[SciPy-Dev] curve_fit() should require initial values for parameters

[Matt Newville]

Hi All,

First, I apologize in advance if this sounds un-appreciative of the efforts
made in scipy and scipy.optimize.  I am a very big fan, and very
appreciative of the work done here.  With lmfit we have tried to take the
"rough edges" from optimization and curve-fitting with python, but we're
very much in favor of building wrappers on top of the core of
scipy.optimize.

Still, many people use `curve_fit` because it comes built-in with scipy, is
well advertised, and is well-suited for simple uses.  It is clearly aimed
at novices and tries to hide many of the complexities of using optimization
routines for curve fitting.  I try to help people with questions about
using `curve_fit` and `scipy.optimize` as well as `lmfit`.  In trying to
help people using `curve_fit`, I've noticed a disturbing trend.  When a
novice or even experienced user asks for help using `curve_fit` because a
fit "doesn't work" there is a very good chance that they did not specify
`p0` for the initial values and that the default behavior of setting all
starting values to 1.0 will cause their fit to fail to converge, or really
to even start.

This failure is understandable to an experienced user, but apparently not
to the novice.  Curve-fitting problems are by nature local solvers and are
always sensitive to initial values. For some problems, parameter values of
1.0 are simply inappropriate, and the numerical derivatives for some
parameters near values of 1.0 will be 0.  Indeed, there is no value that is
always a reasonable starting value for all parameters.  FWIW, in lmfit, we
simply refuse to let a user run a curve-fitting problem without initial
values.  I believe that most other curve-fitting interfaces also require
initial values for all parameters.

Unfortunately, faced with no initial parameter values, `curve_fit` silently
chooses initial values.  It doesn't try to make an informed decision, it
simply chooses '1.0', which can easily be so far off as to prevent a
solution from being found. When this happens, `curve_fit` gives no
information to the user of what the problem is.  Indeed it allows initial
values to not be set, giving the impression that they are not important.
This impression is wrong: initial values are always important. `curve_fit`
is mistaken in having a default starting value.

I've made a Pull Request at https://github.com/scipy/scipy/pull/9701 to fix
this misbehavior, so that `curve_fit()` requires starting values.  It was
suggested there that this topic should be discussed here. I'm happy to do
so.  It was suggested in the github Issue that forcing the user to give
initial values was "annoying".  It was also suggested that a deprecation
cycle would be required. I should say that I don't actually use
`curve_fit()` myself, I'm just trying to help make this commonly used
routine be completely wrong less often.

Cheers,

--Matt Newville
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