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I am studying Mirror descent and nonlinear projected subgradient methods. At page 171 Theorem 4.1., the author claims that the method converges provided $$ \sum_s t_s= \infty , \,\,\,t_k \rightarrow 0 \,\,\,\,\,\text{as} \,\,\,\ k \rightarrow \infty $$ beacuse the right hand side of the following goes to zero:

$$ \min_{1\leq s \leq k} f(x^s) - \min_{x \in X} f(x) \leq \frac{B_{\psi(x^*,x^1)}+(2\sigma)^{-1}\sum_{s=1}^kt_s^2\|f'(x^k)\|_*^2}{\sum_{k=1}^s t_s} $$

My question is that how we know $\sum_{s=1}^kt_s^2\|f'(x^k)\|_*^2$ is bounded provided aforementioned assumption? Although $t_k \rightarrow 0$ does not guarantee that $\sum_{s=1}^kt_s^2\|f'(x^k)\|_*^2$ is bounded.

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Section (b) of Assumption A states that $f$ is $L_f$-Lipschitz. A standard result of optimization asserts that if $f'(x)\in \partial f(x)$, then $\|f'(x)\|_*\leq L_f$ where $\|\cdot\|_*$ denotes the dual norm of $\|\cdot\|$.

Thus $\displaystyle \frac{\sum_{k=1}^n t_k^2\|f'(x^k)\|_*^2}{\sum_{k=1}^n t_k}\leq L_f\frac{\sum_{k=1}^n t_k^2}{\sum_{k=1}^n t_k}$.

Let us show that $\displaystyle \frac{\sum_{k=1}^n t_k^2}{\sum_{k=1}^n t_k} \to 0$. Remember that the $t_n$ are $\geq 0$.

Let $\epsilon >0$. There exists $N$ such that $n\geq N\implies t_n\leq \epsilon$. For $n\geq N$, $$\sum_{k=1}^n t_k^2\leq \sum_{k=1}^N t_k^2 + \epsilon \sum_{k=N+1}^n t_k$$ The sequence $\displaystyle \epsilon \sum_{k=N+1}^n t_k$ (indexed by $n$) diverges to $\infty$, so there exists some $N'>N$ such that $$n\geq N' \implies \sum_{k=1}^N t_k^2\leq\epsilon \sum_{k=N+1}^n t_k $$ For $n\geq N'$, $$\sum_{k=1}^n t_k^2\leq 2\epsilon \sum_{k=N+1}^n t_k\leq 2\epsilon \sum_{k=1}^n t_k$$ and we're done.

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  • $\begingroup$ Thank you so much. It was very instructive. $\endgroup$ – Saeed May 17 at 23:37
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You don't have that $\sum_{s=1}^k t_s^2 \lVert f'(x^k)\rVert^2$ is bounded. However, for any series $\sum_{s=1}^\infty a_s$ that diverges, but the summands converge to zero, you have that $$ \frac{\sum_{s=1}^k a_s^2}{\sum_{s=1}^k a_s} \to 0, \quad \text{for $k\to\infty$}. $$ You can find this for example here.

So in fact, it doesn't really have anything to do with optimization (and just as a comment: it's not really relevant for the paper).

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    $\begingroup$ I do not have $\frac{\sum_{s=1}^k a_s^2}{\sum_{s=1}^k a_s}$, I have $\frac{\sum_{s=1}^k a_s^2b_k^2}{\sum_{s=1}^k a_s}$ where $b_k^2$ is the dual norm of the gradient. How would you handle that? $\endgroup$ – Saeed May 16 at 2:29
  • $\begingroup$ Also, on the cited question, we have have sum of the $a_n$ at the denominator. $\endgroup$ – Saeed May 16 at 2:34
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    $\begingroup$ The gradient is bounded though so you effectively do have the series they posted... $\endgroup$ – Tony S.F. May 16 at 8:17

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