵ)− L∗ ≈ d nσ (2), and depends (to first order) only on the dimensionality d and not on B [17].”
On the Interaction between Norm and Dimensionality: Multiple Regimes in Learning (2010, background): “On the other hand, classical parametric asymptotic analyses (van der Vaart, 1998; Liang et al., 2010) provide answers that depend only on the dimensionality and not the norm.”
A Stochastic Gradient Method with an Exponential Convergence Rate for Finite Training Sets (2012, background): “In low-dimensional settings, the optimal regularization parameter is of the form C/n Liang et al. [2009] where C is a scalar constant, and may thus violate the constraint.”