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Suppose we have the following first-order non-homogeneous recurrence relation $$z_{t+1} \leq \frac{1}{(1+b_1c_t)^2}\big(\left(1+b_2c_t^2 \right)z_t + b_3c_t^2\big) $$

where $t$ is an integer which varies from 0 to $T$, and $b_1$, $b_2$ and $b_3$ are constants which are greater than 0. In the above equation $z_t = \|w_{t+1} - w^{\star}\|$ i.e. above equation shows a relation about how fast is $w_t$ decreasing and will reach to $w^{\star}$. I am trying that the I wanted to solve for $z_T$ such that convergence of $w$ to $w^{\star}$ is $O(\frac{1}{T})$ which can be also written as- $$z_T \leq \frac{d}{T}z_0 + e $$ where $d$ and $e$ are some constants. Is there a way to bound $c_t, b_1, b_2$ and $b_3$ such that the above equation becomes true? I am not sure how to solve the above equation, any pointers would be really helpful.

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  • $\begingroup$ Do you get to choose $c_t$, $b_1$, $b_2$, and $b_3$? Or are they given? If $c_t$ is constant in time, then the convergence is exponential, so it is not clear why you are considering $O(\frac{1}{T})$ convergence. $\endgroup$ – Matt Apr 19 at 20:13
  • $\begingroup$ I get to choose $c_t$. I was thinking of choosing $c_t$ such that it decreases exponentially in time, but that didn't help. $\endgroup$ – Dushyant Sahoo Apr 19 at 20:16
  • $\begingroup$ So we have a freedom to choose any sequence $\{c_t\}$ of positive reals but then we need to show that for any given positive $z_0$, we have $z_T \leq \frac{d}{T}z_0 + e $, right? $\endgroup$ – Alex Ravsky Apr 25 at 10:51
  • $\begingroup$ Yes, you are right $\endgroup$ – Dushyant Sahoo Apr 25 at 17:58
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    $\begingroup$ It is for all $T \geq 0$ $\endgroup$ – Dushyant Sahoo Apr 27 at 6:44
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The answer is positive. Given $b_i$, for simplicity put $c_t=\tfrac 1{b_1t}$ for each $t>0$. Pick any $t\ge\tfrac{2b_2}{b_1^2}$ and any $A\ge \frac{2b_3}{b_1^2}$ such that $z_t\le\tfrac At$. Now it suffices to show that $z_{t+1}\le\tfrac A{t+1}$ and then conclude by induction. It remains to check that $$\frac A{t+1}\ge \frac 1{\left(1+\tfrac 1t\right)^2}\left(\left(1+\frac {b_2}{b_1^2t^2}\right)\frac At+\frac {b_3}{b_1^2t^2}\right)$$ $$\frac A{t+1}\ge \frac 1{t(1+t)^2}\left(At^2+\frac {b_2A}{b_1^2}+\frac{b_3}{b_1^2}t\right)$$

$$At(t+1)\ge At^2+\frac {b_2A}{b_1^2}+\frac{b_3}{b_1^2}t$$

$$At\ge \frac t2A+\frac A2t \ge \frac {b_2}{b_1^2}A+\frac{b_3}{b_1^2}t,$$

which holds by our choice of $t$ and $A$.

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