Hypothesis selection asks for a distribution close to an unknown $P$, given samples from $P$ and access to $n$ candidate hypotheses. We study the computational complexity of achieving statistically optimal sample complexity and approximation constants.

Independence testing is a fundamental problem in statistical inference: given samples from a joint distribution $p$ over multiple random variables, the goal is to determine whether $p$ is a product distribution or is $\epsilon$-far from all product distributions in total variation distance. In the non-parametric finite-sample regime, this task is notoriously expensive, as the minimax sample complexity scales polynomially with the support size. In this work, we move beyond these worst-case limitations by leveraging the framework of augmented distribution testing. We design independence testers that incorporate auxiliary, but potentially untrustworthy, predictive information. Our framework ensures that the tester remains robust, maintaining worst-case validity regardless of the prediction’s quality, while significantly improving sample efficiency when the prediction is accurate. Our main contributions include: (i) a bivariate independence tester for discrete distributions that adaptively reduces sample complexity based on the prediction error; (ii) a generalization to the high-dimensional multivariate setting for testing the independence of $d$ random variables; and (iii) matching minimax lower bounds demonstrating that our testers achieve optimal sample complexity.

A hypothesis testing algorithm is replicable if, when run on two different samples from the same distribution, it produces the same output with high probability. This notion, defined by Impagliazzo, Lei, Pitassi, and Sorell [STOC’22], can increase trust in testing procedures and is deeply related to algorithmic stability, generalization, and privacy. We build general tools to prove lower and upper bounds on the sample complexity of replicable testers, unifying and quantitatively improving upon existing results: (1) We identify a set of canonical properties, and prove that any replicable testing algorithm can be modified to satisfy these properties without worsening accuracy or sample complexity. A canonical replicable algorithm computes a deterministic function of its input (i.e., a test statistic) and thresholds against a uniformly random value in [0, 1]. It is invariant to the order in which the samples are received, and, if the testing problem is “symmetric,” then the algorithm is also invariant to the labeling of the domain elements, resolving an open question by Liu and Ye [NeurIPS’24]. (2) We prove new lower bounds for uniformity, identity, and closeness testing by reducing to the case where the replicable algorithm satisfies these canonical properties. (3) We systematize and improve upon a common strategy for replicable algorithm design based on test statistics with known expectation and bounded variance. Our framework allow testers which have been extensively analyzed in the non-replicable setting to be made replicable with minimal overhead. As direct applications of our framework combined with existing analyses of non-replicable testers, we obtain constant-factor optimal bounds for coin testing and closeness testing and get replicability for free for uniformity testing in a large parameter regime. As replicable coin testing can be used as a black-box to turn any tester into a replicable tester, our results directly imply improved replicable sampling bounds for a myriad of applications beyond the ones specifically studied in this paper. (4) We also give new algorithms and lower bounds for replicable Gaussian mean testing, improving over the previous best sample complexity of Bun, Gaboardi, Hopkins, Impagliazzo, Lei, Pitassi, Sivakumar, and Sorrell [STOC’23].

Model personalization allows a set of individuals, each facing a different learning task, to train models that are more accurate for each person than those they could develop individually. The goals of personalization are captured in a variety of formal frameworks, such as multitask learning and metalearning. Combining data for model personalization poses risks for privacy because the output of an individual’s model can depend on the data of other individuals. In this work we undertake a systematic study of differentially private personalized learning. Our first main contribution is to construct a taxonomy of formal frameworks for private personalized learning. This taxonomy captures different formal frameworks for learning as well as different threat models for the attacker. Our second main contribution is to prove separations between the personalized learning problems corresponding to different choices. In particular, we prove a novel separation between private multitask learning and private metalearning.

