Simulated annealing for deep learning: Why is gradient free statistical learning not in the main stream? In order to define what deep learning is, the learning portion is often listed with backpropagation as a requirement without alternatives in the main stream software libraries and in the literature. There are not many gradient free optimisations are mentioned in deep learning or in general statistical learning. Similarly, in "classical algorithms" (Nonlinear least squares) involves derivatives [1]. In general, gradient free learning in deep learning or classical algorithms are not in the main stream. One promising alternative is simulated annealing [2, 3], so-called 'nature-inspired optimization'.
Is there any inherent theoretical reason that why gradient free deep learning (statistical learning) is not in the main stream? (Or not preferred?)
Notes
[1] Such as Levenberg–Marquardt
[2] Simulated Annealing Algorithm for Deep Learning (2015)
[3] CoolMomentum: a method for stochastic optimization by Langevin dynamics with simulated annealing (2021) Though this is still not fully gradient-free, but does not require auto-differentiation.
Edit 1
Additional references using Ensemble Kalman Filter, showing a derivative free approach:

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*Ensemble Kalman Inversion: A Derivative-Free Technique For Machine Learning Tasks arXiv:1808.03620.

*Ensemble Kalman Filter optimizing Deep Neural Networks: An alternative approach to non-performing Gradient Descent springer (manuscript-pdf)

Edit 2
As far as I gather, Yann LeCun does not consider gradient-free learning as part of deep learning ecosystem. "DL is constructing networks of parameterized functional modules & training them from examples using gradient-based optimization." tweet
Edit 3
Ben Bolker's comment on local geometry definitely deserves to be one of the answers.

The reason we don't use gradient-free methods for training neural nets is simple: gradient-free methods don't work as well as gradient-based methods.
Gradient-based methods converge faster, to better solutions.  Gradient-free methods tend to scale poorly (for instance, one of the papers you cite only tests on MNIST, which is a tiny dataset and task; the other tests on CIFAR-10 but is a gradient-based method) and tend to yield inferior results (for instance, one of the papers you cite reports 97% accuracy on MNIST; but state-of-the-art accuracy on MNIST is well over 99%).
In general, when a gradient is available and the loss surface is not too messy, typically gradient-based methods work better than gradient-free methods.  Gradient-free methods are useful when it is not easy to compute the gradient or when the loss function is not very smooth.
