The conjectured upper bound of k statistical queries on a universe of size T can be released under pure differential privacy with expected worst-coordinate error at the square-root rate suggested by known lower bounds is proved.
Abstract
Nikolov and Ullman asked whether k statistical queries on a universe of size T can be released under pure differential privacy with expected worst-coordinate error at the square-root rate suggested by known lower bounds. We prove their conjectured upper bound. For every database size n and privacy parameter $\varepsilon>0$, there is an $\varepsilon$-differentially private mechanism with expected error $O(\min\{1,\sqrt{\log(2T)\log(2k)/(\varepsilon n)}\})$. This matches the lower-bound dependence in the standard high-dimensional regimes where those bounds apply; the shifted logarithms and outer minimum make the upper bound valid without additional parameter assumptions. The construction starts from a selection-only private multiplicative weights transcript, then replaces its probability mass function by a distance-penalized likelihood envelope. To prove that the modification preserves accuracy, a likelihood-level Maurey argument upper-bounds each Hamming-ball maximum by a small family of auxiliary PMW laws. Renyi moment bounds control nearby balls, a direct mixture bound controls distant balls, and grouping radii at the privacy scale prevents an additional $1/\varepsilon$ factor in the error. The mechanism is information-theoretic. A companion Lean 4 development machine-checks the finite construction, pure privacy after deterministic decoding, and the displayed all-regimes upper bound.
This paper investigates the direct sum question for expected randomized and distributional query complexity. Our main result gives an exact characterization of the amortized expected randomized query complexity. For any total relation $f$ and any error tolerance $\varepsilon \in [0,1]$, we prove \[ \lim_{n \to \infty} \frac{\overline{R}_\varepsilon(f^n)}{n} = (1 - \varepsilon) \overline{R}_0(f). \] Thus the amortization converts bounded-error into zero error with the exact multiplicative factor $1-\varepsilon$. We also prove corresponding liminf/limsup bounds for worst-case randomized and distributional query complexity. These results improve prior direct-sum bounds that were known only up to constant factors or in restricted error regimes, and they resolve an open question posed by Blais and Brody (2019). Additionally for one-sided computation of the function $\operatorname{OR}_n \circ f$, we obtain analogous exact amortized identities for both expected and worst-case cost. As applications, we obtain separations between amortized and single-instance costs, including unbounded separations for distributional complexity and randomized relations, and a quadratic barrier for randomized total functions.
Continual counting under pure differential privacy is one of the simplest and most well-studied problems in the continual observation model. Nevertheless, an asymptotic gap remains between the best known upper and lower bounds for maximum squared error and mean squared error: the upper bound is $O(\epsilon^{-2}\log^3 n)$, while the lower bound is $\Omega(\epsilon^{-2}\log^2 n)$, for both error metrics. The best known constant in the upper bound is achieved by the $k$-ary tree mechanism with the subtraction trick, due to Andersson, Pagh, Steiner, and Torkamani (FORC 2025). In this work, we improve the leading constant in the maximum squared error and the mean squared error. Our approach uses a general matrix factorization mechanism, yielding an improved bound for pure-DP continual counting that does not rely on a tree-based construction. The mechanism starts from a good-quality low-dimensional factorization, obtained via gradient-based optimization, and gives an explicit matrix construction that lifts this factorization to arbitrarily large dimensions, further improving its error guarantees. We offer an efficient algorithmic implementation of our mechanism. On the lower-bound side, we prove an $\Omega(\epsilon^{-2}\log^3 n)$ lower bound for the class of factorizations whose matrices have entries in $\{0,1\}$, matching the upper-bound asymptotics for this class. This class includes the binary tree mechanism and $k$-ary tree mechanisms without the subtraction trick. Extending this lower bound to arbitrary matrix factorizations, and beyond the matrix mechanism altogether, remains an open problem.