Local Differential Privacy (LDP) offers strong privacy guarantees without requiring users to trust external parties. However, LDP applies uniform protection to all data features, including less sensitive ones, which degrades performance of downstream tasks. To overcome this limitation, we propose a Bayesian framework, Bayesian Coordinate Differential Privacy (BCDP), that enables feature-specific privacy quantification. This more nuanced approach complements LDP by adjusting privacy protection according to the sensitivity of each feature, enabling improved performance of downstream tasks without compromising privacy. We characterize the properties of BCDP and articulate its connections with standard non-Bayesian privacy frameworks. We further apply our BCDP framework to the problems of private mean estimation and ordinary least-squares regression. The BCDP-based approach obtains improved accuracy compared to a purely LDP-based approach, without compromising on privacy.

We extend the framework of augmented distribution testing (Aliakbarpour, Indyk, Rubinfeld, and Silwal, NeurIPS 2024) to the differentially private setting. This captures scenarios where a data analyst must perform hypothesis testing tasks on sensitive data, but is able to leverage prior knowledge (public, but possibly erroneous or untrusted) about the data distribution. We design private algorithms in this augmented setting for three flagship distribution testing tasks, \emph{uniformity}, \emph{identity}, and \emph{closeness} testing, whose sample complexity smoothly scales with the claimed quality of the auxiliary information. We complement our algorithms with information-theoretic lower bounds, showing that their sample complexity is optimal (up to logarithmic factors).

Quantum state learning is a fundamental problem in physics and computer science. As near-term quantum devices are error-prone, it is important to design error-resistant algorithms. Apart from device errors, other unexpected factors could also affect the algorithm, such as careless human read-out error, or even a malicious hacker deliberately altering the measurement results. Thus, we want our algorithm to work even in the worst case when things go against our favor.We consider the practical setting of single-copy measurements and propose the $\gamma$-adversarial corruption model where an imaginary adversary can arbitrarily change $\gamma$-fraction of the measurement outcomes. This is stronger than the $\gamma$-bounded SPAM noise model, where the post-measurement state changes by at most $\gamma$ in trace distance. Under our stronger model of corruption, we design an algorithm using non-adaptive measurements that can learn an unknown rank- r state up to $\tilde{O}(\gamma \sqrt{r})$ in trace distance, provided that the number of copies is sufficiently large. We further prove an information-theoretic lower bound of $\Omega(\gamma \sqrt{r})$ for non-adaptive measurements, demonstrating the optimality of our algorithm. Our upper and lower bounds also hold for quantum state testing, where the goal is to test whether an unknown state is equal to a given state or far from it. Our results are intriguingly optimistic and pessimistic at the same time. For general states, the error is dimension-dependent and $\gamma \sqrt{d}$ in the worst case, meaning that only corrupting a very small fraction $(1 / \sqrt{d})$ of the outcomes could totally destroy any non-adaptive learning algorithm. However, for constant-rank states that are useful in many quantum algorithms, it is possible to achieve dimension-independent error, even in the worst-case adversarial setting.

Estimating the density of a distribution from its samples is a fundamental problem in statistics. \emph{Hypothesis selection} addresses the setting where, in addition to a sample set, we are given $n$ candidate distributions---referred to as \emph{hypotheses}---and the goal is to determine which one best describes the underlying data distribution. This problem is known to be solvable very efficiently, requiring roughly $O(\log n)$ samples and running in $\tilde{O}(n)$ time. The quality of the output is measured via the total variation distance to the unknown distribution, and the approximation factor of the algorithm determines how large this distance is compared to the optimal distance achieved by the best candidate hypothesis. It is known that $\alpha = 3$ is the optimal approximation factor for this problem. We study hypothesis selection under the constraint of \emph{differential privacy}. We propose a differentially private algorithm in the central model that runs in nearly linear time with respect to the number of hypotheses, achieves the optimal approximation factor, and incurs only a modest increase in sample complexity, which remains polylogarithmic in $n$. This resolves an open question posed by [Bun, Kamath, Steinke, Wu, NeurIPS 2019]. Prior to our work, existing upper bounds required quadratic time.

