Gautam Kamath

Florian Tramèr, Gautam Kamath, Nicholas Carlini · eth-zurich

The performance of differentially private machine learning can be boosted significantly by leveraging the transfer learning capabilities of non-private models pretrained on large public datasets. We critically review this approach. We primarily question whether the use of large Web-scraped datasets should be viewed as differential-privacy-preserving. We caution that publicizing these models pretrained on Web data as "private" could lead to harm and erode the public's trust in differential privacy as a meaningful definition of privacy. Beyond the privacy considerations of using public data, we further question the utility of this paradigm. We scrutinize whether existing machine learning benchmarks are appropriate for measuring the ability of pretrained models to generalize to sensitive domains, which may be poorly represented in public Web data. Finally, we notice that pretraining has been especially impactful for the largest available models -- models sufficiently large to prohibit end users running them on their own devices. Thus, deploying such models today could be a net loss for privacy, as it would require (private) data to be outsourced to a more compute-powerful third party. We conclude by discussing potential paths forward for the field of private learning, as public pretraining becomes more popular and powerful.

Alex Bie, Gautam Kamath, Guojun Zhang

We show that the canonical approach for training differentially private GANs -- updating the discriminator with differentially private stochastic gradient descent (DPSGD) -- can yield significantly improved results after modifications to training. Specifically, we propose that existing instantiations of this approach neglect to consider how adding noise only to discriminator updates inhibits discriminator training, disrupting the balance between the generator and discriminator necessary for successful GAN training. We show that a simple fix -- taking more discriminator steps between generator steps -- restores parity between the generator and discriminator and improves results. Additionally, with the goal of restoring parity, we experiment with other modifications -- namely, large batch sizes and adaptive discriminator update frequency -- to improve discriminator training and see further improvements in generation quality. Our results demonstrate that on standard image synthesis benchmarks, DPSGD outperforms all alternative GAN privatization schemes. Code: https://github.com/alexbie98/dpgan-revisit.

Da Yu, Gautam Kamath, Janardhan Kulkarni et al.

Differentially private stochastic gradient descent (DP-SGD) is the workhorse algorithm for recent advances in private deep learning. It provides a single privacy guarantee to all datapoints in the dataset. We propose output-specific $(\varepsilon,δ)$-DP to characterize privacy guarantees for individual examples when releasing models trained by DP-SGD. We also design an efficient algorithm to investigate individual privacy across a number of datasets. We find that most examples enjoy stronger privacy guarantees than the worst-case bound. We further discover that the training loss and the privacy parameter of an example are well-correlated. This implies groups that are underserved in terms of model utility simultaneously experience weaker privacy guarantees. For example, on CIFAR-10, the average $\varepsilon$ of the class with the lowest test accuracy is 44.2\% higher than that of the class with the highest accuracy.

Jimmy Z. Di, Jack Douglas, Jayadev Acharya et al.

We introduce camouflaged data poisoning attacks, a new attack vector that arises in the context of machine unlearning and other settings when model retraining may be induced. An adversary first adds a few carefully crafted points to the training dataset such that the impact on the model's predictions is minimal. The adversary subsequently triggers a request to remove a subset of the introduced points at which point the attack is unleashed and the model's predictions are negatively affected. In particular, we consider clean-label targeted attacks (in which the goal is to cause the model to misclassify a specific test point) on datasets including CIFAR-10, Imagenette, and Imagewoof. This attack is realized by constructing camouflage datapoints that mask the effect of a poisoned dataset.

Xin Gu, Gautam Kamath, Zhiwei Steven Wu

Differentially private stochastic gradient descent privatizes model training by injecting noise into each iteration, where the noise magnitude increases with the number of model parameters. Recent works suggest that we can reduce the noise by leveraging public data for private machine learning, by projecting gradients onto a subspace prescribed by the public data. However, given a choice of public datasets, it is not a priori clear which one may be most appropriate for the private task. We give an algorithm for selecting a public dataset by measuring a low-dimensional subspace distance between gradients of the public and private examples. We provide theoretical analysis demonstrating that the excess risk scales with this subspace distance. This distance is easy to compute and robust to modifications in the setting. Empirical evaluation shows that trained model accuracy is monotone in this distance.

Alex Bie, Gautam Kamath, Vikrant Singhal

We initiate the study of differentially private (DP) estimation with access to a small amount of public data. For private estimation of d-dimensional Gaussians, we assume that the public data comes from a Gaussian that may have vanishing similarity in total variation distance with the underlying Gaussian of the private data. We show that under the constraints of pure or concentrated DP, d+1 public data samples are sufficient to remove any dependence on the range parameters of the private data distribution from the private sample complexity, which is known to be otherwise necessary without public data. For separated Gaussian mixtures, we assume that the underlying public and private distributions are the same, and we consider two settings: (1) when given a dimension-independent amount of public data, the private sample complexity can be improved polynomially in terms of the number of mixture components, and any dependence on the range parameters of the distribution can be removed in the approximate DP case; (2) when given an amount of public data linear in the dimension, the private sample complexity can be made independent of range parameters even under concentrated DP, and additional improvements can be made to the overall sample complexity.

Yiwei Lu, Gautam Kamath, Yaoliang Yu

Data poisoning attacks, in which a malicious adversary aims to influence a model by injecting "poisoned" data into the training process, have attracted significant recent attention. In this work, we take a closer look at existing poisoning attacks and connect them with old and new algorithms for solving sequential Stackelberg games. By choosing an appropriate loss function for the attacker and optimizing with algorithms that exploit second-order information, we design poisoning attacks that are effective on neural networks. We present efficient implementations that exploit modern auto-differentiation packages and allow simultaneous and coordinated generation of tens of thousands of poisoned points, in contrast to existing methods that generate poisoned points one by one. We further perform extensive experiments that empirically explore the effect of data poisoning attacks on deep neural networks.

