How to find the perfect regularization number?
Approaches can be any of the following:


Model the noise that you expect, for ex: gaussian (least squares in Minimum variance unbiased estimator for a linear signal model in presence of gaussian noise). Based on this model try and estimate the noise variance, the regularization term should be close to noise varainace.
Deploy machine learning techniques based on signal parameters such as dynamic range, frqeuency, FFT noise floor etc. to estimate noise
Follow a recursive approach with adjusting the regularization to minimize a suitable objective, ex: gradient descent etc.


In general your regularization is as good as how close it is to the actual noise or estimation of noise. This goes back to the idea of whether a frequentist approach is better or Bayesian approach. The answers is if the modelling of the parameter is accurate to it's actual behaviour then the Bayesian approach is better but if you model a parameter with a distribution but it is actually not following that closely then Bayesian would lead to suboptimal result and better go for a frequentist approach.