Inspired by recent advances in notions of spectral approximation of digraphs [Ahm+20], we study spectral algorithms for finding periodic structures in digraphs via the spectrum of a class of rotated Laplacian matrices. This class of Laplacian matrices was previously studied by Lange, Liu, Peyerimhoff, and Post [Lan+15]. We consider a notion of periodicity ratio that generalizes the bipartiteness ratio of Trevisan [Tre09], and show that it is closely related to the spectrum of rotated Laplacian matrices. In particular, if the digraph is strongly connected and represents a Markov chain, this periodicity ratio for a given p ∈ℕ is a quantitative measure of how close this Markov chain is to having periodicity p. We propose and analyze a periodicity-ratio variant of the spectral algorithm by Louis, Raghavendra, Tetali and Vempala [Lou+12]. We show that the algorithm runs in randomized polynomial time and can find many nearly periodic components (i.e, components with small periodicity ratio). This also implies a new higher-order Cheeger-type inequality for periodicity in the spirit of that in [Lou+12; LOT14]. As part of our analysis, we prove a new theorem that upper bounds the probability that the largest magnitudes of two sequences of coordinate-wise correlated complex Gaussian random variables occur at different indices, which may be of independent interest. Previously, an analogous result was known only for real Gaussian random variables.

Random subsampling of edges is a commonly employed technique in graph algorithms, underlying a vast array of modern algorithmic breakthroughs. Unfortunately, using this technique often leads to randomized algorithms with no clear path to derandomization because the analyses rely on a union bound on exponentially many events. In this work, we revisit this goal of derandomizing randomized sampling in graphs. We give several results related to bounded-independence edge subsampling, and in the process of doing so, generalize several of the results of Alon and Nussboim (FOCS 2008), who studied bounded-independence analogues of random graphs (which can be viewed as edge subsamples of the complete graph). Most notably, we show that in graphs with m edges: 1. O(log m)-wise independence suffices for preserving connectivity when sampling at rate 1/2 in a graph with min cut log(m) with probability 1-1/poly(m) (for a sufficiently large constant κ). 2. O(log m)-wise (1/poly(m))-almost independence suffices for ensuring cycle-freeness when sampling at rate 1/2 in a graph with minimum cycle length log(m) with probability 1-1/poly(m) (for a sufficiently large constant κ). 3. If we relax to arbitrary distributions, we show there is an explicit distribution X on {0, 1}^m with marginals ≤ 1/2 generated using O(log(m)loglog(m)) random bits such that in a graph with min cut log(m), a sample from X is still connected with probability 1-1/poly(m). To demonstrate the utility of our results, we revisit the problem of using parallel algorithms to find graphic matroid bases, first studied by Karp, Upfal, and Wigderson (FOCS 1985). We show that the optimal algorithms of Khanna, Putterman, and Song (arxiv 2025) can be explicitly derandomized while maintaining near-optimality.

Many intended uses of differential privacy involve a continual mechanism that is set up to run continuously over a long period of time, making more statistical releases as either queries come in or the dataset is updated. In this paper, we give the first general treatment of privacy against adaptive adversaries for mechanisms that support dataset updates and a variety of queries, all arbitrarily interleaved. It also models a very general notion of neighboring, that includes both event-level and user-level privacy. We prove several concurrent composition theorems for continual mechanisms, which ensure privacy even when an adversary can interleave queries and dataset updates to the different composed mechanisms. Previous concurrent composition theorems for differential privacy were only for the case when the dataset is static, with no adaptive updates. Moreover, we also give the first interactive and continual generalizations of the "parallel composition theorem" for noninteractive differential privacy. Specifically, we show that the analogue of the noninteractive parallel composition theorem holds if either there are no adaptive dataset updates or each of the composed mechanisms satisfies pure differential privacy, but it fails to hold for composing approximately differentially private mechanisms with dataset updates. We then formalize a set of general conditions on a continual mechanism M that runs multiple continual sub-mechanisms such that the privacy guarantees of M follow directly using the above concurrent composition theorems on the sub-mechanisms, without further privacy loss. This enables us to give a simpler and more modular privacy analysis of a recent continual histogram mechanism of Henzinger, Sricharan, and Steiner. In the case of approximate DP, ours is the first proof showing that its privacy holds against adaptive adversaries.

