arXiv subjects

Clément L. Canonne

Publications and source records attributed to Clément L. Canonne.

At least 19 recordsLinked to original sources

arXiv2026

The Quantumly Fast and the Classically Forrious

We study the extremal Forrelation problem, where, provided with oracle access to Boolean functions $f$ and $g$ promised to satisfy either $\operatorname{forr}(f,g)=1$ or $\operatorname{forr}(f,g)=-1$, one must determine (with high probability) which of the two cases holds while performing as few oracle queries as possible. It is well known that this problem can be solved with one quantum query; yet, Girish and Servedio (ITCS 2026) recently showed this problem requires $\widetildeΩ(2^{n/4})$ classical queries, and conjectured the optimal bound to be $\widetildeΘ(2^{n/2})$. By generalizing their construction, we build on their result and prove a non-adaptive lower bound of $Ω(2^{(1/2- o(1))n})$, which matches the conjectured lower bound up to a vanishing constant in the exponent.

cs.CC↗

arXiv2026

NashPG: A Policy Gradient Method with Iteratively Refined Regularization for Finding Nash Equilibria

Finding Nash equilibria in two-player zero-sum imperfect-information games remains a central challenge in multi-agent reinforcement learning. Recent multi-round regularization methods offer a promising direction, yet existing approaches either require full enumeration of the game tree or rely on non-policy-gradient inner solvers that underperform in practice, leaving a scalable policy-gradient-based solution open. In this paper, we propose a novel multi-round regularization procedure and show that it guarantees strictly monotonic reduction in Bregman divergence to Nash equilibria and eventual convergence to one in two-player zero-sum extensive-form games. Guided by this framework, we develop a practical algorithm, Nash Policy Gradient (NashPG), which places the regularization directly in the policy optimization objective and is implemented using standard policy gradient methods. Empirically, NashPG achieves comparable or lower exploitability than prior model-free methods on classic benchmark games and scales to large domains such as Battleship and No-Limit Texas Hold'em, where it attains higher average payoff in head-to-head play.

cs.LG↗

arXiv2026

Entropy Equivalence Testing

We introduce the problem of \emph{entropy equivalence testing} for probability distributions, a relaxation of the well-studied closeness testing problem, where the distribution testing algorithm is now only required to distinguish, given samples from two unknown distributions $p,q$ and a parameter $\varepsilon \in(0,1/2]$, between $p=q$ and $|H(p)-H(q)| \geq \varepsilon$ (where $H$ denotes the Shannon entropy). We provide a time- and sample-efficient algorithm for this task, showing that the optimal sample complexity for this task can be significantly lower than that of closeness testing. As an application, we leverage this result to provide the first non-trivial testing algorithm for (standard) closeness of low-degree \emph{Bayesian networks}, which significantly improves on either the sample or time complexity of a baseline based on full learning.

cs.DS↗

arXiv2026

Distributed Gaussian Mean Testing under Communication Constraints: messages, samples, and coins

We revisit the problem of Gaussian mean testing in a distributed, communication constrained setting, where each of $n$ users independently observes samples from an unknown $d$-dimensional spherical Gaussian distribution $\mathcal{G}(μ,\mathbb{I}_d)$, and can communicate up to $\ell$ bits to a central referee. The referee's goal is then to distinguish between cases (i) $\|μ\|_2 = 0$ versus (ii) $\|μ\|_2\ge \varepsilon$. This problem has been considered in the private- and public-coin settings, when each user holds exactly one sample, or more generally when each holds exactly $m$ samples. In this work, we significantly generalize the question in three directions: when the users only share a small number $s$ of random bits, when each user holds a different number of samples $m_k$, and when each user can send a different number of bits $\ell_k$ to the referee.