Yiwei Lu, Gautam Kamath, Yaoliang Yu

Indiscriminate data poisoning attacks aim to decrease a model's test accuracy by injecting a small amount of corrupted training data. Despite significant interest, existing attacks remain relatively ineffective against modern machine learning (ML) architectures. In this work, we introduce the notion of model poisoning reachability as a technical tool to explore the intrinsic limits of data poisoning attacks towards target parameters (i.e., model-targeted attacks). We derive an easily computable threshold to establish and quantify a surprising phase transition phenomenon among popular ML models: data poisoning attacks can achieve certain target parameters only when the poisoning ratio exceeds our threshold. Building on existing parameter corruption attacks and refining the Gradient Canceling attack, we perform extensive experiments to confirm our theoretical findings, test the predictability of our transition threshold, and significantly improve existing indiscriminate data poisoning baselines over a range of datasets and models. Our work highlights the critical role played by the poisoning ratio, and sheds new insights on existing empirical results, attacks and mitigation strategies in data poisoning.

Shai Ben-David, Alex Bie, Clément L. Canonne et al.

We study the problem of private distribution learning with access to public data. In this setup, which we refer to as public-private learning, the learner is given public and private samples drawn from an unknown distribution $p$ belonging to a class $\mathcal Q$, with the goal of outputting an estimate of $p$ while adhering to privacy constraints (here, pure differential privacy) only with respect to the private samples. We show that the public-private learnability of a class $\mathcal Q$ is connected to the existence of a sample compression scheme for $\mathcal Q$, as well as to an intermediate notion we refer to as list learning. Leveraging this connection: (1) approximately recovers previous results on Gaussians over $\mathbb R^d$; and (2) leads to new ones, including sample complexity upper bounds for arbitrary $k$-mixtures of Gaussians over $\mathbb R^d$, results for agnostic and distribution-shift resistant learners, as well as closure properties for public-private learnability under taking mixtures and products of distributions. Finally, via the connection to list learning, we show that for Gaussians in $\mathbb R^d$, at least $d$ public samples are necessary for private learnability, which is close to the known upper bound of $d+1$ public samples.

Jie Zhang, Debeshee Das, Gautam Kamath et al.

We consider the problem of a training data proof, where a data creator or owner wants to demonstrate to a third party that some machine learning model was trained on their data. Training data proofs play a key role in recent lawsuits against foundation models trained on web-scale data. Many prior works suggest to instantiate training data proofs using membership inference attacks. We argue that this approach is fundamentally unsound: to provide convincing evidence, the data creator needs to demonstrate that their attack has a low false positive rate, i.e., that the attack's output is unlikely under the null hypothesis that the model was not trained on the target data. Yet, sampling from this null hypothesis is impossible, as we do not know the exact contents of the training set, nor can we (efficiently) retrain a large foundation model. We conclude by offering two paths forward, by showing that data extraction attacks and membership inference on special canary data can be used to create sound training data proofs.

Mark Bun, Gautam Kamath, Argyris Mouzakis et al.

We give an example of a class of distributions that is learnable up to constant error in total variation distance with a finite number of samples, but not learnable under $(\varepsilon, δ)$-differential privacy with the same target error. This weakly refutes a conjecture of Ashtiani.

Pranav Subramani, Nicholas Vadivelu, Gautam Kamath

A common pain point in differentially private machine learning is the significant runtime overhead incurred when executing Differentially Private Stochastic Gradient Descent (DPSGD), which may be as large as two orders of magnitude. We thoroughly demonstrate that by exploiting powerful language primitives, including vectorization, just-in-time compilation, and static graph optimization, one can dramatically reduce these overheads, in many cases nearly matching the best non-private running times. These gains are realized in two frameworks: JAX and TensorFlow. JAX provides rich support for these primitives as core features of the language through the XLA compiler. We also rebuild core parts of TensorFlow Privacy, integrating features from TensorFlow 2 as well as XLA compilation, granting significant memory and runtime improvements over the current release version. These approaches allow us to achieve up to 50x speedups in comparison to the best alternatives. Our code is available at https://github.com/TheSalon/fast-dpsgd.

Tosca Lechner, Alex Bie, Gautam Kamath

We consider the question of learnability of distribution classes in the presence of adaptive adversaries -- that is, adversaries capable of intercepting the samples requested by a learner and applying manipulations with full knowledge of the samples before passing it on to the learner. This stands in contrast to oblivious adversaries, who can only modify the underlying distribution the samples come from but not their i.i.d.\ nature. We formulate a general notion of learnability with respect to adaptive adversaries, taking into account the budget of the adversary. We show that learnability with respect to additive adaptive adversaries is a strictly stronger condition than learnability with respect to additive oblivious adversaries.

Valentio Iverson, Gautam Kamath, Argyris Mouzakis

We provide the first $\widetilde{\mathcal{O}}\left(d\right)$-sample algorithm for sampling from unbounded Gaussian distributions under the constraint of $\left(\varepsilon, δ\right)$-differential privacy. This is a quadratic improvement over previous results for the same problem, settling an open question of Ghazi, Hu, Kumar, and Manurangsi.

Gautam Kamath, Alireza F. Pour, Matthew Regehr et al.

