The paper introduces gapped scale-sensitive dimensions to improve learning rate bounds.
problem Improving lower bounds on rates of convergence in statistical and online learning.
method Introducing and analyzing gapped scale-sensitive dimensions for function classes.
result Gapped dimensions lead to stronger lower bounds on offset Rademacher averages.
Improved uniform convergence bound with fat-shattering dimension reduces sample complexity gap.
problem Gap between upper and lower bounds on sample complexity for fat-shattering dimension.
method Provided an improved uniform convergence bound.
result Closed the gap between existing upper and lower bounds on sample complexity.
Study shows exponential gap in sample complexity between noisy and non-noisy recurrent neural networks.
problem Understanding the impact of noise on the sample complexity of recurrent neural networks.
method Analyzing noisy multi-layered sigmoid recurrent neural networks with independent noise and proving lower bounds.
result Exponential gap in sample complexity between noisy and non-noisy networks, even for small noise values.
We use the energy gap result of pure Yang-Mills equation [Feehan P.M.N., Adv. Math. 312 (2017), 547-587, arXiv:1502.00668] to prove another energy gap result of complex Yang-Mills equations [Gagliardo M., Uhlenbeck K., J. Fixed Point Theory Appl. 11 (2012), 185-198, arXiv:1401.7366], when Riemannian manifold X of dim…
simpcomp is an extension (a so called package) to GAP, the well known system for computational discrete algebra. The package enables the user to compute numerous properties of (abstract) simplicial complexes, provides functions to construct new complexes from existing ones and an extensive library of triangulations of …
The study uses Random Matrix Theory to identify structural changes in stock markets during shocks.
problem Understanding structural changes in stock markets during exogenous shocks.
method Random Matrix Theory and complexity gap analysis.
result The complexity gap collapses during shocks, indicating strong synchronization, and widens before shocks, signaling a rich structure.
New algorithm reduces sample complexity for planning in MDPs.
problem Planning in MDPs with unknown transitions.
method MDP-GapE, a trajectory-based MCTS algorithm.
result Proves upper bound on sample complexity in terms of sub-optimality gaps.
Upper and lower bounds on complexity of 3-manifolds from Dehn fillings.
problem Determining the complexity of 3-manifolds obtained by Dehn filling.
method Upper and lower bounds on complexity, characterizing families of fillings, and computing gaps.
result Upper and lower bounds on complexity with a gap of at most 13|T| + 7 for some families of fillings.
We prove a \emph{query complexity} lower bound for approximating the top r dimensional eigenspace of a matrix. We consider an oracle model where, given a symmetric matrix M∈Rd×d, an algorithm Alg is allowed to make T exact queries of the form $\mathsf{w}^{(i)} =…
The paper tackles attributing forecast gaps in complex model suites.
problem Attributing forecast gaps to individual component models in complex model suites.
method Formalized walk analysis, adapted LMDI and Shapley value approaches.
result Developed efficient formulas for gap attribution in practical portfolio-scale examples.
New method for spectral and Bergman kernels under local spectral gap condition.
problem Analyzing spectral and Bergman kernels for complex manifolds.
method Developed a new scaling method to study spectral and Bergman kernels.
result Established pointwise asymptotics of spectral and Bergman kernels.
Computing unlinking number is usually very difficult and complex problem, therefore we define BJ-unlinking number and recall Bernhard-Jablan conjecture stating that the classical unknotting/unlinking number is equal to the BJ-unlinking number. We compute BJ-unlinking number for various families of knots and links for w…
In the classical best arm identification (Best-1-Arm) problem, we are given n stochastic bandit arms, each associated with a reward distribution with an unknown mean. We would like to identify the arm with the largest mean with probability at least 1−δ, using as few samples as possible. Understanding the sample c…
Attributing forecast gaps to component models in complex model suites
problem Attributing forecast gaps between model-suite forecasts and realized outcomes
method Formalizing walk analysis and adapting order-independent attribution frameworks
result Deriving efficient formulas for elementwise and vectorized gap attribution
In our previous work "Characterization of certain homorphic geodesic cycles on Hermitian locally symmetric manifolds of the noncompact type" in "Modern methods in Complex Analysis" Annals of Math. Studies 138 (1995) 85-118, we formulated a conjecture: the so called "gap phenomenon". The purpose of the article is two-fo…
Chaos in cerebellar cells enhances complexity of neural patterns.
problem Understanding how cerebellar granular layer represents complex information.
method Constructed a model of cerebellar granular layer with gap junctions, evaluated using reservoir computing.
result Chaotic dynamics in the cerebellar granular layer produce complex and diverse output patterns.
Paper analyzes complexity of solving nonconvex-strongly-concave problems.
problem Finding approximate stationary points of nonconvex-strongly-concave minimax problems.
method Introduces a generic acceleration scheme to solve crafted subproblems.
result Algorithm nearly matches lower complexity bounds in general setting.
