arXiv subjects

Salil Vadhan

Publications and source records attributed to Salil Vadhan.

At least 19 recordsLinked to original sources

arXiv2026

Bounded-Independence Sampling of Edges for Combinatorial Graph Properties

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 $\geqκ\log(m)$ with probability $1-1/\mathrm{poly}(m)$ (for a sufficiently large constant $κ$). 2. $O(\log m)$-wise $(1/\mathrm{poly}(m))$-almost independence suffices for ensuring cycle-freeness when sampling at rate $1/2$ in a graph with minimum cycle length $\geqκ\log(m)$ with probability $1-1/\mathrm{poly}(m)$ (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 $\leq 1/2$ generated using $O(\log(m)\log\log(m))$ random bits such that in a graph with min cut $\geqκ\log(m)$, a sample from $X$ is still connected with probability $1-1/\mathrm{poly}(m)$. 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.

cs.DS↗

arXiv2026

Finding Nearly-Periodic Components in Digraphs and Markov Chains from the Spectrum of Rotated Laplacian Matrices

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 $p \in \mathbb{N}$ 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.

cs.DS↗

arXiv2026

Generalized and Unified Equivalences between Hardness and Pseudoentropy

Pseudoentropy characterizations give quantitatively precise formulations of the relationship between computational hardness and computational randomness. We prove a unified pseudoentropy characterization that generalizes and strengthens previous results in both uniform and nonuniform models of computation. Our characterization applies to a general family of entropy notions, including Shannon entropy and min-entropy as special cases. Moreover, the characterizations for these different entropy notions can be witnessed simultaneously by a single universal function, which captures both computational hardness and computational randomness. A key technical insight is that weight-restricted calibration, from the recent literature on algorithmic fairness, together with standard computational indistinguishability (known as multiaccuracy in the fairness literature), suffices for proving pseudoentropy characterizations for general entropy notions. To obtain this combination of properties, we prove an enhanced version of the Leakage Simulation Lemma (Jetchev and Pietrzak, 2014), which in turn extends the Complexity Theoretic-Regularity Lemma (Trevisan, Tulsiani, and Vadhan, 2009) from boolean functions to ones over a larger alphabet. Our Enhanced Regularity/Leakage-Simulation Lemma enables us to obtain an exponential improvement in the dependence on the alphabet size compared with the pseudoentropy characterizations of Casacuberta, Dwork, and Vadhan (2024), which are based on the stronger notion of multicalibration. We also show that this exponential dependence on the alphabet size is inevitable for multicalibration and even for the weaker notion of calibrated multiaccuracy.

cs.CC↗

arXiv2026

Tradeoffs in Privacy, Welfare, and Fairness for Facility Location

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.

cs.DS↗

arXiv2026

Concurrent Composition for Differentially Private Continual Mechanisms

Many intended uses of differential privacy involve a $\textit{continual mechanism}$ 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 $\textit{adaptive}$ 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 $\textit{concurrent}$ 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.

cs.DS↗

arXiv2025

Practitioners' Perspectives on a Differential Privacy Deployment Registry

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.

cs.CR↗

arXiv2025

Characterizing the Distinguishability of Product Distributions through Multicalibration

Given a sequence of samples $x_1, \dots , x_k$ promised to be drawn from one of two distributions $X_0, X_1$, a well-studied problem in statistics is to decide $\textit{which}$ 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 $X_0^{\otimes k}$ and $X_1^{\otimes k}$. However, when we restrict our attention to $\textit{efficient distinguishers}$ (i.e., small circuits) of these two distributions, exactly characterizing the ability to distinguish $X_0^{\otimes k}$ and $X_1^{\otimes k}$ 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 $\textit{information-theoretic}$ indistinguishability of some specific, related variables $\widetilde{X}_0$ and $\widetilde{X}_1$. As a consequence, we prove a new, tight characterization of the number of samples $k$ needed to efficiently distinguish $X_0^{\otimes k}$ and $X_1^{\otimes k}$ with constant advantage as \[ k = Θ\left(d_H^{-2}\left(\widetilde{X}_0, \widetilde{X}_1\right)\right), \] which is the inverse of the squared Hellinger distance $d_H$ between two distributions $\widetilde{X}_0$ and $\widetilde{X}_1$ 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.

cs.CR↗

arXiv2025

Sparsest cut and eigenvalue multiplicities on low degree Abelian Cayley graphs

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 $\text{SD}_\varepsilon(G)$: the smallest $k$ such that the subspace spanned by the first $k$ Laplacian eigenvectors contains all but $\varepsilon$ 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 $(1+\varepsilon)$-approximation to the uniform Sparsest Cut of a degree-$d$ Cayley graph over an Abelian group of size $n$ in time $n^{O(1)}\cdot\exp(d/\varepsilon)^{O(d)}$. 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 $\varepsilon$-net of size $\exp(d/\varepsilon)^{O(d)}$ and that the multiplicity of $λ_2$ is bounded by $2^{O(d)}$. The latter bound is tight and improves on a previous bound of $2^{O(d^2)}$ by Lee and Makarychev.

