Tuesday October 19, 2021
RAPPOR (“Randomized Aggregatable Privacy-Preserving Ordinal Response”) is really close to the classic coin-flip randomized response method, with one more layer of randomness. It's “ordinal” in that it's every bit for itself: you can represent histogram bins, but you can't really represent numbers in the usual binary way. They additionally use Bloom filters to represent sets of hashable things, which is neat too.
Randomized response works like this. Did you wash your hands?
- 25% of the time, just say “yes” regardless
- 25% of the time, just say “no” regardless
- 50% of the time, answer truthfully
Those percentages correspond to \( f = 0.5 \) in the RAPPOR paper's notation. They call this the “permanent” step, because it's done once per respondent.
def permanent_randomized(response: bool, f=0.5): randomizer = random.random() # uniform on [0, 1) if 0 <= randomizer < f/2: # probability f/2 return True elif f/2 <= randomizer < f: # probability f/2 return False elif f <= randomizer < 1: # probability 1 - f return response
If you get 59% positive responses with this method, 25% were just automatically saying “yes” and 34% were answering “yes” because they were being honest. Half of people were being honest, so double the 34% to find the overall true positive rate is 68%.
To let individuals report multiple times without losing privacy or being trackable, RAPPOR adds “instantaneous” randomness as well.
def instantaneous_randomized(response: bool, q=0.75, p=0.5): randomizer = random.random() # uniform on [0, 1) if response is True: return randomizer < q # probability q elif response is False: return randomizer < p # probability p
With the complete method, recovering the true rate gets a little more complicated; see first equation in the paper's §4.
def extract_rate(raw_rate: float, f=0.5, q=0.75, p=0.5): return (raw_rate - p - f*q/2 + f*p/2) / ((1 - f)*(q - p))
And it works!
true_rate = 0.68 n = 1_000_000 raw_rate = sum( instantaneous_randomized( permanent_randomized(random.random() < true_rate)) for _ in range(n)) / n extract_rate(raw_rate) ## 0.6807759999999998
RAPPOR is just doing this for any number of bits per respondent. Each bit can have a direct interpretation like “this thing yes for this respondent.”
RAPPOR also introduces a way of having respondents hash strings (say) into Bloom filters, which is kind of neat. They show how to have different groups of respondents use different hash functions and then do some regression to try to overcome noise and false positives. As far as I can tell, you have to already know the set of things you're going to be hashing into the Bloom filter (you won't discover an unknown unknown) but it is more efficient and flexible than defining a separate bit for every candidate string.
RAPPOR is used in Chrome, for example, and if you have really a lot of respondents, it seems pretty good!