We propose an algorithm with improved query-complexity for the problem of hypothesis selection under local differential privacy constraints. Given a set of $k$ probability distributions $Q$, we describe an algorithm that satisfies local differential privacy, performs $\tilde{O}(k^{3/2})$ non-adaptive queries to individuals who each have samples from a probability distribution $p$, and outputs a probability distribution from the set $Q$ which is nearly the closest to $p$. Previous algorithms required either $Ω(k^2)$ queries or many rounds of interactive queries. Technically, we introduce a new object we dub the Scheffé graph, which captures structure of the differences between distributions in $Q$, and may be of more broad interest for hypothesis selection tasks.

William Xu, Yiwei Lu, Yihan Wang et al.

Targeted data poisoning attacks pose an increasingly serious threat due to their ease of deployment and high success rates. These attacks aim to manipulate the prediction for a single test sample in classification models. Unlike indiscriminate attacks that aim to decrease overall test performance, targeted attacks present a unique threat to individual test instances. This threat model raises a fundamental question: what factors make certain test samples more susceptible to successful poisoning than others? We investigate how attack difficulty varies across different test instances and identify key characteristics that influence vulnerability. This paper introduces three predictive criteria for targeted data poisoning difficulty: ergodic prediction accuracy (analyzed through clean training dynamics), poison distance, and poison budget. Our experimental results demonstrate that these metrics effectively predict the varying difficulty of real-world targeted poisoning attacks across diverse scenarios, offering practitioners valuable insights for vulnerability assessment and understanding data poisoning attacks.

Yihan Wang, Yiwei Lu, Xiao-Shan Gao et al.

Availability attacks, or unlearnable examples, are defensive techniques that allow data owners to modify their datasets in ways that prevent unauthorized machine learning models from learning effectively while maintaining the data's intended functionality. It has led to the release of popular black-box tools (e.g., APIs) for users to upload personal data and receive protected counterparts. In this work, we show that such black-box protections can be substantially compromised if a small set of unprotected in-distribution data is available. Specifically, we propose a novel threat model of protection leakage, where an adversary can (1) easily acquire (unprotected, protected) pairs by querying the black-box protections with a small unprotected dataset; and (2) train a diffusion bridge model to build a mapping between unprotected and protected data. This mapping, termed BridgePure, can effectively remove the protection from any previously unseen data within the same distribution. BridgePure demonstrates superior purification performance on classification and style mimicry tasks, exposing critical vulnerabilities in black-box data protection. We suggest that practitioners implement multi-level countermeasures to mitigate such risks.

Shai Ben-David, Alex Bie, Gautam Kamath et al.

We examine the relationship between learnability and robust (or agnostic) learnability for the problem of distribution learning. We show that, contrary to other learning settings (e.g., PAC learning of function classes), realizable learnability of a class of probability distributions does not imply its agnostic learnability. We go on to examine what type of data corruption can disrupt the learnability of a distribution class and what is such learnability robust against. We show that realizable learnability of a class of distributions implies its robust learnability with respect to only additive corruption, but not against subtractive corruption. We also explore related implications in the context of compression schemes and differentially private learnability.

Martin Pawelczyk, Jimmy Z. Di, Yiwei Lu et al.

We revisit the efficacy of several practical methods for approximate machine unlearning developed for large-scale deep learning. In addition to complying with data deletion requests, one often-cited potential application for unlearning methods is to remove the effects of poisoned data. We experimentally demonstrate that, while existing unlearning methods have been demonstrated to be effective in a number of settings, they fail to remove the effects of data poisoning across a variety of types of poisoning attacks (indiscriminate, targeted, and a newly-introduced Gaussian poisoning attack) and models (image classifiers and LLMs); even when granted a relatively large compute budget. In order to precisely characterize unlearning efficacy, we introduce new evaluation metrics for unlearning based on data poisoning. Our results suggest that a broader perspective, including a wider variety of evaluations, are required to avoid a false sense of confidence in machine unlearning procedures for deep learning without provable guarantees. Moreover, while unlearning methods show some signs of being useful to efficiently remove poisoned data without having to retrain, our work suggests that these methods are not yet ``ready for prime time,'' and currently provide limited benefit over retraining.

Wenxin Ding, Gautam Kamath, Weina Wang et al.

Reviewers in peer review are often miscalibrated: they may be strict, lenient, extreme, moderate, etc. A number of algorithms have previously been proposed to calibrate reviews. Such attempts of calibration can however leak sensitive information about which reviewer reviewed which paper. In this paper, we identify this problem of calibration with privacy, and provide a foundational building block to address it. Specifically, we present a theoretical study of this problem under a simplified-yet-challenging model involving two reviewers, two papers, and an MAP-computing adversary. Our main results establish the Pareto frontier of the tradeoff between privacy (preventing the adversary from inferring reviewer identity) and utility (accepting better papers), and design explicit computationally-efficient algorithms that we prove are Pareto optimal.

