The Langevin Algorithm's stationary distribution is shown to be sub-exponential or sub-Gaussian under certain conditions.
problem Understanding the properties of the Langevin Algorithm's stationary distribution.
method Analysis using a rotation-invariant moment generating function (Bessel function) to study the stationary dynamics of the Langevin Algorithm.
result Concentration results for the Langevin Algorithm's stationary distribution π η π_η π η are established, showing it is sub-exponential or sub-Gaussian under convex or strongly convex potential conditions. Expanding on techniques of concentration of measure, we develop a quantitative framework for modeling liquidity risk using convex risk measures. The fundamental objects of study are curves of the form ( ρ ( λ X ) ) λ ≥ 0 (ρ(λX))_{λ\ge 0} ( ρ ( λ X ) ) λ ≥ 0 , where ρ ρ ρ is a convex risk measure and X X X a random variable, and we call such a curve a \emph{liqu…
SGD converges to an invariant distribution with sub-Gaussian or sub-exponential properties.
problem Optimizing smooth and strongly convex objectives using SGD.
method Analysis through Markov chains, focusing on convergence and concentration properties.
result SGD iterates and their invariant limit distribution inherit sub-Gaussian or sub-exponential concentration properties.
Study Finsler metric measure manifolds' concentration properties.
problem Understanding concentration properties in Finsler metric measure manifolds.
method Established relationships with observable diameter, isoperimetric inequalities, and first eigenvalue.
result Derived a Cheng type upper bound estimate for the first closed eigenvalue.
The paper offers a framework to analyze machine learning problems using concentration of measure.
problem Analyzing machine learning algorithms defined by implicit equations.
method Develops a concentration of measure framework to solve convex problems and implicit formulations.
result Provides precise estimations for the first moments of the solution, describing the behavior and performance of machine learning classifiers.
Improved privacy-preserving methods for convex optimization with heavy-tailed data.
problem Privacy-preserving optimization of convex functions with heavy-tailed data.
method Developed algorithms for private mean estimation and convex optimization under concentrated differential privacy constraints.
result Achieved improved upper bounds on excess population risk for convex and strongly convex loss functions.
Tick-by-tick liquidity provision aims to maximize fees and reserves.
problem Maximizing fees and reserves in concentrated liquidity.
method Convex optimization for tick-level liquidity provision.
result Concentrating liquidity around current price is not always best.
Estimating sparse transition matrix from partially observed high-dimensional time series data.
problem Estimating transition matrix from sparse and partially observed high-dimensional time series data.
method Novel concentration result and new quantity for characterizing interactions.
result New theoretical challenges and novel approaches for handling missing data in sparse transition matrix estimation.
The consistency of Fréchet medians is proved for probability measures in proper metric spaces. In the context of Riemannian manifolds, assuming that the probability measure has more than a half mass lying in a convex ball and verifies some concentration conditions, the positions of its Fréchet medians are estimated. It…
Paper analyzes sample complexity for offline f f f -divergence-regularized contextual bandits.
problem Lack of tight analyses for sample complexity in offline reinforcement learning.
method Novel pessimism-based analysis for reverse KL divergence, establishing i l d e O ( ε − 1 ) ilde{O}(ε^{-1}) i l d e O ( ε − 1 ) sample complexity. result Achieves i l d e O ( ε − 1 ) ilde{O}(ε^{-1}) i l d e O ( ε − 1 ) sample complexity for reverse KL divergence, surpassing existing bounds. Quantum kernel methods can lead to trivial models due to exponential concentration of kernel values.
problem Exponential concentration of quantum kernel values can lead to trivial models in QML.
method Analyzing the resources needed to accurately estimate quantum kernel values and identifying four sources of concentration.
result Quantum kernel values can be exponentially concentrated, leading to trivial models.
New bounds for convex clustering under graph connectivity.
problem Understanding clustering performance under different graph connectivity structures.
method Random walks and concentration inequalities for random graph models.
result Improved rates of convergence for centroid recovery.
Frame flows on certain symmetric spaces mix exponentially.
problem Exponential mixing of frame flows in convex cocompact locally symmetric spaces.
method Generalized local non-integrability and non-concentration properties to apply Dolgopyat's method.
result Exponential mixing of frame flows proved for convex cocompact locally symmetric spaces.
Paper develops a new local convexity condition for non-isolated minima in non-convex optimization.
problem Lack of theory for non-isolated minima in non-convex optimization.
method Formulates a new local convexity condition and studies SGD convergence under this condition.
result Shows SGD can converge locally under the new condition.
New method learns shared structures in non-linear tasks.
problem Learning shared linear representations in non-linear tasks.
method Convex optimization with structural assumptions.
result Rank and clustered estimators recover shared structures under certain conditions.