cs.DS↗

arXiv2025

Uniformity Testing under User-Level Local Privacy

We initiate the study of distribution testing under \emph{user-level} local differential privacy, where each of $n$ users contributes $m$ samples from the unknown underlying distribution. This setting, albeit very natural, is significantly more challenging that the usual locally private setting, as for the same parameter $\varepsilon$ the privacy guarantee must now apply to a full batch of $m$ data points. While some recent work consider distribution \emph{learning} in this user-level setting, nothing was known for even the most fundamental testing task, uniformity testing (and its generalization, identity testing). We address this gap, by providing (nearly) sample-optimal user-level LDP algorithms for uniformity and identity testing. Motivated by practical considerations, our main focus is on the private-coin, symmetric setting, which does not require users to share a common random seed nor to have been assigned a globally unique identifier.

cs.DS↗

arXiv2025

With a Little Help From My Friends: Exploiting Probability Distribution Advice in Algorithm Design

We study online algorithms with predictions using distributional advice, a type of prediction that arises when leveraging expert knowledge or historical data. To demonstrate the usefulness and versatility of this framework, we focus on the fundamental problem of online metric matching, considering both the fractional and integral variants. Our main positive result is, for the former, an algorithm achieving the optimal cost under perfect advice, while smoothly defaulting to competitive ratios comparable to advice-free algorithms as the prediction's quality degrades. For the integral matching, we are able to provide an algorithm with essentially the same guarantees, up to an additive sublinear term. We conclude by discussing how our algorithmic framework can be extended to other online optimization problems.

cs.DS↗

arXiv2025

Instance-Optimal Uniformity Testing and Tracking

In the uniformity testing task, an algorithm is provided with samples from an unknown probability distribution over a (known) finite domain, and must decide whether it is the uniform distribution, or, alternatively, if its total variation distance from uniform exceeds some input distance parameter. This question has received a significant amount of interest and its complexity is, by now, fully settled. Yet, we argue that it fails to capture many scenarios of interest, and that its very definition as a gap problem in terms of a prespecified distance may lead to suboptimal performance. To address these shortcomings, we introduce the problem of uniformity tracking, whereby an algorithm is required to detect deviations from uniformity (however they may manifest themselves) using as few samples as possible, and be competitive against an optimal algorithm knowing the distribution profile in hindsight. Our main contribution is a $\operatorname{polylog}(\operatorname{opt})$-competitive uniformity tracking algorithm. We obtain this result by leveraging new structural results on Poisson mixtures, which we believe to be of independent interest.

cs.DS↗

arXiv2025

Zero-Shot Attribution for Large Language Models: A Distribution Testing Approach

A growing fraction of all code is sampled from Large Language Models (LLMs). We investigate the problem of attributing code generated by language models using hypothesis testing to leverage established techniques and guarantees. Given a set of samples $S$ and a suspect model $\mathcal{L}^*$, our goal is to assess the likelihood of $S$ originating from $\mathcal{L}^*$. Due to the curse of dimensionality, this is intractable when only samples from the LLM are given: to circumvent this, we use both samples and density estimates from the LLM, a form of access commonly available. We introduce $\mathsf{Anubis}$, a zero-shot attribution tool that frames attribution as a distribution testing problem. Our experiments on a benchmark of code samples show that $\mathsf{Anubis}$ achieves high AUROC scores ( $\ge0.9$) when distinguishing between LLMs like DeepSeek-Coder, CodeGemma, and Stable-Code using only $\approx 2000$ samples.

cs.LG↗

arXiv2025

Gaussian Mean Testing under Truncation

We consider the task of Gaussian mean testing, that is, of testing whether a high-dimensional vector perturbed by white noise has large magnitude, or is the zero vector. This question, originating from the signal processing community, has recently seen a surge of interest from the machine learning and theoretical computer science community, and is by now fairly well understood. What is much less understood, and the focus of our work, is how to perform this task under truncation: that is, when the observations (i.i.d.\ samples from the underlying high-dimensional Gaussian) are only observed when they fall in an given subset of the domain $\R^d$. This truncation model, previously studied in the context of learning (instead of testing) the mean vector, has a range of applications, in particular in Economics and Social Sciences. As our work shows, sample truncations affect the complexity of the testing task in a rather subtle and surprising way.

math.ST↗

arXiv2025

Local Computation Algorithms for Knapsack: impossibility results, and how to avoid them