Metalearning and multitask learning are two frameworks for solving a group of related learning tasks more efficiently than we could hope to solve each of the individual tasks on their own. In multitask learning, we are given a fixed set of related learning tasks and need to output one accurate model per task, whereas in metalearning we are given tasks that are drawn i.i.d. from a metadistribution and need to output some common information that can be easily specialized to new, previously unseen tasks from the metadistribution. In this work, we consider a binary classification setting where tasks are related by a shared representation, that is, every task $P$ of interest can be solved by a classifier of the form $f_{P} \circ h$ where $h \in \mathcal{H}$ is a map from features to some representation space that is shared across tasks, and $f_{P} \in \mathcal{F}$ is a task-specific classifier from the representation space to labels. The main question we ask in this work is how much data do we need to metalearn a good representation? Here, the amount of data is measured in terms of both the number of tasks $t$ that we need to see and the number of samples $n$ per task. We focus on the regime where the number of samples per task is extremely small. Our main result shows that, in a distribution-free setting where the feature vectors are in $\mathbb{R}^d$, the representation is a linear map from $\mathbb{R}^d \to \mathbb{R}^k$, and the task-specific classifiers are halfspaces in $\mathbb{R}^k$, we can metalearn a representation with error $\varepsilon$ using just $n = k+2$ samples per task, and $d \cdot (1/\varepsilon)^{O(k)}$ tasks. Learning with so few samples per task is remarkable because metalearning would be impossible with $k+1$ samples per task, and because we cannot even hope to learn an accurate task-specific classifier with just $k+2$ samples per task. To obtain this result, we develop a sample-and-task-complexity theory for distribution-free metalearning and multitask learning, which identifies what properties of $\mathcal{F}$ and $\mathcal{H}$ make metalearning possible with few samples per task. Our theory also yields a simple characterization of distribution-free multitask learning. Finally, we give sample-efficient reductions between metalearning and multitask learning, which, when combined with our characterization of multitask learning, give a characterization of metalearning in certain parameter regimes.

We construct differentially private estimators with low sample complexity that estimate the median of an arbitrary distribution over $\mathbb{R}$ satisfying very mild moment conditions. Our result stands in contrast to the surprising negative result of Bun et al. (FOCS 2015) that showed there is no differentially private estimator with any finite sample complexity that returns any non-trivial approximation to the median of an arbitrary distribution.

Hypothesis selection, also known as density estimation, is a fundamental problem in statistics and learning theory. Suppose we are given a sample set from an unknown distribution $P$ and a finite class of candidate distributions (called hypotheses) $\mathcal{H} \coloneqq \{H_1, H_2, \ldots, H_n\}$. The aim is to design an algorithm that selects a distribution $\hat H$ in $\mathcal{H}$ that best fits the data. The algorithm's accuracy is measured based on the distance between $\hat{H}$ and $P$ compared to the distance of the closest distribution in $\mathcal{H}$ to $P$ (denoted by $OPT$). Concretely, we aim for $\|\hat{H} - P\|_{TV}$ to be at most $ \alpha \cdot OPT + \epsilon$ for some small $\epsilon$ and $\alpha$. While it is possible to decrease the value of $\epsilon$ as the number of samples increases, $\alpha$ is an inherent characteristic of the algorithm. In fact, one cannot hope to achieve $\alpha < 3$ even when there are only two candidate hypotheses, unless the number of samples is proportional to the domain size of $P$ [Bousquet, Kane, Moran '19]. Finding the best $\alpha$ has been one of the main focuses of studies of the problem since early work of [Devroye, Lugosi '01]. Prior to our work, no algorithm was known that achieves $\alpha = 3$ in near-linear time. We provide the first algorithm that operates in almost linear time ($\tilde{O}(n/\epsilon^3)$ time) and achieves $\alpha = 3$. This result improves upon a long list of results in hypothesis selection. Previously known algorithms either had worse time complexity, a larger factor $\alpha$, or extra assumptions about the problem setting. In addition to this algorithm, we provide another (almost) linear-time algorithm with better dependency on the additive accuracy parameter $\epsilon$, albeit with a slightly worse accuracy parameter, $\alpha = 4$.

