This paper investigates the average-case time complexity of certifying RIP matrices.
problem Certifying the restricted isometry property (RIP) for large sparsity levels in random Gaussian matrices.
method Analysis of the low-degree likelihood ratio to determine the average-case time complexity.
result Subexponential runtime of NildeΩ(s2/M) is required for certifying RIP matrices. Tackles the computational hardness of HPC detection, conjecturing equivalence to PC detection.
problem Computational hardness of hypergraphic planted clique detection.
method No specific method mentioned; focuses on conjecturing equivalence.
result Equivalence of computational hardness between HPC and PC detection.
The restricted isometry property (RIP) for design matrices gives guarantees for optimal recovery in sparse linear models. It is of high interest in compressed sensing and statistical learning. This property is particularly important for computationally efficient recovery methods. As a consequence, even though it is in …
SLR tackles sparse linear regression problems, showing hardness for efficient algorithms.
problem Sparse linear regression with noisy data and k-sparse solutions.
method Reduction from lattice problems to SLR instances, showing hardness.
result Hardness of SLR instances, even for isotropic Gaussian design matrices.
This paper strengthens the computational separation between multimodal and unimodal learning, showing unimodal learning is hard on typical instances.
problem Theoretical justification for empirical success of multimodal machine learning.
method Introduced a stronger average-case computational separation between unimodal and multimodal learning.
result For typical instances, unimodal learning is computationally hard, while multimodal learning is easy.
Quantum circuits are hard to learn on average.
problem Learning the output distributions of quantum circuits is hard.
method Statistical query model analysis.
result Learning quantum circuits requires exponentially many queries.
This paper develops several average-case reduction techniques to show new hardness results for three central high-dimensional statistics problems, implying a statistical-computational gap induced by robustness, a detection-recovery gap and a universality principle for these gaps. A main feature of our approach is to ma…
New insights link diverse statistical problems via secret leakage planted clique.
problem Statistical-computational gaps in inference problems.
method Secret leakage planted clique as a new hardness assumption for reductions.
result Establishes tight statistical-computational tradeoffs for various problems.
This paper studies the problem of detecting the presence of a small dense community planted in a large Erdős-Rényi random graph G(N,q), where the edge probability within the community exceeds q by a constant factor. Assuming the hardness of the planted clique detection problem, we show that the computatio…
In the past decade, sparse principal component analysis has emerged as an archetypal problem for illustrating statistical-computational tradeoffs. This trend has largely been driven by a line of research aiming to characterize the average-case complexity of sparse PCA through reductions from the planted clique (PC) con…
New analysis shows halting time is predictable for large models, improving optimization efficiency.
problem Understanding the average-case complexity of optimization algorithms for large-scale models.
method Average-case analysis of first-order methods on random least squares and neural networks.
result Halting time is independent of input distribution, leading to tighter convergence rates.
Detects dense subhypergraphs in random hypergraphs using low-degree polynomials.
problem Detecting a planted dense subhypergraph in a random hypergraph model.
method Degree-n^o(1) polynomials of adjacency tensor entries.
result Thresholds for detection in different density regimes.
We investigate the average-case complexity of decision problems for finitely generated groups, in particular the word and membership problems. Using our recent results on ``generic-case complexity'' we show that if a finitely generated group G has the word problem solvable in subexponential time and has a subgroup of…
Graph neural networks struggle to distinguish certain graph structures.
problem Difficulty in distinguishing graphs with graph neural networks.
method Analysis of communication capacity in message-passing model of graph neural networks.
result Capacity of MPNN needs to grow linearly for trees and quadratically for general connected graphs.
Hardness proof for agnostically learning halfspaces from worst-case lattice problems.
problem Agnostically learning halfspaces in the presence of noise.
method Reduction to worst-case lattice problems (GapSVP, SIVP).
result No efficient algorithm can achieve misclassification error better than 1/2 - γ under given hardness assumptions.
We continue the study of statistical/computational tradeoffs in learning robust classifiers, following the recent work of Bubeck, Lee, Price and Razenshteyn who showed examples of classification tasks where (a) an efficient robust classifier exists, in the small-perturbation regime; (b) a non-robust classifier can be l…
Over recent years, devising classification algorithms that are robust to adversarial perturbations has emerged as a challenging problem. In particular, deep neural nets (DNNs) seem to be susceptible to small imperceptible changes over test instances. However, the line of work in provable robustness, so far, has been fo…
Average teaching complexity for locating target regions among halfspace intersections is Θ(d).
problem Teaching the location of a target region among intersections of halfspaces.
method Novel insights from computational geometry to count convex polytopes and faces.
result Average-case teaching complexity is Θ(d), contrasting with Θ(n) worst-case complexity.
