# The meaning of with probability at least 1-\delta

machine learningoptimizationprobabilitystatistics

In the theoretical analysis of some algorithm for stochastic optimization, we often need to prove that something like
$$error\leq\epsilon,~~~~(under~some~conditions)$$
holds with probability at least $$1-\delta$$. But I found that many works use the specific number like $$\delta = 1/4$$ in their result even if the more general conclusion for arbitrary $$\delta\in(0, 1/2]$$ has been made. I was wondering that is there any particular meaning assigned to this specific number.

As long as $$1-\delta>1/2$$ can be achieved, then repeatedly apply the algorithm, and by the law of large numbers, we can make the probability of error $$\le \epsilon$$ as large as possible, so it's not very important what $$\delta$$ is as long as $$\delta<1/2$$. (Although sometimes, it's interesting to optimize $$\delta$$ for specific $$\epsilon$$.)
This is really universal in TCS. For example in the definition of BPP, the number of $$1/3$$ and $$2/3$$ can really be replaced by $$\epsilon$$ and $$1-\epsilon$$ for any $$\epsilon<1/2$$.