Study on Vapnik-Chervonenkis dimension of product intervals in R^d.
problem Combinatorial complexity of product intervals in R^d.
method Vapnik-Chervonenkis geometry approach.
result Vapnik-Chervonenkis dimension of balls in ℓ∞^d equals (3d+1)/2.
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48 results for “Vapnik dimension”
The Vapnik-Chervonenkis (VC) dimension of a collection of subsets of a set is an important combinatorial concept in settings such as discrete geometry and machine learning. In this paper we prove that the VC dimension of the family of -dimensional cubes in is .
Vapnik-Chervonenkis (VC) dimension is a fundamental measure of the generalization capacity of learning algorithms. However, apart from a few special cases, it is hard or impossible to calculate analytically. Vapnik et al. [10] proposed a technique for estimating the VC dimension empirically. While their approach behave…
In this dissertation, I derive a new method to estimate the Vapnik-Chervonenkis Dimension (VCD) for the class of linear functions. This method is inspired by the technique developed by Vapnik et al. Vapnik et al. (1994). My contribution rests on the approximation of the expected maximum difference between two empirical…
For any family of measurable sets in a probability space, we show that either (i) the family has infinite Vapnik-Chervonenkis (VC) dimension or (ii) for every epsilon > 0 there is a finite partition pi such the pi-boundary of each set has measure at most epsilon. Immediate corollaries include the fact that a family wit…
The paper provides risk bounds for learning many response functions using linear regression.
problem Learning many response functions from a single dataset.
method Ordinary least squares regression in a high-dimensional feature space.
result Convergence guarantees on worst-case excess prediction risk for infinite response functions with finite VC dimension.
Optimal sample complexity for contrastive learning of distances.
problem Minimum labeled tuples needed for high accuracy in learning distances.
method Analyzes sample complexity in various distance settings, proving tight bounds.
result Almost optimal bound on sample complexity for learning distances.
New algorithm for learning functions with bounds on error and sample complexity.
problem Learning -valued functions in a prediction model.
method General-purpose algorithm with upper and lower bounds on expected error and sample complexity.
result Improved bounds on sample complexity and agnostic learning conditions.
This research sets limits on how complex multi-class learning problems can be.
problem Understanding the complexity of multi-class classification problems.
method Established upper bounds on Natarajan dimensions for specific function classes.
result Upper bounds on Natarajan dimensions for multi-class decision trees, random forests, and neural networks.
Linear classifiers in product space forms improve scRNA-seq data classification.
problem Linear classification in products of Euclidean, spherical, and hyperbolic spaces.
method Novel formulations of linear classifiers on Riemannian manifolds, proving expressive power, and formalizing perceptron and SVM classifiers.
result Linear classifiers in product space forms have the same expressive power as in Euclidean space of the same dimension.
Deep Heaviside networks are limited but can be improved with connections or linear neurons.
problem Limited expressivity of deep Heaviside networks.
method Including skip connections or linear activation neurons improves expressivity.
result Lower and upper bounds for VC dimensions and approximation rates are derived.
Study on VC dimension of GCNNs with input resolution effects.
problem Understanding the generalization capabilities of GCNNs.
method Derived upper and lower bounds for VC dimension, analyzed factors affecting it.
result Extended previous results on VC dimension of GCNNs, providing insights into input resolution dependence.
Study on Privileged ERM showing limitations and providing capacity analysis.
problem Improving classification accuracy with privileged information.
method Theoretical analysis of Privileged ERM using VC dimension and generalization bounds.
result Worst-case guarantees for Privileged ERM cannot improve over standard ERM unless privileged information capacity is similar or smaller.
In Statistical Learning, the Vapnik-Chervonenkis (VC) dimension is an important combinatorial property of classifiers. To our knowledge, no theoretical results yet exist for the VC dimension of edited nearest-neighbour (1NN) classifiers with reference set of fixed size. Related theoretical results are scattered in the …
The recently proposed Minimal Complexity Machine (MCM) finds a hyperplane classifier by minimizing an exact bound on the Vapnik-Chervonenkis (VC) dimension. The VC dimension measures the capacity of a learning machine, and a smaller VC dimension leads to improved generalization. On many benchmark datasets, the MCM gene…
Adversarial attacks during the testing phase of neural networks pose a challenge for the deployment of neural networks in security critical settings. These attacks can be performed by adding noise that is imperceptible to humans on top of the original data. By doing so, an attacker can create an adversarial sample, whi…
Study tests whether trade-off functions are above or below benchmarks using finite samples.
problem Testing trade-off functions between unknown distributions.
method Identifies a condition for nontrivial testing, constructs a test with error guarantees, and inverts the test for confidence bands.
result Finite-sample testing is possible under specific structural assumptions about rejection regions.