Samuel B. Hopkins, Gautam Kamath, Mahbod Majid

We give the first polynomial-time algorithm to estimate the mean of a $d$-variate probability distribution with bounded covariance from $\tilde{O}(d)$ independent samples subject to pure differential privacy. Prior algorithms for this problem either incur exponential running time, require $Ω(d^{1.5})$ samples, or satisfy only the weaker concentrated or approximate differential privacy conditions. In particular, all prior polynomial-time algorithms require $d^{1+Ω(1)}$ samples to guarantee small privacy loss with "cryptographically" high probability, $1-2^{-d^{Ω(1)}}$, while our algorithm retains $\tilde{O}(d)$ sample complexity even in this stringent setting. Our main technique is a new approach to use the powerful Sum of Squares method (SoS) to design differentially private algorithms. SoS proofs to algorithms is a key theme in numerous recent works in high-dimensional algorithmic statistics -- estimators which apparently require exponential running time but whose analysis can be captured by low-degree Sum of Squares proofs can be automatically turned into polynomial-time algorithms with the same provable guarantees. We demonstrate a similar proofs to private algorithms phenomenon: instances of the workhorse exponential mechanism which apparently require exponential time but which can be analyzed with low-degree SoS proofs can be automatically turned into polynomial-time differentially private algorithms. We prove a meta-theorem capturing this phenomenon, which we expect to be of broad use in private algorithm design. Our techniques also draw new connections between differentially private and robust statistics in high dimensions. In particular, viewed through our proofs-to-private-algorithms lens, several well-studied SoS proofs from recent works in algorithmic robust statistics directly yield key components of our differentially private mean estimation algorithm.

Jayadev Acharya, Ayush Jain, Gautam Kamath et al.

We study the problem of robustly estimating the parameter $p$ of an Erdős-Rényi random graph on $n$ nodes, where a $γ$ fraction of nodes may be adversarially corrupted. After showing the deficiencies of canonical estimators, we design a computationally-efficient spectral algorithm which estimates $p$ up to accuracy $\tilde O(\sqrt{p(1-p)}/n + γ\sqrt{p(1-p)} /\sqrt{n}+ γ/n)$ for $γ< 1/60$. Furthermore, we give an inefficient algorithm with similar accuracy for all $γ<1/2$, the information-theoretic limit. Finally, we prove a nearly-matching statistical lower bound, showing that the error of our algorithms is optimal up to logarithmic factors.

Shubhankar Mohapatra, Sajin Sasy, Xi He et al.

Hyperparameter optimization is a ubiquitous challenge in machine learning, and the performance of a trained model depends crucially upon their effective selection. While a rich set of tools exist for this purpose, there are currently no practical hyperparameter selection methods under the constraint of differential privacy (DP). We study honest hyperparameter selection for differentially private machine learning, in which the process of hyperparameter tuning is accounted for in the overall privacy budget. To this end, we i) show that standard composition tools outperform more advanced techniques in many settings, ii) empirically and theoretically demonstrate an intrinsic connection between the learning rate and clipping norm hyperparameters, iii) show that adaptive optimizers like DPAdam enjoy a significant advantage in the process of honest hyperparameter tuning, and iv) draw upon novel limiting behaviour of Adam in the DP setting to design a new and more efficient optimizer.

Gautam Kamath, Argyris Mouzakis, Vikrant Singhal et al.

We give the first polynomial-time, polynomial-sample, differentially private estimator for the mean and covariance of an arbitrary Gaussian distribution $\mathcal{N}(μ,Σ)$ in $\mathbb{R}^d$. All previous estimators are either nonconstructive, with unbounded running time, or require the user to specify a priori bounds on the parameters $μ$ and $Σ$. The primary new technical tool in our algorithm is a new differentially private preconditioner that takes samples from an arbitrary Gaussian $\mathcal{N}(0,Σ)$ and returns a matrix $A$ such that $A ΣA^T$ has constant condition number.

Christian Covington, Xi He, James Honaker et al.

We present a method for producing unbiased parameter estimates and valid confidence intervals under the constraints of differential privacy, a formal framework for limiting individual information leakage from sensitive data. Prior work in this area is limited in that it is tailored to calculating confidence intervals for specific statistical procedures, such as mean estimation or simple linear regression. While other recent work can produce confidence intervals for more general sets of procedures, they either yield only approximately unbiased estimates, are designed for one-dimensional outputs, or assume significant user knowledge about the data-generating distribution. Our method induces distributions of mean and covariance estimates via the bag of little bootstraps (BLB) and uses them to privately estimate the parameters' sampling distribution via a generalized version of the CoinPress estimation algorithm. If the user can bound the parameters of the BLB-induced parameters and provide heavier-tailed families, the algorithm produces unbiased parameter estimates and valid confidence intervals which hold with arbitrarily high probability. These results hold in high dimensions and for any estimation procedure which behaves nicely under the bootstrap.

Da Yu, Saurabh Naik, Arturs Backurs et al.

We give simpler, sparser, and faster algorithms for differentially private fine-tuning of large-scale pre-trained language models, which achieve the state-of-the-art privacy versus utility tradeoffs on many standard NLP tasks. We propose a meta-framework for this problem, inspired by the recent success of highly parameter-efficient methods for fine-tuning. Our experiments show that differentially private adaptations of these approaches outperform previous private algorithms in three important dimensions: utility, privacy, and the computational and memory cost of private training. On many commonly studied datasets, the utility of private models approaches that of non-private models. For example, on the MNLI dataset we achieve an accuracy of $87.8\%$ using RoBERTa-Large and $83.5\%$ using RoBERTa-Base with a privacy budget of $ε= 6.7$. In comparison, absent privacy constraints, RoBERTa-Large achieves an accuracy of $90.2\%$. Our findings are similar for natural language generation tasks. Privately fine-tuning with DART, GPT-2-Small, GPT-2-Medium, GPT-2-Large, and GPT-2-XL achieve BLEU scores of 38.5, 42.0, 43.1, and 43.8 respectively (privacy budget of $ε= 6.8,δ=$ 1e-5) whereas the non-private baseline is $48.1$. All our experiments suggest that larger models are better suited for private fine-tuning: while they are well known to achieve superior accuracy non-privately, we find that they also better maintain their accuracy when privacy is introduced.

