I want to say that "the value decreases at a rate at least x" (i.e. faster than or equal to x). Does the phrase "the value decreases at minimum rate x" mean the same thing? If not, is there any other way to rephrase it?
Minimum means at least, so "the value decreases at minimum rate x" can mean what you want. However, I'd advise against it, since it could also be understood to say "the value decreases at the minimum rate, which is x", i.e. that it never decreases any slower or faster than x.
Therefore, I suggest:
The minimum rate at which the value decreases is x.
This unequivocably means that the rate of decrease is at least x, but possibly more.
An undesired interpretation is likely to occur for "The value decreases at minimum rate x": "The value decreases as slowly as it can, which is rate x". I'd avoid the word minimum and instead say, "The value decreases at a rate no less than x".