arXiv subjects

Kobbi Nissim

Publications and source records attributed to Kobbi Nissim.

52 records · Page 3Linked to original sources

arXiv2017

Private Incremental Regression

Data is continuously generated by modern data sources, and a recent challenge in machine learning has been to develop techniques that perform well in an incremental (streaming) setting. In this paper, we investigate the problem of private machine learning, where as common in practice, the data is not given at once, but rather arrives incrementally over time. We introduce the problems of private incremental ERM and private incremental regression where the general goal is to always maintain a good empirical risk minimizer for the history observed under differential privacy. Our first contribution is a generic transformation of private batch ERM mechanisms into private incremental ERM mechanisms, based on a simple idea of invoking the private batch ERM procedure at some regular time intervals. We take this construction as a baseline for comparison. We then provide two mechanisms for the private incremental regression problem. Our first mechanism is based on privately constructing a noisy incremental gradient function, which is then used in a modified projected gradient procedure at every timestep. This mechanism has an excess empirical risk of $\approx\sqrt{d}$, where $d$ is the dimensionality of the data. While from the results of [Bassily et al. 2014] this bound is tight in the worst-case, we show that certain geometric properties of the input and constraint set can be used to derive significantly better results for certain interesting regression problems.

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arXiv2016

Adaptive Learning with Robust Generalization Guarantees

The traditional notion of generalization---i.e., learning a hypothesis whose empirical error is close to its true error---is surprisingly brittle. As has recently been noted in [DFH+15b], even if several algorithms have this guarantee in isolation, the guarantee need not hold if the algorithms are composed adaptively. In this paper, we study three notions of generalization---increasing in strength---that are robust to postprocessing and amenable to adaptive composition, and examine the relationships between them. We call the weakest such notion Robust Generalization. A second, intermediate, notion is the stability guarantee known as differential privacy. The strongest guarantee we consider we call Perfect Generalization. We prove that every hypothesis class that is PAC learnable is also PAC learnable in a robustly generalizing fashion, with almost the same sample complexity. It was previously known that differentially private algorithms satisfy robust generalization. In this paper, we show that robust generalization is a strictly weaker concept, and that there is a learning task that can be carried out subject to robust generalization guarantees, yet cannot be carried out subject to differential privacy. We also show that perfect generalization is a strictly stronger guarantee than differential privacy, but that, nevertheless, many learning tasks can be carried out subject to the guarantees of perfect generalization.

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arXiv2015

Simultaneous Private Learning of Multiple Concepts

We investigate the direct-sum problem in the context of differentially private PAC learning: What is the sample complexity of solving $k$ learning tasks simultaneously under differential privacy, and how does this cost compare to that of solving $k$ learning tasks without privacy? In our setting, an individual example consists of a domain element $x$ labeled by $k$ unknown concepts $(c_1,\ldots,c_k)$. The goal of a multi-learner is to output $k$ hypotheses $(h_1,\ldots,h_k)$ that generalize the input examples. Without concern for privacy, the sample complexity needed to simultaneously learn $k$ concepts is essentially the same as needed for learning a single concept. Under differential privacy, the basic strategy of learning each hypothesis independently yields sample complexity that grows polynomially with $k$. For some concept classes, we give multi-learners that require fewer samples than the basic strategy. Unfortunately, however, we also give lower bounds showing that even for very simple concept classes, the sample cost of private multi-learning must grow polynomially in $k$.

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arXiv2015

On the Generalization Properties of Differential Privacy

A new line of work, started with Dwork et al., studies the task of answering statistical queries using a sample and relates the problem to the concept of differential privacy. By the Hoeffding bound, a sample of size $O(\log k/α^2)$ suffices to answer $k$ non-adaptive queries within error $α$, where the answers are computed by evaluating the statistical queries on the sample. This argument fails when the queries are chosen adaptively (and can hence depend on the sample). Dwork et al. showed that if the answers are computed with $(ε,δ)$-differential privacy then $O(ε)$ accuracy is guaranteed with probability $1-O(δ^ε)$. Using the Private Multiplicative Weights mechanism, they concluded that the sample size can still grow polylogarithmically with the $k$. Very recently, Bassily et al. presented an improved bound and showed that (a variant of) the private multiplicative weights algorithm can answer $k$ adaptively chosen statistical queries using sample complexity that grows logarithmically in $k$. However, their results no longer hold for every differentially private algorithm, and require modifying the private multiplicative weights algorithm in order to obtain their high probability bounds. We greatly simplify the results of Dwork et al. and improve on the bound by showing that differential privacy guarantees $O(ε)$ accuracy with probability $1-O(δ\log(1/ε)/ε)$. It would be tempting to guess that an $(ε,δ)$-differentially private computation should guarantee $O(ε)$ accuracy with probability $1-O(δ)$. However, we show that this is not the case, and that our bound is tight (up to logarithmic factors).