The differentially private (DP) facility location problem seeks to determine a socially optimal placement for a public facility while ensuring that each participating agent's location remains private. To privatize its input data, a DP mechanism must inject noise into its output distribution, producing a placement that will have lower expected social welfare than the optimal spot for the facility. The privacy-induced welfare loss can be viewed as the "cost of privacy," illustrating a tradeoff between social welfare and privacy that has been the focus of prior work. Yet, the imposition of privacy also induces a third consideration that has not been similarly studied: fairness in how the "cost of privacy" is distributed across individuals. For instance, a mechanism may satisfy DP with minimal social welfare loss, yet still be undesirable if that loss falls entirely on one individual. In this paper, we quantify this new notion of unfairness and design mechanisms for facility location that attempt to simultaneously optimize across privacy, social welfare, and fairness. We first derive an impossibility result, showing that privacy and fairness cannot be simultaneously guaranteed over all possible datasets that could represent the locations of individuals in a population. We then consider a relaxation that still requires worst-case DP, but only seeks fairness and social welfare over smaller, more "realistic-looking" families of datasets. For this relaxation, we construct a DP mechanism and demonstrate that it is simultaneously optimal (or, for a harder family of datasets, near-optimal up to small factors) on fairness and social welfare. This suggests that while there is a tradeoff between privacy and each of social welfare and fairness, there is no additional tradeoff when we consider all three objectives simultaneously, provided that the population data is sufficiently natural.

To promote shared learning and accountability around differential privacy (DP) deployments, Dwork, Kohli, and Mulligan [1] proposed a public-facing repository (“registry”) of real-world DP deployments. The DP community has recently started working toward realizing this vision. We contribute to this effort by (1) developing a holistic, hierarchical schema to describe any given DP deployment and (2) designing and implementing an interactive interface to act as a registry where practitioners can access information about past DP deployments. We (3) populate our interface with 21 real-world DP deployments and (4) conduct an exploratory user study with DP practitioners ($\mathbf{n} \boldsymbol{=} \mathbf{1 6}$) to understand how they would use the registry, as well as what challenges and opportunities they foresee around its adoption. We find that participants were enthusiastic about the registry as a valuable resource for evaluating prior deployments and making future deployments. They also identified several opportunities for the registry, including that it can become a “hub” for the community and support broader communication around DP (e.g., to legal teams). At the same time, they identified challenges around the registry gaining adoption, including the effort and risk involved with making implementation choices public and moderating the quality of entries. Based on our findings, we offer recommendations for encouraging adoption and increasing the registry's value not only to DP practitioners, but also to policymakers, data users, and data subjects.

A recent paper of Braverman, Cohen, and Garg (STOC'18) introduced the concept of a weighted pseudorandom generator (WPRG), which amounts to a pseudorandom generator (PRG) whose outputs are accompanied with (potentially negative) real coefficients that scale the acceptance probabilities of any potential distinguisher. They gave an explicit construction of WPRGs for ordered branching programs whose seed length has a better dependence on the error parameter s than the classic PRG construction of Nisan (STOC'90 and Combinatorica 1992). In this article we give an explicit construction of WPRGs that achieve parameters that are impossible to achieve by a PRG. In particular, we construct a WPRG for ordered permutation branching programs of unbounded width with a single accept state that has seed length O(log(3/2) n) for error parameter epsilon = 1/poly(n), where n is the input length. In contrast, recent work of Hoza et al. (ITCS'21) shows that any PRG for this model requires seed length Omega(log(2) =) to achieve error epsilon= 1/poly(=). As a corollary, we obtain explicit WPRGs with seed length O(log(3/2 )=) and error s = 1/poly(=) for ordered permutation branching programs of width F = poly(=) with an arbitrary number of accept states. Previously, seed length o(log(2 )=) was only known when both the width and the reciprocal of the error are subpolynomial, i. e., omega = n(0(1)) and epsilon = 1/n(0(1)) (Braverman, Rao, Raz, Yehudayoff, FOCS'10 and SICOMP 2014). The starting point for our work was the recent family of space-efficient algorithms for estimating random-walk probabilities in directed graphs by Ahmadinejad, Kelner, Murtagh, Peebles, Sidford, and Vadhan (FOCS'20), which are based on spectral graph theory and space-efficient Laplacian solvers. We interpret these algorithms as giving WPRGs with large seed length, which we then derandomize to obtain our results. We also note that this approach gives a simpler proof of the original result of Braverman, Cohen, and Garg, as independently discovered by Cohen, Doron, Renard, Sberlo, and Ta-Shma (CCC'21).

