GNA optimally identifies the best arm with small gaps.
problem Best arm identification in fixed-budget settings.
method Generalized Neyman Allocation (GNA) for asymptotically locally minimax optimal BAI.
result GNA's worst-case bounds match the lower and upper bounds in the small-gap regime.
Study best arm identification with contextual info, achieving optimal misidentification probability.
problem Identify the best treatment arm with minimal misidentification probability in a small gap scenario.
method Developed RS-AIPW strategy that matches lower bound of misidentification probability in the small-gap regime.
result RS-AIPW strategy is asymptotically optimal for best arm identification.
The study finds arbitrarily small spectral gaps for random hyperbolic surfaces with many cusps.
problem Understanding spectral gaps of random hyperbolic surfaces with many cusps.
method Analysis of moduli spaces of hyperbolic surfaces with Weil-Petersson metric.
result Arbitrarily small spectral gaps are observed as the number of cusps grows slower than the genus.
New theorem shows curvature concentration depends linearly on volume ratio.
problem Gap theorem for nonnegative Ricci curvature manifolds with small curvature concentration.
method Exhibited Ricci flow solution with faster than 1/t curvature decay.
result Curvature concentration depends linearly on asymptotic volume ratio.
We extend the well-known Sacks-Uhlenbeck energy gap result (1981) for harmonic maps from closed Riemann surfaces into closed Riemannian manifolds from the case of maps with small energy (thus near a constant map), to the case of harmonic maps with high absolute energy but small energy relative to a reference harmonic m…
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…
Optimal strategy found for identifying best arm in bandits with small gap.
problem Best arm identification in two-armed bandits with a fixed budget and small gap.
method Neyman allocation rule augmented with inverse probability weighting.
result Proposed strategy is asymptotically optimal when gap is small.
Fundamental gap vanishes for convex domains in hyperbolic space.
problem Behavior of fundamental gap in convex domains in hyperbolic space.
method Proof for Laplace operator with Dirichlet boundary conditions.
result Fundamental gap can be arbitrarily small for domains of any diameter.
New SDP algorithm recovers large clusters in SBM with small clusters of any size.
problem Graph clustering in SBM with large and small clusters.
method Semidefinite programming (SDP) with novel techniques to handle small clusters.
result Proves exact recovery of large clusters regardless of small cluster sizes.
Negative curvature restricts the gap between the first and second eigenvalues of convex domains.
problem The fundamental gap of convex domains is limited by negative curvature.
method Adapted from Bourni et. al. (2022) for Riemannian manifolds with negative sectional curvature.
result The product of the fundamental gap and the square of the diameter can be arbitrarily small in domains with negative curvature.
Random feature model shows slow self-correction of generalization gap.
problem Slow deterioration of generalization error in random feature model.
method Examined the dynamic behavior of gradient descent in the model's resonance regime.
result Gradient descent exhibits a self-correction mechanism, reducing generalization gap over time.
Paper tackles small eigen-gap estimation and inference for noisy symmetric matrices.
problem Estimating eigenvectors with small eigen-gap and fine-grained statistical reasoning.
method Eigen-decomposition of asymmetric data matrix, distribution-free procedures, adaptive to heteroscedastic noise.
result Minimax optimal under Gaussian noise, confidence intervals for eigenvalues, small eigen-gap handling.
Paper addresses eigenvector perturbation in small eigen-gap scenarios.
problem Fine-grained behavior of eigenvectors in the presence of small eigen-gaps.
method Develops de-biased estimators for linear functions of an unknown eigenvector.
result Achieves minimax lower bounds for a family of scenarios, even with small eigen-gaps.
Flat space for manifolds with tiny curvature.
problem Understanding manifolds with curvature concentration.
method Analyzing non-compact manifolds with non-negative Ricci curvature and small curvature concentration.
result Manifolds with curvature concentration are flat.
We present a separation property for the gaps in the length spectrum of a compact Riemannian manifold with negative curvature. In arbitrary small neighborhoods of the metric for some suitable topology, we show that there are negatively curved metrics with a length spectrum exponentially separated from below. This prope…
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.