In many applications of relational learning, the available data can be seen as a sample from a larger relational structure (e.g. we may be given a small fragment from some social network). In this paper we are particularly concerned with scenarios in which we can assume that (i) the domain elements appearing in the giv…
The paper provides bounds for regression schemes using nonstationary training samples.
problem Developing confidence intervals for nonparametric regression with nonstationary data.
method The approach involves Rademacher and Vapnik-Chervonenkis theories to analyze the cost and optimality of regression schemes.
result The paper establishes nonasymptotic bounds for regression schemes and optimality in -distance.
New neural network class reduces VC dimension, leading to better generalization.
problem VC theory struggles with explaining small generalization errors in overparametrized neural networks.
method Developed hyperplane arrangement neural networks (HANNs) and used sample compression analysis.
result HANNs can have significantly smaller VC dimension than the number of weights, yet remain highly expressive.
In this paper, we present a Longstaff-Schwartz-type algorithm for optimal stopping time problems based on the Brownian motion filtration. The algorithm is based on Leão, Ohashi and Russo and, in contrast to previous works, our methodology applies to optimal stopping problems for fully non-Markovian and non-semimartinga…
Paper proposes a new confidence dimension to measure DNN generalization.
problem Measuring the generalization ability of deep neural networks is challenging.
method Introduces confidence dimension (CD) based on Hoeffding's inequality and VC-dimension.
result CD provides a feasible framework to calculate the upper bound of generalization.
Reducing network complexity has been a major research focus in recent years with the advent of mobile technology. Convolutional Neural Networks that perform various vision tasks without memory overhaul is the need of the hour. This paper focuses on qualitative and quantitative analysis of reducing the network complexit…
Introduces greedy feature selection for classifier-dependent feature ranking.
problem Feature selection for classification tasks.
method Greedy feature selection, identifying the most important feature at each step based on the selected classifier.
result Theoretical and numerical benefits of greedy feature selection.
Deep ReLU networks generalize well with few parameters.
problem Generalization of overparametrized deep neural networks.
method Explicit bounds on test error independent of overparametrization and VC dimension.
result Generalization error is independent of network architecture and overparametrization.
Study extends GNN VC dimension bounds to Pfaffian activation functions.
problem Bounding GNN VC dimension for new activation functions.
method Pfaffian function theory applied to GNNs with sigmoid and hyperbolic tangent activations.
result Bounds on GNN VC dimension for various architectures and graph properties.
Contradistinguisher learns to distinguish target domain without aligning source and target domains.
problem Difficulty in aligning source and target domains for domain adaptation.
method Direct approach to unsupervised domain adaptation that learns contrastive features and improves classification performance.
result Achieves state-of-the-art performance on Office-31 and VisDA-2017 datasets.
In this article, we derive concentration inequalities for the cross-validation estimate of the generalization error for subagged estimators, both for classification and regressor. General loss functions and class of predictors with both finite and infinite VC-dimension are considered. We slightly generalize the formali…
WL meet VCcs.LG
The paper connects GNNs to VC dimension theory to study their generalization performance.
problem Understanding GNNs' ability to make meaningful predictions beyond the training set.
method Using Vapnik-Chervonenkis (VC) dimension theory in two settings: no upper bound on graph order and known upper bound.
result Tight connections between GNNs' bitlength, number of colors, and VC dimension in different settings.
Improves conformal prediction by combining multiple score functions and optimizing weights.
problem Limitations of single-score conformal predictors in multi-class classification.
method Combines multiple score functions and optimizes weights to minimize prediction set size.
result Consistently outperforms single-score conformal predictors while maintaining valid coverage.
We investigate the use of Deep Neural Networks for the classification of image datasets where texture features are important for generating class-conditional discriminative representations. To this end, we first derive the size of the feature space for some standard textural features extracted from the input dataset an…
The existence of evasion attacks during the test phase of machine learning algorithms represents a significant challenge to both their deployment and understanding. These attacks can be carried out by adding imperceptible perturbations to inputs to generate adversarial examples and finding effective defenses and detect…
We consider the fundamental question of learnability of a hypotheses class in the supervised learning setting and in the general learning setting introduced by Vladimir Vapnik. We survey classic results characterizing learnability in term of suitable notions of complexity, as well as more recent results that establish …
Paper extends nonparametric regression bounds for dependent -mixing samples.
problem Analyzing error in nonparametric regression with dependent data.
method Extends uniform deviation inequalities from independent to dependent -mixing samples.
result Derives generalization bounds for nonparametric regression with dependent data.
