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Journal articles on the topic 'Generalization bounds'

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Author: Grafiati

Published: 4 June 2021

Last updated: 31 January 2023

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1

Cohn, David, and Gerald Tesauro. "How Tight Are the Vapnik-Chervonenkis Bounds?" Neural Computation 4, no. 2 (March 1992): 249–69. http://dx.doi.org/10.1162/neco.1992.4.2.249.

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We describe a series of numerical experiments that measure the average generalization capability of neural networks trained on a variety of simple functions. These experiments are designed to test the relationship between average generalization performance and the worst-case bounds obtained from formal learning theory using the Vapnik-Chervonenkis (VC) dimension (Blumer et al. 1989; Haussler et al. 1990). Recent statistical learning theories (Tishby et al. 1989; Schwartz et al. 1990) suggest that surpassing these bounds might be possible if the spectrum of possible generalizations has a “gap” near perfect performance. We indeed find that, in some cases, the average generalization is significantly better than the VC bound: the approach to perfect performance is exponential in the number of examples m, rather than the 1/m result of the bound. However, in these cases, we have not found evidence of the gap predicted by the above statistical theories. In other cases, we do find the 1/m behavior of the VC bound, and in these cases, the numerical prefactor is closely related to the prefactor contained in the bound.

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2

Nedovic, M., and Lj Cvetkovic. "Norm bounds for the inverse and error bounds for linear complementarity problems for {P1,P2}-Nekrasov matrices." Filomat 35, no. 1 (2021): 239–50. http://dx.doi.org/10.2298/fil2101239n.

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{P1,P2}-Nekrasov matrices represent a generalization of Nekrasov matrices via permutations. In this paper, we obtained an error bound for linear complementarity problems for fP1; P2g-Nekrasov matrices. Numerical examples are given to illustrate that new error bound can give tighter results compared to already known bounds when applied to Nekrasov matrices. Also, we presented new max-norm bounds for the inverse of {P1,P2}-Nekrasov matrices in the block case, considering two different types of block generalizations. Numerical examples show that new norm bounds for the block case can give tighter results compared to already known bounds for the point-wise case.

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3

Nedovic, M. "Norm bounds for the inverse for generalized Nekrasov matrices in point-wise and block case." Filomat 35, no. 8 (2021): 2705–14. http://dx.doi.org/10.2298/fil2108705n.

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Lower-semi-Nekrasov matrices represent a generalization of Nekrasov matrices. For the inverse of lower-semi-Nekrasov matrices, a max-norm bound is proposed. Numerical examples are given to illustrate that new norm bound can give tighter results compared to already known bounds when applied to Nekrasov matrices. Also, we presented new max-norm bounds for the inverse of lower-semi-Nekrasov matrices in the block case. We considered two types of block generalizations and illustrated the results with numerical examples.

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4

Rubab, Faiza, Hira Nabi, and Asif R. Khan. "GENERALIZATION AND REFINEMENTS OF JENSEN INEQUALITY." Journal of Mathematical Analysis 12, no. 5 (October 31, 2021): 1–27. http://dx.doi.org/10.54379/jma-2021-5-1.

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We give generalizations and refinements of Jensen and Jensen− Mercer inequalities by using weights which satisfy the conditions of Jensen and Jensen− Steffensen inequalities. We also give some refinements for discrete and integral version of generalized Jensen−Mercer inequality and shown to be an improvement of the upper bound for the Jensen’s difference given in [32]. Applications of our work include new bounds for some important inequalities used in information theory, and generalizing the relations among means.

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5

Liu, Tongliang, Dacheng Tao, and Dong Xu. "Dimensionality-Dependent Generalization Bounds for k-Dimensional Coding Schemes." Neural Computation 28, no. 10 (October 2016): 2213–49. http://dx.doi.org/10.1162/neco_a_00872.

