Gradient success/failure flag

[wupeifan]

Currently, the solution object will return a flag indicating whether the solution computations are successful or not. If not, for B-K and other linear algebra related errors, the exact error message will be returned; otherwise just a general failure flag. This flag is used in the downstream sampling, as the log-posterior will be set automatically as -Inf when the solution doesn't return a success sign. But there are cases that the solutions are fine and the gradients are not. With the current patch in the code, it skips the gradient computation step.

I think what should be done is to add the gradient failure flag to the solutions and check it downstream in sampling.

[jlperla]

So I think that we should basically take https://github.com/HighDimensionalEconLab/DifferentiableStateSpaceModels.jl/blob/main/src/generate_perturbation_derivatives.jl#L306-L309 and https://github.com/HighDimensionalEconLab/DifferentiableStateSpaceModels.jl/blob/main/src/generate_perturbation_derivatives.jl#L387-L391

and make them something like

    sol = generate_perturbation(m, p_d, p_f, Val(2); cache, settings)
    if (sol.retcode == :Success)
        grad_ret = generate_perturbation_derivatives!(m, p_d, p_f, cache)
        if grad_ret != :Success
             sol = FirstOrderPerturbationSolution(:Failure, m, cache)
       end
    end

or something like that. If this causes some type-stability issues then you could instead do it at the bottom, and then it is https://github.com/HighDimensionalEconLab/DifferentiableStateSpaceModels.jl/blob/main/src/generate_perturbation_derivatives.jl#L378 becomes

return (grad_ret == :Success ? sol : FirstOrderPerturbationSolution(:Failure, m, cache)),  generate_perturbation_pb

or something like that. And if this doesn't have type stability I think there is something else we could try later.

[wupeifan]

Right. I totally forget that sol is just fetching elements from cache.

[wupeifan]

@jlperla I'll change the :failure to :GradientFailure