The paper improves error bounds for Bayesian quadrature in noisy settings.
problem Improving error bounds for Bayesian quadrature in noisy settings.
method Develops a two-step meta-algorithm to relate average-case quadrature error to L2-function approximation error. result Provides new average-case results for various kernels and noise settings.
AMP algorithms can be efficiently simulated by SDPs even with corrupted data.
problem Optimizing average-case optimization problems with corrupted data.
method Local statistics hierarchy semidefinite programs (SDPs) simulate AMP algorithms robustly.
result Robust guarantees for many AMP algorithms are offered, contrasting with strong lower bounds for SDPs.
Pairwise comparison data arises in many domains, including tournament rankings, web search, and preference elicitation. Given noisy comparisons of a fixed subset of pairs of items, we study the problem of estimating the underlying comparison probabilities under the assumption of strong stochastic transitivity (SST). We…
Proposes a new framework for balancing average- and worst-case performance in machine learning.
problem Robustness issues in machine learning, especially in safety-critical domains.
method Probabilistic robustness framework that balances average- and worst-case performance.
result Effective algorithm balances average- and worst-case performance with lower computational cost.
Transformers excel at sparse token selection, surpassing FCNs in both worst and average cases.
problem Sparse token selection task
method One-layer transformer trained with gradient descent
result Transformers learn sparse token selection and exhibit strong out-of-distribution length generalization
Survey on using low-degree polynomials to assess statistical tasks complexity.
problem Understanding the complexity of statistical tasks using polynomial functions.
method Applying low-degree polynomials to measure the complexity of statistical tasks, including detection, recovery, and estimation.
result Low-degree polynomials provide a framework to predict and explain statistical-computational tradeoffs.
We study least squares linear regression over N uncorrelated Gaussian features that are selected in order of decreasing variance. When the number of selected features p is at most the sample size n, the estimator under consideration coincides with the principal component regression estimator; when p>n, the esti…
This paper establishes a statistical versus computational trade-off for solving a basic high-dimensional machine learning problem via a basic convex relaxation method. Specifically, we consider the {\em Sparse Principal Component Analysis} (Sparse PCA) problem, and the family of {\em Sum-of-Squares} (SoS, aka Lasserre/…
Study on InstaHide's security, linking to phase retrieval problem.
problem Security of InstaHide scheme for private dataset sharing.
method Design of a provable algorithm for private vector recovery.
result Private vectors can be recovered using synthetic vectors and public vectors.
New findings on tensor decomposition complexity, showing polynomial functions can estimate the largest component under certain conditions.
problem The complexity of tensor decomposition, especially for low-degree polynomials.
method Modeling a slightly larger component in a random tensor decomposition and using polynomial functions to estimate it.
result Polynomial functions can accurately estimate the largest component when r≪n3/2 but fail when r≫n3/2. New framework analyzes SGD dynamics in large samples and dimensions.
problem Analyzing stochastic gradient descent in large-scale settings.
method Inspired by random matrix theory, new framework for fixed stepsize and finite sum settings.
result SGD dynamics become deterministic in the large sample and dimensional limit, governed by a Volterra integral equation.
Analysis of momentum methods on quadratic models, showing SGD's superiority.
problem Analysis of stochastic gradient algorithms with momentum on quadratic models.
method Inspired by random matrix theory, exact characterization of loss values.
result Stochastic heavy-ball momentum does not improve over SGD in the strongly convex setting.
Options are generally learned by using an inaccurate environment model (or simulator), which contains uncertain model parameters. While there are several methods to learn options that are robust against the uncertainty of model parameters, these methods only consider either the worst case or the average (ordinary) case…
New findings show some designs can't be robustly recovered in linear regression.
problem Robust linear regression with adversarial corruption.
method Investigated well-spreadness of design matrices for robust recovery.
result Certified well-spreadness of random matrices efficiently, but hardness for small observations.
