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تسجيل مستخدم جديدSome convergent results for Backtracking Gradient Descent method on Banach spaces
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Tuyen Trung Truong
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Our main result concerns the following condition: {bf Condition C.} Let $X$ be a Banach space. A $C^1$ function $f:Xrightarrow mathbb{R}$ satisfies Condition C if whenever ${x_n}$ weakly converges to $x$ and $lim _{nrightarrowinfty}|| abla f(x_n)||=0$, then $ abla 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 $f:Xrightarrow mathbb{R}$ be a $C^2$ function which satisfies Condition C. Moreover, we assume that for every bounded set $Ssubset X$, then $sup _{xin S}|| abla ^2f(x)||<infty$. We choose a random point $x_0in X$ and construct by the Local Backtracking GD procedure (which depends on $3$ hyper-parameters $alpha ,beta ,delta _0$, see later for details) the sequence $x_{n+1}=x_n-delta (x_n) abla f(x_n)$. Then we have: 1) Every cluster point of ${x_n}$, in the {bf weak} topology, is a critical point of $f$. 2) Either $lim _{nrightarrowinfty}f(x_n)=-infty$ or $lim _{nrightarrowinfty}||x_{n+1}-x_n||=0$. 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 $mathcal{B}cap A ot= emptyset$, 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.
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