We study distributed testing of $\mathrm{Ber}(\alpha)$ versus $\mathrm{Ber}(\beta)$ in the broadcast, or shared-blackboard, model. For protocols with constant advantage, we characterise up to universal constant factors the information complexity under either hypothesis for every pair $\beta<\alpha$. The characterisation shows that the two information costs can be quite different and identifies three parameter regimes, with optimal protocols based respectively on clean samples, a noisy binary symmetric channel, and an asymmetric $Z$-channel. The lower bounds rely on a novel mixed Hellinger--Jensen--Shannon inequality that may be of independent interest. We also characterise the constant-advantage information complexity of testing arbitrary discrete distributions via an optimisation problem over channels, and show that binary-output channels suffice. We obtain bounds for bounded likelihood-ratio distributions, and give general upper bounds in terms of $\chi^2$ divergence. As applications, we recover the broadcast-model set-disjointness lower bound, and derive stronger lower bounds in the multi-pass streaming setting for some problems considered in prior work.
We study the problem of releasing counting-query outputs through a stochastic mechanism that is both consistent and \((\epsilon,\delta)\)-differentially private. Consistency requires the released value to lie within the feasible range of the query, while utility is measured by the worst-case probability of error. We first derive a closed-form expression for the minimum achievable error probability and obtain an explicit optimal mechanism. By exploiting the active differential privacy constraints satisfied by this mechanism, we then characterize the entire class of optimal mechanisms via a propagation argument, identifying the structural properties shared by all optimizers. We next extend the framework to the setting in which the privacy mechanism is cascaded with an arbitrary fixed stochastic transformation representing a predetermined portion of the communication medium between the source and the destination. We first establish necessary and sufficient conditions under which this partial fixation of the medium incurs no loss in utility. We then derive upper and lower bounds on the optimal achievable performance based on convex mixing and spectral perturbation. Finally, we apply the theory to (M)-ary phase-shift keying (PSK) transmission over an additive white Gaussian noise (AWGN) channel and show that uncoded transmission is effectively optimal in the high-privacy regime.
Pujol and Desfontaines asked whether a private histogram can allow more error on larger counts and use that slack to protect members of larger groups more strongly. We study this question for fixed disjoint groups under add-or-remove-one adjacency. The privacy budget $v(n)$ depends on the affected count, is nonincreasing, and must bound both R\'enyi-divergence directions at every order. This is the count-dependent form of zero-concentrated differential privacy (zCDP) studied here. The original strict relative-error condition is impossible at count zero. We therefore make the boundary tolerance explicit by requiring $\mathbb{E}\lvert\widehat{x}_i-x_i\rvert<r\max\{x_i,1\}$, without changing the requirement at any positive count. Our main result determines the best dependence on group size. For the upper bound, we directly specialize an existing shifted-transformation framework. The resulting shifted-log Gaussian mechanism has a certified budget $v(n)=O_r(n^{-2})$. Conversely, for every fixed $0<r<1$, any mechanism satisfying the same positive-count utility requirement and count-dependent zCDP must have $v(n)=\Omega_r(n^{-2})$. Thus the inverse-square rate is optimal under the repaired formulation. A many-count information argument further places the leading coefficient in the large-count-then-small-error limit between $\pi/(4e^2)$ and $1/\pi$, a factor below three. At $r=1$, a data-independent release meets the repaired criterion with zero privacy loss.
We characterize the exact rate--exponent tradeoff for new classes of one-way distributed hypothesis testing problems by demonstrating that a recent upper bound, derived via the auxiliary-receiver technique, coincides with known lower bounds. We achieve this by relaxing the upper bound on the type-II error exponent into a form that shares the same inner functional as Han's lower bound, differing only in the outer rate constraint. Furthermore, we prove that this upper bound is tight for testing against dependence and for the doubly symmetric binary source (DSBS) with crossover probabilities $\kappa_0$ under the null hypothesis and $\kappa_1$ under the alternative hypothesis, provided $0<\kappa_1<\kappa_0<\frac12$. The characterization of the exact error exponent for the DSBS source holds for every communication rate $R \geq 0$.
Zhenduo Wen, Amin Gohari, Michèle A. Wigger· 1 citation
