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Learning convex bodies under uniform distribution

W. Kern
1992 Information Processing Letters  
Learning convex bodies under uniform distribution. Information Processing Letters 43 (lY92) 35-3').  ...  We prove that the class of convex bodies contained in a fixed (prescribed) bounded region R c lQd is PAC-learnable, if the positive examples are drawn according to the uniform distribution Df on the target  ...  class & of convex bodies contained in a fixed bounded region R c IWd under uniform distribution on the positive examples. The sample size 3.1 [l].  ... 
doi:10.1016/0020-0190(92)90026-r fatcat:oopdnspkefet5nwppmpyhrb6qq

Information Theoretic Guarantees for Empirical Risk Minimization with Applications to Model Selection and Large-Scale Optimization

Ibrahim M. Alabdulmohsin
2018 International Conference on Machine Learning  
We prove that under the Axiom of Choice, the existence of an ERM learning rule with a vanishing mutual information is equivalent to the assertion that the loss class has a finite VC dimension, thus bridging  ...  information theory with statistical learning theory.  ...  However, the true goal behind stochastic convex optimization in the machine learning setting is not to compute the empirical risk minimizer ĥ per se but to estimate h .  ... 
dblp:conf/icml/Alabdulmohsin18 fatcat:bvl5jvusajdnhg43ku4ghqq4cq

Learnability, Stability and Uniform Convergence

Shai Shalev-Shwartz, Ohad Shamir, Nathan Srebro, Karthik Sridharan
2010 Journal of machine learning research  
We show that in this setting, there are non-trivial learning problems where uniform convergence does not hold, empirical risk minimization fails, and yet they are learnable using alternative mechanisms  ...  In this paper, we consider the General Learning Setting (introduced by Vapnik), which includes most statistical learning problems as special cases.  ...  To justify the necessity of uniform convergence even in the General Learning Setting, Vapnik attempted to show that in this setting, learnability with the ERM learning rule is equivalent to uniform convergence  ... 
dblp:journals/jmlr/Shalev-ShwartzSSS10 fatcat:uxm6usdfkzafri6456myrfq4oy

Learning Geometric Concepts via Gaussian Surface Area

Adam R. Klivans, Ryan O'Donnell, Rocco A. Servedio
2008 2008 49th Annual IEEE Symposium on Foundations of Computer Science  
We study the learnability of sets in R n under the Gaussian distribution, taking Gaussian surface area as the "complexity measure" of the sets being learned.  ...  These results together show that Gaussian surface area essentially characterizes the computational complexity of learning under the Gaussian distribution.  ...  I(f )/ǫ, and hence can be learned under the uniform distribution in time n I(f )/ǫ .  ... 
doi:10.1109/focs.2008.64 dblp:conf/focs/KlivansOS08 fatcat:dns7sl4qhbhbbcyepzw7lve5mu

The Perceptron Algorithm is Fast for Nonmalicious Distributions

Eric B. Baum
1990 Neural Computation  
In an appendix we show that, for uniform distributions, some classes of infinite V-C dimension including convex sets and a class of nested differences of convex sets are learnable.  ...  Under this definition, the Perceptron algorithm is shown to be a distribution independent learning algorithm.  ...  For example, for the uniform distribution, the class of convex sets and a class of nested differences of convex sets ( both of which trivially have infinite V -C dimension) are shown to be learnable in  ... 
doi:10.1162/neco.1990.2.2.248 fatcat:a56lp3tgfjefhp66fk6zfxa6bi

Stability and Generalization of Decentralized Stochastic Gradient Descent

Tao Sun, Dongsheng Li, Bao Wang
We verify our theoretical findings by using a variety of decentralized settings and benchmark machine learning models.  ...  Leveraging this formulation together with (non)convex optimization theory, we establish the first stability and generalization guarantees for the decentralized stochastic gradient descent.  ...  The uniform stability of empirical risk minimization (ERM) under strongly convex objectives is considered by Bousquet and Elisseeff (2002) .  ... 
doi:10.1609/aaai.v35i11.17173 fatcat:4aqdj6l7r5hw5jwongxjzimud4

Robust Optimization for Non-Convex Objectives [article]

Robert Chen, Brendan Lucier, Yaron Singer, Vasilis Syrgkanis
2017 arXiv   pre-print
We show that de-randomizing this solution is NP-hard in general, but can be done for a broad class of statistical learning tasks.  ...  We develop a reduction from robust improper optimization to Bayesian optimization: given an oracle that returns α-approximate solutions for distributions over objectives, we compute a distribution over  ...  Indeed, let D be any distribution over F, and suppose f i is any function with maximum probability under D. Then the set S = {a i1 , . . . , a ik } maximizes expected value under D.  ... 
arXiv:1707.01047v1 fatcat:fgmmjek7uzdarpdl22fbl77hna

