We relate two notions of local error for integration schemes on Riemannian homogeneous spaces, and show how to derive global error estimates from such local bounds. In doing so, we prove for the first time that the Lie-Butcher theory of Lie group integrators leads to global error estimates.
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48 results for “global errors”
How PC-based Methods Err: Towards Better Reporting of Assumption Violations and Small Sample Errorsstat.ML
The study analyzes and mitigates errors in PC-based causal discovery methods.
problem Errors in PC-based causal discovery methods can lead to incorrect graphs.
method The study introduces coherency scores to detect assumption violations and small sample errors in PC-based methods.
result The coherency scores can detect errors that other methods cannot, bridging between global and local error detection.
Accelerates convergence in global non-convex optimization with reversible diffusion.
problem Global non-convex optimization challenges.
method Utilizes reversible diffusion processes with adaptive diffusion coefficients.
result Accelerated convergence with reduced discretization error.
The problem of estimation error of Expected Shortfall is analyzed, with a view of its introduction as a global regulatory risk measure.
We study the error landscape of deep linear and nonlinear neural networks with the squared error loss. Minimizing the loss of a deep linear neural network is a nonconvex problem, and despite recent progress, our understanding of this loss surface is still incomplete. For deep linear networks, we present necessary and s…
JSRT improves regression tree performance by incorporating global node information.
problem Regression tree performance relies on local node means, ignoring global node information.
method Proposes JSRT by integrating global mean information from different nodes.
result Demonstrates superior performance and efficiency compared to other regression tree methods.
Analyzes error sources in global feature effect estimation methods.
problem Unexplored error sources in global feature effect estimation methods.
method Systematic, estimator-level analysis of bias and variance.
result Holdout data is theoretically cleanest, but estimation variance depends on sample size and model characteristics.
Global optimization for low-dimensional switching linear regression and bounded-error estimationcs.LG
The paper provides global optimization algorithms for two particularly difficult nonconvex problems raised by hybrid system identification: switching linear regression and bounded-error estimation. While most works focus on local optimization heuristics without global optimality guarantees or with guarantees valid only…
This letter presents an improved version of diffusion least mean ppower (LMP) algorithm for distributed estimation. Instead of sum of mean square errors, a weighted sum of mean square error is defined as the cost function for global and local cost functions of a network of sensors. The weight coefficients are updated b…
Adaptive method improves prediction intervals with global coverage guarantees and local error distribution.
problem Global coverage guarantees of conformal regression are often violated by local error distributions.
method Adaptive Conformal Regression with Jackknife+ Rescaled Scores
result Improves local coverage without sacrificing global coverage, especially in low-data regimes.
The study analyzes how covariance estimation errors affect the global minimum-variance portfolio under heavy-tailed distributions.
problem The impact of covariance estimation errors on the global minimum-variance portfolio under heavy-tailed distributions.
method Characterization of covariance-estimation error's effect on GMVP suboptimality, derivation of regret identity and bound, application to heavy-tailed returns.
result The decision geometry of GMVP regret is invariant to a (p-1)-dimensional projection of the error matrix, with invariance to the covariance-scale direction as an exact special case.
A hybrid loss framework improves time series forecasting by balancing global and component errors.
problem Current time series methods may prioritize less significant sub-series, leading to forecasting bias.
method Proposes a hybrid loss framework combining global and component losses, dynamically adjusting weights.
result Improves time series forecasting performance by 0.5-2% on average.
GAAVI offers anytime-valid tests for CMF global null and contrasts.
problem Inference on the conditional mean function for high confidence decisions.
method Asymptotic anytime-valid tests for CMF global null and contrasts.
result Achieves asymptotic type-I error guarantees, power one, and optimal sample complexity.
Supervised training of neural networks for classification is typically performed with a global loss function. The loss function provides a gradient for the output layer, and this gradient is back-propagated to hidden layers to dictate an update direction for the weights. An alternative approach is to train the network …
Replica exchange Langevin diffusion accelerates nonconvex optimization.
problem Nonconvex optimization challenges in machine learning.
method Replica exchange Langevin diffusion, discretization analysis.
result Replica exchange accelerates convergence to global minima.
