Our main result concerns the following condition: {\bf Condition C.} Let $X$ be a Banach space. A $C^1$ function $f:X\rightarrow \mathbb{R}$ satisfies Condition C if whenever $\{x_n\}$ weakly converges to $x$ and $\lim _{n\rightarrow\infty}||\nabla f(x_n)||=0$, then $\nabla f(x)=0$. We assume that there is given a canonical isomorphism between $X$ and its dual $X^*$, for example when $X$ is a Hilbert space. {\bf Theorem.} Let $X$ be a reflexive, complete Banach space and $f:X\rightarrow \mathbb{R}$ be a $C^2$ function which satisfies Condition C. Moreover, we assume that for every bounded set $S\subset X$, then $\sup _{x\in S}||\nabla ^2f(x)||<\infty$. We choose a random point $x_0\in X$ and construct by the Local Backtracking GD procedure (which depends on $3$ hyper-parameters $\alpha ,\beta ,\delta _0$, see later for details) the sequence $x_{n+1}=x_n-\delta (x_n)\nabla f(x_n)$. Then we have: 1) Every cluster point of $\{x_n\}$, in the {\bf weak} topology, is a critical point of $f$. 2) Either $\lim _{n\rightarrow\infty}f(x_n)=-\infty$ or $\lim _{n\rightarrow\infty}||x_{n+1}-x_n||=0$. 3) Here we work with the weak topology. Let $\mathcal{C}$ be the set of critical points of $f$. Assume that $\mathcal{C}$ has a bounded component $A$. Let $\mathcal{B}$ be the set of cluster points of $\{x_n\}$. If $\mathcal{B}\cap A\not= \emptyset$, then $\mathcal{B}\subset A$ and $\mathcal{B}$ is connected. 4) Assume that $X$ is separable. Then for generic choices of $\alpha ,\beta ,\delta _0$ and the initial point $x_0$, if the sequence $\{x_n\}$ converges - in the {\bf weak} topology, then the limit point cannot be a saddle point.

Let $z=(x,y)$ be coordinates for the product space $\mathbb{R}^{m_1}\times \mathbb{R}^{m_2}$. Let $f:\mathbb{R}^{m_1}\times \mathbb{R}^{m_2}\rightarrow \mathbb{R}$ be a $C^1$ function, and $\nabla f=(\partial _xf,\partial _yf)$ its gradient. Fix $0<\alpha <1$. For a point $(x,y) \in \mathbb{R}^{m_1}\times \mathbb{R}^{m_2}$, a number $\delta >0$ satisfies Armijo's condition at $(x,y)$ if the following inequality holds: \begin{eqnarray*} f(x-\delta \partial _xf,y-\delta \partial _yf)-f(x,y)\leq -\alpha \delta (||\partial _xf||^2+||\partial _yf||^2). \end{eqnarray*} When $f(x,y)=f_1(x)+f_2(y)$ is a coordinate-wise sum map, we propose the following {\bf coordinate-wise} Armijo's condition. Fix again $0<\alpha <1$. A pair of positive numbers $\delta _1,\delta _2>0$ satisfies the coordinate-wise variant of Armijo's condition at $(x,y)$ if the following inequality holds: \begin{eqnarray*} [f_1(x-\delta _1\nabla f_1(x))+f_2(y-\delta _2\nabla f_2(y))]-[f_1(x)+f_2(y)]\leq -\alpha (\delta _1||\nabla f_1(x)||^2+\delta _2||\nabla f_2(y)||^2). \end{eqnarray*} We then extend results in our recent previous results, on Backtracking Gradient Descent and some variants, to this setting. We show by an example the advantage of using coordinate-wise Armijo's condition over the usual Armijo's condition.

The main result of this paper is: {\bf Theorem.} Let $f:\mathbb{R}^k\rightarrow \mathbb{R}$ be a $C^{1}$ function, so that $\nabla f$ is locally Lipschitz continuous. Assume moreover that $f$ is $C^2$ near its generalised saddle points. Fix real numbers $\delta_0>0$ and $0<\alpha <1$. Then there is a smooth function $h:\mathbb{R}^k\rightarrow (0,\delta_0]$ so that the map $H:\mathbb{R}^k\rightarrow \mathbb{R}^k$ defined by $H(x)=x-h(x)\nabla f(x)$ has the following property: (i) For all $x\in \mathbb{R}^k$, we have $f(H(x)))-f(x)\leq -\alpha h(x)||\nabla f(x)||^2$. (ii) For every $x_0\in \mathbb{R}^k$, the sequence $x_{n+1}=H(x_n)$ either satisfies $\lim_{n\rightarrow\infty}||x_{n+1}-x_n||=0$ or $ \lim_{n\rightarrow\infty}||x_n||=\infty$. Each cluster point of $\{x_n\}$ is a critical point of $f$. If moreover $f$ has at most countably many critical points, then $\{x_n\}$ either converges to a critical point of $f$ or $\lim_{n\rightarrow\infty}||x_n||=\infty$. (iii) There is a set $\mathcal{E}_1\subset \mathbb{R}^k$ of Lebesgue measure $0$ so that for all $x_0\in \mathbb{R}^k\backslash \mathcal{E}_1$, the sequence $x_{n+1}=H(x_n)$, {\bf if converges}, cannot converge to a {\bf generalised} saddle point. (iv) There is a set $\mathcal{E}_2\subset \mathbb{R}^k$ of Lebesgue measure $0$ so that for all $x_0\in \mathbb{R}^k\backslash \mathcal{E}_2$, any cluster point of the sequence $x_{n+1}=H(x_n)$ is not a saddle point, and more generally cannot be an isolated generalised saddle point. Some other results are proven.