Arithmetic spaces simplified to simplicial complexes.
problem Understanding the complexity of arithmetic locally symmetric spaces.
method Homotopy equivalence to a simplicial complex with linearly bounded simplices, using a strengthened Margulis collar lemma.
result Arithmetic locally symmetric spaces are homotopy equivalent to simplicial complexes with linearly bounded simplices.
Improved sample complexity for Gaussian Mixture Models using Pair Correlation Factor.
problem Understanding the sample complexity of Gaussian Mixture Models.
method Introducing Pair Correlation Factor (PCF) to measure clustering of component means and improving sample complexity bounds.
result The Pair Correlation Factor (PCF) more accurately determines the difficulty of parameter recovery in Gaussian Mixture Models.
Improved gap-dependent bounds for reinforcement learning with linear approximations.
problem Achieving nearly minimax-optimal performance with linear function approximation.
method Developed and analyzed the LSVI-UCB++ algorithm and its concurrent variant.
result First gap-dependent regret bound for nearly minimax-optimal algorithm LSVI-UCB++.
New complexity measure explains neural network generalization gap.
problem Understanding the generalization gap between neural networks and linear models.
method Introducing a new complexity measure for functions that governs PAC-Bayes bounds and relates to neural network complexity.
result Demonstrates a separation in sample complexity between 2 and 4-layer neural networks for periodic functions.
Estimates covariance matrices using Markov chain Monte Carlo with improved sample complexity.
problem Complexity of covariance matrix estimation for Gibbs distributions.
method Uses Markov chain Monte Carlo with conditions on the chain's spectral gap and Poincaré inequality.
result Achieves similar sample complexity as i.i.d. samples with better query complexity.
Optimal privacy-preserving algorithm for solving saddle point problems.
problem Solving convex-concave stochastic saddle point problems under differential privacy constraints.
method Recursive regularization technique repurposed for saddle point problems, achieving strong gap rate of O(1/√n + √d/nε).
result Achieves nearly optimal strong gap rate of O(1/√n + √d/nε) with gradient complexity O(min{n^2ε^(1.5)/√d, n^(3/2)}).
Improved online Q-learning for MDPs with concentration bounds.
problem Online Q-learning in infinite-horizon discounted MDPs with sublinear regret for large gaps.
method Smoothed εn-Greedy exploration scheme combining εn-greedy and Boltzmann exploration, analyzed using concentration bounds for contractive Markovian stochastic approximation. result Near-ildeO(N9/10) regret bound for Smoothed εn-Greedy exploration scheme. New bound limits generalization gap for large models, independent of model complexity.
problem Understanding generalization gap in large-scale machine learning models.
method Established a model-independent upper bound for generalization gap using Rényi entropy.
result Generalization gap can be maintained with arbitrarily large models if data entropy is sufficient.
Factorial moments are convenient tools in particle physics to characterize the multiplicity distributions when phase-space resolution (Δ) becomes small. They include all correlations within the system of particles and represent integral characteristics of any correlation between these particles. In this letter, we sh…
New theorem improves spectral gap for sampling from mixture distributions.
problem Sampling from multimodal distributions with simulated tempering.
method Introduced a decomposition theorem for the restricted spectral gap of simulated tempering.
result Lower bound on the restricted spectral gap for mixture distributions.
Study identifies a Strategic Gap in market efficiency due to AI-driven timing and complexity in disclosure.
problem Market inefficiency due to structural influence of disclosure timing and complexity.
method Introduces Autonomous Disclosure Regulator, a multi-node AI framework to audit disclosure complexity and unpredictability.
result Companies use confusing language and unpredictable timing to slow down market learning, creating a 60% Structural Gap.
Data visualization and interaction with large data sets is known to be essential and critical in many businesses today, and the same applies to research and teaching, in this case, when exploring large and complex mathematical objects. GAP is a computer algebra system for computational discrete algebra with an emphasis…
New method identifies Condorcet winner in dueling bandits with improved sample complexity.
problem Identifying Condorcet winner in noisy pairwise comparisons.
method Exploits full gap matrix Δ to improve sample complexity.
result Improves sample complexity guarantees by leveraging informative comparisons.
Fluctuation scaling is observed phenomenon from complex networks through finance to ecology. It means that the variance and the mean of a specific quantity are related as $\ev{σ^2|n}\propto \ev{n|A}^{2α}$ with 1/2≥α≥1 when a parameter A (usually the system size) is varied. A can be the strength of the nod…
Study quantile multi-armed bandits for identifying the best arm with a specified quantile level.
problem Identifying the arm with the highest quantile in multi-armed bandits with private rewards.
method Proposed a (non-private) and differentially private successive elimination algorithms for best-arm identification.
result The proposed algorithms are essentially optimal for quantile bandit problems, with finite sample complexity even for distributions with infinite support-size.