We present a new way of constructing an ensemble classifier, named the Guided Random Forest (GRAF) in the sequel. GRAF extends the idea of building oblique decision trees with localized partitioning to obtain a global partitioning. We show that global partitioning bridges the gap between decision trees and boosting alg…
We show that if is a convex class of functions that is -subgaussian, the error rate of learning problems generated by independent noise is equivalent to a fixed point determined by `local' covering estimates of the class, rather than by the gaussian averages. To that end, we establish new sharp upper and lower e…
Optimal Robust Estimation under Local and Global Corruptions: Stronger Adversary and Smaller Errorcs.DS
New algorithm optimizes robust estimation under mixed local and global corruptions.
problem Combining local and global corruptions in robust statistics.
method Information-theoretic approach using sliced-Wasserstein metric.
result Optimal error achieved in polynomial time for stronger local perturbations.
Temporal-difference learning (TD), coupled with neural networks, is among the most fundamental building blocks of deep reinforcement learning. However, due to the nonlinearity in value function approximation, such a coupling leads to nonconvexity and even divergence in optimization. As a result, the global convergence …
Neuroscientific theory suggests that dopaminergic neurons broadcast global reward prediction errors to large areas of the brain influencing the synaptic plasticity of the neurons in those regions. We build on this theory to propose a multi-agent learning framework with spiking neurons in the generalized linear model (G…
Study compares local and global models for hierarchical forecasting accuracy.
problem Challenges in hierarchical time series forecasting, especially in accuracy and information utilisation.
method Developed and evaluated local and global forecasting models (GFMs) to exploit cross-series and cross-hierarchies information.
result Global Forecasting Models (GFMs) outperform local models in hierarchical forecasting accuracy and computational efficiency.
We study streaming principal component analysis (PCA), that is to find, in space, the top eigenvectors of a hidden matrix with online vectors drawn from covariance matrix . We provide convergence for Oja's algorithm which is popularly used in practice but lacks t…
We apply ideas from viscosity theory to establish the existence of a unique global weak solution to the generalized Kahler-Ricci flow in the setting of commuting complex structures. Our results are restricted to the case of a smooth manifold with smooth background data. We discuss the possibility of extending these res…
This technical note extends recent results on the computational complexity of globally minimizing the error of piecewise-affine models to the related problem of minimizing the error of switching linear regression models. In particular, we show that, on the one hand the problem is NP-hard, but on the other hand, it admi…
GIV methodology extends instrumental variable estimation for high-dimensional data.
problem Estimating structural parameters in high-dimensional models with endogeneity and latent factors.
method Extends GIV methodology to large N and T, treats factors and loadings as unknown, and uses additional instruments for efficiency.
result Efficiency gains and negligible sampling errors in estimated instrument and factors.
Neural networks can approximate complex stochastic equations well.
problem Approximating general stochastic differential equations.
method Identified neural network classes approximating continuous functions.
result Neural stochastic differential equations can approximate general stochastic differential equations arbitrarily well.
Paper tackles fairness in CCA by minimizing correlation disparity error.
problem Fairness issues in CCA.
method Framework to minimize correlation disparity error in CCA.
result Reduces correlation disparity error without sacrificing CCA accuracy.
New algorithm trains deep neural networks without global optimization.
problem Training deep neural networks efficiently and without global optimization.
method Uses random complex exponential activation functions and Markov Chain Monte Carlo sampling.
result Consistently attains theoretical approximation rate for residual networks.
SeqRF straightens generative model flows to speed up sampling.
problem High global truncation error in ODE-based solvers for generative models.
method SeqRF, a learning technique that straightens the probability flow.
result Significantly improved sampling speed and synthesis quality.
Estimating Covariance for Global Minimum Variance Portfolio: A Decision-Focused Learning Approachq-fin.PM
Paper uses DFL to optimize portfolio risk and outperforms conventional methods.
problem Optimizing portfolio risk and return under uncertainty.
method Decision-focused learning (DFL) to derive global minimum variance portfolio (GMVP).
result DFL-based methods consistently deliver superior decision performance in portfolio optimization.