Local Computation Algorithms (LCA), as introduced by Rubinfeld, Tamir, Vardi, and Xie (2011), are a type of ultra-efficient algorithms which, given access to a (large) input for a given computational task, are required to provide fast query access to a consistent output solution, without maintaining a state between queries. This paradigm of computation in particular allows for hugely distributed algorithms, where independent instances of a given LCA provide consistent access to a common output solution. The past decade has seen a significant amount of work on LCAs, by and large focusing on graph problems. In this paper, we initiate the study of Local Computation Algorithms for perhaps the archetypal combinatorial optimization problem, Knapsack. We first establish strong impossibility results, ruling out the existence of any non-trivial LCA for Knapsack as several of its relaxations. We then show how equipping the LCA with additional access to the Knapsack instance, namely, weighted item sampling, allows one to circumvent these impossibility results, and obtain sublinear-time and query LCAs. Our positive result draws on a connection to the recent notion of reproducibility for learning algorithms (Impagliazzo, Lei, Pitassi, and Sorrell, 2022), a connection we believe to be of independent interest for the design of LCAs.

cs.DS↗

arXiv2024

The Discrete Gaussian for Differential Privacy

A key tool for building differentially private systems is adding Gaussian noise to the output of a function evaluated on a sensitive dataset. Unfortunately, using a continuous distribution presents several practical challenges. First and foremost, finite computers cannot exactly represent samples from continuous distributions, and previous work has demonstrated that seemingly innocuous numerical errors can entirely destroy privacy. Moreover, when the underlying data is itself discrete (e.g., population counts), adding continuous noise makes the result less interpretable. With these shortcomings in mind, we introduce and analyze the discrete Gaussian in the context of differential privacy. Specifically, we theoretically and experimentally show that adding discrete Gaussian noise provides essentially the same privacy and accuracy guarantees as the addition of continuous Gaussian noise. We also present an simple and efficient algorithm for exact sampling from this distribution. This demonstrates its applicability for privately answering counting queries, or more generally, low-sensitivity integer-valued queries.

cs.DS↗

arXiv2024

Uniformity testing when you have the source code

We study quantum algorithms for verifying properties of the output probability distribution of a classical or quantum circuit, given access to the source code that generates the distribution. We consider the basic task of uniformity testing, which is to decide if the output distribution is uniform on $[d]$ or $ε$-far from uniform in total variation distance. More generally, we consider identity testing, which is the task of deciding if the output distribution equals a known hypothesis distribution, or is $ε$-far from it. For both problems, the previous best known upper bound was $O(\min\{d^{1/3}/ε^{2},d^{1/2}/ε\})$. Here we improve the upper bound to $O(\min\{d^{1/3}/ε^{4/3}, d^{1/2}/ε\})$, which we conjecture is optimal.

quant-ph↗

arXiv2024

Locally Private Histograms in All Privacy Regimes

Frequency estimation, a.k.a. histograms, is a workhorse of data analysis, and as such has been thoroughly studied under differentially privacy. In particular, computing histograms in the \emph{local} model of privacy has been the focus of a fruitful recent line of work, and various algorithms have been proposed, achieving the order-optimal $\ell_\infty$ error in the high-privacy (small $\varepsilon$) regime while balancing other considerations such as time- and communication-efficiency. However, to the best of our knowledge, the picture is much less clear when it comes to the medium- or low-privacy regime (large $\varepsilon$), despite its increased relevance in practice. In this paper, we investigate locally private histograms, and the very related distribution learning task, in this medium-to-low privacy regime, and establish near-tight (and somewhat unexpected) bounds on the $\ell_\infty$ error achievable. As a direct corollary of our results, we obtain a protocol for histograms in the \emph{shuffle} model of differential privacy, with accuracy matching previous algorithms but significantly better message and communication complexity. Our theoretical findings emerge from a novel analysis, which appears to improve bounds across the board for the locally private histogram problem. We back our theoretical findings by an empirical comparison of existing algorithms in all privacy regimes, to assess their typical performance and behaviour beyond the worst-case setting.

