Paper proves first non-trivial PTF testing lower bounds for NGCA.
problem Proving lower bounds against PTF tests is challenging.
method Developed tools to prove PTF testing lower bounds for NGCA.
result First non-trivial PTF testing lower bounds for NGCA.
Optimal sequential testing for Markovian data with lower and upper bounds.
problem Sequential hypothesis testing for Markovian data.
method Non-asymptotic lower bounds and optimal test design.
result Optimal test matches lower bound asymptotically.
Paper develops a framework to derive lower bounds on FDR and FNR in multiple testing.
problem Addressing the tradeoffs between FDR and FNR in multiple testing.
method Derandomization approach to derive lower bounds for various models.
result Meta-theorem providing a general recipe for obtaining lower bounds.
In this work, we consider the sample complexity required for testing the monotonicity of distributions over partial orders. A distribution p p p over a poset is monotone if, for any pair of domain elements x x x and y y y such that x ⪯ y x \preceq y x ⪯ y , p ( x ) ≤ p ( y ) p(x) \leq p(y) p ( x ) ≤ p ( y ) . To understand the sample complexity of this problem, we intro…
New bounds for sequential tests under power-one error levels.
problem Determining stopping times for sequential tests with power-one error levels.
method Proved two lower bounds for stopping times under specific conditions.
result Upper and lower bounds for sequential tests are shown to be tight.
Paper studies signal detection in noisy environments with limited communication.
problem Signal detection in Gaussian noise with 1-bit communication constraints.
method Derives lower bounds and exhibits optimal testing strategies.
result Optimal distributed testing strategies attain the derived lower bound.
A new method uses randomized trials to estimate the strength of unobserved confounding.
problem Unobserved confounding compromises causal conclusions from non-randomized studies.
method Designs a statistical test to detect unobserved confounding strength and estimates a lower bound.
result Estimates an asymptotically valid lower bound on unobserved confounding strength.
New lower bounds for linear classification problems in high dimensions.
problem Linear classification problems in high-dimensional spaces.
method Reduction from hardness conjectures for Affine Degeneracy testing and k-Sum problems.
result Matching lower bounds of Ω(n^d) and respectively Ω(1/ε^d) for Maximum Halfspace Discrepancy problem.
Study robust hypothesis testing under Hellinger distance, proving lower bounds and providing tests.
problem Testing close variants of specified distributions robustly to Hellinger distance.
method Lower bound on slack factor, testing with Hellinger balls, symmetric chi-squared distance analysis.
result Lower bound on slack factor quantifies robustness under misspecification.
In statistical inference problems, we wish to obtain lower bounds on the minimax risk, that is to bound the performance of any possible estimator. A standard technique to obtain risk lower bounds involves the use of Fano's inequality. In an information-theoretic setting, it is known that Fano's inequality typically doe…
Transductive learning considers a training set of m m m labeled samples and a test set of u u u unlabeled samples, with the goal of best labeling that particular test set. Conversely, inductive learning considers a training set of m m m labeled samples drawn iid from P ( X , Y ) P(X,Y) P ( X , Y ) , with the goal of best labeling any future sample…
Paper tests DPPs for diversity models, distinguishing them from other distributions.
problem Testing whether a given distribution is a Determinantal Point Process (DPP) or far from any DPP.
method Proposes the first algorithm for DPP testing and establishes a lower bound on sample complexity.
result Establishes a matching lower bound on the sample complexity of DPP testing.
Linear independence testing is a fundamental information-theoretic and statistical problem that can be posed as follows: given n n n points { ( X i , Y i ) } i = 1 n \{(X_i,Y_i)\}^n_{i=1} {( X i , Y i ) } i = 1 n from a p + q p+q p + q dimensional multivariate distribution where X i ∈ R p X_i \in \mathbb{R}^p X i ∈ R p and Y i ∈ R q Y_i \in\mathbb{R}^q Y i ∈ R q , determine whether a T X a^T X a T X and b T Y b^T Y b T Y are uncorrela…
Improved GNN simulation of WL test with exponentially lower complexity.
problem Improving the complexity of simulating the Weisfeiler-Lehman test with GNNs.
method Exponentially lower complexity simulation of WL test using GNNs with polylogarithmic parameters and O(log n) bits feature vectors.
result Near-optimal construction with logarithmic lower bounds for feature vector length and neural network size.
The paper designs tests for comparing ranked preference data and finds significant differences.
problem Comparing pairwise comparison and ranking data in various applications.
method Developed two-sample tests for pairwise comparison and ranking data, proving upper and lower bounds.
result Upper and lower bounds show tightness of the proposed tests, and significant differences in preferences were found.
Study on policy testing in MDPs with lower bounds and new algorithm.
problem Deciding if policy value exceeds a threshold with limited samples.
method Derived lower bound, proposed new algorithm, reformulated problem, used policy optimization in reversed MDP.
result New algorithm outperforms existing methods in policy testing.