We solve ReLU regression with efficient approximations for various distributions.
problem Finding the best fitting ReLU function with square loss from unknown distributions.
method Introduced efficient constant-factor approximation algorithm and polynomial-time approximation scheme.
result First constant-factor approximation algorithm for ReLU regression with weak concentration conditions.
Simple analysis for fast rates in empirical minimization with concave losses and convex regularization.
problem Fast rates in empirical minimization with concave losses and convex regularization.
method Simple analysis using covering number and concentration inequality.
result First result of fast rates with high probability for exponential concave empirical risk minimization.
SCOPE estimator improves covariance and precision matrix estimation.
problem Estimating covariance and precision matrices accurately.
method Distributionally robust optimization with convex spectral divergence.
result SCOPE estimator reduces spectral bias and improves condition number.
Paper addresses concentration of distances for fractional quasi p-norms, identifying conditions for concentration and anti-concentration.
problem Understanding concentration of distances for fractional quasi p-norms in high dimensions.
method Analyzes conditions for concentration and anti-concentration of distances for fractional quasi p-norms.
result Identifies conditions for concentration and anti-concentration of fractional quasi p-norms, ruling out some approaches and specifying conditions for control.
A new depth measure for non-convex data supports, faster than halfspace depth.
problem Non-convex data supports in multivariate statistics.
method Extending halfspace depth to Reproducing Kernel Hilbert Space (RKHS).
result The new depth measure is consistent and can be computed faster.
Unified technique for sequential estimation of convex divergences.
problem Estimating convex divergences between distributions.
method Martingale methods and maximal inequalities for reverse submartingales.
result Valid time-uniform confidence sequences for arbitrary stopping times.
Study shows how to control jump-diffusion processes with stable feedback controls in reinforcement learning.
problem Control jump-diffusion processes with unknown coefficients in reinforcement learning.
method Lipschitz continuous optimal feedback controls, stability analysis of forward-backward SDEs, least-squares algorithm.
result Achieves O ( N ln N ) O(\sqrt{N\ln N}) O ( N ln N ) regret for linear-convex learning problems with jumps. We show that the cone-volume measure of a convex body with centroid at the origin satisfies the subspace concentration condition. This implies, among others, a conjectured best possible inequality for the U \mathrm{U} U -functional of a convex body. For both results we provide stronger versions in the sense of stability i…
New algorithm robustly optimizes data streams with heavy-tailed or infinite variance samples.
problem Optimizing data streams with heavy-tailed or infinite variance samples.
method Gradient quantile clipping for SGD, leveraging Markov chain connections.
result Algorithm converges to a concentrated distribution with high probability bounds.
Algorithm learns halfspaces in noisy data efficiently.
problem Learning halfspaces with Tsybakov noise.
method Novel semi-definite programming and online convex optimization.
result First non-trivial PAC learning algorithm for Tsybakov noise.
Estimates barycenter in geodesic spaces with finite sample bounds.
problem Estimating the barycenter of a distribution in geodesic spaces.
method Finite sample error bounds, Hoeffding- and Bernstein-type concentration inequalities, efficient algorithms.
result Statistical guarantees for efficient barycenter computation.
Deep models can't generate heavy-tailed samples well.
problem Understanding the limitations of deep generative models in generating samples with heavy tails.
method Unified framework using concentration of measure and convex geometry, Gromov-Levy inequality.
result Deep generative models are not universal generators and can only produce concentrated samples with light tails.
The paper studies reward concentration in MDPs, covering asymptotic and non-asymptotic settings.
problem Reward concentration in Markov Decision Processes (MDPs).
method Unified approach to reward concentration in MDPs, including asymptotic and non-asymptotic bounds.
result Rate-equivalent definitions of regret for learning policies.
Paper explores robustness of CCS model for matrix completion.
problem Robustness of cross-concentrated sampling model against sparse outliers.
method Proposes Robust CUR Completion (RCURC) algorithm for efficient non-convex iterative matrix completion.
result Empirical validation of RCURC's efficiency and robustness in synthetic and real datasets.
The paper provides concentration inequalities for Markov chain variance estimators.
problem Estimating the variance of Markov chains with concentration properties.
method Martingale decomposition method for uniformly geometrically ergodic Markov chains.
result Explicit control of the p-th moment of the OBM estimator difference and dependence on p and mixing time.
The paper studies Dirac operators and their solutions concentrating near singular sets.
problem Understanding concentration properties of solutions to Dirac equations.
method Analyzes Dirac operators of the form D ε = D + ε − 1 A D_\varepsilon= D+\varepsilon^{-1}\mathcal A D ε = D + ε − 1 A and their solutions. result Solutions concentrate exponentially near the locus where the rank of ker ( A ) \ker(\mathcal A) ker ( A ) jumps. New axioms justify ES without NRC, linking it to mean-ES portfolio selection.
problem Economic axioms for portfolio risk assessment and mean-ES portfolio selection.
method Introducing concentration aversion as an alternative to NRC, establishing axiomatic foundations.
result Concentration aversion uniquely characterizes the family of ES and provides new formulas.