cs.DS↗

arXiv2025

Singular Values Versus Expansion in Directed and Undirected Graphs

We relate the nontrivial singular values $σ_2,\ldots,σ_n$ 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 $(1+δ)\cdot |S|$ 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.

math.CO↗

arXiv2025

Securing Unbounded Differential Privacy Against Timing Attacks

Recent works 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 (a) incurs a constant additive increase in error probability (and thus does not provide vanishing error as $n\to\infty$) (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. In this work, we overcome these limitations. Specifically, we show that the error required for pure JOT-DP in the unbounded setting depends on the model of computation. In a randomized RAM model where the dataset size $n$ is given (or can be computed in constant time) and we can generate random numbers (not just random bits) in constant time, polynomially small error probability is necessary and sufficient. If $n$ is not given or we only have a random-bit generator, an (arbitrarily small) constant error probability is necessary and sufficient. The aforementioned positive results are proven by efficient procedures to convert any pure JOT-DP program $P$ in the upper-bounded setting to a pure JOT-DP program $P'$ in the unbounded setting, such that the output distribution of $P'$ is $γ$-close in total variation distance to that of $P$, where $γ$ is either an arbitrarily small constant or polynomially small, depending on the model of computation.

cs.CR↗

arXiv2025

Concurrent Composition for Interactive Differential Privacy with Adaptive Privacy-Loss Parameters

In this paper, we study the concurrent composition of interactive mechanisms with adaptively chosen privacy-loss parameters. In this setting, the adversary can interleave queries to existing interactive mechanisms, as well as create new ones. We prove that every valid privacy filter and odometer for noninteractive mechanisms extends to the concurrent composition of interactive mechanisms if privacy loss is measured using $(ε, δ)$-DP, $f$-DP, or Rényi DP of fixed order. Our results offer strong theoretical foundations for enabling full adaptivity in composing differentially private interactive mechanisms, showing that concurrency does not affect the privacy guarantees. We also provide an implementation for users to deploy in practice.

cs.CR↗

arXiv2024

Differentially Private Release and Learning of Threshold Functions

We prove new upper and lower bounds on the sample complexity of $(ε, δ)$ differentially private algorithms for releasing approximate answers to threshold functions. A threshold function $c_x$ over a totally ordered domain $X$ evaluates to $c_x(y) = 1$ if $y \le x$, and evaluates to $0$ otherwise. We give the first nontrivial lower bound for releasing thresholds with $(ε,δ)$ differential privacy, showing that the task is impossible over an infinite domain $X$, and moreover requires sample complexity $n \ge Ω(\log^*|X|)$, which grows with the size of the domain. Inspired by the techniques used to prove this lower bound, we give an algorithm for releasing thresholds with $n \le 2^{(1+ o(1))\log^*|X|}$ samples. This improves the previous best upper bound of $8^{(1 + o(1))\log^*|X|}$ (Beimel et al., RANDOM '13). Our sample complexity upper and lower bounds also apply to the tasks of learning distributions with respect to Kolmogorov distance and of properly PAC learning thresholds with differential privacy. The lower bound gives the first separation between the sample complexity of properly learning a concept class with $(ε,δ)$ differential privacy and learning without privacy. For properly learning thresholds in $\ell$ dimensions, this lower bound extends to $n \ge Ω(\ell \cdot \log^*|X|)$. To obtain our results, we give reductions in both directions from releasing and properly learning thresholds and the simpler interior point problem. Given a database $D$ of elements from $X$, the interior point problem asks for an element between the smallest and largest elements in $D$. We introduce new recursive constructions for bounding the sample complexity of the interior point problem, as well as further reductions and techniques for proving impossibility results for other basic problems in differential privacy.

cs.CR↗

arXiv2024

A Framework for Differential Privacy Against Timing Attacks

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.

cs.CR↗

arXiv2024

"I inherently just trust that it works": Investigating Mental Models of Open-Source Libraries for Differential Privacy

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.

cs.HC↗

arXiv2024

Membership Inference Attacks and Privacy in Topic Modeling

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.

cs.CR↗

arXiv2024

Complexity-Theoretic Implications of Multicalibration

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.

cs.CC↗

arXiv2024

Programming Frameworks for Differential Privacy

Many programming frameworks have been introduced to support the development of differentially private software applications. In this chapter, we survey some of the conceptual ideas underlying these frameworks in a way that we hope will be helpful for both practitioners and researchers. For practitioners, the survey can provide a starting point for understanding what features may be valuable when selecting a programming framework. For researchers, it can help organize existing work in a unified way and provide context for understanding new features in future frameworks.

cs.CR↗

arXiv2024

Analyzing the Differentially Private Theil-Sen Estimator for Simple Linear Regression

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.

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