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The k-dimensional coding schemes refer to a collection of methods that attempt to represent data using a set of representative k-dimensional vectors and include nonnegative matrix factorization, dictionary learning, sparse coding, k-means clustering, and vector quantization as special cases. Previous generalization bounds for the reconstruction error of the k-dimensional coding schemes are mainly dimensionality-independent. A major advantage of these bounds is that they can be used to analyze the generalization error when data are mapped into an infinite- or high-dimensional feature space. However, many applications use finite-dimensional data features. Can we obtain dimensionality-dependent generalization bounds for k-dimensional coding schemes that are tighter than dimensionality-independent bounds when data are in a finite-dimensional feature space? Yes. In this letter, we address this problem and derive a dimensionality-dependent generalization bound for k-dimensional coding schemes by bounding the covering number of the loss function class induced by the reconstruction error. The bound is of order [Formula: see text], where m is the dimension of features, k is the number of the columns in the linear implementation of coding schemes, and n is the size of sample, [Formula: see text] when n is finite and [Formula: see text] when n is infinite. We show that our bound can be tighter than previous results because it avoids inducing the worst-case upper bound on k of the loss function. The proposed generalization bound is also applied to some specific coding schemes to demonstrate that the dimensionality-dependent bound is an indispensable complement to the dimensionality-independent generalization bounds.

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6

Pereira, Rajesh, and Mohammad Ali Vali. "Generalizations of the Cauchy and Fujiwara Bounds for Products of Zeros of a Polynomial." Electronic Journal of Linear Algebra 31 (February 5, 2016): 565–71. http://dx.doi.org/10.13001/1081-3810.3333.

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The Cauchy bound is one of the best known upper bounds for the modulus of the zeros of a polynomial. The Fujiwara bound is another useful upper bound for the modulus of the zeros of a polynomial. In this paper, compound matrices are used to derive a generalization of both the Cauchy bound and the Fujiwara bound. This generalization yields upper bounds for the modulus of the product of $m$ zeros of the polynomial.

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7

Parrondo, J. M. R., and C. Van den Broeck. "Vapnik-Chervonenkis bounds for generalization." Journal of Physics A: Mathematical and General 26, no. 9 (May 7, 1993): 2211–23. http://dx.doi.org/10.1088/0305-4470/26/9/016.

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8

Freund, Yoav, Yishay Mansour, and Robert E. Schapire. "Generalization bounds for averaged classifiers." Annals of Statistics 32, no. 4 (August 2004): 1698–722. http://dx.doi.org/10.1214/009053604000000058.

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9

Papadatos, N., and V. Papathanasiou. "A generalization of variance bounds." Statistics & Probability Letters 28, no. 2 (June 1996): 191–94. http://dx.doi.org/10.1016/0167-7152(95)00117-4.

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10

Kochedykov, D. A. "Combinatorial shell bounds for generalization ability." Pattern Recognition and Image Analysis 20, no. 4 (December 2010): 459–73. http://dx.doi.org/10.1134/s1054661810040061.

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11

Balle, Borja, and Mehryar Mohri. "Generalization bounds for learning weighted automata." Theoretical Computer Science 716 (March 2018): 89–106. http://dx.doi.org/10.1016/j.tcs.2017.11.023.

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12

Jose, Sharu Theresa, and Osvaldo Simeone. "Information-Theoretic Generalization Bounds for Meta-Learning and Applications." Entropy 23, no. 1 (January 19, 2021): 126. http://dx.doi.org/10.3390/e23010126.

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Meta-learning, or “learning to learn”, refers to techniques that infer an inductive bias from data corresponding to multiple related tasks with the goal of improving the sample efficiency for new, previously unobserved, tasks. A key performance measure for meta-learning is the meta-generalization gap, that is, the difference between the average loss measured on the meta-training data and on a new, randomly selected task. This paper presents novel information-theoretic upper bounds on the meta-generalization gap. Two broad classes of meta-learning algorithms are considered that use either separate within-task training and test sets, like model agnostic meta-learning (MAML), or joint within-task training and test sets, like reptile. Extending the existing work for conventional learning, an upper bound on the meta-generalization gap is derived for the former class that depends on the mutual information (MI) between the output of the meta-learning algorithm and its input meta-training data. For the latter, the derived bound includes an additional MI between the output of the per-task learning procedure and corresponding data set to capture within-task uncertainty. Tighter bounds are then developed for the two classes via novel individual task MI (ITMI) bounds. Applications of the derived bounds are finally discussed, including a broad class of noisy iterative algorithms for meta-learning.

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13

Abou–Moustafa, Karim, and Csaba Szepesvári. "An Exponential Tail Bound for the Deleted Estimate." Proceedings of the AAAI Conference on Artificial Intelligence 33 (July 17, 2019): 3143–50. http://dx.doi.org/10.1609/aaai.v33i01.33013143.