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arXiv2015

Algorithmic Stability for Adaptive Data Analysis

Adaptivity is an important feature of data analysis---the choice of questions to ask about a dataset often depends on previous interactions with the same dataset. However, statistical validity is typically studied in a nonadaptive model, where all questions are specified before the dataset is drawn. Recent work by Dwork et al. (STOC, 2015) and Hardt and Ullman (FOCS, 2014) initiated the formal study of this problem, and gave the first upper and lower bounds on the achievable generalization error for adaptive data analysis. Specifically, suppose there is an unknown distribution $\mathbf{P}$ and a set of $n$ independent samples $\mathbf{x}$ is drawn from $\mathbf{P}$. We seek an algorithm that, given $\mathbf{x}$ as input, accurately answers a sequence of adaptively chosen queries about the unknown distribution $\mathbf{P}$. How many samples $n$ must we draw from the distribution, as a function of the type of queries, the number of queries, and the desired level of accuracy? In this work we make two new contributions: (i) We give upper bounds on the number of samples $n$ that are needed to answer statistical queries. The bounds improve and simplify the work of Dwork et al. (STOC, 2015), and have been applied in subsequent work by those authors (Science, 2015, NIPS, 2015). (ii) We prove the first upper bounds on the number of samples required to answer more general families of queries. These include arbitrary low-sensitivity queries and an important class of optimization queries. As in Dwork et al., our algorithms are based on a connection with algorithmic stability in the form of differential privacy. We extend their work by giving a quantitatively optimal, more general, and simpler proof of their main theorem that stability implies low generalization error. We also study weaker stability guarantees such as bounded KL divergence and total variation distance.

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arXiv2015

Learning Privately with Labeled and Unlabeled Examples

A private learner is an algorithm that given a sample of labeled individual examples outputs a generalizing hypothesis while preserving the privacy of each individual. In 2008, Kasiviswanathan et al. (FOCS 2008) gave a generic construction of private learners, in which the sample complexity is (generally) higher than what is needed for non-private learners. This gap in the sample complexity was then further studied in several followup papers, showing that (at least in some cases) this gap is unavoidable. Moreover, those papers considered ways to overcome the gap, by relaxing either the privacy or the learning guarantees of the learner. We suggest an alternative approach, inspired by the (non-private) models of semi-supervised learning and active-learning, where the focus is on the sample complexity of labeled examples whereas unlabeled examples are of a significantly lower cost. We consider private semi-supervised learners that operate on a random sample, where only a (hopefully small) portion of this sample is labeled. The learners have no control over which of the sample elements are labeled. Our main result is that the labeled sample complexity of private learners is characterized by the VC dimension. We present two generic constructions of private semi-supervised learners. The first construction is of learners where the labeled sample complexity is proportional to the VC dimension of the concept class, however, the unlabeled sample complexity of the algorithm is as big as the representation length of domain elements. Our second construction presents a new technique for decreasing the labeled sample complexity of a given private learner, while roughly maintaining its unlabeled sample complexity. In addition, we show that in some settings the labeled sample complexity does not depend on the privacy parameters of the learner.

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arXiv2014

Private Learning and Sanitization: Pure vs. Approximate Differential Privacy

We compare the sample complexity of private learning [Kasiviswanathan et al. 2008] and sanitization~[Blum et al. 2008] under pure $ε$-differential privacy [Dwork et al. TCC 2006] and approximate $(ε,δ)$-differential privacy [Dwork et al. Eurocrypt 2006]. We show that the sample complexity of these tasks under approximate differential privacy can be significantly lower than that under pure differential privacy. We define a family of optimization problems, which we call Quasi-Concave Promise Problems, that generalizes some of our considered tasks. We observe that a quasi-concave promise problem can be privately approximated using a solution to a smaller instance of a quasi-concave promise problem. This allows us to construct an efficient recursive algorithm solving such problems privately. Specifically, we construct private learners for point functions, threshold functions, and axis-aligned rectangles in high dimension. Similarly, we construct sanitizers for point functions and threshold functions. We also examine the sample complexity of label-private learners, a relaxation of private learning where the learner is required to only protect the privacy of the labels in the sample. We show that the VC dimension completely characterizes the sample complexity of such learners, that is, the sample complexity of learning with label privacy is equal (up to constants) to learning without privacy.