We relate the nontrivial singular values σ_2,…,σ_n of the normalized adjacency matrix of an Eulerian directed graph to combinatorial measures of graph expansion: 1. We introduce a new directed analogue of conductance ϕ_dir, and prove a Cheeger-like inequality showing that ϕ_dir is bounded away from 0 iff σ_2 is bounded away from 1. In undirected graphs, this can be viewed as a unification of the standard Cheeger Inequality and Trevisan's Cheeger Inequality for the smallest eigenvalue. 2. We prove a singular-value analogue of the Higher-Order Cheeger Inequalities, giving a combinatorial characterization of when σ_k is bounded away from 1. 3. We tighten the relationship between σ_2 and vertex expansion, proving that if a d-regular graph G with the property that all sets S of size at most n/2 have at least (1+δ)· |S| out-neighbors, then 1-σ_2=Ω(δ^2/d). This bound is tight and saves a factor of d over the previously known relationship.

Differential privacy (DP) – a principled approach to producing statistical data products with strong, mathematically provable privacy guarantees for the individuals in the underlying dataset – has seen substantial adoption in practice over the past decade. Applying DP requires making several implementation decisions, each with significant impacts on data privacy and/or utility. Hence, to promote shared learning and accountability around DP deployments, Dwork, Kohli, and Mulligan (2019) proposed a public-facing repository ("registry") of DP deployments. The DP community has recently started to work toward realizing this vision. We contribute to this effort by (1) developing a holistic, hierarchical schema to describe any given DP deployment and (2) designing and implementing an interactive interface to act as a registry where practitioners can access information about past DP deployments. We (3) populate our interface with 21 real-world DP deployments and (4) conduct an exploratory user study with DP practitioners (n=16) to understand how they would use the registry, as well as what challenges and opportunities they foresee around its adoption. We find that participants were enthusiastic about the registry as a valuable resource for evaluating prior deployments and making future deployments. They also identified several opportunities for the registry, including that it can become a "hub" for the community and support broader communication around DP (e.g., to legal teams). At the same time, they identified challenges around the registry gaining adoption, including the effort and risk involved with making implementation choices public and moderating the quality of entries. Based on our findings, we offer recommendations for encouraging adoption and increasing the registry's value not only to DP practitioners, but also to policymakers, data users, and data subjects.

Pseudoentropy characterizations provide a quantitatively precise demonstration of the close relationship between computational hardness and computational randomness. We prove a unified pseudoentropy characterization that generalizes and strengthens previous results for both uniform and non-uniform models of computation. Our characterization holds for a general family of entropy notions that encompasses the common notions of Shannon entropy and min entropy as special cases. Moreover, we show that the characterizations for different entropy notions can be simultaneously achieved by a single, universal function that simultaneously witnesses computational hardness and computational randomness. A key technical insight of our work is that the notion of weight-restricted calibration from the recent literature on algorithm fairness, along with standard computational indistinguishability (known as multiaccuracy in the fairness literature), suffices for proving pseudoentropy characterizations for general entropy notions. This demonstrates the power of weight-restricted calibration to enhance the classic Complexity-Theoretic Regularity Lemma (Trevisan, Tulsiani, and Vadhan, 2009) and Leakage Simulation Lemma (Jetchev and Pietrzak, 2014) and allows us to achieve an exponential improvement in the complexity dependency on the alphabet size compared to the pseudoentropy characterizations by Casacuberta, Dwork, and Vadhan (2024) based on the much stronger notion of multicalibration. We show that the exponential dependency on the alphabet size is inevitable for multicalibration as well as for the weaker notion of calibrated multiaccuracy.

Whether or not the Sparsest Cut problem admits an efficient O(1)-approximation algorithm is a fundamental algorithmic question with connections to geometry and the Unique Games Conjecture. Revisiting spectral algorithms for Sparsest Cut, we present a novel, simple algorithm that combines eigenspace enumeration with a new algorithm for the Cut Improvement problem. The runtime of our algorithm is parametrized by a quantity that we call the solution dimension SD epsilon(G): the smallest k such that the subspace spanned by the first k Laplacian eigenvectors contains all but epsilon fraction of a sparsest cut. Our algorithm matches the guarantees of prior methods based on the threshold-rank paradigm, while also extending beyond them. To illustrate this, we study its performance on low degree Cayley graphs over Abelian groups - canonical examples of graphs with poor expansion properties. We prove that low degree Abelian Cayley graphs have small solution dimension, yielding an algorithm that computes a (1 + epsilon)-approximation to the uniform Sparsest Cut of a degree- d Cayley graph over an Abelian group of size n in time n(O(1)) center dot exp{(d/epsilon)(O(d))}. Along the way to bounding the solution dimension of Abelian Cayley graphs, we analyze their sparse cuts and spectra, proving that the collection of O(1)-approximate sparsest cuts has an e-net of size exp{(d/epsilon)(O(d))} and that the multiplicity of lambda(2) is bounded by 2(O(d)). The latter bound is tight and improves on a previous bound of 2(O(d2)) by Lee and Makarychev.

