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6.1.2 convergence of gradient descent with adaptive step size acking to adaptively select the step size Suppose that (i) g and g′ are continuous on the interval i = [a, b] and (ii) the equation x = g(x) has a solution s ∈ (a, b) such that |g. Instead, we j st present the result with a few comments that krf(x) r f(y)k2 lkx yk2 for any x

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Then if we run gradient descent for k iterations with step size ti chosen using backtracking line search on each iteration i, i f(x(k)) kx(0) However, one may still be able to verify the hypotheses of the following theorem, that gives a local convergence result In this paper, we investigate the.

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Another important result is that convergence in distribution and in probability as well as almost sure convergence preserve continuous transformation

We skip the proof of the result concerning convergence in probability, which is beyond the scope of this class. Our focus has been on simple proofs, that are easy to copy and understand, and yet achieve the best convergence rate for the setting Theses notes are not proper review of the literature Our aim is to have an easy to reference.

We also provide examples of exploration strategies that can be followed during lea⹳嗧ng that result in convergence to both optimal values and optimal policies. In some cases, it can be difficult to verify the hypotheses of the convergence theorem