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Added by
Ilya Mironov
Publication date
2018
fields
Informatics Engineering
and research's language is
English
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Many commonly used learning algorithms work by iteratively updating an intermediate solution using one or a few data points in each iteration. Analysis of differential privacy for such algorithms often involves ensuring privacy of each step and then reasoning about the cumulative privacy cost of the algorithm. This is enabled by composition theorems for differential privacy that allow releasing of all the intermediate results. In this work, we demonstrate that for contractive iterations, not releasing the intermediate results strongly amplifies the privacy guarantees. We describe several applications of this new analysis technique to solving convex optimization problems via noisy stochastic gradient descent. For example, we demonstrate that a relatively small number of non-private data points from the same distribution can be used to close the gap between private and non-private convex optimization. In addition, we demonstrate that we can achieve guarantees similar to those obtainable using the privacy-amplification-by-sampling technique in several natural settings where that technique cannot be applied.
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Sensitive statistics are often collected across sets of users, with repeated collection of reports done over time. For example, trends in users private preferences or software usage may be monitored via such reports. We study the collection of such statistics in the local differential privacy (LDP) model, and describe an algorithm whose privacy cost is polylogarithmic in the number of changes to a users value. More fundamentally---by building on anonymity of the users reports---we also demonstrate how the privacy cost of our LDP algorithm can actually be much lower when viewed in the central model of differential privacy. We show, via a new and general privacy amplification technique, that any permutation-invariant algorithm satisfying $varepsilon$-local differential privacy will satisfy $(O(varepsilon sqrt{log(1/delta)/n}), delta)$-central differential privacy. By this, we explain how the high noise and $sqrt{n}$ overhead of LDP protocols is a consequence of them being significantly more private in the central model. As a practical corollary, our results imply that several LDP-based industrial deployments may have much lower privacy cost than their advertised $varepsilon$ would indicate---at least if reports are anonymized.
Recent work of Erlingsson, Feldman, Mironov, Raghunathan, Talwar, and Thakurta [EFMRTT19] demonstrates that random shuffling amplifies differential privacy guarantees of locally randomized data. Such amplification implies substantially stronger privacy guarantees for systems in which data is contributed anonymously [BEMMRLRKTS17] and has lead to significant interest in the shuffle model of privacy [CSUZZ19; EFMRTT19]. We show that random shuffling of $n$ data records that are input to $varepsilon_0$-differentially private local randomizers results in an $(O((1-e^{-varepsilon_0})sqrt{frac{e^{varepsilon_0}log(1/delta)}{n}}), delta)$-differentially private algorithm. This significantly improves over previous work and achieves the asymptotically optimal dependence in $varepsilon_0$. Our result is based on a new approach that is simpler than previous work and extends to approximate differential privacy with nearly the same guarantees. Importantly, our work also yields an algorithm for deriving tighter bounds on the resulting $varepsilon$ and $delta$ as well as Renyi differential privacy guarantees. We show numerically that our algorithm gets to within a small constant factor of the optimal bound. As a direct corollary of our analysis we derive a simple and nearly optimal algorithm for frequency estimation in the shuffle model of privacy. We also observe that our result implies the first asymptotically optimal privacy analysis of noisy stochastic gradient descent that applies to sampling without replacement.
We develop theory for using heuristics to solve computationally hard problems in differential privacy. Heuristic approaches have enjoyed tremendous success in machine learning, for which performance can be empirically evaluated. However, privacy guarantees cannot be evaluated empirically, and must be proven --- without making heuristic assumptions. We show that learning problems over broad classes of functions can be solved privately and efficiently, assuming the existence of a non-private oracle for solving the same problem. Our first algorithm yields a privacy guarantee that is contingent on the correctness of the oracle. We then give a reduction which applies to a class of heuristics which we call certifiable, which allows us to convert oracle-dependent privacy guarantees to worst-case privacy guarantee that hold even when the heuristic standing in for the oracle might fail in adversarial ways. Finally, we consider a broad class of functions that includes most classes of simple boolean functions studied in the PAC learning literature, including conjunctions, disjunctions, parities, and discrete halfspaces. We show that there is an efficient algorithm for privately constructing synthetic data for any such class, given a non-private learning oracle. This in particular gives the first oracle-efficient algorithm for privately generating synthetic data for contingency tables. The most intriguing question left open by our work is whether or not every problem that can be solved differentially privately can be privately solved with an oracle-efficient algorithm. While we do not resolve this, we give a barrier result that suggests that any generic oracle-efficient reduction must fall outside of a natural class of algorithms (which includes the algorithms given in this paper).
We design differentially private algorithms for the bandit convex optimization problem in the projection-free setting. This setting is important whenever the decision set has a complex geometry, and access to it is done efficiently only through a linear optimization oracle, hence Euclidean projections are unavailable (e.g. matroid polytope, submodular base polytope). This is the first differentially-private algorithm for projection-free bandit optimization, and in fact our bound of $widetilde{O}(T^{3/4})$ matches the best known non-private projection-free algorithm (Garber-Kretzu, AISTATS `20) and the best known private algorithm, even for the weaker setting when projections are available (Smith-Thakurta, NeurIPS `13).
Much of the literature on differential privacy focuses on item-level privacy, where loosely speaking, the goal is to provide privacy per item or training example. However, recently many practical applications such as federated learning require preserving privacy for all items of a single user, which is much harder to achieve. Therefore understanding the theoretical limit of user-level privacy becomes crucial. We study the fundamental problem of learning discrete distributions over $k$ symbols with user-level differential privacy. If each user has $m$ samples, we show that straightforward applications of Laplace or Gaussian mechanisms require the number of users to be $mathcal{O}(k/(malpha^2) + k/epsilonalpha)$ to achieve an $ell_1$ distance of $alpha$ between the true and estimated distributions, with the privacy-induced penalty $k/epsilonalpha$ independent of the number of samples per user $m$. Moreover, we show that any mechanism that only operates on the final aggregate counts should require a user complexity of the same order. We then propose a mechanism such that the number of users scales as $tilde{mathcal{O}}(k/(malpha^2) + k/sqrt{m}epsilonalpha)$ and hence the privacy penalty is $tilde{Theta}(sqrt{m})$ times smaller compared to the standard mechanisms in certain settings of interest. We further show that the proposed mechanism is nearly-optimal under certain regimes. We also propose general techniques for obtaining lower bounds on restricted differentially private estimators and a lower bound on the total variation between binomial distributions, both of which might be of independent interest.
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