This problem involves identifying a density function that accurately represents the distribution of a given dataset.

Hypothesis selection, also known as density estimation, is a fundamental problem in statistics and learning theory. Suppose we are given a sample set from an unknown distribution $P$ and a finite class of candidate distributions (called hypotheses) $\mathcal{H} \coloneqq \{H_1, H_2, \ldots, H_n\}$. The aim is to design an algorithm that selects a distribution $\hat H$ in $\mathcal{H}$ that best fits the data. The algorithm's accuracy is measured based on the distance between $\hat{H}$ and $P$ compared to the distance of the closest distribution in $\mathcal{H}$ to $P$ (denoted by $OPT$).

Hypothesis selection is a fundamental problem in learning theory and statistics. Given a dataset and a finite set of candidate distributions, the goal is to select a distribution that matches the data as well as possible. More specifically, suppose we have sample access to an unknown distribution $P$ over a domain $\mathcal{X}$ that we know is well-approximated by one of a a class of $n$ distributions (a.k.a.

In the hypothesis selection problem, we are given sample and query access to finite set of candidate distributions (hypotheses), $\mathcal{H} = \{H_1, \ldots, H_n\}$, and samples from an unknown distribution $P$, both over a domain $\mathcal{X}$. The goal is to output a distribution $Q$ whose distance to $P$ is comparable to that of the nearest hypothesis in $\mathcal{H}$. Specifically, if the minimum distance is $\mathsf{OPT}$, we aim to output $Q$ such that, with probability at least $1-δ$, its total variation distance to $P$ is at most $C \cdot \mathsf{OPT} + \varepsilon$. The optimal approximation for proper algorithms (where $Q \in \mathcal{H}$) is $C=3$ using $Θ(\log(n/δ)/\varepsilon^2)$ samples from $P$ and for improper algorithms (where $Q$ is not necessarily in $\mathcal{H}$) is $C=2$ using $\tildeΘ(\log(n/δ)/\varepsilon^2)$ samples from $P$. In the improper setting, the algorithm achieving $C=2$ [Bousquet, Braverman, Kol, Efremenko, Moran, FOCS 2021] runs in time which grows polynomially with $|\mathcal{X}|$ -- it does not run in finite time for real-valued distributions. A promising path towards improved runtime is to consider improper algorithms which output a mixture $Q$ of the hypotheses as such a distribution can be represented in $n$ words of memory. We show (1) a lower bound that no algorithm which outputs a mixture can achieve approximation better than $C = 3-2/n$ unless the number of samples is polynomial in $|\mathcal{X}|$, as well as (2) an algorithm which runs in time $\text{poly}(n)$ and achieves the same approximation guarantee. In the proper setting, [Aliakbarpour, Bun, Smith, NeurIPS 2024] provided an algorithm with $C=3$ running in $\tilde{O}(n/(δ^3\varepsilon^3))$ time. We improve this time complexity to $\tilde{O}(n/(δ\varepsilon^2))$, significantly reducing the dependence on the confidence and error parameters.

Multimodal abductive reasoning--the generation and selection of explanatory hypotheses from partial observations--is a cornerstone of intelligence. Current evaluations of this ability in vision-language models (VLMs) are largely confined to static, single-agent tasks. Inspired by Dixit, we introduce DixitWorld, a comprehensive evaluation suite designed to deconstruct this challenge. DIXITWORLD features two core components: DixitArena, a dynamic, multi-agent environment that evaluates both hypothesis generation (a "storyteller" crafting cryptic clues) and hypothesis selection ("listeners" choosing the target image from decoys) under imperfect information; and DixitBench, a static QA benchmark that isolates the listener's task for efficient, controlled evaluation. Results from DixitArena reveal distinct, role-dependent behaviors: smaller open-source models often excel as creative storytellers, producing imaginative yet less discriminative clues, whereas larger proprietary models demonstrate superior overall performance, particularly as listeners. Performance on DixitBench strongly correlates with listener results in DixitArena, validating it as a reliable proxy for hypothesis selection. Our findings reveal a key trade-off between generative creativity and discriminative understanding in multimodal abductive reasoning, a central challenge for developing more balanced and capable vision-language agents.

Neural Information Processing Systems http://nips.cc/

We propose an algorithm with improved query-complexity for the problem of hypothesis selection under local differential privacy constraints. Given a set of $k$ probability distributions $Q$, we describe an algorithm that satisfies local differential privacy, performs $\tilde{O}(k^{3/2})$ non-adaptive queries to individuals who each have samples from a probability distribution $p$, and outputs a probability distribution from the set $Q$ which is nearly the closest to $p$. Previous algorithms required either $Ω(k^2)$ queries or many rounds of interactive queries. Technically, we introduce a new object we dub the Scheffé graph, which captures structure of the differences between distributions in $Q$, and may be of more broad interest for hypothesis selection tasks.

Estimating the density of a distribution from its samples is a fundamental problem in statistics. Hypothesis selection addresses the setting where, in addition to a sample set, we are given $n$ candidate distributions -- referred to as hypotheses -- and the goal is to determine which one best describes the underlying data distribution. This problem is known to be solvable very efficiently, requiring roughly $O(\log n)$ samples and running in $\tilde{O}(n)$ time. The quality of the output is measured via the total variation distance to the unknown distribution, and the approximation factor of the algorithm determines how large this distance is compared to the optimal distance achieved by the best candidate hypothesis. It is known that $α= 3$ is the optimal approximation factor for this problem. We study hypothesis selection under the constraint of differential privacy. We propose a differentially private algorithm in the central model that runs in nearly-linear time with respect to the number of hypotheses, achieves the optimal approximation factor, and incurs only a modest increase in sample complexity, which remains polylogarithmic in $n$. This resolves an open question posed by [Bun, Kamath, Steinke, Wu, NeurIPS 2019]. Prior to our work, existing upper bounds required quadratic time.

Hypothesis selection is a fundamental problem in learning theory and statistics. Given a dataset and a finite set of candidate distributions, the goal is to select a distribution that matches the data as well as possible. More specifically, suppose we have sample access to an unknown distribution P over a domain \mathcal{X} that we know is well-approximated by one of a a class of n distributions (a.k.a. The goal is to design an algorithm that outputs a distribution \hat{H} \in \mathcal{H} whose total variation distance from P is nearly minimal.In this work, we study the hypothesis selection problem under memory constraints. We consider a model where samples from P are presented in a stream and we access each sample x via PDF-comparison'' queries that allow us to compare the probability densities of any pair of hypothesesat the domain point x (i.e., is H_i(x) H_j(x)?).