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There is an accumulating evidence in the literature that stability of learning algorithms is a key characteristic that permits a learning algorithm to generalize. Despite various insightful results in this direction, there seems to be an overlooked dichotomy in the type of stability-based generalization bounds we have in the literature. On one hand, the literature seems to suggest that exponential generalization bounds for the estimated risk, which are optimal, can be only obtained through stringent, distribution independent and computationally intractable notions of stability such as uniform stability. On the other hand, it seems that weaker notions of stability such as hypothesis stability, although it is distribution dependent and more amenable to computation, can only yield polynomial generalization bounds for the estimated risk, which are suboptimal. In this paper, we address the gap between these two regimes of results. In particular, the main question we address here is whether it is possible to derive exponential generalization bounds for the estimated risk using a notion of stability that is computationally tractable and distribution dependent, but weaker than uniform stability. Using recent advances in concentration inequalities, and using a notion of stability that is weaker than uniform stability but distribution dependent and amenable to computation, we derive an exponential tail bound for the concentration of the estimated risk of a hypothesis returned by a general learning rule, where the estimated risk is expressed in terms of the deleted estimate. Interestingly, we note that our final bound has similarities to previous exponential generalization bounds for the deleted estimate, in particular, the result of Bousquet and Elisseeff (2002) for the regression case.

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Gassner, Niklas, Marcus Greferath, Joachim Rosenthal, and Violetta Weger. "Bounds for Coding Theory over Rings." Entropy 24, no. 10 (October 16, 2022): 1473. http://dx.doi.org/10.3390/e24101473.

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Coding theory where the alphabet is identified with the elements of a ring or a module has become an important research topic over the last 30 years. It has been well established that, with the generalization of the algebraic structure to rings, there is a need to also generalize the underlying metric beyond the usual Hamming weight used in traditional coding theory over finite fields. This paper introduces a generalization of the weight introduced by Shi, Wu and Krotov, called overweight. Additionally, this weight can be seen as a generalization of the Lee weight on the integers modulo 4 and as a generalization of Krotov's weight over the integers modulo 2s for any positive integer s. For this weight, we provide a number of well-known bounds, including a Singleton bound, a Plotkin bound, a sphere-packing bound and a Gilbert–Varshamov bound. In addition to the overweight, we also study a well-known metric on finite rings, namely the homogeneous metric, which also extends the Lee metric over the integers modulo 4 and is thus heavily connected to the overweight. We provide a new bound that has been missing in the literature for homogeneous metric, namely the Johnson bound. To prove this bound, we use an upper estimate on the sum of the distances of all distinct codewords that depends only on the length, the average weight and the maximum weight of a codeword. An effective such bound is not known for the overweight.

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15

Cao, Yuan, and Quanquan Gu. "Generalization Error Bounds of Gradient Descent for Learning Over-Parameterized Deep ReLU Networks." Proceedings of the AAAI Conference on Artificial Intelligence 34, no. 04 (April 3, 2020): 3349–56. http://dx.doi.org/10.1609/aaai.v34i04.5736.

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Empirical studies show that gradient-based methods can learn deep neural networks (DNNs) with very good generalization performance in the over-parameterization regime, where DNNs can easily fit a random labeling of the training data. Very recently, a line of work explains in theory that with over-parameterization and proper random initialization, gradient-based methods can find the global minima of the training loss for DNNs. However, existing generalization error bounds are unable to explain the good generalization performance of over-parameterized DNNs. The major limitation of most existing generalization bounds is that they are based on uniform convergence and are independent of the training algorithm. In this work, we derive an algorithm-dependent generalization error bound for deep ReLU networks, and show that under certain assumptions on the data distribution, gradient descent (GD) with proper random initialization is able to train a sufficiently over-parameterized DNN to achieve arbitrarily small generalization error. Our work sheds light on explaining the good generalization performance of over-parameterized deep neural networks.

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16

Wu, Liang, Antoine Ledent, Yunwen Lei, and Marius Kloft. "Fine-grained Generalization Analysis of Vector-Valued Learning." Proceedings of the AAAI Conference on Artificial Intelligence 35, no. 12 (May 18, 2021): 10338–46. http://dx.doi.org/10.1609/aaai.v35i12.17238.