Recent works [9, 25] have started to theoretically investigate how we can protect differentially private programs against timing attacks, by making the joint distribution the output and the runtime differentially private (JOT-DP). However, the existing approaches to JOT-DP have some limitations, particularly in the setting of unbounded DP (which protects the size of the dataset and applies to arbitrarily large datasets). First, the known conversion of pure DP programs to pure JOT-DP programs in the unbounded setting [9] (a) incurs a constant additive increase in error probability (and thus does not provide vanishing error as n→∞ ) (b) produces JOT-DP programs that fail to preserve the computational efficiency of the original pure DP program and (c) is analyzed in a toy computational model in which the runtime is defined to be the number of coin flips. For approximate JOT-DP, an efficient conversion with vanishing error in the RAM model is known [17, 25], but only applies to programs that run in O(n) time on datasets of size n, as linear runtime is implied by “timing stability,” the timing analogue of global sensitivity. In this work, we overcome these limitations. Specifically:

In this paper, we study differentially private point and confidence interval estimators for simple linear regression. Motivated by recent work that highlights the strong empirical performance of an algorithm based on robust statistics, DPTheilSen, we provide a rigorous, finite-sample analysis of its privacy and accuracy properties, offer guidance on setting hyperparameters, and show how to produce differentially private confidence intervals to accompany its point estimates.

We initiate the study of the randomness complexity of differential privacy, i.e., how many random bits an algorithm needs in order to generate accurate differentially private releases. As a test case, we focus on the task of releasing the results of d counting queries, or equivalently all one-way marginals on a d-dimensional dataset with boolean attributes. While standard differentially private mechanisms for this task have randomness complexity that grows linearly with d, we show that, surprisingly, only log(2) d + O(1) random bits (in expectation) suffice to achieve an error that depends polynomially on d (and is independent of the size n of the dataset), and furthermore this is possible with pure, unbounded differential privacy and privacy-loss parameter epsilon = 1/ poly(d). Conversely, we show that at least log(2) d - O(1) random bits are also necessary for nontrivial accuracy, even with approximate, bounded DP, provided the privacy-loss parameters satisfy epsilon, delta <= 1/ poly(d). We obtain our results by establishing a close connection between the randomness complexity of differentially private mechanisms and the geometric notion of "deterministic rounding schemes" recently introduced and studied by Vander Woude et al. (2022, 2023).

Given a sequence of samples x(1),..., x(k) promised to be drawn from one of two distributions X-0, X-1, a well-studied problem in statistics is to decide which distribution the samples are from. Information theoretically, the maximum advantage in distinguishing the two distributions given k samples is captured by the total variation distance between X-circle times k (0) and X-circle times k (1). However, when we restrict our attention to efficient distinguishers (i.e., small circuits) of these two distributions, exactly characterizing the ability to distinguish X-circle times k (0) and X-circle times k (1) is more involved and less understood. In this work, we give a general way to reduce bounds on the computational indistinguishability of X-0 and X-1 to bounds on the information-theoretic indistinguishability of some specific, related variables X-0(similar to) and X-1(similar to). As a consequence, we prove a new, tight characterization of the number of samples k needed to efficiently distinguish X-circle times k (0) and X-circle times k (1) with constant advantage as k = Theta (d(-2) (H) (X-0(similar to), X-1(similar to)) ), which is the inverse of the squared Hellinger distance d(H) between two distributions X-0(similar to) and X-1(similar to) that are computationally indistinguishable from X-0 and X-1. Likewise, our framework can be used to re-derive a result of Halevi and Rabin (TCC 2008) and Geier (TCC 2022), proving nearly-tight bounds on how computational indistinguishability scales with the number of samples for arbitrary product distributions. At the heart of our work is the use of the Multicalibration Theorem (Hebert-Johnson, Kim, Reingold, Rothblum 2018) in a way inspired by recent work of Casacuberta, Dwork, and Vadhan (STOC 2024). Multicalibration allows us to relate the computational indistinguishability of X-0, X-1 to the statistical indistinguishability of X-0(similar to), X-1(similar to) (for lower bounds on k) and construct explicit circuits to distinguish between X-0(similar to), X-1(similar to) and consequently X-0, X-1 (for upper bounds on k).

