Finite quotients of braid groups have large sizes.
problem Understanding the size of finite quotients of braid groups.
method Derived lower bounds on the size of quotients.
result Lower bounds are superexponential in the number of strands.
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48 results for “finite size”
Learnable multiclass hypothesis classes don't always have a sample compression scheme of fixed size.
problem The limitation of sample compression schemes for multiclass hypothesis classes.
method Analysis of DS dimension and sample compression schemes.
result Learnable multiclass hypothesis classes do not always have a sample compression scheme of fixed size.
Study reveals finite-size effects and sensitivity to random numbers in Levy-Levy-Solomon model.
problem Finite-size effects and sensitivity to random numbers in Levy-Levy-Solomon model.
method Simulations and analysis of Levy-Levy-Solomon model with different random number generators and stopping criteria.
result Low-quality pseudo random number generators significantly impact simulation results.
Fewer obstructions for small graphs in knotless embedding.
problem Finite obstructions for knotless embedding of small graphs.
method Analysis of graph minor obstructions and knotless embedding.
result There are only three obstructions for size 23 graphs, significantly fewer than previously known.
We consider the roughness properties of NYSE (New York Stock Exchange) stock-price fluctuations. The statistical properties of the data are relatively homogeneous within the same day but the large jumps between different days prevent the extension of the analysis to large times. This leads to intrinsic finite size effe…
Finite market size as a source of extreme wealth inequality and market instabilitycond-mat.stat-mech
We study the finite-size effects in some scaling systems, and show that the finite number of agents N leads to a cut-off in the upper value of the Pareto law for the relative individual wealth. The exponent of the Pareto law obtained in stochastic multiplicative market models is crucially affected by the fact that …
Study non-monotonic loss functions in CRC, achieving valid risk control with large calibration samples.
problem Non-monotonic loss functions in CRC, violating existing theory's monotonicity assumption.
method Finite grid selection, calibration sample size analysis, Lipschitz continuity, monotonicity, distribution shift.
result Valid CRC achieved with large calibration samples, optimal excess risk rate of .
Evaluating accuracy of community detection using the relative normalized mutual informationphysics.soc-ph
The Normalized Mutual Information (NMI) has been widely used to evaluate the accuracy of community detection algorithms. However in this article we show that the NMI is seriously affected by systematic errors due to finite size of networks, and may give a wrong estimate of performance of algorithms in some cases. We gi…
Finite element method applied to Leland's model for option pricing with transaction costs.
problem Option pricing with transaction costs using Leland's model.
method Spatial finite element models based on P1 and/or P2 elements combined with a Crank-Nicolson-type temporal scheme.
result Results compare favorably with finite difference methods in the literature.
In the information-based paradigm of inference, model selection is performed by selecting the candidate model with the best estimated predictive performance. The success of this approach depends on the accuracy of the estimate of the predictive complexity. In the large-sample-size limit of a regular model, the predicti…
On Simply Connected Quandlesmath.GR
The paper characterizes simply connected quandles using cocycles with prime values.
problem Characterizing simply connected quandles.
method Using cocycles with values in abelian groups of prime size.
result Classification of simply connected quandles of sizes and for .
Study 2-complexes' homology properties and torsion growth.
problem Quantitative connections between 1-cycle filling inequalities and homology complexities.
method Geometric lower bounds on first homology of finite covers.
result Geometric lower bound on first homology size of finite covers.
Emergence of a finite-size-scaling function in the supervised learning of the Ising phase transitioncond-mat.stat-mech
A neural network model predicts the critical point of the Ising phase transition.
problem Predicting the critical point of the Ising phase transition using supervised learning.
method Proposed a minimal one-free-parameter neural network model to describe the supervised learning problem for the Ising model.
result Just one free parameter is enough to describe the universal finite-size-scaling function in the network output.
The study reveals a transition in neural network performance from infinite-width to variance-limited behavior as dataset size increases.
problem Understanding the transition from infinite-width to variance-limited behavior in neural networks.
method Empirical study of the transition from infinite-width to variance-limited behavior as a function of sample size and network width.
result The critical sample size \( P^* \) is approximately \( \sqrt{N} \) for polynomial regression with ReLU networks.
Improved variance reduction for Riemannian non-convex optimization with adaptive batch size.
problem Optimizing non-convex functions on Riemannian manifolds.
method Batch size adaptation in R-SVRG, R-SRG, and R-SPIDER.
result Achieves lower total complexities for various non-convex functions.
