Paper connects free-energy and low-degree hardness in high-dimensional statistics.
problem High-dimensional statistical inference problems are computationally hard.
method Defines a free-energy criterion and connects it to low-degree hardness.
result Establishes connection between free-energy and low-degree hardness for Gaussian models.
The paper explores when linear system identification is hard or easy, especially for under-actuated systems.
problem Statistical hardness of learning linear systems, especially under-actuated or under-excited systems.
method Using tools from minimax theory and recent statistical tools for finite sample analysis of system identification.
result The controllability index of linear systems affects the sample complexity of identification, making some systems hard to learn.
Binary classification is a common statistical learning problem in which a model is estimated on a set of covariates for some outcome indicating the membership of one of two classes. In the literature, there exists a distinction between hard and soft classification. In soft classification, the conditional class probabil…
We consider stochastic gradient descent (SGD) for least-squares regression with potentially several passes over the data. While several passes have been widely reported to perform practically better in terms of predictive performance on unseen data, the existing theoretical analysis of SGD suggests that a single pass i…
Paper explores limits of high-order clustering with planted structures.
problem Statistical and computational limits of high-order clustering with planted structures.
method Developed methods for detection and recovery of clusters, identified signal-to-noise ratio boundaries.
result Sharp boundaries of signal-to-noise ratio for statistical and computational feasibility.
The use of M-estimators in generalized linear regression models in high dimensional settings requires risk minimization with hard L0 constraints. Of the known methods, the class of projected gradient descent (also known as iterative hard thresholding (IHT)) methods is known to offer the fastest and most scalable sol…
Combinatorial auctions are formulated as frustrated lattice gases on sparse random graphs, allowing the determination of the optimal revenue by methods of statistical physics. Transitions between computationally easy and hard regimes are found and interpreted in terms of the geometric structure of the space of solution…
In this article supervised learning problems are solved using soft rule ensembles. We first review the importance sampling learning ensembles (ISLE) approach that is useful for generating hard rules. The soft rules are then obtained with logistic regression from the corresponding hard rules. In order to deal with the p…
Study reward-free RL in non-linear settings, improving efficiency and removing assumptions.
problem Improving sample efficiency in reward-free reinforcement learning for non-linear function approximation.
method Proposed RFOLIVE algorithm for minimal structural assumptions, analyzed hardness results for reward-free and reward-aware exploration.
result Statistical efficiency and hardness results under various structural assumptions, no need for reachability or explorability assumptions.
Study shows exponential sample complexity for stabilizing certain linear systems.
problem Statistical hardness of learning to stabilize linear time-invariant systems.
method Analysis of sample complexity and co-stabilizability using robust control ideas.
result Sample complexity increases exponentially with system dimension.
New algorithm learns sparse GLMs for binary outcomes efficiently.
problem Sparse modeling of binary outcomes in high-dimensional data.
method Iterative hard thresholding algorithm (BIHT) for sparse GLMs.
result BIHT achieves statistical optimality for logistic regression.
New lower bounds show learning intersections of halfspaces is hard even for a few halfspaces.
problem Learning intersections of halfspaces in polynomial time under standard assumptions.
method Unified connection to parallel pancakes distribution for proving hardness.
result Learning ω(loglogN) halfspaces in dimension N requires super-polynomial time under standard assumptions. 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.
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.
WEINCE improves contrastive learning by correcting softmax biases.
problem Softmax in InfoNCE can lead to misaligned statistical assumptions in contrastive learning.
method WEINCE uses anchor-wise online batch statistics to blend softmax logits with an endpoint shortfall correction.
result WEINCE yields consistent improvements in frozen-feature evaluation across five vision benchmarks.
Research aims to bridge statistical learning to causal models in AI.
problem Challenges in machine learning and AI related to causality.
method Transition from statistical learning to causal models.
result Progress in AI may require advances in causal modeling.
Introduces a continuous version of LWE problem.
problem Hardness of learning mixtures of Gaussians.
method Polynomial-time quantum reduction from CLWE to lattice problems.
result CLWE shares hardness with LWE.
Trading system uses NP-hard optimization to select stocks for high Sharpe ratio trading.
problem Finding profitable, uncorrelated stocks for high Sharpe ratio trading.
method NP-hard combinatorial optimization using Ising machine and simulated bifurcation algorithm.
result Trading strategy with FPGA-based system achieves 164 μs response latency.
Unified approach to tensor PCA and related problems using tensor cumulants.
problem Statistical inference on invariant distributions, particularly tensor PCA.
method Definition and analysis of tensor cumulants to unify and extend previous results.
result Unified explanation of hardness and subexponential-time algorithms for tensor PCA.
Noise Sensitivity Exponent controls statistical-computational gaps in learning.
problem Understanding when learning is statistically possible yet computationally hard in high-dimensional statistics.
method Investigating statistical-computational gaps in single- and multi-index models using Noise Sensitivity Exponent.
result Noise Sensitivity Exponent governs statistical-computational gaps in high-dimensional learning.
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.
Paper develops algorithms to maximize AUC in imbalanced classification.
problem Maximizing AUC in imbalanced classification problems.
method Developed stochastic hard thresholding algorithms to reformulate U-statistics as ERM.
result Proposed algorithm achieves linear convergence rate.
This research develops an efficient reinforcement learning method for undercomplete POMDPs.
problem Learning undercomplete Partially Observable Markov Decision Processes (POMDPs) is computationally hard.
method OOM-UCB algorithm for episodic finite undercomplete POMDPs.
result Achieves optimal sample complexity of ildeO(1/ε2) for finding an ε-optimal policy. Paper compares hard and soft EM for BN learning from incomplete data.
problem Learning BNs from incomplete data using EM algorithms.
method Investigates the impact of imputation vs. belief propagation in hard and soft EM.
result A decision tree can guide practitioners in choosing the best EM algorithm.
