Add mean_tweedie_deviance and particular cases

[0urobor0s]

This PR also adds the particular cases:

• mean_poisson_deviance
• mean_gamma_deviance

And updates mean_square_error to use mean_tweedie_deviance as well.

[msluszniak]

We cannot use runtime checks, because they would fail when function would be called in another defn

[josevalim]

You can test this by calling Nx.Defn.jit(&mean_tweedie_deviance/3).(y_true, y_pred, 1). You will see that jit returns an expression and you can't convert it to number.

[0urobor0s]

Do you mean this:

iex> Nx.Defn.jit(&mean_tweedie_deviance/3).(y_true, y_pred, 1)
** (Protocol.UndefinedError) protocol String.Chars not implemented for #Nx.Tensor<
  s64

  Nx.Defn.Expr
  parameter a:2   s64
> of type Nx.Tensor (a struct). This protocol is implemented for the following type(s): Atom, BitString, Complex, Date, DateTime, Float, Hex.Solver.Assignment, Hex.Solver.Constraints.Empty, Hex.Solver.Constraints.Range, Hex.Solver.Constraints.Union, Hex.Solver.Incompatibility, Hex.Solver.PackageRange, Hex.Solver.Term, Integer, List, NaiveDateTime, Time, URI, Version, Version.Requirement
    (elixir 1.14.5) lib/string/chars.ex:3: String.Chars.impl_for!/1
    (elixir 1.14.5) lib/string/chars.ex:22: String.Chars.to_string/1
    (scholar 0.2.1) lib/scholar/metrics/regression.ex:161: Scholar.Metrics.Regression.mean_tweedie_deviance/3
    (nx 0.7.0-dev) lib/nx/defn/compiler.ex:158: Nx.Defn.Compiler.runtime_fun/3
    (nx 0.7.0-dev) lib/nx/defn/evaluator.ex:82: Nx.Defn.Evaluator.precompile/3
    (nx 0.7.0-dev) lib/nx/defn/evaluator.ex:60: Nx.Defn.Evaluator.__compile__/4
    (nx 0.7.0-dev) lib/nx/defn/evaluator.ex:53: Nx.Defn.Evaluator.__jit__/5
    iex:4: (file)

I thought that by using the deftransform, it could be done. That assumption from reading the docs, was the ideia behing the check_tweedie_deviance_power returning a tensor "error value". But from what I understand now, what you mean is that this happens when a defn function calls the mean_tweedie_deviance defined with deftransform.

If so, do you propose any idea/solution for it. In this case, the values need to be checked in order to guarantee that the results "can be calculated" as to have probabilistic meaning.

PS: For the alternative, would it be feasible to return tensor of different shapes depending on the error?

[josevalim]

You need to imagine that defn is running inside the GPU. Inside the GPU you can't have different shapes and you can't have exceptions. The best you can do is return NaN to signal that something went wrong OR have a separate function, that is a regular def, that receives a tensor to check it. In this case, the best is to assume you receive valid inputs and, if not, the value is unspecified.

[0urobor0s]

I think having the checks might be useful, at least for sanity check. I've added an option to allow for that check to happen in a def function, which by default is false (no check is made). Which means that by default, if the incorrect inputs for the given power value are passed to these functions, there is no guarantee of a result (with some NaNs appearing). Let me know if this can be done like this, or if you have any other concerns.

[0urobor0s]

I was following the opts approach, but if possible I agree. In that case would mean_tweedie_deviance! be a def function?

[josevalim]

Yeah, it would be a def!

[0urobor0s]

Should a note be added, for mean_tweedie_deviance!, as in Nx.to_number/1?
Note: This function cannot be used in defn.

[josevalim]

That would be great, yeah!

[0urobor0s]

Ok, added it as well.

[josevalim]

We can remove this clause as it is the same as the last?

[msluszniak]

Actually, they are not the same. In the first one, there is max(y_true,0)

[josevalim]

I think I liked your previous implementation more, where you did the checking in Nx, and returned an integer. :)

[josevalim]

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