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Many fundamental machine learning tasks can be formulated as a problem of learning with vector-valued functions, where we learn multiple scalar-valued functions together. Although there is some generalization analysis on different specific algorithms under the empirical risk minimization principle, a unifying analysis of vector-valued learning under a regularization framework is still lacking. In this paper, we initiate the generalization analysis of regularized vector-valued learning algorithms by presenting bounds with a mild dependency on the output dimension and a fast rate on the sample size. Our discussions relax the existing assumptions on the restrictive constraint of hypothesis spaces, smoothness of loss functions and low-noise condition. To understand the interaction between optimization and learning, we further use our results to derive the first generalization bounds for stochastic gradient descent with vector-valued functions. We apply our general results to multi-class classification and multi-label classification, which yield the first bounds with a logarithmic dependency on the output dimension for extreme multi-label classification with the Frobenius regularization. As a byproduct, we derive a Rademacher complexity bound for loss function classes defined in terms of a general strongly convex function.

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17

Wang, Shusen. "A Sharper Generalization Bound for Divide-and-Conquer Ridge Regression." Proceedings of the AAAI Conference on Artificial Intelligence 33 (July 17, 2019): 5305–12. http://dx.doi.org/10.1609/aaai.v33i01.33015305.

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We study the distributed machine learning problem where the n feature-response pairs are partitioned among m machines uniformly at random. The goal is to approximately solve an empirical risk minimization (ERM) problem with the minimum amount of communication. The divide-and-conquer (DC) method, which was proposed several years ago, lets every worker machine independently solve the same ERM problem using its local feature-response pairs and the driver machine combine the solutions. This approach is in one-shot and thereby extremely communication-efficient. Although the DC method has been studied by many prior works, reasonable generalization bound has not been established before this work.For the ridge regression problem, we show that the prediction error of the DC method on unseen test samples is at most ε times larger than the optimal. There have been constantfactor bounds in the prior works, their sample complexities have a quadratic dependence on d, which does not match the setting of most real-world problems. In contrast, our bounds are much stronger. First, our 1 + ε error bound is much better than their constant-factor bounds. Second, our sample complexity is merely linear with d.

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18

Caro, Matthias C., Elies Gil-Fuster, Johannes Jakob Meyer, Jens Eisert, and Ryan Sweke. "Encoding-dependent generalization bounds for parametrized quantum circuits." Quantum 5 (November 17, 2021): 582. http://dx.doi.org/10.22331/q-2021-11-17-582.

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A large body of recent work has begun to explore the potential of parametrized quantum circuits (PQCs) as machine learning models, within the framework of hybrid quantum-classical optimization. In particular, theoretical guarantees on the out-of-sample performance of such models, in terms of generalization bounds, have emerged. However, none of these generalization bounds depend explicitly on how the classical input data is encoded into the PQC. We derive generalization bounds for PQC-based models that depend explicitly on the strategy used for data-encoding. These imply bounds on the performance of trained PQC-based models on unseen data. Moreover, our results facilitate the selection of optimal data-encoding strategies via structural risk minimization, a mathematically rigorous framework for model selection. We obtain our generalization bounds by bounding the complexity of PQC-based models as measured by the Rademacher complexity and the metric entropy, two complexity measures from statistical learning theory. To achieve this, we rely on a representation of PQC-based models via trigonometric functions. Our generalization bounds emphasize the importance of well-considered data-encoding strategies for PQC-based models.

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19

Wu, Xinxing, and Guosheng Hu. "Generalization Bounds for Coregularized Multiple Kernel Learning." Computational Intelligence and Neuroscience 2018 (November 1, 2018): 1–13. http://dx.doi.org/10.1155/2018/1853517.

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Multiple kernel learning (MKL) as an approach to automated kernel selection plays an important role in machine learning. Some learning theories have been built to analyze the generalization of multiple kernel learning. However, less work has been studied on multiple kernel learning in the framework of semisupervised learning. In this paper, we analyze the generalization of multiple kernel learning in the framework of semisupervised multiview learning. We apply Rademacher chaos complexity to control the performance of the candidate class of coregularized multiple kernels and obtain the generalization error bound of coregularized multiple kernel learning. Furthermore, we show that the existing results about multiple kennel learning and coregularized kernel learning can be regarded as the special cases of our main results in this paper.

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20

Cao, Qiong, Zheng-Chu Guo, and Yiming Ying. "Generalization bounds for metric and similarity learning." Machine Learning 102, no. 1 (June 20, 2015): 115–32. http://dx.doi.org/10.1007/s10994-015-5499-7.