New method reduces over-parametrization in neural networks, ensuring sparsity and finite network size.
problem Over-parametrization leads to too many active neurons in neural networks, especially with large data.
method Investigates a nonconvex regularization method for shallow ReLU networks.
result Locally optimal networks are finite even with infinite data, maintaining approximation guarantees and network size bounds.
Policy gradient methods achieve linear convergence in simple MDPs.
problem Analyzing convergence rates of policy gradient methods in finite MDPs.
method Connections with policy iteration to show linear convergence with large step-sizes.
result Policy gradient methods succeed with large step-sizes and achieve linear rate of convergence.
We investigate finite-time decoupled convergence in nonlinear two-time-scale stochastic approximation.
problem Achieving decoupled convergence in nonlinear two-time-scale stochastic approximation.
method Nested local linearity assumption, suitable step size selection, convergence analysis of matrix cross term, fourth-order moment convergence rates.
result Finite-time decoupled convergence rates can be achieved in nonlinear two-time-scale stochastic approximation with proper step size selection.
The paper improves methods for estimating set size using samples.
problem Estimating the size of a set from a uniform sample.
method Refines estimators using the birthday problem and maximum of sample.
result Develops a general theory for non-asymptotic error bounds.
We study the sample complexity of private synthetic data generation over an unbounded sized class of statistical queries, and show that any class that is privately proper PAC learnable admits a private synthetic data generator (perhaps non-efficient). Previous work on synthetic data generators focused on the case that …
Field theory for optimal signal propagation in ResNetscond-mat.dis-nn
Field theory explains optimal scaling in ResNets for signal propagation.
problem Understanding optimal scaling parameter for ResNet performance.
method Finite-size field theory for ResNets to study signal propagation and scaling.
result Analytical expressions for optimal scaling parameter, independent of other hyperparameters.
Study on improving the linear two-time-scale stochastic approximation method with a restarting scheme.
problem Characterizing and optimizing the finite-time complexity of linear two-time-scale stochastic approximation.
method Analysis of mean square errors, introduction of a restarting scheme to improve performance.
result The method achieves an exact convergence to the desired solution with improved complexity under time-varying step sizes.
Estimates neural representation dimensionality from small sample sizes.
problem Estimating neural representation dimensionality from limited data.
method Proposed a bias-corrected estimator for participation ratio of eigenvalues.
result The estimator is more accurate with finite samples and noise.
We consider the dynamics of a linear stochastic approximation algorithm driven by Markovian noise, and derive finite-time bounds on the moments of the error, i.e., deviation of the output of the algorithm from the equilibrium point of an associated ordinary differential equation (ODE). We obtain finite-time bounds on t…
Study shows finite agent equilibrium converges to mean-field limit in asset pricing.
problem Asset pricing equilibrium in markets with finite vs infinite agents.
method Existence of finite agent equilibrium and strong convergence to mean-field limit.
result Finite agent equilibrium converges to mean-field limit under suitable conditions.
Privacy affects how much data is needed for CVaR optimization.
problem Privacy constraints impact the effective sample size for CVaR optimization.
method Analyzes the privacy-relevant sample size and decomposes CVaR excess risk.
result The effective private tail sample size is εnτ, affecting CVaR learning rates.
Non-asymptotic analysis of the performance of the penalized least trimmed squares in sparse modelsstat.ML
This study analyzes LTS in sparse models with finite sample error bounds.
problem Robust regression in high-dimensional sparse models with limited data.
method Non-asymptotic analysis of LTS error bounds.
result Established finite sample error bounds for LTS in sparse models.
The growth dynamics of German business firmscond-mat.stat-mech
We determine the distribution of size and growthrates of German business firms in 1987-1997. We find a log-normal size distribution. The distribution of growth rates has fat tails. It can be fitted to an exponential in a narrow central region and is dominated by finite-sample-size effects far in its wings. We study the…
Explicit size distributions of failure cascades redefine systemic risk on finite networksphysics.soc-ph
How big is the risk that a few initial failures of nodes in a network amplify to large cascades that span a substantial share of all nodes? Predicting the final cascade size is critical to ensure the functioning of a system as a whole. Yet, this task is hampered by uncertain or changing parameters and missing informati…
Let G be the identity component of SO(n,1), acting linearly on a finite dimensional real vector space V. Consider a vector w_0 in V such that the stabilizer of w_0 is a symmetric subgroup of G or the stabilizer of the line Rw_0 is a parabolic subgroup of G. For any non-elementary discrete subgroup Gamma of G with w_0Ga…
We prove (without using Federer's structure theorem) that a finite-mass flat chain over any coefficient group is rectifiable if and only if almost all of its 0-dimensional slices are rectifiable. This implies that every flat chain of finite mass and finite size is rectifiable. It also leads to a simple necessary and su…
New TD method stabilizes average-reward learning.
problem Stability issues in average-reward TD learning.
method Implicit fixed point update for average-reward TD().
result Improved numerical stability and broader step-size range.
