Convergence rate of the smoothing-induced bias term

Determine the asymptotic convergence rate, as the smoothing parameter s_n tends to infinity, of the bias term \overline{\theta}_{\mathrm{sig}} − \overline{\theta} arising from the sigmoid-smoothed moment function m_{\mathrm{sig}}(W,\boldsymbol{\gamma}) in the debiased machine learning estimator, for specified distributions of the treatment effect difference \overline{\tau}(X), and characterize how this rate depends on the behavior of \overline{\tau}(X) in a neighborhood of zero.

Background

The paper introduces a sigmoid-based smoothing of indicator functions to enable debiased machine learning inference when nuisance parameters appear inside indicators. This smoothing induces a bias term because the indicator is approximated by a smooth function.

Proposition 2 expresses \overline{\theta}_{\mathrm{sig}} − \overline{\theta} as an integral involving the logistic density and the distribution of the treatment effect difference \overline{\tau}(X), highlighting that the magnitude and rate of this bias depend on the distribution of \overline{\tau}(X), especially near the cutoff at zero. The authors note that, even with knowledge of the distribution of \overline{\tau}(X), the precise convergence rate of the bias term remains unclear.

Because the optimal choice of the smoothing parameter s_n in Theorem 1 balances bias and variance via a worst-case bound under a margin assumption, a sharp characterization of the convergence rate of \overline{\theta}_{\mathrm{sig}} − \overline{\theta} would directly inform tuning and potentially yield tighter inference.