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21

Kuznetsov, Vitaly, and Mehryar Mohri. "Generalization bounds for non-stationary mixing processes." Machine Learning 106, no. 1 (October 3, 2016): 93–117. http://dx.doi.org/10.1007/s10994-016-5588-2.

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22

Choi, Daeshik. "A generalization of the Chung-Erdős inequality." Studia Scientiarum Mathematicarum Hungarica 55, no. 2 (June 2018): 213–30. http://dx.doi.org/10.1556/012.2018.55.2.1386.

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23

Pečarić, Josip, Anamarija Perušić, and Ksenija Smoljak. "Cerone’s Generalizations of Steffensen’s Inequality." Tatra Mountains Mathematical Publications 58, no. 1 (March 1, 2014): 53–75. http://dx.doi.org/10.2478/tmmp-2014-0006.

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Abstract In this paper, generalizations of Steffensen’s inequality with bounds involving any two subintervals motivated by Cerone’s generalizations are given. Furthermore, weaker conditions for Cerone’s generalization as well as for new generalizations obtained in this paper are given. Moreover, functionals defined as the difference between the left-hand and the right-hand side of these generalizations are studied and new Stolarsky type means related to them are obtained.

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24

Zhang, Tong. "Leave-One-Out Bounds for Kernel Methods." Neural Computation 15, no. 6 (June 1, 2003): 1397–437. http://dx.doi.org/10.1162/089976603321780326.

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In this article, we study leave-one-out style cross-validation bounds for kernel methods. The essential element in our analysis is a bound on the parameter estimation stability for regularized kernel formulations. Using this result, we derive bounds on expected leave-one-out cross-validation errors, which lead to expected generalization bounds for various kernel algorithms. In addition, we also obtain variance bounds for leave-oneout errors. We apply our analysis to some classification and regression problems and compare them with previous results.

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25

Bagnall, Alexander, and Gordon Stewart. "Certifying the True Error: Machine Learning in Coq with Verified Generalization Guarantees." Proceedings of the AAAI Conference on Artificial Intelligence 33 (July 17, 2019): 2662–69. http://dx.doi.org/10.1609/aaai.v33i01.33012662.

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We present MLCERT, a novel system for doing practical mechanized proof of the generalization of learning procedures, bounding expected error in terms of training or test error. MLCERT is mechanized in that we prove generalization bounds inside the theorem prover Coq; thus the bounds are machine checked by Coq’s proof checker. MLCERT is practical in that we extract learning procedures defined in Coq to executable code; thus procedures with proved generalization bounds can be trained and deployed in real systems. MLCERT is well documented and open source; thus we expect it to be usable even by those without Coq expertise. To validate MLCERT, which is compatible with external tools such as TensorFlow, we use it to prove generalization bounds on neural networks trained using TensorFlow on the extended MNIST data set.

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26

Ma, Zhi-Hao, Zhi-Hua Chen, Shuai Han, Shao-Ming Feii, and Simone Severin. "Improved bounds on negativity of superpositions." Quantum Information and Computation 12, no. 11&12 (November 2012): 983–88. http://dx.doi.org/10.26421/qic12.11-12-6.

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We consider an alternative formula for the negativity based on a simple generalization of the concurrence. We use the formula to bound the amount of entanglement in a superposition of two bipartite pure states of arbitrary dimension. Various examples indicate that our bounds are tighter than the previously known results.

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27

Gao, Wei, and Yun Gang Zhang. "Generalization Bounds for Certain Class of Ranking Algorithm." Advanced Materials Research 267 (June 2011): 456–61. http://dx.doi.org/10.4028/www.scientific.net/amr.267.456.

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The quality of ranking determines the success or failure of information retrieval and the goal of ranking is to learn a real-valued ranking function that induces a ranking or ordering over an instance space. We focus on a ranking setting which uses truth function to label each pair of instances and the ranking preferences are given randomly from some distributions on the set of possible undirected edge sets of a graph. The contribution of this paper is the given generalization bounds for such ranking algorithm via strong and weak stability. Such stabilities have lower demand than uniform stability and fit for more real applications.

