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On the Complexity of Labeled Datasets [article]

Rodrigo Fernandes de Mello
2021 arXiv   pre-print
The Statistical Learning Theory (SLT) provides the foundation to ensure that a supervised algorithm generalizes the mapping f: 𝒳→𝒴 given f is selected from its search space bias ℱ.  ...  However, the analytical solution of the Shattering coefficient is still an open problem since the first studies by Vapnik and Chervonenkis in 1962, which we address on specific datasets, in this paper,  ...  Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of Itaú Unibanco SA, CNPq, nor FAPESP.  ... 
arXiv:1911.05461v3 fatcat:lb2gcscd2vcrnmlih36ogcfwjy

Computing the Shattering Coefficient of Supervised Learning Algorithms [article]

Rodrigo Fernandes de Mello, Moacir Antonelli Ponti, Carlos Henrique Grossi Ferreira
2018 arXiv   pre-print
This paper proves the Shattering coefficient for any Hilbert space H containing the input space X and discusses its effects in terms of learning guarantees for supervised machine algorithms.  ...  The Statistical Learning Theory (SLT) provides the theoretical guarantees for supervised machine learning based on the Empirical Risk Minimization Principle (ERMP).  ...  SHATTERING COEFFICIENT The Shattering coefficient N (F, 2n) of any h-dimensional Hilbert space H being classified with a single (h − 1)-  ... 
arXiv:1805.02627v4 fatcat:bbglwnkidbb7lkkgzxer3a3rqu

Data driven semi-supervised learning [article]

Maria-Florina Balcan, Dravyansh Sharma
2021 arXiv   pre-print
We expect some of the tools and techniques we develop along the way to be of interest beyond semi-supervised learning, for data driven algorithms for combinatorial problems more generally.  ...  Over the past decades, several elegant graph-based semi-supervised learning algorithms for how to infer the labels of the unlabeled examples given the graph and a few labeled examples have been proposed  ...  We provide asymptotically tight upper and lower bounds on the pseudodimension of learning the best parameter from a parameterized family of semi-supervised learning algorithms, each algorithm corresponding  ... 
arXiv:2103.10547v4 fatcat:tqvet3eg2nbnnmrlq7tinsofse

The Learnability of Unknown Quantum Measurements [article]

Hao-Chung Cheng, Min-Hsiu Hsieh, Ping-Cheng Yeh
2015 arXiv   pre-print
Our main result in the paper is that, for learning an unknown quantum measurement, the upper bound, given by the fat-shattering dimension, is linearly proportional to the dimension of the underlying Hilbert  ...  This kind of problem is, arguably, one of the most fundamental problems in statistical learning theory and the bounds for practical settings can be completely characterised by a simple measure of complexity  ...  procedure of supervised, unsupervised and SVM algorithms.  ... 
arXiv:1501.00559v1 fatcat:5wtqou5pi5e6hgcu54n4rphyce

A Survey on Semi-Supervised Learning for Delayed Partially Labelled Data Streams [article]

Heitor Murilo Gomes, Maciej Grzenda, Rodrigo Mello, Jesse Read, Minh Huong Le Nguyen, Albert Bifet
2021 arXiv   pre-print
To address the learning problems associated with such data, one can ignore the unlabelled data and focus only on the labelled data (supervised learning); use the labelled data and attempt to leverage the  ...  unlabelled data (semi-supervised learning); or assume some labels will be available on request (active learning).  ...  In that sense, let us assume the Shattering coefficient function N (F , 2 ) = 2 for a specific semi-supervised algorithm working on some -dimensional Hilbert input space, thus defining the maximal number  ... 
arXiv:2106.09170v1 fatcat:nn4ja4ptsndwngyocjne2tp4ae

Activized Learning: Transforming Passive to Active with Improved Label Complexity [article]

Steve Hanneke
2011 arXiv   pre-print
We further provide a general characterization of the magnitudes of these improvements in terms of a novel generalization of the disagreement coefficient.  ...  We study the theoretical advantages of active learning over passive learning.  ...  Background The term active learning refers to a family of supervised learning protocols, characterized by the ability of the learning algorithm to pose queries to a teacher, who has access to the target  ... 
arXiv:1108.1766v1 fatcat:hldmtf567vdg5aa7zjpllk4n4u

A general agnostic active learning algorithm

Sanjoy Dasgupta, Daniel J. Hsu, Claire Monteleoni
2007 Neural Information Processing Systems  
Our algorithm extends the simple scheme of Cohn, Atlas, and Ladner [1] to the agnostic setting, using reductions to supervised learning that harness generalization bounds in a simple but subtle manner.  ...  We present an agnostic active learning algorithm for any hypothesis class of bounded VC dimension under arbitrary data distributions.  ...  ensures the nth shatter coefficient S(H, n) is at most O(n d ) by Sauer's lemma).  ... 
dblp:conf/nips/DasguptaHM07 fatcat:3dy4fwfjq5fopnbtt6f6ckvfou