Geodesic spheres in certain symmetric spaces are quantitatively stable under small perturbations.
problem Stability of geodesic spheres in symmetric spaces under perturbations.
method Quantitative stability analysis using spectral gap of the Laplacian on geodesic spheres.
result Geodesic spheres are uniformly stable with respect to small C1-volume preserving perturbations. In this paper we construct compact manifolds with fixed boundary geometry which admit Riemannian metrics of unit volume with arbitrarily large Steklov spectral gap. We also study the effect of localized conformal deformations that fix the boundary geometry. For instance, we prove that it is possible to make the spectra…
This note fixes a small gap in Kerckhoff's proof that the limit set of the handlebody set has measure zero.
We study the spectrum of the Finsler--Laplace operator for regular Hilbert geometries, defined by convex sets with C2 boundaries. We show that for an n-dimensional geometry, the spectral gap is bounded above by (n−1)2/4, which we prove to be the infimum of the essential spectrum. We also construct examples of c…
Study on AI-driven modeling for high burnup accident-tolerant fuels in SMRs.
problem Design and optimization of high burnup accident-tolerant fuels for SMRs.
method Artificial intelligence and multi-scale modeling (neutronics, thermal hydraulics, fuel performance).
result Demonstrated the effectiveness of AI in modeling and optimizing SMR fuels.
Random hyperbolic surfaces have a spectral gap that approaches 1/4 as genus grows.
problem Estimating the spectral gap of random hyperbolic surfaces.
method Analyzing the Weil-Petersson measure on moduli spaces of metrics.
result The spectral gap of random hyperbolic surfaces converges to 1/4 as the genus increases.
It is known (E.L. Green (1997), O. Post (2003)) that for an arbitrary m∈N one can construct a periodic non-compact Riemannian manifold M with at least m gaps in the spectrum of the corresponding Laplace-Beltrami operator −ΔM. In this work we want not only to produce a new type of periodic manifolds …
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.
Background: Deep learning models are typically trained using stochastic gradient descent or one of its variants. These methods update the weights using their gradient, estimated from a small fraction of the training data. It has been observed that when using large batch sizes there is a persistent degradation in genera…
Given a collection of entities (or nodes) in a network and our intermittent observations of activities from each entity, an important problem is to learn the hidden edges depicting directional relationships among these entities. Here, we study causal relationships (excitations) that are realized by a multivariate Hawke…
Study small perturbations on low energy Laplace eigenfunctions.
problem Understanding small changes in low energy Laplace eigenfunctions.
method Investigates nodal geometry and topology, focusing on low frequency regimes and small perturbations.
result Highlight interesting aspects of spectral theory and nodal phenomena tied to ground state/low energy eigenfunctions.
The study provides energy estimates for Willmore surfaces and derives a gap statement.
problem Analyzing the tracefree curvature of Willmore surfaces.
method Proves ε-regularity result for tracefree curvature with bounded second fundamental form.
result Derives a gap statement for surfaces of the specified type.
Framework explains deep learning generalization by comparing real and ideal worlds.
problem Understanding why deep models generalize well in practice.
method Integrates real-world empirical loss with ideal population loss to decompose test error.
result The gap between real and ideal worlds is small in deep learning, suggesting robust optimization leads to good generalization.
Uniform counting formulas for orthogeodesics in Kleinian groups converge.
problem Counting orthogeodesics in Kleinian groups converging to a limit.
method Spectral gap of the limit manifold and geodesic flow mixing property.
result Asymptotically uniform counting formulas for orthogeodesics.
New study on neural network calibration, linking it to generalization gap.
problem Neural networks lack strong guarantees on calibration.
method Decomposed calibration error into train set and generalization gap.
result Models with small generalization gap are well-calibrated.
Ricci flow controls curvature on manifolds with bounds.
problem Controlling curvature on manifolds with given bounds.
method Ricci flow with curvature bounds and entropy controls.
result Global curvature control at positive times for manifolds.