Clément L. Canonne, Ayush Jain, Gautam Kamath et al.

We revisit the problem of tolerant distribution testing. That is, given samples from an unknown distribution $p$ over $\{1, \dots, n\}$, is it $\varepsilon_1$-close to or $\varepsilon_2$-far from a reference distribution $q$ (in total variation distance)? Despite significant interest over the past decade, this problem is well understood only in the extreme cases. In the noiseless setting (i.e., $\varepsilon_1 = 0$) the sample complexity is $Θ(\sqrt{n})$, strongly sublinear in the domain size. At the other end of the spectrum, when $\varepsilon_1 = \varepsilon_2/2$, the sample complexity jumps to the barely sublinear $Θ(n/\log n)$. However, very little is known about the intermediate regime. We fully characterize the price of tolerance in distribution testing as a function of $n$, $\varepsilon_1$, $\varepsilon_2$, up to a single $\log n$ factor. Specifically, we show the sample complexity to be \[\tilde Θ\left(\frac{\sqrt{n}}{\varepsilon_2^{2}} + \frac{n}{\log n} \cdot \max \left\{\frac{\varepsilon_1}{\varepsilon_2^2},\left(\frac{\varepsilon_1}{\varepsilon_2^2}\right)^{\!\!2}\right\}\right),\] providing a smooth tradeoff between the two previously known cases. We also provide a similar characterization for the problem of tolerant equivalence testing, where both $p$ and $q$ are unknown. Surprisingly, in both cases, the main quantity dictating the sample complexity is the ratio $\varepsilon_1/\varepsilon_2^2$, and not the more intuitive $\varepsilon_1/\varepsilon_2$. Of particular technical interest is our lower bound framework, which involves novel approximation-theoretic tools required to handle the asymmetry between $\varepsilon_1$ and $\varepsilon_2$, a challenge absent from previous works.

Gautam Kamath, Xingtu Liu, Huanyu Zhang

We study stochastic convex optimization with heavy-tailed data under the constraint of differential privacy (DP). Most prior work on this problem is restricted to the case where the loss function is Lipschitz. Instead, as introduced by Wang, Xiao, Devadas, and Xu \cite{WangXDX20}, we study general convex loss functions with the assumption that the distribution of gradients has bounded $k$-th moments. We provide improved upper bounds on the excess population risk under concentrated DP for convex and strongly convex loss functions. Along the way, we derive new algorithms for private mean estimation of heavy-tailed distributions, under both pure and concentrated DP. Finally, we prove nearly-matching lower bounds for private stochastic convex optimization with strongly convex losses and mean estimation, showing new separations between pure and concentrated DP.

Ayush Sekhari, Jayadev Acharya, Gautam Kamath et al.

We study the problem of unlearning datapoints from a learnt model. The learner first receives a dataset $S$ drawn i.i.d. from an unknown distribution, and outputs a model $\widehat{w}$ that performs well on unseen samples from the same distribution. However, at some point in the future, any training datapoint $z \in S$ can request to be unlearned, thus prompting the learner to modify its output model while still ensuring the same accuracy guarantees. We initiate a rigorous study of generalization in machine unlearning, where the goal is to perform well on previously unseen datapoints. Our focus is on both computational and storage complexity. For the setting of convex losses, we provide an unlearning algorithm that can unlearn up to $O(n/d^{1/4})$ samples, where $d$ is the problem dimension. In comparison, in general, differentially private learning (which implies unlearning) only guarantees deletion of $O(n/d^{1/2})$ samples. This demonstrates a novel separation between differential privacy and machine unlearning.

Ishaq Aden-Ali, Hassan Ashtiani, Gautam Kamath

We provide sample complexity upper bounds for agnostically learning multivariate Gaussians under the constraint of approximate differential privacy. These are the first finite sample upper bounds for general Gaussians which do not impose restrictions on the parameters of the distribution. Our bounds are near-optimal in the case when the covariance is known to be the identity, and conjectured to be near-optimal in the general case. From a technical standpoint, we provide analytic tools for arguing the existence of global "locally small" covers from local covers of the space. These are exploited using modifications of recent techniques for differentially private hypothesis selection. Our techniques may prove useful for privately learning other distribution classes which do not possess a finite cover.

Sourav Biswas, Yihe Dong, Gautam Kamath et al.

We present simple differentially private estimators for the mean and covariance of multivariate sub-Gaussian data that are accurate at small sample sizes. We demonstrate the effectiveness of our algorithms both theoretically and empirically using synthetic and real-world datasets -- showing that their asymptotic error rates match the state-of-the-art theoretical bounds, and that they concretely outperform all previous methods. Specifically, previous estimators either have weak empirical accuracy at small sample sizes, perform poorly for multivariate data, or require the user to provide strong a priori estimates for the parameters.

Gautam Kamath, Jonathan Ullman

Differentially private statistical estimation has seen a flurry of developments over the last several years. Study has been divided into two schools of thought, focusing on empirical statistics versus population statistics. We suggest that these two lines of work are more similar than different by giving examples of methods that were initially framed for empirical statistics, but can be applied just as well to population statistics. We also provide a thorough coverage of recent work in this area.