Activized Learning with Uniform Classification Noise

Liu Yang, Steve Hanneke
2013 International Conference on Machine Learning  
We prove that for any VC class, it is possible to transform any passive learning algorithm into an active learning algorithm with strong asymptotic improvements in label complexity for every nontrivial  ...  This generalizes a similar result proven by (Hanneke, 2009; 2012) for the realizable case, and is the first result establishing that such general improvement guarantees are possible in the presence of  ...  It is therefore important to try to reduce the total number of labels needed for supervised learning.  ... 
dblp:conf/icml/YangH13 fatcat:w4ye3ymbrrdtxlky6vgsfv66ka

Theoretical learning guarantees applied to acoustic modeling

Christopher D. Shulby, Martha D. Ferreira, Rodrigo F. de Mello, Sandra M. Aluisio
2019 Journal of the Brazilian Computer Society  
Furthermore, we present proof that our algorithm does adhere to the learning guarantees provided by the statistical learning theory.  ...  More importantly, we isolate the performance of the acoustic model and provide results on both the frame and phoneme level, considering the true robustness of the model.  ...  Also, we thank Capes for the grant PROEX-7901561/D to MF.  ... 
doi:10.1186/s13173-018-0081-3 fatcat:vdxoozswkjfznkhwyfjpt3runy

The VC-Dimension of Similarity Hypotheses Spaces [article]

Mark Herbster, Paul Rubenstein, James Townsend
2015 arXiv   pre-print
Given a set X and a function h:X⟶{0,1} which labels each element of X with either 0 or 1, we may define a function h^(s) to measure the similarity of pairs of points in X according to h.  ...  This idea can be extended to a set of functions, or hypothesis space H⊆{0,1}^X by defining a similarity hypothesis space H^(s):={h^(s):h∈H}. We show that vc-dimension(H^(s)) ∈Θ(vc-dimension(H)).  ...  In particular we would like thank Ruth Urner for proving an initial motivating upper bound of d(H (s) ) ≤ 2d(H) log(2d(H)).  ... 
arXiv:1502.07143v1 fatcat:utlfcoh6dja5hevnciaue64zla

Finite Sample Complexity of Rare Pattern Anomaly Detection

Md Amran Siddiqui, Alan Fern, Thomas G. Dietterich, Shubhomoy Das
2016 Conference on Uncertainty in Artificial Intelligence  
However, compared to supervised learning, there has been very little work aimed at understanding the sample complexity of anomaly detection.  ...  In analogy with the PAC framework for supervised learning, we develop sample complexity results that relate the complexity of the pattern space to the data requirements needed for PAC guarantees.  ...  Acknowledgements This work is partially supported by the Future of Life Institute and DARPA under contract number FA8650-15-C-7557 and W911NF-11-C-0088.  ... 
dblp:conf/uai/SiddiquiFDD16 fatcat:d2ikhytdgvbijfqtvt6qeeeyuu

Bounds for the VC Dimension of 1NN Prototype Sets [article]

Iain A. D. Gunn, Ludmila I. Kuncheva
2019 arXiv   pre-print
We discuss the implications of these bounds for the size of training set needed to learn such a classifier to a given accuracy.  ...  In Statistical Learning, the Vapnik-Chervonenkis (VC) dimension is an important combinatorial property of classifiers.  ...  While preparing the paper for publication, IG received support from the European Union's Horizon 2020 research and Innovation programme under grant agreement No 731593.  ... 
arXiv:1902.02660v1 fatcat:ikfbxwkrtbao7itqd6csfsrsuq

Approximation beats concentration? An approximation view on inference with smooth radial kernels [article]

Mikhail Belkin
2018 arXiv   pre-print
We also investigate the fitting capacity of kernels, giving explicit bounds on the fat shattering dimension of the balls in Reproducing Kernel Hilbert spaces.  ...  We analyze eigenvalue decay of kernels operators and matrices, properties of eigenfunctions/eigenvectors and "Fourier" coefficients of functions in the kernel space restricted to a discrete set of data  ...  He would like to acknowledge NSF funding and the hospitality of the Simons Institute for the Theory of Computing, where early discussions took place.  ... 
arXiv:1801.03437v2 fatcat:bp3imztybzhg7c6qu3nj7htyjq

Robust Phoneme Recognition with Little Data

Christopher Dane Shulby, Martha Dais Ferreira, Rodrigo F. De Mello, Sandra Maria Aluisio, Michael Wagner
2019 Symposium on Languages, Applications and Technologies  
and calculating the shattering coefficient for the algorithms used.  ...  We also beat the best replication study of the state of the art with a 28% FER.  ...  Additionally, Vapnik proved a bound for supervised learning algorithms considering the shattering coefficient N (F, 2n).  ... 
doi:10.4230/oasics.slate.2019.4 dblp:conf/slate/ShulbyFMA19 fatcat:lesl5ro5pjh3beo5i4ewvhg3qm

PAC-learning in the presence of evasion adversaries [article]

Daniel Cullina, Arjun Nitin Bhagoji, Prateek Mittal
2018 arXiv   pre-print
The existence of evasion attacks during the test phase of machine learning algorithms represents a significant challenge to both their deployment and understanding.  ...  In this paper, we step away from the attack-defense arms race and seek to understand the limits of what can be learned in the presence of an evasion adversary.  ...  In this paper, we take a more fundamental approach to understanding the robustness of supervised classification algorithms by extending well-understood results for supervised batch learning in statistical  ... 
arXiv:1806.01471v2 fatcat:nvwuhgloqzhp5hckltscsfo4gi
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