cs.DS↗

arXiv2024

Communication with Imperfectly Shared Randomness

The communication complexity of many fundamental problems reduces greatly when the communicating parties share randomness that is independent of the inputs to the communication task. Natural communication processes (say between humans) however often involve large amounts of shared correlations among the communicating players, but rarely allow for perfect sharing of randomness. Can the communication complexity benefit from shared correlations as well as it does from shared randomness? This question was considered mainly in the context of simultaneous communication by Bavarian et al. (ICALP 2014). In this work we study this problem in the standard interactive setting and give some general results. In particular, we show that every problem with communication complexity of $k$ bits with perfectly shared randomness has a protocol using imperfectly shared randomness with complexity $\exp(k)$ bits. We also show that this is best possible by exhibiting a promise problem with complexity $k$ bits with perfectly shared randomness which requires $\exp(k)$ bits when the randomness is imperfectly shared. Along the way we also highlight some other basic problems such as compression, and agreement distillation, where shared randomness plays a central role and analyze the complexity of these problems in the imperfectly shared randomness model. The technical highlight of this work is the lower bound that goes into the result showing the tightness of our general connection. This result builds on the intuition that communication with imperfectly shared randomness needs to be less sensitive to its random inputs than communication with perfectly shared randomness. The formal proof invokes results about the small-set expansion of the noisy hypercube and an invariance principle to convert this intuition to a proof, thus giving a new application domain for these fundamental results.

cs.CC↗

arXiv2024

Learning bounded-degree polytrees with known skeleton

We establish finite-sample guarantees for efficient proper learning of bounded-degree polytrees, a rich class of high-dimensional probability distributions and a subclass of Bayesian networks, a widely-studied type of graphical model. Recently, Bhattacharyya et al. (2021) obtained finite-sample guarantees for recovering tree-structured Bayesian networks, i.e., 1-polytrees. We extend their results by providing an efficient algorithm which learns $d$-polytrees in polynomial time and sample complexity for any bounded $d$ when the underlying undirected graph (skeleton) is known. We complement our algorithm with an information-theoretic sample complexity lower bound, showing that the dependence on the dimension and target accuracy parameters are nearly tight.

cs.LG↗

arXiv2023

Simpler Distribution Testing with Little Memory

We consider the question of distribution testing (specifically, uniformity and closeness testing) in the streaming setting, \ie under stringent memory constraints. We improve on the results of Diakonikolas, Gouleakis, Kane, and Rao (2019) by providing considerably simpler algorithms, which remove some restrictions on the range of parameters and match their lower bounds.

cs.DS↗

arXiv2023

Private independence testing across two parties

We introduce $π$-test, a privacy-preserving algorithm for testing statistical independence between data distributed across multiple parties. Our algorithm relies on privately estimating the distance correlation between datasets, a quantitative measure of independence introduced in Székely et al. [2007]. We establish both additive and multiplicative error bounds on the utility of our differentially private test, which we believe will find applications in a variety of distributed hypothesis testing settings involving sensitive data.

math.ST↗

arXiv2023

Private Distribution Testing with Heterogeneous Constraints: Your Epsilon Might Not Be Mine

Private closeness testing asks to decide whether the underlying probability distributions of two sensitive datasets are identical or differ significantly in statistical distance, while guaranteeing (differential) privacy of the data. As in most (if not all) distribution testing questions studied under privacy constraints, however, previous work assumes that the two datasets are equally sensitive, i.e., must be provided the same privacy guarantees. This is often an unrealistic assumption, as different sources of data come with different privacy requirements; as a result, known closeness testing algorithms might be unnecessarily conservative, "paying" too high a privacy budget for half of the data. In this work, we initiate the study of the closeness testing problem under heterogeneous privacy constraints, where the two datasets come with distinct privacy requirements. We formalize the question and provide algorithms under the three most widely used differential privacy settings, with a particular focus on the local and shuffle models of privacy; and show that one can indeed achieve better sample efficiency when taking into account the two different "epsilon" requirements.

cs.DS↗