Unified framework for adaptive learning systems using consolidation and expansion operations.
problem Managing the balance between consolidating known knowledge and expanding into new evidence in adaptive learning systems.
method Introduces Consolidation-Expansion Operator Mechanics (OpMech) with the order-gap metric to control the balance.
result The order-gap signal provides real-time control and termination guarantees for adaptive learning systems.
Sharp stability of Alexandrov's theorem for C1 domains in the small-excess regime
problem Stability of Alexandrov's theorem for C1 domains in the small-excess regime method Combines a BV version of Fuglede's spectral-gap argument, a star-shaped rearrangement for sets of finite perimeter, quantitative estimates for the part of the boundary contained in the tentacles, and a polyhedral approximation argument for the non-graphical region result Sharp stability estimate in a genuinely non-parametric regime
In this short note, using Günther's volume comparison theorem and Yokota's gap theorem on complete shrinking gradient Ricci solitons, we prove that for any complete shrinking gradient Ricci soliton (Mn,g,f) with sectional curvature K(g)<A and Volf(M)≥v for some uniform constant A,v, there exists…
Study spectral properties of sub-Laplacians in Carnot groups.
problem Spectral properties of sub-Laplacians in Carnot groups.
method Proved pure point spectrum and spectral gap; applied to small ball problem and heat content.
result Proved existence of spectral gap and pure point spectrum.
In an effort to better understand the different ways in which the discount factor affects the optimization process in reinforcement learning, we designed a set of experiments to study each effect in isolation. Our analysis reveals that the common perception that poor performance of low discount factors is caused by (to…
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.
In this paper we establish a gap phenomenon for immersed surfaces with arbitrary codimension, topology and boundaries that satisfy one of a family of systems of fourth-order anisotropic geometric partial differential equations. Examples include Willmore surfaces, stationary solitons for the surface diffusion flow, and …
New bounds on hyperbolic surfaces' properties using linear programming.
problem Finding bounds on various geometric and spectral properties of hyperbolic surfaces.
method Adapted linear programming methods from sphere packings to hyperbolic surfaces.
result Obtained new upper and lower bounds on multiple properties of hyperbolic surfaces.
Paper establishes a universal growth rate for smooth surrogate losses in classification.
problem Analyzing growth rates of consistency bounds for various surrogate losses.
method Proves square-root growth rate for smooth margin-based losses; extends to multi-class classification.
result Demonstrates a universal square-root growth rate for smooth comp-sum and constrained losses.
Study on Ricci flow on 4-spheres, proving standard sphere convergence.
problem Characterizing and understanding Ricci flow on 4-spheres.
method Investigation of integral conformal invariants, analysis of flow properties.
result Established monotonic decay of certain curvature norms, leading to standard sphere convergence.
Detection of dense cycles in graphs reveals a gap between easy detection and hard recovery.
problem Detecting and recovering dense cycles in Erdős-Rényi graphs.
method Characterization of computational thresholds for detection and recovery using low-degree polynomial algorithms.
result A gap exists between the detection and recovery thresholds for certain parameter regimes.
The paper proves parabolic gap theorems for Yang-Mills energy.
problem Yang-Mills energy and instantons on various manifolds.
method Parabolic Yang-Mills flow and Morrey norms.
result Spaces of connections with Yang-Mills energy less than a certain threshold deformation-retract onto spaces of instantons.
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)} =…
Proves product metrics are Yamabe metrics under small flat torus conditions.
problem Yamabe metrics on product spaces with small flat tori.
method Extends earlier results to Type~I and Type~II Yamabe constants, Q-curvature problems, and isoperimetric-ratio type problems. result Product metrics are Yamabe metrics for sufficiently small flat tori.
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. Sparse attention model reduces long-context inference time with exponential accuracy guarantees.
problem Efficiently processing long-context queries in large language models.
method Formalizes attention as a projection onto key vectors, analyzes entropic relaxation, and introduces Vashista Sparse Attention.
result Sparse attention concentrates on a constant-size active face, leading to exponential decay of inactive tokens' mass and linear scaling of active face error.