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28

Liu, Tongliang, Dacheng Tao, Mingli Song, and Stephen J. Maybank. "Algorithm-Dependent Generalization Bounds for Multi-Task Learning." IEEE Transactions on Pattern Analysis and Machine Intelligence 39, no. 2 (February 1, 2017): 227–41. http://dx.doi.org/10.1109/tpami.2016.2544314.

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29

Lei, Yunwen, Urun Dogan, Ding-Xuan Zhou, and Marius Kloft. "Data-Dependent Generalization Bounds for Multi-Class Classification." IEEE Transactions on Information Theory 65, no. 5 (May 2019): 2995–3021. http://dx.doi.org/10.1109/tit.2019.2893916.

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30

Vorontsov, K. V. "Combinatorial probability and the tightness of generalization bounds." Pattern Recognition and Image Analysis 18, no. 2 (June 2008): 243–59. http://dx.doi.org/10.1134/s1054661808020090.

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31

Kinouchi, O., and N. Caticha. "Lower bounds on generalization errors for drifting rules." Journal of Physics A: Mathematical and General 26, no. 22 (November 21, 1993): 6161–71. http://dx.doi.org/10.1088/0305-4470/26/22/017.

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32

Oneto, Luca, Sandro Ridella, and Davide Anguita. "Differential privacy and generalization: Sharper bounds with applications." Pattern Recognition Letters 89 (April 2017): 31–38. http://dx.doi.org/10.1016/j.patrec.2017.02.006.

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33

Li, Cong, Michael Georgiopoulos, and Georgios C. Anagnostopoulos. "Multitask Classification Hypothesis Space With Improved Generalization Bounds." IEEE Transactions on Neural Networks and Learning Systems 26, no. 7 (July 2015): 1468–79. http://dx.doi.org/10.1109/tnnls.2014.2347054.

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34

Bu, Yuheng, Shaofeng Zou, and Venugopal V. Veeravalli. "Tightening Mutual Information-Based Bounds on Generalization Error." IEEE Journal on Selected Areas in Information Theory 1, no. 1 (May 2020): 121–30. http://dx.doi.org/10.1109/jsait.2020.2991139.

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35

Cao, Feilong, Yufang Liu, and Weiguo Zhang. "Generalization Bounds of Regularization Algorithm with Gaussian Kernels." Neural Processing Letters 39, no. 2 (March 30, 2013): 179–94. http://dx.doi.org/10.1007/s11063-013-9298-5.

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Guo, Zheng-Chu, and Yiming Ying. "Guaranteed Classification via Regularized Similarity Learning." Neural Computation 26, no. 3 (March 2014): 497–522. http://dx.doi.org/10.1162/neco_a_00556.

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Learning an appropriate (dis)similarity function from the available data is a central problem in machine learning, since the success of many machine learning algorithms critically depends on the choice of a similarity function to compare examples. Despite many approaches to similarity metric learning that have been proposed, there has been little theoretical study on the links between similarity metric learning and the classification performance of the resulting classifier. In this letter, we propose a regularized similarity learning formulation associated with general matrix norms and establish their generalization bounds. We show that the generalization error of the resulting linear classifier can be bounded by the derived generalization bound of similarity learning. This shows that a good generalization of the learned similarity function guarantees a good classification of the resulting linear classifier. Our results extend and improve those obtained by Bellet, Habrard, and Sebban ( 2012 ). Due to the techniques dependent on the notion of uniform stability (Bousquet & Elisseeff, 2002 ), the bound obtained there holds true only for the Frobenius matrix-norm regularization. Our techniques using the Rademacher complexity (Bartlett & Mendelson, 2002 ) and its related Khinchin-type inequality enable us to establish bounds for regularized similarity learning formulations associated with general matrix norms, including sparse L1-norm and mixed (2,1)-norm.

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37

Bozin, Vladimir, and Miodrag Mateljevic. "Bounds for Jacobian of harmonic injective mappings in n-dimensional space." Filomat 29, no. 9 (2015): 2119–24. http://dx.doi.org/10.2298/fil1509119b.

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Using normal family arguments, we show that the degree of the first nonzero homogenous polynomial in the expansion of n dimensional Euclidean harmonic K-quasiconformal mapping around an internal point is odd, and that such a map from the unit ball onto a bounded convex domain, with K < 3n-1, is co-Lipschitz. Also some generalizations of this result are given, as well as a generalization of Heinz?s lemma for harmonic quasiconformal maps in Rn and related results.

