# Upper bound via standard manipulation in proof of semi-private learning

I have been reading a paper on private learning [1]. In the proof of lemma 3.3. they claim that $$2\left(\frac{2e n_\text{pub}}{d}\right)^{2d}e^{-\alpha n_\text{pub}/4}$$ is upper bounded by $$\beta$$ via standard manipulation when $$n_\text{pub}=O\left(\frac{d\log(1/\alpha)+\log(1/\beta)}{\alpha}\right)$$, $$\alpha, \beta \in (0, 1)$$, and $$d>0$$ is an integer. This step is essential in proving that an algorithm achieves semi-private learning. However, I do not see why this value is upper bounded by $$\beta$$. Any help as to why this is the case would be greatly appreciated. I have tried setting $$n_\text{pub}$$ to different values that satisfy $$n_\text{pub}=O\left(\frac{d\log(1/\alpha)+\log(1/\beta)}{\alpha}\right)$$, but I still can't seem to make it work.

1. Limits of Private Learning with Access to Public Data by N. Alon et al (2019, 25 Oct)
• Alright @D.W. I have added a better citation of the paper now. Commented Oct 23, 2023 at 11:47