New Sauer inequality improves multiclass hypothesis class bounds.
problem Bounding the size of multiclass hypothesis classes.
method Polynomial method and combinatorial parameters (DS, list-DS dimensions).
result Sharp Sauer inequality with optimal polynomial dependence on list size and alphabet size.
New bounds show agnostic multiclass learning depends on two dimensions: Natarajan and Daniely-Shalev-Shwartz.
problem Understanding sample complexity in multiclass classification with agnostic learning.
method Developed a novel online procedure based on a self-adaptive multiplicative-weights algorithm.
result Agnostic sample complexity bounds are in the form of DS^(1.5)/ε + Nat/ε^2, nearly tight up to a √DS factor.
Rademacher complexity and spin glasses: A link between the replica and statistical theories of learningcond-mat.dis-nn
Statistical learning theory provides bounds of the generalization gap, using in particular the Vapnik-Chervonenkis dimension and the Rademacher complexity. An alternative approach, mainly studied in the statistical physics literature, is the study of generalization in simple synthetic-data models. Here we discuss the c…
SVR analyzed within RQ framework for risk management.
problem Risk management in stochastic optimization.
method Risk Quadrangle (RQ) theory applied to SVR.
result SVR formulations as minimization of Vapnik error and CVaR norm.
New algorithm reduces sample complexity for multi-distribution learning.
problem Achieving data-efficient multi-distribution learning with robustness and fairness.
method Proposes a novel algorithm with sample complexity (d+k)/varepsilon^2 for Vapnik-Chervonenkis (VC) dimension d, matching lower bounds.
result Algorithm matches best-known lower bound and resolves open problems in COLT 2023.
Statistical Regeneration Guarantees of the Wasserstein Autoencoder with Latent Space Consistencystat.ML
This paper provides statistical guarantees for WAE's latent space regeneration.
problem Lack of statistical analysis for Autoencoders, especially WAE.
method Utilizes Vapnik Chervonenkis (VC) theory and Optimal Transport of measures under the Wasserstein metric.
result WAE achieves the target distribution in the latent space and regenerates the input distribution.
In Ben-David et al.'s "Learnability Can Be Undecidable," they prove an independence result in theoretical machine learning. In particular, they define a new type of learnability, called Estimating The Maximum (EMX) learnability. They argue that this type of learnability fits in with other notions such as PAC learnabili…
The paper tackles extrapolation in extreme regions of regression problems.
problem Extrapolation on the tails of covariates in continuous regression problems.
method Statistical regression on a subsample of furthest observations, focusing on their angular components, using multivariate regular variation theory.
result Quantifies predictive performance on tail regions in terms of excess risk, presenting it as a finite sample risk bound with a bias-variance decomposition.
Distillation (Hinton et al., 2015) and privileged information (Vapnik & Izmailov, 2015) are two techniques that enable machines to learn from other machines. This paper unifies these two techniques into generalized distillation, a framework to learn from multiple machines and data representations. We provide theoretica…
Chemical networks outperform spiking neural networks in classification tasks.
problem Learning tasks with spiking neural networks require hidden layers, which are computationally expensive.
method Used deterministic mass-action kinetics to prove chemical reaction networks without hidden layers can solve tasks previously solved by spiking neural networks.
result A chemical reaction network without hidden layers outperforms a spiking neural network with hidden layers in a handwritten digit classification task.
An Approach to One-Bit Compressed Sensing Based on Probably Approximately Correct Learning Theorystat.ML
In this paper, the problem of one-bit compressed sensing (OBCS) is formulated as a problem in probably approximately correct (PAC) learning. It is shown that the Vapnik-Chervonenkis (VC-) dimension of the set of half-spaces in generated by -sparse vectors is bounded below by and above by…
Paper combines RL with policy regularization for inventory policies.
problem Optimizing inventory policies using RL and dynamic programming.
method Hybrid approach combining RL with policy regularization.
result Generalization guarantees for inventory policies using VC theory.
Deep networks can efficiently approximate functions on curved manifolds.
problem Approximating functions and their derivatives on complex, curved domains.
method Proved constant-depth ReLU networks can approximate functions in Sobolev spaces on manifolds.
result Deep networks with bounded weights can approximate functions in to an error of using parameters.
Many machine learning algorithms assume that all input samples are independently and identically distributed from some common distribution on either the input space X, in the case of unsupervised learning, or the input and output space X x Y in the case of supervised and semi-supervised learning. In the last number of …