Clément L. Canonne, Gautam Kamath, Thomas Steinke

A key tool for building differentially private systems is adding Gaussian noise to the output of a function evaluated on a sensitive dataset. Unfortunately, using a continuous distribution presents several practical challenges. First and foremost, finite computers cannot exactly represent samples from continuous distributions, and previous work has demonstrated that seemingly innocuous numerical errors can entirely destroy privacy. Moreover, when the underlying data is itself discrete (e.g., population counts), adding continuous noise makes the result less interpretable. With these shortcomings in mind, we introduce and analyze the discrete Gaussian in the context of differential privacy. Specifically, we theoretically and experimentally show that adding discrete Gaussian noise provides essentially the same privacy and accuracy guarantees as the addition of continuous Gaussian noise. We also present an simple and efficient algorithm for exact sampling from this distribution. This demonstrates its applicability for privately answering counting queries, or more generally, low-sensitivity integer-valued queries.

Wanrong Zhang, Gautam Kamath, Rachel Cummings

In hypothesis testing, a false discovery occurs when a hypothesis is incorrectly rejected due to noise in the sample. When adaptively testing multiple hypotheses, the probability of a false discovery increases as more tests are performed. Thus the problem of False Discovery Rate (FDR) control is to find a procedure for testing multiple hypotheses that accounts for this effect in determining the set of hypotheses to reject. The goal is to minimize the number (or fraction) of false discoveries, while maintaining a high true positive rate (i.e., correct discoveries). In this work, we study False Discovery Rate (FDR) control in multiple hypothesis testing under the constraint of differential privacy for the sample. Unlike previous work in this direction, we focus on the online setting, meaning that a decision about each hypothesis must be made immediately after the test is performed, rather than waiting for the output of all tests as in the offline setting. We provide new private algorithms based on state-of-the-art results in non-private online FDR control. Our algorithms have strong provable guarantees for privacy and statistical performance as measured by FDR and power. We also provide experimental results to demonstrate the efficacy of our algorithms in a variety of data environments.

Huanyu Zhang, Gautam Kamath, Janardhan Kulkarni et al.

We consider the problem of learning Markov Random Fields (including the prototypical example, the Ising model) under the constraint of differential privacy. Our learning goals include both structure learning, where we try to estimate the underlying graph structure of the model, as well as the harder goal of parameter learning, in which we additionally estimate the parameter on each edge. We provide algorithms and lower bounds for both problems under a variety of privacy constraints -- namely pure, concentrated, and approximate differential privacy. While non-privately, both learning goals enjoy roughly the same complexity, we show that this is not the case under differential privacy. In particular, only structure learning under approximate differential privacy maintains the non-private logarithmic dependence on the dimensionality of the data, while a change in either the learning goal or the privacy notion would necessitate a polynomial dependence. As a result, we show that the privacy constraint imposes a strong separation between these two learning problems in the high-dimensional data regime.

Gautam Kamath, Vikrant Singhal, Jonathan Ullman

We give new upper and lower bounds on the minimax sample complexity of differentially private mean estimation of distributions with bounded $k$-th moments. Roughly speaking, in the univariate case, we show that $n = Θ\left(\frac{1}{α^2} + \frac{1}{α^{\frac{k}{k-1}}\varepsilon}\right)$ samples are necessary and sufficient to estimate the mean to $α$-accuracy under $\varepsilon$-differential privacy, or any of its common relaxations. This result demonstrates a qualitatively different behavior compared to estimation absent privacy constraints, for which the sample complexity is identical for all $k \geq 2$. We also give algorithms for the multivariate setting whose sample complexity is a factor of $O(d)$ larger than the univariate case.

Sivakanth Gopi, Gautam Kamath, Janardhan Kulkarni et al.

We initiate the study of hypothesis selection under local differential privacy. Given samples from an unknown probability distribution $p$ and a set of $k$ probability distributions $\mathcal{Q}$, we aim to output, under the constraints of $\varepsilon$-local differential privacy, a distribution from $\mathcal{Q}$ whose total variation distance to $p$ is comparable to the best such distribution. This is a generalization of the classic problem of $k$-wise simple hypothesis testing, which corresponds to when $p \in \mathcal{Q}$, and we wish to identify $p$. Absent privacy constraints, this problem requires $O(\log k)$ samples from $p$, and it was recently shown that the same complexity is achievable under (central) differential privacy. However, the naive approach to this problem under local differential privacy would require $\tilde O(k^2)$ samples. We first show that the constraint of local differential privacy incurs an exponential increase in cost: any algorithm for this problem requires at least $Ω(k)$ samples. Second, for the special case of $k$-wise simple hypothesis testing, we provide a non-interactive algorithm which nearly matches this bound, requiring $\tilde O(k)$ samples. Finally, we provide sequentially interactive algorithms for the general case, requiring $\tilde O(k)$ samples and only $O(\log \log k)$ rounds of interactivity. Our algorithms are achieved through a reduction to maximum selection with adversarial comparators, a problem of independent interest for which we initiate study in the parallel setting. For this problem, we provide a family of algorithms for each number of allowed rounds of interaction $t$, as well as lower bounds showing that they are near-optimal for every $t$. Notably, our algorithms result in exponential improvements on the round complexity of previous methods.

Clément L. Canonne, Xi Chen, Gautam Kamath et al.

We give a nearly-optimal algorithm for testing uniformity of distributions supported on $\{-1,1\}^n$, which makes $\tilde O (\sqrt{n}/\varepsilon^2)$ queries to a subcube conditional sampling oracle (Bhattacharyya and Chakraborty (2018)). The key technical component is a natural notion of random restriction for distributions on $\{-1,1\}^n$, and a quantitative analysis of how such a restriction affects the mean vector of the distribution. Along the way, we consider the problem of mean testing with independent samples and provide a nearly-optimal algorithm.

