arXiv subjects

Bardiya Aryanfard

Publications and source records attributed to Bardiya Aryanfard.

3 recordsLinked to original sources

arXiv2026

Connectivity in a New Bounded-Space Dynamic Model

We introduce a new model for dynamic algorithms in which the algorithm has read-only oracle access to a dynamically changing input and random access to its own bounded workspace. It is a dynamic analogue of the time-space tradeoffs for static inputs studied in complexity theory, where only writable memory counts as space. We study dynamic undirected $s$-$t$ connectivity: after every edge update to an $n$-vertex graph, an algorithm with $S$ words of workspace must report whether $s$ and $t$ are connected. The graph is accessed via adjacency list queries (degree, $j$-th neighbor) or adjacency matrix queries. Our algorithms are randomized; our lower bounds hold for randomized algorithms with $cn$ bits of workspace, $c>0$ a small constant ($cn/\log n$ for the worst-case matrix bound). We prove: (1) With adjacency lists, worst-case time $\tildeΘ(n^2/S^2)$ per insertion is necessary and sufficient for incremental algorithms. (2) With an adjacency matrix, worst-case time $\tildeΩ(n^2/S)$ per insertion is necessary for incremental algorithms, and sufficient for $S\ge\sqrt{n}$. (3) With amortization, incremental algorithms achieve $\tilde O(n/S)$ time per insertion with adjacency lists and $\tilde O(n^2/S)$ with an adjacency matrix, while $\tildeΩ(\sqrt{n/S})$ and $\tildeΩ(n\sqrt{n/S})$, respectively, are necessary. (4) Fully dynamic algorithms need amortized time $\tildeΩ(n/S)$ per update with adjacency lists and $\tildeΩ(n^2/S)$ with an adjacency matrix, even on forests, where both bounds are tight (lists: under bounded degree). Conventional dynamic connectivity admits polylogarithmic update time with $Ω(n)$ words of space. In contrast, our lower bounds show that polynomial update time is necessary whenever $S\le n^{1-\varepsilon}$, even for a single pair of vertices and even for incremental algorithms with amortized guarantees.

cs.DS↗

arXiv2026

On Differentially Private Mechanisms for Linear Regression

We study the classic least-squares linear regression problem under differential privacy (DP) in the batch and the continual observation setting. In the latter setting, the mechanism receives a stream of $n$ data points $(x_i,y_i)$ ($x_i\in\mathbb{R}^d$ and $y_i\in\mathbb{R}$) and must output an estimate of the optimal parameters at each time step $i$. Although differentially private linear regression has been extensively studied in the batch setting, the problem remains underexplored for the continual setting, with prior work being restricted to constrained linear regression and update time polynomial in $n$. In this work, we introduce ContinualAdaSSP, the first mechanism for unconstrained linear regression under continual differential privacy. Our mechanism's suboptimality gap is proportional to $\sqrt{d}\operatorname{polylog}(n)$. It requires $O(d^2\log{n})$ space and $O(d^2\log{n}+d^ω)$ update time, where $d$ is the dimension of the input space. We also study fundamental limits of differentially private linear regression by showing the first lower bounds for the box-constrained least-squares linear regression problem. Our lower bounds are proportional to $\min(n,\frac{\sqrt{d}}{\log^{1.5}d})$ for $(1,O(\frac{1}{n}))$-DP, and $\min(n,\frac{d}ε)$ for $ε$-DP, and are derived using a reduction from a fingerprinting code problem from [Peter et al., 2024], providing new insights into the inherent hardness of this canonical task. Our experimental evaluation on real-world datasets shows that the accuracy of our continual observation mechanism clearly outperforms any other known differentially private mechanism for this problem such as rerunning the batch algorithm at each time step.

cs.LG↗

arXiv2025

Improved Lower Bounds for Privacy under Continual Release

We study the problem of continually releasing statistics of an evolving dataset under differential privacy. In the event-level setting, we show the first polynomial lower bounds on the additive error for insertions-only graph problems such as maximum matching, degree histogram and $k$-core. This is an exponential improvement on the polylogarithmic lower bounds of Fichtenberger et al.[ESA 2021] for the former two problems, and are the first continual release lower bounds for the latter. Our results run counter to the intuition that the difference between insertions-only vs fully dynamic updates causes the gap between polylogarithmic and polynomial additive error. We show that for maximum matching and $k$-core, allowing small multiplicative approximations is what brings the additive error down to polylogarithmic. Beyond graph problems, our techniques also show that polynomial additive error is unavoidable for Simultaneous Norm Estimation in the insertions-only setting. When multiplicative approximations are allowed, we circumvent this lower bound by giving the first continual mechanism with polylogarithmic additive error under $(1+ζ)$ multiplicative approximations, for $ζ>0$, for estimating all monotone symmetric norms simultaneously. In the item-level setting, we show polynomial lower bounds on the product of the multiplicative and the additive error of continual mechanisms for a large range of graph problems. To the best of our knowledge, these are the first lower bounds for any differentially private continual release mechanism with multiplicative error. To obtain this, we prove a new lower bound on the product of multiplicative and additive error for 1-Way-Marginals, from which we reduce to continual graph problems. This generalizes the lower bounds of Hardt and Talwar[STOC 2010] and Bun et al.[STOC 2014] on the additive error for mechanisms with no multiplicative error.

cs.DS↗