The exponential mechanism is a technique for designing differentially private algorithms. It was developed by Frank McSherry[1] and Kunal Talwar[2] in 2007. Their work was recognized as a co-winner of the 2009 PET Award for Outstanding Research in Privacy Enhancing Technologies.[3]
Most of the initial research in the field of differential privacy revolved around real-valued functions which have relatively low sensitivity to change in the data of a single individual and whose usefulness is not hampered by small additive perturbations. A natural question is what happens in the situation when one wants to preserve more general sets of properties. The exponential mechanism helps to extend the notion of differential privacy to address these issues. Moreover, it describes a class of mechanisms that includes all possible differentially private mechanisms.
In very generic terms, a privacy mechanism maps a set of inputs from domain to a range . The map may be randomized, in which case each element of the domain corresponds to a probability distribution over the range . The privacy mechanism makes no assumption about the nature of and apart from a base measure on . Let us define a function . Intuitively this function assigns a score to the pair , where and . The score reflects the appeal of the pair , i.e. the higher the score, the more appealing the pair is.
Given the input , the mechanism's objective is to return an such that the function is approximately maximized. To achieve this, set up the mechanism as follows: Definition: For any function , and a base measure over , define:
Choose with probability proportional to , where .
This definition implies the fact that the probability of returning an increases exponentially with the increase in the value of . Ignoring the base measure then the value which maximizes has the highest probability. Moreover, this mechanism is differentially private. Proof of this claim will follow. One technicality that should be kept in mind is that in order to properly define the should be finite.
Now, if a single change in changes by at most then the numerator can change at most by a factor of and the denominator minimum by a factor of . Thus, the ratio of the new probability density (i.e. with new ) and the earlier one is at most .
We would ideally want the random draws of from the mechanism to nearly maximize . If we consider to be then we can show that the probability of the mechanism deviating from is low, as long as there is a sufficient mass (in terms of ) of values with value close to the optimum.
Lemma: Let and , we have is at most . The probability is taken over .
Proof: The probability is at most , as the denominator can be at most one. Since both the probabilities have the same normalizing term so,
The value of is at most one, and so this bound implies the lemma statement.
Theorem (Accuracy): For those values of , we have .
Proof: It follows from the previous lemma that the probability of the score being at least is . By hypothesis, . Substituting the value of we get this probability to be at least . Multiplying with yields the desired bound.
We can assume for to be less than or equal to one in all the computations, because we can always normalize with .
Definition (Usefulness): A mechanism[permanent dead link] is -useful for queries in class with probability , if and every dataset , for , .
Informally, it means that with high probability the query will behave in a similar way on the original dataset and on the synthetic dataset .
Consider a common problem in Data Mining. Assume there is a database with entries. Each entry consist of -tuples of the form where . Now, a user wants to learn a linear halfspace of the form . In essence the user wants to figure out the values of such that maximum number of tuples in the database satisfy the inequality. The algorithm we describe below can generate a synthetic database which will allow the user to learn (approximately) the same linear half-space while querying on this synthetic database. The motivation for such an algorithm being that the new database will be generated in a differentially private manner and thus assure privacy to the individual records in the database .
In this section we show that it is possible to release a dataset which is useful for concepts from a polynomial VC-Dimension class and at the same time adhere to -differential privacy as long as the size of the original dataset is at least polynomial on the VC-Dimension of the concept class. To state formally:
Theorem: For any class of functions and any dataset such that
we can output an -useful dataset that preserves -differential privacy. As we had mentioned earlier the algorithm need not be efficient.
One interesting fact is that the algorithm which we are going to develop generates a synthetic dataset whose size is independent of the original dataset; in fact, it only depends on the VC-dimension of the concept class and the parameter . The algorithm outputs a dataset of size
Lemma: Given any dataset there exists a dataset of size such that .
Proof:
We know from the uniform convergence theorem that
where probability is over the distribution of the dataset.
Thus, if the RHS is less than one then we know for sure that the data set exists. To bound the RHS to less than one we need , where is some positive constant. Since we stated earlier that we will output a dataset of size , so using this bound on we get . Hence the lemma.
Now we invoke the exponential mechanism.
Definition: For any function and input dataset , the exponential mechanism outputs each dataset with probability proportional to .
From the exponential mechanism we know this preserves -differential privacy. Let's get back to the proof of the Theorem.
We define .
To show that the mechanism satisfies the -usefulness, we should show that it outputs some dataset with with probability .
There are at most output datasets and the probability that is at most proportional to . Thus by union bound, the probability of outputting any such dataset is at most proportional to .
Again, we know that there exists some dataset for which . Therefore, such a dataset is output with probability at least proportional to .
Let the event that the exponential mechanism outputs some dataset such that .
the event that the exponential mechanism outputs some dataset such that .
Now setting this quantity to be at least , we find that it suffices to have
In the above example of the usage of exponential mechanism, one can output a synthetic dataset in a differentially private manner and can use the dataset to answer queries with good accuracy. Other private mechanisms, such as posterior sampling,[6] which returns parameters rather than datasets, can be made equivalent to the exponential one.[7]
Apart from the setting of privacy, the exponential mechanism has also been studied in the context of auction theory and classification algorithms.[8] In the case of auctions the exponential mechanism helps to achieve a truthful auction setting.
^Shiva Prasad Kasiviswanathan, Homin K. Lee, Kobbi Nissim, Sofya Raskhodnikova, Adam Smith. What Can We Learn Privately? Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer Science. arXiv:0803.0924
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