Gautam Kamath, Or Sheffet, Vikrant Singhal et al.

Learning the parameters of Gaussian mixture models is a fundamental and widely studied problem with numerous applications. In this work, we give new algorithms for learning the parameters of a high-dimensional, well separated, Gaussian mixture model subject to the strong constraint of differential privacy. In particular, we give a differentially private analogue of the algorithm of Achlioptas and McSherry. Our algorithm has two key properties not achieved by prior work: (1) The algorithm's sample complexity matches that of the corresponding non-private algorithm up to lower order terms in a wide range of parameters. (2) The algorithm does not require strong a priori bounds on the parameters of the mixture components.

Mark Bun, Gautam Kamath, Thomas Steinke et al.

We provide a differentially private algorithm for hypothesis selection. Given samples from an unknown probability distribution $P$ and a set of $m$ probability distributions $\mathcal{H}$, the goal is to output, in a $\varepsilon$-differentially private manner, a distribution from $\mathcal{H}$ whose total variation distance to $P$ is comparable to that of the best such distribution (which we denote by $α$). The sample complexity of our basic algorithm is $O\left(\frac{\log m}{α^2} + \frac{\log m}{α\varepsilon}\right)$, representing a minimal cost for privacy when compared to the non-private algorithm. We also can handle infinite hypothesis classes $\mathcal{H}$ by relaxing to $(\varepsilon,δ)$-differential privacy. We apply our hypothesis selection algorithm to give learning algorithms for a number of natural distribution classes, including Gaussians, product distributions, sums of independent random variables, piecewise polynomials, and mixture classes. Our hypothesis selection procedure allows us to generically convert a cover for a class to a learning algorithm, complementing known learning lower bounds which are in terms of the size of the packing number of the class. As the covering and packing numbers are often closely related, for constant $α$, our algorithms achieve the optimal sample complexity for many classes of interest. Finally, we describe an application to private distribution-free PAC learning.

Clément L. Canonne, Gautam Kamath, Audra McMillan et al.

In this work we present novel differentially private identity (goodness-of-fit) testers for natural and widely studied classes of multivariate product distributions: Gaussians in $\mathbb{R}^d$ with known covariance and product distributions over $\{\pm 1\}^{d}$. Our testers have improved sample complexity compared to those derived from previous techniques, and are the first testers whose sample complexity matches the order-optimal minimax sample complexity of $O(d^{1/2}/α^2)$ in many parameter regimes. We construct two types of testers, exhibiting tradeoffs between sample complexity and computational complexity. Finally, we provide a two-way reduction between testing a subclass of multivariate product distributions and testing univariate distributions, and thereby obtain upper and lower bounds for testing this subclass of product distributions.

Clément L. Canonne, Gautam Kamath, Audra McMillan et al.

Hypothesis testing plays a central role in statistical inference, and is used in many settings where privacy concerns are paramount. This work answers a basic question about privately testing simple hypotheses: given two distributions $P$ and $Q$, and a privacy level $\varepsilon$, how many i.i.d. samples are needed to distinguish $P$ from $Q$ subject to $\varepsilon$-differential privacy, and what sort of tests have optimal sample complexity? Specifically, we characterize this sample complexity up to constant factors in terms of the structure of $P$ and $Q$ and the privacy level $\varepsilon$, and show that this sample complexity is achieved by a certain randomized and clamped variant of the log-likelihood ratio test. Our result is an analogue of the classical Neyman-Pearson lemma in the setting of private hypothesis testing. We also give an application of our result to the private change-point detection. Our characterization applies more generally to hypothesis tests satisfying essentially any notion of algorithmic stability, which is known to imply strong generalization bounds in adaptive data analysis, and thus our results have applications even when privacy is not a primary concern.

Gautam Kamath, Christos Tzamos

We investigate distribution testing with access to non-adaptive conditional samples. In the conditional sampling model, the algorithm is given the following access to a distribution: it submits a query set $S$ to an oracle, which returns a sample from the distribution conditioned on being from $S$. In the non-adaptive setting, all query sets must be specified in advance of viewing the outcomes. Our main result is the first polylogarithmic-query algorithm for equivalence testing, deciding whether two unknown distributions are equal to or far from each other. This is an exponential improvement over the previous best upper bound, and demonstrates that the complexity of the problem in this model is intermediate to the the complexity of the problem in the standard sampling model and the adaptive conditional sampling model. We also significantly improve the sample complexity for the easier problems of uniformity and identity testing. For the former, our algorithm requires only $\tilde O(\log n)$ queries, matching the information-theoretic lower bound up to a $O(\log \log n)$-factor. Our algorithm works by reducing the problem from $\ell_1$-testing to $\ell_\infty$-testing, which enjoys a much cheaper sample complexity. Necessitated by the limited power of the non-adaptive model, our algorithm is very simple to state. However, there are significant challenges in the analysis, due to the complex structure of how two arbitrary distributions may differ.

Gautam Kamath, Jerry Li, Vikrant Singhal et al.

We present novel, computationally efficient, and differentially private algorithms for two fundamental high-dimensional learning problems: learning a multivariate Gaussian and learning a product distribution over the Boolean hypercube in total variation distance. The sample complexity of our algorithms nearly matches the sample complexity of the optimal non-private learners for these tasks in a wide range of parameters, showing that privacy comes essentially for free for these problems. In particular, in contrast to previous approaches, our algorithm for learning Gaussians does not require strong a priori bounds on the range of the parameters. Our algorithms introduce a novel technical approach to reducing the sensitivity of the estimation procedure that we call recursive private preconditioning.