Barren plateaus are not an average-case phenomenon, but a highly non-unique problem.
problem Avoiding barren plateaus in neural network training
method First-moment framework for initialization strategies
result Many families of inequivalent initialization strategies can avoid concentration
The Lasso performs well in ultra-sparse linear models with finite support size.
problem Performance analysis of Lasso in ultra-sparse linear models.
method Novel application of replica method from statistical physics, rigorous analysis of average case performance.
result Average performance of Lasso assessed without scaling assumptions, offering sample complexity bounds.
New attacks exploit neural network energy and latency, increasing costs by 10-200x.
problem Energy and latency vulnerabilities in neural networks.
method Crafted sponge examples to maximize energy and latency.
result Adversaries can increase energy consumption by 10-200x and delay decisions.
Online TERM improves robustness and fairness in streaming data.
problem Streaming data's lack of worst-case fairness and robustness in ERM.
method Proposes an online TERM formulation to balance average-case accuracy with worst-case fairness and robustness.
result Negative tilting effectively suppresses outlier influence, positive tilting improves recall with minimal precision loss.
We augment adversarial training (AT) with worst case adversarial training (WCAT) which improves adversarial robustness by 11% over the current state-of-the-art result in the ℓ2 norm on CIFAR-10. We obtain verifiable average case and worst case robustness guarantees, based on the expected and maximum values of the…
Backward exploration reduces sample complexity in policy evaluation.
problem Empirical policy evaluation in reinforcement learning.
method Backward exploration algorithms from high-cost states.
result Reduced average-case sample complexity to O(logS). This work explores the trade-offs between stability and accuracy in statistical estimation.
problem Understanding the statistical cost of algorithmic stability.
method Statistical decision-theoretic perspective, focusing on worst-case and average-case stability.
result Optimal stable estimators for mean estimation and regression settings are developed, revealing trade-offs between stability and accuracy.
We develop a method to factorize symmetric sparse Boolean matrices efficiently.
problem Finding a symmetric factorization of a given matrix into a sparse, Boolean matrix.
method Polynomial-time algorithm based on bootstrapping higher-order information and tensor decomposition.
result A matrix with full column rank can be recovered with high probability when the matrix size is sufficiently large.
Estimates isotonic functions under unknown permutations, achieving optimal statistical and computational efficiency.
problem Estimating isotonic functions with unknown permutations in multiway comparison data.
method Mirsky partition estimator for minimax optimal and adaptive estimation.
result Achieves optimal worst-case statistical performance and computational efficiency.
This article attempts to place the emergence of probabilistic numerics as a mathematical-statistical research field within its historical context and to explore how its gradual development can be related both to applications and to a modern formal treatment. We highlight in particular the parallel contributions of Sul'…
New algorithms optimize spectral risk measures, improving interpolation between average and worst-case performance.
problem Optimizing spectral risk measures for learning systems.
method Developed stochastic algorithms to optimize spectral risk measures by characterizing their subdifferential and addressing challenges like biasedness of subgradient estimates and non-smoothness.
result Our approach outperforms out-of-the-box stochastic subgradient and dual averaging methods in optimizing spectral risk measures.
This article considers algorithmic and statistical aspects of linear regression when the correspondence between the covariates and the responses is unknown. First, a fully polynomial-time approximation scheme is given for the natural least squares optimization problem in any constant dimension. Next, in an average-case…
Lower class selectivity makes networks more robust to natural perturbations but more vulnerable to adversarial attacks.
problem Understanding how class selectivity affects robustness to different types of perturbations in neural networks.
method Investigated the relationship between class selectivity and robustness to natural and adversarial perturbations in neural networks.
result Lower class selectivity increases robustness to natural perturbations but decreases robustness to adversarial attacks.
Efficiently transforms Gaussian data to simulate various target distributions.
problem Generating observations from different target distributions given a single Gaussian observation.
method Designs computationally efficient procedures to approximate target distributions.
result Establishes reduction-based computational lower bounds for high-dimensional statistical models.
We build a theoretical framework for designing and understanding practical meta-learning methods that integrates sophisticated formalizations of task-similarity with the extensive literature on online convex optimization and sequential prediction algorithms. Our approach enables the task-similarity to be learned adapti…
A new algorithm avoids worst-case outcomes in risky contexts.
problem Risk-averse behavior in contextual bandits is challenging.
method Developed a first risk-averse contextual bandit algorithm with online regret guarantees.
result First algorithm with an online regret guarantee for risk-averse contextual bandits.