Differential privacy (DP) is a promising framework for privacy-preserving data science, but recent studies have exposed challenges in bringing this theoretical framework for privacy into practice. These tensions are particularly salient in the context of open-source software libraries for DP data analysis, which are emerging tools to help data stewards and analysts build privacy-preserving data pipelines for their applications. While there has been significant investment into such libraries, we need further inquiry into the role of these libraries in promoting understanding of and trust in DP, and in turn, the ways in which design of these open-source libraries can shed light on the challenges of creating trustworthy data infrastructures in practice. In this study, we use qualitative methods and mental models approaches to analyze the differences between conceptual models used to design open-source DP libraries and mental models of DP held by users. Through a two-stage study design involving formative interviews with 5 developers of open-source DP libraries and user studies with 17 data analysts, we find that DP libraries often struggle to bridge the gaps between developer and user mental models. In particular, we highlight the tension DP libraries face in maintaining rigorous DP implementations and facilitating user interaction. We conclude by offering practical recommendations for further development of DP libraries.

The standard definition of differential privacy (DP) ensures that a mechanism's output distribution on adjacent datasets is indistinguishable. However, real-world implementations of DP can, and often do, reveal information through their runtime distributions, making them susceptible to timing attacks. In this work, we establish a general framework for ensuring differential privacy in the presence of timing side channels. We define a new notion of timing privacy, which captures programs that remain differentially private to an adversary that observes the program's runtime in addition to the output. Our framework enables chaining together component programs that are timing-stable followed by a random delay to obtain DP programs that achieve timing privacy. Importantly, our definitions allow for measuring timing privacy and output privacy using different privacy measures. We illustrate how to instantiate our framework by giving programs for standard DP computations in the RAM and Word RAM models of computation. Furthermore, we show how our framework can be realized in code through a natural extension of the OpenDP Programming Framework.

Recent research shows that large language models are susceptible to privacy attacks that infer aspects of the training data. However, it is unclear if simpler generative models, like topic models, share similar vulnerabilities. In this work, we propose an attack against topic models that can confidently identify members of the training data in Latent Dirichlet Allocation. Our results suggest that the privacy risks associated with generative modeling are not restricted to large neural models. Additionally, to mitigate these vulnerabilities, we explore differentially private (DP) topic modeling. We propose a framework for private topic modeling that incorporates DP vocabulary selection as a pre-processing step, and show that it improves privacy while having limited effects on practical utility.

We present connections between the recent literature on multigroup fairness for prediction algorithms and classical results in computational complexity. Multiaccurate predictors are correct in expectation on each member of an arbitrary collection of pre-specified sets. Multicalibrated predictors satisfy a stronger condition: they are calibrated on each set in the collection. Multiaccuracy is equivalent to a regularity notion for functions defined by Trevisan, Tulsiani, and Vadhan (2009). They showed that, given a class $F$ of (possibly simple) functions, an arbitrarily complex function $g$ can be approximated by a low-complexity function $h$ that makes a small number of oracle calls to members of $F$, where the notion of approximation requires that $h$ cannot be distinguished from $g$ by members of $F$. This complexity-theoretic Regularity Lemma is known to have implications in different areas, including in complexity theory, additive number theory, information theory, graph theory, and cryptography. Starting from the stronger notion of multicalibration, we obtain stronger and more general versions of a number of applications of the Regularity Lemma, including the Hardcore Lemma, the Dense Model Theorem, and the equivalence of conditional pseudo-min-entropy and unpredictability. For example, we show that every boolean function (regardless of its hardness) has a small collection of disjoint hardcore sets, where the sizes of those hardcore sets are related to how balanced the function is on corresponding pieces of an efficient partition of the domain.

We give a deterministic, nearly logarithmic-space algorithm for mild spectral sparsification of undirected graphs. Given a weighted, undirected graph $G$ on $n$ vertices described by a binary string of length $N$, an integer $k\leq \log n$, and an error parameter $\epsilon > 0$, our algorithm runs in space $\tilde{O}(k\log (N\cdot w_{\mathrm{max}}/w_{\mathrm{min}}))$ where $w_{\mathrm{max}}$ and $w_{\mathrm{min}}$ are the maximum and minimum edge weights in $G$, and produces a weighted graph $H$ with $\tilde{O}(n^{1+2/k}/\epsilon^2)$ edges that spectrally approximates $G$, in the sense of Spielmen and Teng [ST04], up to an error of $\epsilon$. Our algorithm is based on a new bounded-independence analysis of Spielman and Srivastava's effective resistance based edge sampling algorithm [SS08] and uses results from recent work on space-bounded Laplacian solvers [MRSV17]. In particular, we demonstrate an inherent tradeoff (via upper and lower bounds) between the amount of (bounded) independence used in the edge sampling algorithm, denoted by $k$ above, and the resulting sparsity that can be achieved.