We consider the problem of hypothesis testing for discrete distributions. In the standard model, where we have sample access to an underlying distribution $p$, extensive research has established optimal bounds for uniformity testing, identity testing (goodness of fit), and closeness testing (equivalence or two-sample testing). We explore these problems in a setting where a predicted data distribution, possibly derived from historical data or predictive machine learning models, is available. We demonstrate that such a predictor can indeed reduce the number of samples required for all three property testing tasks. The reduction in sample complexity depends directly on the predictor’s quality, measured by its total variation distance from $p$. A key advantage of our algorithms is their adaptability to the precision of the prediction. Specifically, our algorithms can self-adjust their sample complexity based on the accuracy of the available prediction, operating without any prior knowledge of the estimation’s accuracy (i.e. they are consistent). Additionally, we never use more samples than the standard approaches require, even if the predictions provide no meaningful information (i.e. they are also robust). We provide lower bounds to indicate that the improvements in sample complexity achieved by our algorithms are information-theoretically optimal. Furthermore, experimental results show that the performance of our algorithms on real data significantly exceeds our worst-case guarantees for sample complexity, demonstrating the practicality of our approach.

Understanding the shape of a distribution of data is of interest to people in a great variety of fields, as it may affect the types of algorithms used for that data. We study one such problem in the framework of {\em distribution property testing}, characterizing the number of samples required to to distinguish whether a distribution has a certain property or is far from having that property. In particular, given samples from a distribution, we seek to characterize the tail of the distribution, that is, understand how many elements appear infrequently. We develop an algorithm based on a careful bucketing scheme that distinguishes light-tailed distributions from non-light-tailed ones with respect to a definition based on the hazard rate, under natural smoothness and ordering assumptions. We bound the number of samples required for this test to succeed with high probability in terms of the parameters of the problem, showing that it is polynomial in these parameters. Further, we prove a hardness result that implies that this problem cannot be solved without any assumptions.

Hypothesis selection is a fundamental problem in learning theory and statistics. Given a dataset and a finite set of candidate distributions, the goal is to select a distribution that matches the data as well as possible. More specifically, suppose we have sample access to an unknown distribution $P$ over a domain $\mathcal{X}$ that we know is well-approximated by one of a a class of $n$ distributions (a.k.a. hypotheses), $\mathcal{H} \coloneqq \{H_1, H_2, \ldots, H_n\}$. The goal is to design an algorithm that outputs a distribution $\hat{H} \in \mathcal{H}$ whose total variation distance from $P$ is nearly minimal. In this work, we study the hypothesis selection problem under memory constraints. We consider a model where samples from $P$ are presented in a stream and we access each sample $x$ via ``PDF-comparison'' queries that allow us to compare the probability densities of any pair of hypotheses at the domain point $x$ (i.e., is $H_i(x) < H_j(x)$?). This model allows us to study how much memory is needed at any point in time to store information about the portion of the stream seen so far. Our main result is an algorithm that achieves a nearly optimal tradeoff between memory usage and the number of samples required. In particular, given $b$ bits of memory (for $b$ roughly between $\log n$ and $n$), our algorithm solves the hypothesis selection problem with $s$ samples, where $b \cdot s = O(n \log n)$. This result is optimal up to an $O(\log n)$ factor, for all $b$.

Recent work of Acharya et al.~(NeurIPS 2019) showed how to estimate the entropy of a distribution $\mathcal D$ over an alphabet of size $k$ up to $\pm\epsilon$ additive error by streaming over $(k/\epsilon^3) \cdot \text{polylog}(1/\epsilon)$ i.i.d.\ samples and using only $O(1)$ words of memory. In this work, we give a new constant memory scheme that reduces the sample complexity to $(k/\epsilon^2)\cdot \text{polylog}(1/\epsilon)$. We conjecture that this is optimal up to $\text{polylog}(1/\epsilon)$ factors.