The Hedge Algorithm on a Continuum

Walid Krichene, Maximilian Balandat, Claire J. Tomlin, Alexandre M. Bayen
2015 International Conference on Machine Learning  
Finally, we propose a generalization to the dual averaging method on the set of Lebesgue-continuous distributions over S.  ...  We consider an online optimization problem on a compact subset S ⊂ R n (not necessarily convex), in which a decision maker chooses, at each iteration t, a probability distribution x (t) over S, and seeks  ...  The reference measure is the Lebesgue measure λ, and the initial distribution x (0) is the Lebesgueuniform distribution over S, i.e. x (0) (s) = 1 λ(S) . Sublinear Regret on Convex Sets Lemma 3.  ... 
dblp:conf/icml/KricheneBTB15 fatcat:aeqcnkwj5nahlppjm3zauezlcm

Stability and Risk Bounds of Iterative Hard Thresholding

Xiaotong Yuan, Ping Li
2021 International Conference on Artificial Intelligence and Statistics  
excess risk; and 2) a fast rate of order Õ(n −1 k(log 3 (n) + log(p))) can be derived for strongly convex risk function under certain strong-signal conditions.  ...  From the perspective of statistical learning theory, another fundamental question is how well the I-HT estimation would perform on unseen samples.  ...  ) under Grant No.61876090 and No.61936005.  ... 
dblp:conf/aistats/YuanL21 fatcat:uvunsleqizakdmrrg5uamhxc4q

Testing convexity of figures under the uniform distribution

Piotr Berman, Meiram Murzabulatov, Sofya Raskhodnikova
2018 Random structures & algorithms (Print)  
Our testing algorithm runs in time O( −4/3 ) and thus beats the Ω( −3/2 ) sample lower bound for learning convex figures under the uniform distribution from [26] .  ...  It shows that, with uniform samples, we can check if a set is approximately convex much faster than we can find an approximate representation of a convex set.  ...  uniform samples can be done faster than learning convex figures under the uniform distribution.  ... 
doi:10.1002/rsa.20797 fatcat:y3rvb2yev5bbfi23jm3wj3ok5q

PAC-Bayesian Collective Stability

Ben London, Bert Huang, Ben Taskar, Lise Getoor
2014 International Conference on Artificial Intelligence and Statistics  
We then derive a generalization bound for a class of structured predictors with variably convex inference, which suggests a novel learning objective that optimizes collective stability.  ...  Recent results have shown that the generalization error of structured predictors decreases with both the number of examples and the size of each example, provided the data distribution has weak dependence  ...  Government is authorized to reproduce and distribute reprints for governmental purposes notwithstanding any copyright annotation thereon.  ... 
dblp:conf/aistats/LondonHTG14 fatcat:ljo74muzfzckhoi5ib2shw2tfa

Making risk minimization tolerant to label noise

Aritra Ghosh, Naresh Manwani, P.S. Sastry
2015 Neurocomputing  
assume the classes to be separable under noise-free data distribution.  ...  We prove a sufficient condition on a loss function for the risk minimization under that loss to be tolerant to uniform label noise.  ...  Under balanced training set, symmetric classes with uniform densities, SVM performs moderately well under noise.  ... 
doi:10.1016/j.neucom.2014.09.081 fatcat:hycmsefwz5galkc6bdvkc7ewvi

Stochastic Convex Optimization

Shai Shalev-Shwartz, Ohad Shamir, Nathan Srebro, Karthik Sridharan
2009 Annual Conference Computational Learning Theory  
Our results demonstrate that the celebrated theorem of Alon et al on the equivalence of learnability and uniform convergence does not extend to Vapnik's General Setting of Learning, that in the General  ...  Inspired by recent regret bounds for online convex optimization, we study stochastic convex optimization, and uncover a surprisingly different situation in the more general setting: although the stochastic  ...  And as we mentioned, learnability (under the standard supervised learning model) is in fact equivalent to a uniform convergence property.  ... 
dblp:conf/colt/Shalev-ShwartzSSS09 fatcat:5o2nz6pspjhcdch4uw6kc7wtpi

Learning with Noisy Labels

Nagarajan Natarajan, Inderjit S. Dhillon, Pradeep Ravikumar, Ambuj Tewari
2013 Neural Information Processing Systems  
Second, by leveraging a reduction of risk minimization under noisy labels to classification with weighted 0-1 loss, we suggest the use of a simple weighted surrogate loss, for which we are able to obtain  ...  Learning with convex losses has been addressed only under limiting assumptions like separability or uniform noise rates [Manwani and Sastry, 2013] .  ...  To the best of our knowledge, we are the first to provide guarantees for risk minimization under random label noise in the general setting of convex surrogates, without any assumptions on the true distribution  ... 
dblp:conf/nips/NatarajanDRT13 fatcat:mv52dz3jsfbotje6ricpjutwq4

One Size Fits All: Can We Train One Denoiser for All Noise Levels? [article]

Abhiram Gnansambandam, Stanley H. Chan
2020 arXiv   pre-print
For estimators with non-convex admissible sets such as deep neural networks, our dual formulation converges to a solution of the convex relaxation.  ...  We derive a dual ascent algorithm to determine the optimal sampling distribution of which the convergence is guaranteed as long as the set of admissible estimators is closed and convex.  ...  Acknowledgement The work is supported, in part, by the US National Science Foundation under grants CCF-1763896 and CCF-1718007.  ... 
arXiv:2005.09627v3 fatcat:4l3woql4uvffddwssg4q4optli
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