Study bounds variance modulation function for K-spider distributions.
problem Bounding variance modulation function for K-spider distributions.
method Used folded moments and total probabilities of spider legs.
result Gave an interval for the variance modulation function.
Kurtosis is seen as a measure of the discrepancy between the observed data and a Gaussian distribution and is defined when the 4th moment is finite. In this work an empirical study is conducted to investigate the behaviour of the sample estimate of kurtosis with respect to sample size and the tail index when applied to…
The paper analyzes the expected size of conformal prediction sets.
problem Lack of finite-sample analysis and guarantees for prediction set sizes.
method Theoretical quantification and empirical computation of expected set size.
result Derives point estimates and high-probability interval bounds for prediction set size.
Develops a simple model to understand learning curves for arbitrary power laws.
problem Lack of theoretical understanding of scaling laws in machine learning.
method Analyzes a toy model to determine if learning curves are universal or depend on data distribution.
result Determines that learning curves can exhibit for arbitrary power .
It was proved in 1998 by Ben-David and Litman that a concept space has a sample compression scheme of size d if and only if every finite subspace has a sample compression scheme of size d. In the compactness theorem, measurability of the hypotheses of the created sample compression scheme is not guaranteed; at the same…
Exact distribution of split conformal prediction coverage found.
problem Determining the reliability of prediction sets in batch mode.
method Analysis of exchangeable data to find universal distribution of empirical coverage.
result Exact distribution of empirical coverage is universal and determined by nominal miscoverage level and calibration sample size.
Deep learning method improves regression accuracy.
problem Nonparametric regression challenges.
method Over-parametrized deep neural networks with logistic activation, gradient descent, special topology, random initialization, and data-dependent learning rate.
result Theoretical bound on error and improved finite sample performance.
Functorial semi-norms on singular homology give refined "size" information on singular homology classes. A fundamental example is the l^1-semi-norm. We show that there exist finite functorial semi-norms on singular homology that are exotic in the sense that they are not carried by the l^1-semi-norm.
Proposes SVI for covariate-shift generalization with sparse variable independence.
problem Covariate-shift generalization with limited data and unstable variables.
method Introduces sparsity constraint and combines reweighting and selection in an iterative way.
result Improves covariate-shift generalization performance on synthetic and real-world datasets.
The paper explores properties of continuous actions on manifolds, proving bounds on subgroup size and fixed points.
problem Properties of continuous finite group actions on topological manifolds.
method Analyzes properties including Jordan property and almost fixed point property, proving bounds on subgroup size.
result Existence of a constant C such that for any continuous action of a finite group G on a manifold X, there is a subgroup H with [G:H] ≤ C and a fixed point.
Stochastic gradient descent (SGD) is almost ubiquitously used for training non-convex optimization tasks. Recently, a hypothesis proposed by Keskar et al. [2017] that large batch methods tend to converge to sharp minimizers has received increasing attention. We theoretically justify this hypothesis by providing new pro…
New theorem bounds group quotient size to subgroups index.
problem Understanding subgroup structure in hyperbolic groups.
method Proved quotient size bounds on Eilenberg-MacLane spaces.
result One-ended hyperbolic groups cannot have isomorphic finite-index subgroups.
This work introduces COLA, a strategy to aggregate conformal prediction sets efficiently.
problem Efficiently combining multiple conformity scores to reduce prediction set size.
method Introduces COnfidence-Level Allocation (COLA) to optimally allocate confidence levels across sets.
result COLA achieves smaller prediction sets than state-of-the-art methods while maintaining valid coverage.
This manuscript studies statistical properties of linear classifiers obtained through minimization of an unregularized convex risk over a finite sample. Although the results are explicitly finite-dimensional, inputs may be passed through feature maps; in this way, in addition to treating the consistency of logistic reg…
We present a plausible micro-founded model for the previously postulated power law finite time singular form of the crash hazard rate in the Johansen-Ledoit-Sornette model of rational expectation bubbles. The model is based on a percolation picture of the network of traders and the concept that clusters of connected tr…
Enhanced DFO using adaptive batch-based FD estimates.
problem Derivative-free optimization with imprecise gradient estimates.
method Adaptive batch-based finite difference estimation and dynamic sampling strategy.
result Algorithm achieves convergence rate similar to KW and SPSA methods.