An Euler discretization of the Langevin diffusion is known to converge to the global minimizers of certain convex and non-convex optimization problems. We show that this property holds for any suitably smooth diffusion and that different diffusions are suitable for optimizing different classes of convex and non-convex …
A central challenge to many fields of science and engineering involves minimizing non-convex error functions over continuous, high dimensional spaces. Gradient descent or quasi-Newton methods are almost ubiquitously used to perform such minimizations, and it is often thought that a main source of difficulty for these l…
Policy gradient methods are among the most effective methods in challenging reinforcement learning problems with large state and/or action spaces. However, little is known about even their most basic theoretical convergence properties, including: if and how fast they converge to a globally optimal solution or how they …
In this paper we define a small variation of the Taylor method and a formula for the global error of this new numerical method that allows us to keep track of the round-off error and does not require previous knowledge of the exact solution. As an application we provide a rigorous proof of the construction/existence of…
We analyze the loss landscape and expressiveness of practical deep convolutional neural networks (CNNs) with shared weights and max pooling layers. We show that such CNNs produce linearly independent features at a "wide" layer which has more neurons than the number of training samples. This condition holds e.g. for the…
We propose SEARNN, a novel training algorithm for recurrent neural networks (RNNs) inspired by the "learning to search" (L2S) approach to structured prediction. RNNs have been widely successful in structured prediction applications such as machine translation or parsing, and are commonly trained using maximum likelihoo…
We study distributed learning with the least squares regularization scheme in a reproducing kernel Hilbert space (RKHS). By a divide-and-conquer approach, the algorithm partitions a data set into disjoint data subsets, applies the least squares regularization scheme to each data subset to produce an output function, an…
Global Convergence and Generalization Bound of Gradient-Based Meta-Learning with Deep Neural Netscs.LG
GBML with deep nets converges globally and generalizes well.
problem Theoretical guarantees for few-shot learning with deep nets.
method Proving global convergence and generalization bounds for GBML with over-parameterized DNNs.
result GBML with over-parameterized DNNs converges globally to the optimum at a linear rate and achieves good generalization.
Federated learning models are analyzed through game theory to determine optimal model sharing.
problem Agents with different data distributions face a choice between local or global models in federated learning.
method The problem is analyzed using coalitional game theory and hedonic game theory, considering different degrees of customization in model sharing.
result Exact expected MSE values are derived for linear regression and mean estimation problems, and stable partitions of players into coalitions are analyzed.
New framework analyzes deep learning optimization with finite width networks, revealing generalization gaps and excess risks.
problem Analyzing generalization error of deep learning with finite width networks.
method Formulating neural network training as transportation map estimation and analyzing via infinite dimensional Langevin dynamics.
result Achieves fast learning rate and minimax optimal rates for classification and regression problems.
Current system thermal-hydraulic codes have limited credibility in simulating real plant conditions, especially when the geometry and boundary conditions are extrapolated beyond the range of test facilities. This paper proposes a data-driven approach, Feature Similarity Measurement FFSM), to establish a technical basis…
Optimized AIS scheme reduces bias and MSE for general proposals.
problem Performing Monte Carlo integration with general proposals.
method Global optimization of χ²-divergence using stochastic gradient Langevin dynamics.
result Explicit theoretical guarantees for uniform-in-time MSE reduction.
Learning a Class of Mixed Linear Regressions: Global Convergence under General Data Conditionsstat.ML
Paper addresses global convergence of MLR estimation under weak data conditions.
problem Learning mixed linear regression models with general data conditions.
method Two-step recursive identification algorithm using least squares and EM principles.
result Global convergence and optimal clustering performance established under general data conditions.
Study of regularized least squares in RKKS with indefinite kernels.
problem Asymptotic properties of regularized least squares with indefinite kernels in RKKS.
method Introducing a bounded hyper-sphere constraint, theoretical demonstration of globally optimal solution, modified error decomposition techniques, matrix perturbation theory.
result Derivation of learning rates in RKKS, same as RKHS under certain conditions.
The paper explores privacy-preserving methods for counting unique elements in distributed settings.
problem Counting unique elements in a distributed setting while maintaining privacy.
method Analyzes and proves lower bounds for differentially private protocols in various settings.
result Achieves optimal error bounds for multi-message shuffle protocols in estimating distinct elements.
This paper studies how label noise affects Federated Learning.
problem The impact of label noise on Federated Learning.
method The paper derives an upper bound for the generalization error and conducts experiments on MNIST and CIFAR-10 datasets.
result The global model accuracy decreases linearly with increasing label noise, consistent with theoretical analysis.
Neural networks, a central tool in machine learning, have demonstrated remarkable, high fidelity performance on image recognition and classification tasks. These successes evince an ability to accurately represent high dimensional functions, but rigorous results about the approximation error of neural networks after tr…
We examine the squared error loss landscape of shallow linear neural networks. We show---with significantly milder assumptions than previous works---that the corresponding optimization problems have benign geometric properties: there are no spurious local minima and the Hessian at every saddle point has at least one ne…