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Found 3 papers, 2 papers with code
Asymptotic behaviour of learning rates in Armijo's condition
This complements the first author's results on Unbounded Backtracking GD, and shows that in case of convergence to a non-degenerate critical point the behaviour of Unbounded Backtracking GD is not too different from that of usual Backtracking GD.
A fast and simple modification of Newton's method helping to avoid saddle points
The main result of this paper roughly says that if $f$ is $C^3$ (can be unbounded from below) and a sequence $\{x_n\}$, constructed by the New Q-Newton's method from a random initial point $x_0$, {\bf converges}, then the limit point is a critical point and is not a saddle point, and the convergence rate is the same as that of Newton's method.
Backtracking gradient descent method for general $C^1$ functions, with applications to Deep Learning
Then either $\lim _{n\rightarrow\infty}||z_n||=\infty$ or $\{z_n\}$ converges to a critical point of $f$.