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38

Ujevič, Nenad. "Generalization of the Corrected Mid-point Rule and Error Bounds." Computational Methods in Applied Mathematics 5, no. 1 (2005): 97–104. http://dx.doi.org/10.2478/cmam-2005-0005.

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39

Feng, Yunlong, Shao-Gao Lv, Hanyuan Hang, and Johan A. K. Suykens. "Kernelized Elastic Net Regularization: Generalization Bounds, and Sparse Recovery." Neural Computation 28, no. 3 (March 2016): 525–62. http://dx.doi.org/10.1162/neco_a_00812.

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Kernelized elastic net regularization (KENReg) is a kernelization of the well-known elastic net regularization (Zou & Hastie, 2005 ). The kernel in KENReg is not required to be a Mercer kernel since it learns from a kernelized dictionary in the coefficient space. Feng, Yang, Zhao, Lv, and Suykens ( 2014 ) showed that KENReg has some nice properties including stability, sparseness, and generalization. In this letter, we continue our study on KENReg by conducting a refined learning theory analysis. This letter makes the following three main contributions. First, we present refined error analysis on the generalization performance of KENReg. The main difficulty of analyzing the generalization error of KENReg lies in characterizing the population version of its empirical target function. We overcome this by introducing a weighted Banach space associated with the elastic net regularization. We are then able to conduct elaborated learning theory analysis and obtain fast convergence rates under proper complexity and regularity assumptions. Second, we study the sparse recovery problem in KENReg with fixed design and show that the kernelization may improve the sparse recovery ability compared to the classical elastic net regularization. Finally, we discuss the interplay among different properties of KENReg that include sparseness, stability, and generalization. We show that the stability of KENReg leads to generalization, and its sparseness confidence can be derived from generalization. Moreover, KENReg is stable and can be simultaneously sparse, which makes it attractive theoretically and practically.

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Lv, Shao-Gao. "Refined Generalization Bounds of Gradient Learning over Reproducing Kernel Hilbert Spaces." Neural Computation 27, no. 6 (June 2015): 1294–320. http://dx.doi.org/10.1162/neco_a_00739.

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Gradient learning (GL), initially proposed by Mukherjee and Zhou ( 2006 ) has been proved to be a powerful tool for conducting variable selection and dimensional reduction simultaneously. This approach presents a nonparametric version of a gradient estimator with positive definite kernels without estimating the true function itself, so that the proposed version has wide applicability and allows for complex effects between predictors. In terms of theory, however, existing generalization bounds for GL depend on capacity-independent techniques, and the capacity of kernel classes cannot be characterized completely. Thus, this letter considers GL estimators that minimize the empirical convex risk. We prove generalization bounds for such estimators with rates that are faster than previous results. Moreover, we provide a novel upper bound for Rademacher chaos complexity of order two, which also plays an important role in general pairwise-type estimations, including ranking and score problems.

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Meng, Juan, Guyu Hu, Dong Li, Yanyan Zhang, and Zhisong Pan. "Generalization Bounds Derived IPM-Based Regularization for Domain Adaptation." Computational Intelligence and Neuroscience 2016 (2016): 1–8. http://dx.doi.org/10.1155/2016/7046563.

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Domain adaptation has received much attention as a major form of transfer learning. One issue that should be considered in domain adaptation is the gap between source domain and target domain. In order to improve the generalization ability of domain adaption methods, we proposed a framework for domain adaptation combining source and target data, with a new regularizer which takes generalization bounds into account. This regularization term considers integral probability metric (IPM) as the distance between the source domain and the target domain and thus can bound up the testing error of an existing predictor from the formula. Since the computation of IPM only involves two distributions, this generalization term is independent with specific classifiers. With popular learning models, the empirical risk minimization is expressed as a general convex optimization problem and thus can be solved effectively by existing tools. Empirical studies on synthetic data for regression and real-world data for classification show the effectiveness of this method.

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Poliquin, Guillaume. "Principal frequency of the p-Laplacian and the inradius of Euclidean domains." Journal of Topology and Analysis 07, no. 03 (May 15, 2015): 505–11. http://dx.doi.org/10.1142/s1793525315500211.

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We study the lower bounds for the principal frequency of the p-Laplacian on N-dimensional Euclidean domains. For p > N, we obtain a lower bound for the first eigenvalue of the p-Laplacian in terms of its inradius, without any assumptions on the topology of the domain. Moreover, we show that a similar lower bound can be obtained if p > N - 1 assuming the boundary is connected. This result can be viewed as a generalization of the classical bounds for the first eigenvalue of the Laplace operator on simply connected planar domains.