Ilias Diakonikolas, Gautam Kamath, Daniel M. Kane et al.

In high dimensions, most machine learning methods are brittle to even a small fraction of structured outliers. To address this, we introduce a new meta-algorithm that can take in a base learner such as least squares or stochastic gradient descent, and harden the learner to be resistant to outliers. Our method, Sever, possesses strong theoretical guarantees yet is also highly scalable -- beyond running the base learner itself, it only requires computing the top singular vector of a certain $n \times d$ matrix. We apply Sever on a drug design dataset and a spam classification dataset, and find that in both cases it has substantially greater robustness than several baselines. On the spam dataset, with $1\%$ corruptions, we achieved $7.4\%$ test error, compared to $13.4\%-20.5\%$ for the baselines, and $3\%$ error on the uncorrupted dataset. Similarly, on the drug design dataset, with $10\%$ corruptions, we achieved $1.42$ mean-squared error test error, compared to $1.51$-$2.33$ for the baselines, and $1.23$ error on the uncorrupted dataset.

Jayadev Acharya, Gautam Kamath, Ziteng Sun et al.

We develop differentially private methods for estimating various distributional properties. Given a sample from a discrete distribution $p$, some functional $f$, and accuracy and privacy parameters $α$ and $\varepsilon$, the goal is to estimate $f(p)$ up to accuracy $α$, while maintaining $\varepsilon$-differential privacy of the sample. We prove almost-tight bounds on the sample size required for this problem for several functionals of interest, including support size, support coverage, and entropy. We show that the cost of privacy is negligible in a variety of settings, both theoretically and experimentally. Our methods are based on a sensitivity analysis of several state-of-the-art methods for estimating these properties with sublinear sample complexities.

Steve Hanneke, Adam Kalai, Gautam Kamath et al.

A generative model may generate utter nonsense when it is fit to maximize the likelihood of observed data. This happens due to "model error," i.e., when the true data generating distribution does not fit within the class of generative models being learned. To address this, we propose a model of active distribution learning using a binary invalidity oracle that identifies some examples as clearly invalid, together with random positive examples sampled from the true distribution. The goal is to maximize the likelihood of the positive examples subject to the constraint of (almost) never generating examples labeled invalid by the oracle. Guarantees are agnostic compared to a class of probability distributions. We show that, while proper learning often requires exponentially many queries to the invalidity oracle, improper distribution learning can be done using polynomially many queries.

Constantinos Daskalakis, Nishanth Dikkala, Gautam Kamath

We prove near-tight concentration of measure for polynomial functions of the Ising model under high temperature. For any degree $d$, we show that a degree-$d$ polynomial of a $n$-spin Ising model exhibits exponential tails that scale as $\exp(-r^{2/d})$ at radius $r=\tildeΩ_d(n^{d/2})$. Our concentration radius is optimal up to logarithmic factors for constant $d$, improving known results by polynomial factors in the number of spins. We demonstrate the efficacy of polynomial functions as statistics for testing the strength of interactions in social networks in both synthetic and real world data.

Constantinos Daskalakis, Gautam Kamath, John Wright

Given samples from an unknown distribution $p$ and a description of a distribution $q$, are $p$ and $q$ close or far? This question of "identity testing" has received significant attention in the case of testing whether $p$ and $q$ are equal or far in total variation distance. However, in recent work, the following questions have been been critical to solving problems at the frontiers of distribution testing: -Alternative Distances: Can we test whether $p$ and $q$ are far in other distances, say Hellinger? -Tolerance: Can we test when $p$ and $q$ are close, rather than equal? And if so, close in which distances? Motivated by these questions, we characterize the complexity of distribution testing under a variety of distances, including total variation, $\ell_2$, Hellinger, Kullback-Leibler, and $χ^2$. For each pair of distances $d_1$ and $d_2$, we study the complexity of testing if $p$ and $q$ are close in $d_1$ versus far in $d_2$, with a focus on identifying which problems allow strongly sublinear testers (i.e., those with complexity $O(n^{1 - γ})$ for some $γ> 0$ where $n$ is the size of the support of the distributions $p$ and $q$). We provide matching upper and lower bounds for each case. We also study these questions in the case where we only have samples from $q$ (equivalence testing), showing qualitative differences from identity testing in terms of when tolerance can be achieved. Our algorithms fall into the classical paradigm of $χ^2$-statistics, but require crucial changes to handle the challenges introduced by each distance we consider. Finally, we survey other recent results in an attempt to serve as a reference for the complexity of various distribution testing problems.

Ilias Diakonikolas, Gautam Kamath, Daniel M. Kane et al.

We study the fundamental problem of learning the parameters of a high-dimensional Gaussian in the presence of noise -- where an $\varepsilon$-fraction of our samples were chosen by an adversary. We give robust estimators that achieve estimation error $O(\varepsilon)$ in the total variation distance, which is optimal up to a universal constant that is independent of the dimension. In the case where just the mean is unknown, our robustness guarantee is optimal up to a factor of $\sqrt{2}$ and the running time is polynomial in $d$ and $1/ε$. When both the mean and covariance are unknown, the running time is polynomial in $d$ and quasipolynomial in $1/\varepsilon$. Moreover all of our algorithms require only a polynomial number of samples. Our work shows that the same sorts of error guarantees that were established over fifty years ago in the one-dimensional setting can also be achieved by efficient algorithms in high-dimensional settings.