We obtain sharp bounds on the performance of Empirical Risk Minimization performed in a convex class and with respect to the squared loss, without assuming that class members and the target are bounded functions or have rapidly decaying tails. Rather than resorting to a concentration-based argument, the method used her…
The study uses heat flow to analyze properties of Laplace eigenfunctions on manifolds and domains.
problem Analyzing mass concentration and nodal domains of Laplace eigenfunctions.
method Heat diffusion technique to study eigenfunctions and their nodal sets.
result Discovers new insights into the decay and behavior of Laplace eigenfunctions.
The paper shows how solutions of perturbed Dirac operators concentrate near singular sets.
problem Understanding concentration of solutions for perturbed Dirac operators.
method Analyzing the algebraic criterion on $(c, \A)$ and spectral properties of deformed Laplacians.
result Proves an index localization theorem based on spectral separation properties.
Sharp Gaussian isoperimetry proven along Ricci flow.
problem Proving sharp Gaussian isoperimetric inequality for Ricci flow.
method Using monotonicity formula to prove inequality.
result Exact Gaussian enlargement theorem and concentration estimates.
The quantification of diversification benefits due to risk aggregation plays a prominent role in the (regulatory) capital management of large firms within the financial industry. However, the complexity of today's risk landscape makes a quantifiable reduction of risk concentration a challenging task. In the present pap…
Net-Trim simplifies neural networks by pruning layers efficiently.
problem Simplifying trained deep neural networks for efficiency.
method Net-Trim is a convex post-processing technique that prunes neural network layers.
result Net-Trim can find a network with a limited number of nonzero terms from a small number of samples.
This paper studies node embeddings of networks, revealing their geometric properties.
problem Understanding the geometric properties of node embeddings in random networks.
method Characterization of ergodic limits, generalization, and convex relaxations of random walk node embedding objectives.
result The optimal node embedding Grammians have rank 1 for a nuclear norm relaxation of the non-randomized objective.
Study on free boundary minimal surfaces, focusing on curvature concentration and geometric convergence.
problem Understanding the limit behavior of free boundary minimal hypersurfaces with curvature concentration.
method Detailed blow-up analysis and quantization identity derivation for total curvature functional.
result Derivation of a constraint relating topology of limit hypersurfaces and their blow-up models.
It is well known that isoperimetric inequalities imply in a very general measure-metric-space setting appropriate concentration inequalities. The former bound the boundary measure of sets as a function of their measure, whereas the latter bound the measure of sets separated from sets having half the total measure, as a…
This paper empirically measures intrinsic robustness of image classification models.
problem Understanding the robustness of image classification models under small perturbations.
method Empirical measurement of concentration in concrete datasets, using ℓ ∞ \ell_\infty ℓ ∞ and ℓ 2 \ell_2 ℓ 2 perturbations. result Empirical estimates of intrinsic robustness for various image classification benchmarks.
Proves formula for reconstruction performance in generalized linear models.
problem Analyzing reconstruction performance in generalized linear models with arbitrary bounded spectrum.
method Message passing algorithms and dynamical system stability analysis.
result Analytical formula confirms replica method conjecture for convex models.
We investigate the m m m -relative entropy, which stems from the Bregman divergence, on weighted Riemannian and Finsler manifolds. We prove that the displacement K K K -convexity of the m m m -relative entropy is equivalent to the combination of the nonnegativity of the weighted Ricci curvature and the K K K -convexity of the weig…
We introduce a class of generalized relative entropies (inspired by the Bregman divergence in information theory) on the Wasserstein space over a weighted Riemannian or Finsler manifold. We prove that the convexity of all the entropies in this class is equivalent to the combination of the nonnegative weighted Ricci cur…
Paper shows how gradient concentration helps in learning from inexact data.
problem Learning from inexact and stochastic training data.
method Combines probabilistic gradient concentration with inexact optimization techniques.
result Derives sharp test error guarantees for learning.
For convex co-compact hyperbolic manifolds Γ \ H n + 1 Γ\backslash \mathbb{H}^{n+1} Γ\ H n + 1 for which the dimension of the limit set satisfies δ Γ < n / 2 δ_Γ< n/2 δ Γ < n /2 , we show that the high-frequency Eisenstein series associated to a point ξ ξ ξ "at infinity" concentrate microlocally on a measure supported by (the closure of) the set of points in the …
Topological constraints improve neural network generalization.
problem Improving generalization in neural networks with limited data.
method Imposing topological constraints on internal representations of neural networks.
result Topological constraints lead to better mass concentration around training instances, improving generalization.