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Martin, W. J., and D. R. Stinson. "A Generalized Rao Bound for Ordered Orthogonal Arrays and (t, m, s)-Nets." Canadian Mathematical Bulletin 42, no. 3 (September 1, 1999): 359–70. http://dx.doi.org/10.4153/cmb-1999-042-x.

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AbstractIn this paper, we provide a generalization of the classical Rao bound for orthogonal arrays, which can be applied to ordered orthogonal arrays and (t, m, s)-nets. Application of our new bound leads to improvements in many parameter situations to the strongest bounds (i.e., necessary conditions) for existence of these objects.

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Yao, Mingchen, Chao Zhang, and Wei Wu. "Learning Bounds of ERM Principle for Sequences of Time-Dependent Samples." Discrete Dynamics in Nature and Society 2015 (2015): 1–8. http://dx.doi.org/10.1155/2015/826812.

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Many generalization results in learning theory are established under the assumption that samples are independent and identically distributed (i.i.d.). However, numerous learning tasks in practical applications involve the time-dependent data. In this paper, we propose a theoretical framework to analyze the generalization performance of the empirical risk minimization (ERM) principle for sequences of time-dependent samples (TDS). In particular, we first present the generalization bound of ERM principle for TDS. By introducing some auxiliary quantities, we also give a further analysis of the generalization properties and the asymptotical behaviors of ERM principle for TDS.

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Ying, Yiming, and Colin Campbell. "Rademacher Chaos Complexities for Learning the Kernel Problem." Neural Computation 22, no. 11 (November 2010): 2858–86. http://dx.doi.org/10.1162/neco_a_00028.

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We develop a novel generalization bound for learning the kernel problem. First, we show that the generalization analysis of the kernel learning problem reduces to investigation of the suprema of the Rademacher chaos process of order 2 over candidate kernels, which we refer to as Rademacher chaos complexity. Next, we show how to estimate the empirical Rademacher chaos complexity by well-established metric entropy integrals and pseudo-dimension of the set of candidate kernels. Our new methodology mainly depends on the principal theory of U-processes and entropy integrals. Finally, we establish satisfactory excess generalization bounds and misclassification error rates for learning gaussian kernels and general radial basis kernels.

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ENGEL, A. "UNIFORM CONVERGENCE BOUNDS FOR LEARNING FROM EXAMPLES." Modern Physics Letters B 08, no. 27 (November 20, 1994): 1683–708. http://dx.doi.org/10.1142/s021798499400162x.

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Basic notions of learning from examples in feed-forward neural networks are reviewed with special emphasis on the relation between the different approaches. First classical results of mathematical statistics on uniform bounds for the convergence of the learning error to the generalization error are discussed from a physical point of view. Recent work of statistical mechanics on the generalization ability of large networks of formal neurons is shown to reproduce and extend these results. In particular for simple architectures the tightness of the convergence bounds as well as the relation between the typical and the worst case performance can be determined in the thermodynamic limit. Several interesting questions remain open.

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Vaz, Yule, Rodrigo Fernandes de Mello, and Carlos Henrique Grossi Ferreira. "Coarse-refinement dilemma: On generalization bounds for data clustering." Expert Systems with Applications 184 (December 2021): 115399. http://dx.doi.org/10.1016/j.eswa.2021.115399.

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Krishnapuram, B., L. Carin, M. A. T. Figueiredo, and A. J. Hartemink. "Sparse multinomial logistic regression: fast algorithms and generalization bounds." IEEE Transactions on Pattern Analysis and Machine Intelligence 27, no. 6 (June 2005): 957–68. http://dx.doi.org/10.1109/tpami.2005.127.

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Feeman, Timothy G., and Jesse Frey. "Efficiency bounds for a generalization of ranked-set sampling." Communications in Statistics - Theory and Methods 45, no. 3 (March 24, 2015): 739–56. http://dx.doi.org/10.1080/03610926.2013.835418.

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Kochedykov, D. "A combinatorial approach to hypothesis similarity in generalization bounds." Pattern Recognition and Image Analysis 21, no. 4 (December 2011): 616–29. http://dx.doi.org/10.1134/s1054661811040109.

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