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arXiv2014

Characterizing the Sample Complexity of Private Learners

In 2008, Kasiviswanathan et al. defined private learning as a combination of PAC learning and differential privacy. Informally, a private learner is applied to a collection of labeled individual information and outputs a hypothesis while preserving the privacy of each individual. Kasiviswanathan et al. gave a generic construction of private learners for (finite) concept classes, with sample complexity logarithmic in the size of the concept class. This sample complexity is higher than what is needed for non-private learners, hence leaving open the possibility that the sample complexity of private learning may be sometimes significantly higher than that of non-private learning. We give a combinatorial characterization of the sample size sufficient and necessary to privately learn a class of concepts. This characterization is analogous to the well known characterization of the sample complexity of non-private learning in terms of the VC dimension of the concept class. We introduce the notion of probabilistic representation of a concept class, and our new complexity measure RepDim corresponds to the size of the smallest probabilistic representation of the concept class. We show that any private learning algorithm for a concept class C with sample complexity m implies RepDim(C)=O(m), and that there exists a private learning algorithm with sample complexity m=O(RepDim(C)). We further demonstrate that a similar characterization holds for the database size needed for privately computing a large class of optimization problems and also for the well studied problem of private data release.

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arXiv2014

Redrawing the Boundaries on Purchasing Data from Privacy-Sensitive Individuals

We prove new positive and negative results concerning the existence of truthful and individually rational mechanisms for purchasing private data from individuals with unbounded and sensitive privacy preferences. We strengthen the impossibility results of Ghosh and Roth (EC 2011) by extending it to a much wider class of privacy valuations. In particular, these include privacy valuations that are based on (ε, δ)-differentially private mechanisms for non-zero δ, ones where the privacy costs are measured in a per-database manner (rather than taking the worst case), and ones that do not depend on the payments made to players (which might not be observable to an adversary). To bypass this impossibility result, we study a natural special setting where individuals have mono- tonic privacy valuations, which captures common contexts where certain values for private data are expected to lead to higher valuations for privacy (e.g. having a particular disease). We give new mech- anisms that are individually rational for all players with monotonic privacy valuations, truthful for all players whose privacy valuations are not too large, and accurate if there are not too many players with too-large privacy valuations. We also prove matching lower bounds showing that in some respects our mechanism cannot be improved significantly.

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arXiv2012

Privacy-Aware Mechanism Design

In traditional mechanism design, agents only care about the utility they derive from the outcome of the mechanism. We look at a richer model where agents also assign non-negative dis-utility to the information about their private types leaked by the outcome of the mechanism. We present a new model for privacy-aware mechanism design, where we only assume an upper bound on the agents' loss due to leakage, as opposed to previous work where a full characterization of the loss was required. In this model, under a mild assumption on the distribution of how agents value their privacy, we show a generic construction of privacy-aware mechanisms and demonstrate its applicability to electronic polling and pricing of a digital good.

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arXiv2011

Approximately Optimal Mechanism Design via Differential Privacy

In this paper we study the implementation challenge in an abstract interdependent values model and an arbitrary objective function. We design a mechanism that allows for approximate optimal implementation of insensitive objective functions in ex-post Nash equilibrium. If, furthermore, values are private then the same mechanism is strategy proof. We cast our results onto two specific models: pricing and facility location. The mechanism we design is optimal up to an additive factor of the order of magnitude of one over the square root of the number of agents and involves no utility transfers. Underlying our mechanism is a lottery between two auxiliary mechanisms: with high probability we actuate a mechanism that reduces players' influence on the choice of the social alternative, while choosing the optimal outcome with high probability. This is where the recent notion of differential privacy is employed. With the complementary probability we actuate a mechanism that is typically far from optimal but is incentive compatible. The joint mechanism inherits the desired properties from both.

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arXiv2011

Distributed Private Data Analysis: On Simultaneously Solving How and What

We examine the combination of two directions in the field of privacy concerning computations over distributed private inputs - secure function evaluation (SFE) and differential privacy. While in both the goal is to privately evaluate some function of the individual inputs, the privacy requirements are significantly different. The general feasibility results for SFE suggest a natural paradigm for implementing differentially private analyses distributively: First choose what to compute, i.e., a differentially private analysis; Then decide how to compute it, i.e., construct an SFE protocol for this analysis. We initiate an examination whether there are advantages to a paradigm where both decisions are made simultaneously. In particular, we investigate under which accuracy requirements it is beneficial to adapt this paradigm for computing a collection of functions including binary sum, gap threshold, and approximate median queries. Our results imply that when computing the binary sum of $n$ distributed inputs then: * When we require that the error is $o(\sqrt{n})$ and the number of rounds is constant, there is no benefit in the new paradigm. * When we allow an error of $O(\sqrt{n})$, the new paradigm yields more efficient protocols when we consider protocols that compute symmetric functions. Our results also yield new separations between the local and global models of computations for private data analysis.