New algorithm identifies good arms with fewer samples when thresholds are close.
problem Good arm identification in bandit problems with small threshold gaps.
method Proposes lil'HDoC algorithm to improve GAI under small threshold gaps.
result Sample complexity of first λ output arm is nearly identical to HDoC algorithm when thresholds are close.
New algorithms improve contextual bandit performance by adapting to problem difficulty.
problem Improving contextual bandit performance on problems with varying difficulty.
method Introducing complexity measures and oracle-efficient algorithms.
result Achieves optimal instance-dependent regret bounds for rich policy classes.
New analysis shows PE in Transformers increases generalization gap and vulnerability.
problem Understanding the impact of PE on Transformer generalization and robustness.
method Generalization analysis and adversarial Rademacher bounds for a single-layer Transformer with trainable PE.
result PE systematically enlarges the generalization gap and makes models more vulnerable to attacks.
GACELA fills long gaps in musical audio with a GAN and context conditioning.
problem Restoring long gaps in musical audio with varying complexity and duration.
method Generative adversarial network (GAN) with five parallel discriminators and context conditioning.
result Reduced artifacts in inpaintings from unacceptable to mildly disturbing.
UCB algorithm's arm-sampling behavior is revealed, leading to new insights and proofs.
problem Optimizing multi-armed bandit algorithms for worst-case scenarios.
method Analysis of UCB algorithm's arm-sampling behavior and process-level characterization.
result UCB's arm-sampling rates are asymptotically deterministic, regardless of problem complexity.
The infinitesimal symmetry algebra of any Cartan geometry has maximum dimension realized by the flat model, but often this dimension drops significantly when considering non-flat geometries, so a gap phenomenon arises. For general (regular, normal) parabolic geometries of type (G,P), we use Tanaka theory to derive a un…
In this note, we fill in a gap in the literature by proving that the Teichmueller modular groups (mapping class groups) are not Poincare duality groups and the complexes of curves of surfaces have infinite homotopy type (i.e. are not homotopy equivalent to a finite CW-complex).
The paper improves L2-estimates for Dirac-Dolbeault operators on complex manifolds.
problem Improving L2-estimates for Dirac-Dolbeault operators on complex manifolds. method Generalized classical method to handle mixed curvature cases and provided bounds on error terms.
result Full asymptotic expansion for Bergman kernel obtained.
We consider the problem of estimating from sample paths the absolute spectral gap γ∗ of a reversible, irreducible and aperiodic Markov chain (Xt)t∈N over a finite state space Ω. We propose the UCPI (Upper Confidence Power Iteration) algorithm for this problem, a low-complexity algorithm …
The effectiveness of model-based versus model-free methods is a long-standing question in reinforcement learning (RL). Motivated by recent empirical success of RL on continuous control tasks, we study the sample complexity of popular model-based and model-free algorithms on the Linear Quadratic Regulator (LQR). We show…
We present new lower bounds on the complexity of Dehn surgery manifolds of knots, using our recent result on the Cheeger-Gromov rho invariants and triangulations. As an application, we give explicit examples of closed hyperbolic 3-manifolds with fixed first homology for which the gap between the Gromov norm and the com…
Paper tightens lower bounds on decentralized training complexity.
problem Understanding and optimizing iteration complexity in decentralized training.
method Proved a tight lower bound on iteration complexity and proposed DeTAG algorithm.
result DeTAG achieves the theoretical lower bound with only a logarithmic gap.
The paper explores how simplicity leads to better out-of-distribution generalization in models.
problem Understanding the theoretical principles behind out-of-distribution (OOD) generalization in modern models.
method Examining diffusion models in image generation to analyze compositional generalization abilities and develop a theoretical framework for simplicity-based OOD generalization.
result The true, generalizable model corresponds to the simplest among consistent models, and this simplicity can be quantified and used to establish sample complexity guarantees.
Study shows gap between de Rham and symplectic-Bott-Chern harmonic forms for specific almost-Kähler manifolds.
problem Understanding the gap between de Rham and symplectic-Bott-Chern harmonic forms on specific almost-Kähler manifolds.
method Analyzing the space of de Rham harmonic forms and symplectic-Bott-Chern harmonic forms on closed almost-Kähler manifolds.
result The second non-HLC degree measures the gap between de Rham and symplectic-Bott-Chern harmonic forms.
Uniform spectral gap for convex cocompact hyperbolic surfaces and expanders.
problem Spectral gap for convex cocompact hyperbolic surfaces and their covers.
method Using thermodynamic formalism for twisted Selberg zeta functions.
result Uniform resonance-free regions for convex cocompact hyperbolic surfaces and expanders.
We suggest a general oracle-based framework that captures different parallel stochastic optimization settings described by a dependency graph, and derive generic lower bounds in terms of this graph. We then use the framework and derive lower bounds for several specific parallel optimization settings, including delayed …