Aggregating data is fundamental to data analytics, data exploration, and OLAP. Approximate query processing (AQP) techniques are often used to accelerate computation of aggregates using samples, for which confidence intervals (CIs) are widely used to quantify the associated error. CIs used in practice fall into two categories: techniques that are tight but not correct, i.e., they yield tight intervals but only offer asymptoticguarantees,makingthem unreliable, or techniques that are correct but not tight, i.e., they offer rigorous guarantees, but are overly conservative, leading to confidence intervals that are too loose to be useful. In this paper, we develop a CI technique that is both correct and tighter than traditional approaches. Starting from conservative CIs, we identify two issues they often face: pessimistic mass allocation (PMA) and phantom outlier sensitivity (PHOS). By developing a novel range-trimming technique for eliminating PHOS and pairing it with known CI techniques without PMA, we develop a technique for computing CIs with strong guarantees that requires fewer samples for the same width. We implement our techniques underneath a sampling-optimized in-memory column store and show how they accelerate queries involving aggregates on real datasets with typical speedups on the order of 10× over both traditional AQP-with-guarantees and exact methods, all while obeying accuracy constraints.

We investigate the local differential privacy (LDP) guarantees of a randomized privacy mechanism via its contraction properties. We first show that LDP constraints can be equivalently cast in terms of the contraction coefficient of the$\mathsf{E}_{\gamma}$-divergence. We then use this equivalent formula to express LDP guarantees of privacy mechanisms in terms of contraction coefficients of arbitrary$f$-divergences. When combined with standard estimation-theoretic tools (such as Le Cam's and Fano's converse methods), this result allows us to study the trade-off between privacy and utility in several testing and minimax and Bayesian estimation problems.

We propose a new setting for testing properties of distributions while receiving samples from several distributions, but few samples per distribution. Given samples from s distributions, p_1, p_2, …, p_s, we design testers for the following problems: (1) Uniformity Testing: Testing whether all the p_i’s are uniform or ε-far from being uniform in ℓ_1-distance (2) Identity Testing: Testing whether all the p_i’s are equal to an explicitly given distribution q or ε-far from q in ℓ_1-distance, and (3) Closeness Testing: Testing whether all the p_i’s are equal to a distribution q which we have sample access to, or ε-far from q in ℓ_1-distance. By assuming an additional natural condition about the source distributions, we provide sample optimal testers for all of these problems.

Determinantal point processes (DPPs) are popular probabilistic models of diversity. In this paper, we investigate DPPs from a new perspective: property testing of distributions. Given sample access to an unknown distribution $q$ over the subsets of a ground set, we aim to distinguish whether $q$ is a DPP distribution or $\epsilon$-far from all DPP distributions in $\ell_1$-distance. In this work, we propose the first algorithm for testing DPPs. Furthermore, we establish a matching lower bound on the sample complexity of DPP testing. This lower bound also extends to showing a new hardness result for the problem of testing the more general class of log-submodular distributions.

There has been significant study on the sample complexity of testing properties of distributions over large domains. For many properties, it is known that the sample complexity can be substantially smaller than the domain size. For example, over a domain of size $n$, distinguishing the uniform distribution from distributions that are far from uniform in $\ell_1$-distance uses only $O(\sqrt{n})$ samples. However, the picture is very different in the presence of arbitrary noise, even when the amount of noise is quite small. In this case, one must distinguish if samples are coming from a distribution that is $\epsilon$-close to uniform from the case where the distribution is $(1-\epsilon)$-far from uniform. The latter task requires nearly linear in $n$ samples (Valiant, 2008; Valiant and Valiant, 2017a). In this work, we present a noise model that on one hand is more tractable for the testing problem, and on the other hand represents a rich class of noise families. In our model, the noisy distribution is a mixture of the original distribution and noise, where the latter is known to the tester either explicitly or via sample access; the form of the noise is also known \emph{a priori}. Focusing on the identity and closeness testing problems leads to the following mixture testing question: Given samples of distributions $p, q_1,q_2$, can we test if $p$ is a mixture of $q_1$ and $q_2$? We consider this general question in various scenarios that differ in terms of how the tester can access the distributions, and show that indeed this problem is more tractable. Our results show that the sample complexity of our testers are exactly the same as for the classical non-mixture case.