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arXiv2010

Impossibility of Differentially Private Universally Optimal Mechanisms

The notion of a universally utility-maximizing privacy mechanism was recently introduced by Ghosh, Roughgarden, and Sundararajan [STOC 2009]. These are mechanisms that guarantee optimal utility to a large class of information consumers, simultaneously, while preserving Differential Privacy [Dwork, McSherry, Nissim, and Smith, TCC 2006]. Ghosh et al. have demonstrated, quite surprisingly, a case where such a universally-optimal differentially-private mechanisms exists, when the information consumers are Bayesian. This result was recently extended by Gupte and Sundararajan [PODS 2010] to risk-averse consumers. Both positive results deal with mechanisms (approximately) computing a single count query (i.e., the number of individuals satisfying a specific property in a given population), and the starting point of our work is a trial at extending these results to similar settings, such as sum queries with non-binary individual values, histograms, and two (or more) count queries. We show, however, that universally-optimal mechanisms do not exist for all these queries, both for Bayesian and risk-averse consumers. For the Bayesian case, we go further, and give a characterization of those functions that admit universally-optimal mechanisms, showing that a universally-optimal mechanism exists, essentially, only for a (single) count query. At the heart of our proof is a representation of a query function $f$ by its privacy constraint graph $G_f$ whose edges correspond to values resulting by applying $f$ to neighboring databases.

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arXiv2010

What Can We Learn Privately?

Learning problems form an important category of computational tasks that generalizes many of the computations researchers apply to large real-life data sets. We ask: what concept classes can be learned privately, namely, by an algorithm whose output does not depend too heavily on any one input or specific training example? More precisely, we investigate learning algorithms that satisfy differential privacy, a notion that provides strong confidentiality guarantees in contexts where aggregate information is released about a database containing sensitive information about individuals. We demonstrate that, ignoring computational constraints, it is possible to privately agnostically learn any concept class using a sample size approximately logarithmic in the cardinality of the concept class. Therefore, almost anything learnable is learnable privately: specifically, if a concept class is learnable by a (non-private) algorithm with polynomial sample complexity and output size, then it can be learned privately using a polynomial number of samples. We also present a computationally efficient private PAC learner for the class of parity functions. Local (or randomized response) algorithms are a practical class of private algorithms that have received extensive investigation. We provide a precise characterization of local private learning algorithms. We show that a concept class is learnable by a local algorithm if and only if it is learnable in the statistical query (SQ) model. Finally, we present a separation between the power of interactive and noninteractive local learning algorithms.

cs.LG↗

arXiv2006

Hard Instances of the Constrained Discrete Logarithm Problem

The discrete logarithm problem (DLP) generalizes to the constrained DLP, where the secret exponent $x$ belongs to a set known to the attacker. The complexity of generic algorithms for solving the constrained DLP depends on the choice of the set. Motivated by cryptographic applications, we study sets with succinct representation for which the constrained DLP is hard. We draw on earlier results due to Erdös et al. and Schnorr, develop geometric tools such as generalized Menelaus' theorem for proving lower bounds on the complexity of the constrained DLP, and construct sets with succinct representation with provable non-trivial lower bounds.

math.NT↗

arXiv2001

Communication Complexity and Secure Function Evaluation

We suggest two new methodologies for the design of efficient secure protocols, that differ with respect to their underlying computational models. In one methodology we utilize the communication complexity tree (or branching for f and transform it into a secure protocol. In other words, "any function f that can be computed using communication complexity c can be can be computed securely using communication complexity that is polynomial in c and a security parameter". The second methodology uses the circuit computing f, enhanced with look-up tables as its underlying computational model. It is possible to simulate any RAM machine in this model with polylogarithmic blowup. Hence it is possible to start with a computation of f on a RAM machine and transform it into a secure protocol. We show many applications of these new methodologies resulting in protocols efficient either in communication or in computation. In particular, we exemplify a protocol for the "millionaires problem", where two participants want to compare their values but reveal no other information. Our protocol is more efficient than previously known ones in either communication or computation.

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