We study the fundamental tradeoffs between computational tractability and statistical accuracy for a general family of hypothesis testing problems with combinatorial structures. Based upon an oracle model of computation, which captures the interactions between algorithms and data, we establish a general lower bound tha…
Upper and lower bounds on regret for noisy optimization of Brownian motion.
problem Optimizing a one-dimensional Brownian motion with noisy observations.
method Upper bound uses confidence bounds and Markov property; lower bound uses hypothesis testing reduction.
result Upper and lower bounds are tight up to a factor of O ( ( log T ) 1.5 ) O((\log T)^{1.5}) O (( log T ) 1.5 ) . The study examines conditions for achieving a simple lower bound in estimating mean from samples.
problem Achieving a simple lower bound for estimating the mean of a distribution.
method Analyzes conditions for nearly attaining Le Cam's two-point testing lower bound for mean estimation.
result An algorithm nearly attains the two-point testing rate for mixtures of symmetric, log-concave distributions with a common mean.
Optimal distributed testing under communication constraints with shared randomness.
problem Signal detection in a distributed system with limited communication.
method Derivation of minimax testing errors, distributed testing algorithms, and theoretical lower bounds.
result Consistent nonparametric distributed testing is possible even with minimal communication.
We show that for the problem of testing if a matrix A ∈ F n × n A \in F^{n \times n} A ∈ F n × n has rank at most d d d , or requires changing an ε ε ε -fraction of entries to have rank at most d d d , there is a non-adaptive query algorithm making O ~ ( d 2 / ε ) \widetilde{O}(d^2/ε) O ( d 2 / ε ) queries. Our algorithm works for any field F F F . This improves upon the previous…
Gradient descent fails to learn simple neural networks efficiently.
problem Learning one-layer neural networks efficiently using gradient descent.
method Gradient descent and statistical query algorithms.
result Superpolynomial lower bounds for learning one-layer neural networks.
Investigates tight PAC-Bayes bounds for small datasets.
problem Tightening PAC-Bayes bounds for small data.
method Generic PAC-Bayes theorem, meta-learning, synthetic tasks.
result PAC-Bayes bounds are competitive with Chernoff bounds but not as tight.
New method tests causal relationships from data without needing to learn the entire graph.
problem Testing if a causal graph belongs to a specific Markov equivalence class from observational data.
method Established bounds on the number of independence tests required and provided an algorithm that matches these bounds.
result Testing requires exponentially less independence tests compared to learning, especially in graphs with high in-degrees and small clique sizes.
Detecting a planted submatrix in random matrices with non-asymptotic methods.
problem Detecting a planted submatrix in random matrices with non-zero entries.
method Established minimax lower bounds and derived optimal tests for distinguishing the null and alternative hypotheses.
result Non-asymptotic upper and lower bounds match for any configuration of matrix dimensions.
In the context of sparse principal component detection, we bring evidence towards the existence of a statistical price to pay for computational efficiency. We measure the performance of a test by the smallest signal strength that it can detect and we propose a computationally efficient method based on semidefinite prog…
A new test optimizes detecting small communities in large networks.
problem Detecting small communities in large networks.
method Using Sinkhorn's theorem and a degree-corrected block model (DCBM), the study optimizes the SgnQ test for this challenging setting.
result The SgnQ test is optimal for detecting communities larger than √n, achieving the computational lower bound (CLB).
This work sets theoretical limits on meta-learning performance.
problem Understanding the difficulty of adapting machine learning models to real-world data distributions.
method Information-theoretic lower bounds on convergence rates for meta-learning algorithms.
result Theoretical bounds on parameter estimation error for hierarchical Bayesian models of meta-learning.
Statistical query algorithms and low-degree tests are nearly equivalent in high-dimensional hypothesis testing.
problem High-dimensional hypothesis testing and information-computation gaps.
method Analysis of statistical query framework and low-degree polynomials.
result Statistical query algorithms and low-degree polynomials are almost equivalent in power under mild conditions.
The paper improves regret lower bounds for communicating MDPs.
problem Regret lower bounds for communicating MDPs.
method Lower bound proof and optimization problem formulation.
result Regret lower bound becomes significantly more complex in communicating MDPs.
Develops a hypothesis testing framework for generalized Thurstone models.
problem Determining whether pairwise comparison data fits a generalized Thurstone model.
method Introduces separation distance and derives upper and lower bounds for testing.
result Critical threshold for testing depends on observation graph topology and scales as Θ ( ( n k ) − 1 / 2 ) Θ((nk)^{-1/2}) Θ (( nk ) − 1/2 ) for complete graphs. We propose a new family of combinatorial inference problems for graphical models. Unlike classical statistical inference where the main interest is point estimation or parameter testing, combinatorial inference aims at testing the global structure of the underlying graph. Examples include testing the graph connectivity…
Stochastic Bayesian Neural Network improves scalability and performance.
problem Challenges in calculating posterior distribution in Bayesian Neural Networks.
method Maximizes Evidence Lower Bound using Stochastic Evidence Lower Bound objective function.
result Demonstrates improved performance and scalability over previous algorithms.
