Perhaps your mental laziness is an unconcious way in which your mind is telling you that you are not in the right track. Since you grew up in a high pressure it seems that now you need an opportunity to live, for some time, under low pressure conditions in order to obtain a new vision about what to do in the future.
The general case require some work and conditions. But to give a hint, the case of only one parameter is an application of the mean value theorem (1).
Suppose a model (y = f(p,x) ) with only one parameter p0 and an exact point (x0,y0) (that is y0=f(p0,x)) and a data point (x0,y1) such that
y1-y0=error in the data. And that there is a value p1 of the parameter such that f(p1,x0) = y1, then y1 - y0 = f(p1,x0) - f(p0,x0) = f'(sigma) . (p1-p0), so that
p1-p0 = (y1-y0)/f'(sigma) that is (error in the parameter) = (error in the data)/(derivative with respect to the parameter) where sigma is between p0 and p1. The general case is a generalization of this idea using the mean value inequality.
I suppose that the s-curve with 3 parameters that the author is talking about is the logistic function. In general, if you consider a differentiable function of three parameters and try to determine and interval for the values of the parameters of that model then the length of that interval is bound by the ratio of the error in the data over the derivative with respect the parameter. For example estimating the parameter k (wikipedia logistic growth rate) with points such that x near x0 = (wikipedia midpoint of the sigmoid) is hard, since the derivative of the function with respect to k at x=x0 is zero. So mathematically this seems to be a well known fact when one try to estimate parameters from datapoints.
Hello, thanks for the book. It seems we'll have to learn RUTILS. Anyway in machine learning and other fields python and java have huge libraries. I think the algorithms part is the best since lisp libraries can't compete with other libraries like the more than 8000 packages for R. Don't know what happened to clasp (C++ Lisp), and other gnu scientific libraries, ffi. Other languages and libraries are moving fast (Nim, Julia, Cristal, Kotlin, Rust), but lisp macros are still strong. I would like to predict a bright future for Lisp but I don't see anyway it can compete with all the other options.
Edited: googling I found someone recommended the book the slight edge, and a review is here that contains the main ideas: https://www.amazon.es/Slight-Edge-Jeff-Olson/dp/1626340463#c..., look for Daisy's review.