A new algorithm optimizes graph problems faster and more accurately.
problem Hard optimization problems on graphs.
method Gumbel-softmax technique with gradient descent and evolution strategy.
result High-quality solutions obtained with less time.
Neural networks struggle with learning fixed parities.
problem Difficulty of learning fixed parities with neural networks.
method Using perturbed gradient descent on one-hidden-layer ReLU networks.
result Training neural networks on fixed parities fails to produce meaningful results.
Efficient synthetic data generation improves model performance on tabular data.
problem Improving model robustness and performance with scarce or low-quality data.
method Hardness characterization to identify high-value training points, generating synthetic data only from these points.
result Synthetic data generated from hardest points outperforms non-targeted methods on tabular datasets.
Paper develops error rates for physics-informed learning, comparing it to data-driven methods.
problem Understanding the trade-off between soft penalties and hard constraints in PISL.
method Develops complexity-dependent error rates using the small-ball method.
result Physics-informed estimators have comparable error rates to hard constrained methods, differing only by constants.
Noise makes learning linear thresholds hard, but algorithms can still learn near-optimal thresholds.
problem Learning linear thresholds in noisy data.
method Exploiting natural assumptions on data-generating process.
result Efficient learning of near-optimal linear thresholds is still possible with small data even in the presence of noise.
Recent convolutional neural networks (CNNs) have led to impressive performance but often suffer from poor calibration. They tend to be overconfident, with the model confidence not always reflecting the underlying true ambiguity and hardness. In this paper, we propose angular visual hardness (AVH), a score given by the …
Gradient descent achieves optimal learning for elliptic PDEs via Sobolev norms.
problem Learning elliptic PDEs from noisy data.
method Gradient descent on Sobolev norm objective functions.
result Gradient descent achieves statistical optimality for elliptic PDEs.
Study shows it's impossible to count communities without finding them.
problem Determining the number and sizes of communities in random graph models.
method Hypothesis testing between models with different community structures, using low-degree polynomial framework.
result Testing between two different planted distributions is as hard as finding the communities.
Optimized Franz-Parisi criterion matches SQ lower bounds for various statistical models.
problem Understanding computational hardness in statistical inference.
method Proposed and refined Franz-Parisi criterion, established equivalence with SQ lower bounds.
result Optimized Franz-Parisi criterion is equivalent to Statistical Query (SQ) lower bounds.
In this paper, we propose a general framework for tensor singular value decomposition (tensor SVD), which focuses on the methodology and theory for extracting the hidden low-rank structure from high-dimensional tensor data. Comprehensive results are developed on both the statistical and computational limits for tensor …
Learning to control linear systems is statistically hard, especially for underactuated systems.
problem Statistical difficulty of learning to control linear systems, especially underactuated ones.
method Utilized minimax lower bounds and structural assumptions to prove learning complexity can be exponential.
result Learning complexity can be at most exponential with the controllability index of the system.
Score matching offers efficient estimation for certain distributions.
problem Estimating probability distributions with intractable constants.
method Score matching as an alternative to maximum likelihood.
result Score matching is computationally and statistically efficient for certain distributions.
Reduces learning periodic neural networks to lattice problems, proving hardness under cryptographic assumptions.
problem Learning single periodic neurons in noisy environments.
method Reduction to worst-case lattice problems, using LLL algorithm.
result Polynomial-time algorithms for learning these functions are hard under cryptographic assumptions.
New lower bounds show sparse recovery is hard even with multiple preconditioners.
problem Sparse recovery with ill-conditioned designs is hard for certain algorithms.
method Constructing a single signal distribution that multiple preconditioned Lasso programs fail on.
result Standard sparse random designs are robust to erasures, aiding sparse recovery.
Near-optimal SQ hardness shows learning halfspaces with Massart noise is hard.
problem Learning halfspaces with Massart noise in the presence of label corruption.
method Statistical Query (SQ) model analysis.
result No efficient SQ algorithm can achieve better than Ω(η) error, even for optimal noise levels. This paper is about two related decision theoretic problems, nonparametric two-sample testing and independence testing. There is a belief that two recently proposed solutions, based on kernels and distances between pairs of points, behave well in high-dimensional settings. We identify different sources of misconception…
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.
Intensive algorithmic efforts have been made to enable the rapid improvements of certificated robustness for complex ML models recently. However, current robustness certification methods are only able to certify under a limited perturbation radius. Given that existing pure data-driven statistical approaches have reache…
We propose a novel approach for estimating the difficulty and transferability of supervised classification tasks. Unlike previous work, our approach is solution agnostic and does not require or assume trained models. Instead, we estimate these values using an information theoretic approach: treating training labels as …
Paper tackles hard shape constraints in kernel machines.
problem Enforcing shape requirements in a hard fashion is challenging.
method Tightened second-order cone constrained reformulation for kernel machines.
result Performance guarantees and efficiency demonstrated in various applications.
Paper proves MDS NP-hard and provides a PTAS.
problem Theoretical limitations of MDS objective function.
method Proves NP-hardness and provides a PTAS approximation algorithm.
result Minimizing Kamada-Kawai objective is NP-hard.
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 …
Variable selection in linear models plays a pivotal role in modern statistics. Hard-thresholding methods such as l0 regularization are theoretically ideal but computationally infeasible. In this paper, we propose a new approach, called the LAGS, short for "least absulute gradient selector", to this challenging yet i…
Gradient descent struggles to learn equivariant neural networks, even with symmetries.
problem Learning equivariant neural networks via gradient descent is hard.
method Lower bounds for various equivariant neural network classes.
result Gradient descent struggles to learn equivariant neural networks, even with symmetries.