In this work, we consider the sample complexity required for testing the monotonicity of distributions over partial orders. A distribution $p$ over a poset is {\em monotone} if, for any pair of domain elements $x$ and $y$ such that $x \preceq y$, $p(x) \leq p(y)$. To understand the sample complexity of this problem, we introduce a new property called \emph{bigness} over a finite domain, where the distribution is $T$-big if the minimum probability for any domain element is at least $T$. We establish a lower bound of $\Omega(n/\log n)$ for testing bigness of distributions on domains of size $n$. We then build on these lower bounds to give $\Omega(n/\log{n})$ lower bounds for testing monotonicity over a matching poset of size $n$ and significantly improved lower bounds over the hypercube poset. We give sublinear sample complexity bounds for testing bigness and for testing monotonicity over the matching poset. We then give a number of tools for analyzing upper bounds on the sample complexity of the monotonicity testing problem.

Statistical tests are at the heart of many scientific tasks. To validate their hypothesis, researchers in medical and social sciences use individuals' data. The sensitivity of participants' data requires the design of statistical tests that ensure the privacy of the individuals in the most efficient way. In this paper, we use the framework of property testing to design algorithms to test the properties of the distribution that the data is drawn from with respect to differential privacy. In particular, we investigate testing two fundamental properties of distributions: (1) testing the equivalence of two distributions when we have unequal numbers of samples from the two distributions. (2) Testing independence of two random variables. In both cases, we show that our testers achieve near optimal sample complexity (up to logarithmic factors). Moreover, our dependence on the privacy parameter is an additive term, which indicates that differential privacy can be obtained in most regimes of parameters for free.

We study the fundamental problems of identity and equivalence testing over a discrete population from random samples. Our goal is to develop efficient testers while guaranteeing differential privacy to the individuals of the population. We provide sample-efficient differentially private testers for these problems. Our theoretical results significantly improve over the best known algorithms for identity testing, and are the first results for private equivalence testing. The conceptual message of our work is that there exist private hypothesis testers that are nearly as sample-efficient as their non-private counterparts. We perform an experimental evaluation of our algorithms on synthetic data. Our experiments illustrate that our private testers achieve small type I and type II errors with sample size sublinear in the domain size of the underlying distributions.

Data visualization is an effective mechanism for identifying trends, insights, and anomalies in data. On large datasets, however, generating visualizations can take a long time, delaying the extraction of insights, hampering decision making, and reducing exploration time. One solution is to use online sampling-based schemes to generate visualizations faster while improving the displayed estimates incrementally, eventually converging to the exact visualization computed on the entire data. However, the intermediate visualizations are approximate, and often fluctuate drastically, leading to potentially incorrect decisions. We propose sampling-based incremental visualization algorithms that reveal the "salient" features of the visualization quickly---with a 46× speedup relative to baselines---while minimizing error, thus enabling rapid and error-free decision making. We demonstrate that these algorithms are optimal in terms of sample complexity, in that given the level of interactivity, they generate approximations that take as few samples as possible. We have developed the algorithms in the context of an incremental visualization tool, titled I nc V isage , for trendline and heatmap visualizations. We evaluate the usability of I nc V isage via user studies and demonstrate that users are able to make effective decisions with incrementally improving visualizations, especially compared to vanilla online-sampling based schemes.

We consider the problem of learning distributions in the presence of irrelevant features. This problem is formalized by introducing a new notion of \emphk-junta distributions. Informally, a distribution \mathcalD over the domain \mathcalX^n is a \emphk-junta distribution with respect to another distribution \mathcalU over the same domain if there is a set J ⊆[n] of size |J| \le k that captures the difference between \mathcal D and \mathcalU. We show that it is possible to learn k-junta distributions with respect to the uniform distribution over the Boolean hypercube {0,1}^n in time \poly(n^k, 1/ε). This result is obtained via a new Fourier-based learning algorithm inspired by the Low-Degree Algorithm of Linial, Mansour, and Nisan (1993). We also consider the problem of testing whether an unknown distribution is a k-junta distribution with respect to the uniform distribution. We give a nearly-optimal algorithm for this task. Both the analysis of the algorithm and the lower bound showing its optimality are obtained by establishing connections between the problem of testing junta distributions and testing uniformity of weighted collections of distributions.