New SQ lower bounds show learning mixtures of bounded covariance Gaussians is hard.
problem Learning mixtures of Gaussians with bounded covariance matrices is hard.
method Statistical Query (SQ) lower bounds.
result Any SQ algorithm requires complexity at least d Ω ( 1 / ε ) d^{Ω(1/ε)} d Ω ( 1/ ε ) for learning mixtures of bounded covariance Gaussians. Lower bounds on Bayes risk for realizable models derived using information theory.
problem Deriving lower bounds on Bayes risk for realizable machine learning models.
method Information-theoretic analysis using rate-distortion theory and mutual information.
result Lower bounds on Bayes risk for realizable models, matching known bounds up to logarithmic factors.
The main result of this paper shows that "test configurations" give new lower bounds on the L 2 L^{2} L 2 norm of the scalar curvature on a Kahler manifold. This is closely analogous to the analysis of the Yang-Mills functional over Riemann surfaces by Atiyah and Bott. The proof uses asymptotic approximation by finite-dimens…
Sharp bounds derived for test error of finite-rank kernel ridge regression.
problem Loose bounds on test error for finite-rank kernels in machine learning.
method Sharp non-asymptotic upper and lower bounds for KRR test error.
result Tighter bounds on finite-rank KRR test error, valid for any regularization parameters.
Survey on statistical inference under memory constraints.
problem Effect of memory limitations on statistical inference performance.
method Review of state-of-the-art in several canonical problems.
result Identification of fundamental building blocks and useful techniques.
The paper develops a new method to test if two multidimensional distributions are equivalent or significantly different.
problem Testing equivalence of multidimensional distributions with sub-linear sample complexity.
method Uses generalized A_k distance and Ramsey theory to develop a computationally efficient closeness tester.
result First sub-linear sample complexity closeness tester for multidimensional distributions.
New bounds for Neyman-Pearson region using f f f -divergences.
problem Bounding the Neyman-Pearson region for hypothesis testing.
method Establishing novel lower and upper bounds using f f f -divergences. result Best possible lower bound for the Neyman-Pearson boundary using hockey-stick f f f -divergences. The aim of this paper is give a simple proof of some results in \cite{Jun Ling-2006-IJM} and \cite{JunLing-2007-AGAG}, which are very deep studies in the sharp lower bound of the first eigenvalue in the Laplacian operator on compact Riemannian manifolds with nonnegative Ricci curvature. We also get a result about lower…
We formalize the problem of detecting a community in a network into testing whether in a given (random) graph there is a subgraph that is unusually dense. We observe an undirected and unweighted graph on N nodes. Under the null hypothesis, the graph is a realization of an Erdös-Rényi graph with probability p0. Under th…
New method uses almost orthonormal bases to prove low-degree lower bounds in complex statistical models.
problem Proving statistical-computational gaps in high-dimensional models with planted structures.
method Constructing an almost orthonormal polynomial basis under the planted distribution.
result Established new low-degree lower bounds for various complex models.
We partially confirm an old conjecture of Donaldson that if there exists a cscK metrics in a given Kähler class, then there is no degenerated geodesic ray which is tamed by a bounded ambient geometry unless it parallels to a holomorphic line consists of cscK metrics only. We also prove that for simple test configuratio…
Robust covariance testing requires significantly more samples in contaminated data.
problem Testing the covariance matrix of a high-dimensional Gaussian in the presence of contamination.
method We study the problem in the Huber's contamination model, distinguishing between the identity matrix and matrices far from it in Frobenius norm.
result The sample complexity of covariance testing increases dramatically to Ω ( d 2 ) Ω(d^2) Ω ( d 2 ) in the contaminated setting. Study designs experiments to identify causal graph structure with cycles and latent confounders.
problem Identify causal graph structure with cycles and latent confounders.
method Established lower bounds, developed CI and do see tests algorithms, and proved tightness.
result Proposed algorithms can recover all causal edges except for double adjacent bidirected edges.
Sum-of-Squares lower bound shows NGCA requires more samples than known algorithms.
problem Finding a non-Gaussian direction in a high-dimensional dataset.
method Sum-of-Squares (SoS) framework to prove lower bounds.
result First super-constant degree SoS lower bound for NGCA.
No algorithm outperforms uniform sampling in A/B testing.
problem Identifying the best arm in A/B testing with fixed budget.
method Introducing consistent and stable algorithms, deriving lower bounds, and proving optimality of uniform sampling.
result No algorithm performs better than uniform sampling in A/B testing.