A relaxed randomized Kaczmarz algorithm is investigated in a least squares regression setting by a learning theory approach. When the sampling values are accurate and the regression function (conditional means) is linear, such an algorithm has been well studied in the community of non-uniform sampling. In this paper, we are mainly interested in the different case of either noisy random measurements or a nonlinear regression function. In this case, we show that relaxation is needed. A necessary and sufficient condition on the sequence of relaxation parameters or step sizes for the convergence of the algorithm in expectation is presented. Moreover, polynomial rates of convergence, both in expectation and in probability, are provided explicitly. As a result, the almost sure convergence of the algorithm is proved by applying the Borel-Cantelli Lemma.
{"title":"Learning theory of randomized Kaczmarz algorithm","authors":"Junhong Lin, Ding-Xuan Zhou","doi":"10.5555/2789272.2912105","DOIUrl":"https://doi.org/10.5555/2789272.2912105","url":null,"abstract":"A relaxed randomized Kaczmarz algorithm is investigated in a least squares regression setting by a learning theory approach. When the sampling values are accurate and the regression function (conditional means) is linear, such an algorithm has been well studied in the community of non-uniform sampling. In this paper, we are mainly interested in the different case of either noisy random measurements or a nonlinear regression function. In this case, we show that relaxation is needed. A necessary and sufficient condition on the sequence of relaxation parameters or step sizes for the convergence of the algorithm in expectation is presented. Moreover, polynomial rates of convergence, both in expectation and in probability, are provided explicitly. As a result, the almost sure convergence of the algorithm is proved by applying the Borel-Cantelli Lemma.","PeriodicalId":14794,"journal":{"name":"J. Mach. Learn. Res.","volume":"188 1","pages":"3341-3365"},"PeriodicalIF":0.0,"publicationDate":"2015-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73265887","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
CEKA is a software package for developers and researchers to mine the wisdom of crowds. It makes the entire knowledge discovery procedure much easier, including analyzing qualities of workers, simulating labeling behaviors, inferring true class labels of instances, filtering and correcting mislabeled instances (noise), building learning models and evaluating them. It integrates a set of state-of-the-art inference algorithms, a set of general noise handling algorithms, and abundant functions for model training and evaluation. CEKA is written in Java with core classes being compatible with the well-known machine learning tool WEKA, which makes the utilization of the functions in WEKA much easier.
{"title":"CEKA: a tool for mining the wisdom of crowds","authors":"J. Zhang, V. Sheng, B. Nicholson, Xindong Wu","doi":"10.5555/2789272.2912090","DOIUrl":"https://doi.org/10.5555/2789272.2912090","url":null,"abstract":"CEKA is a software package for developers and researchers to mine the wisdom of crowds. It makes the entire knowledge discovery procedure much easier, including analyzing qualities of workers, simulating labeling behaviors, inferring true class labels of instances, filtering and correcting mislabeled instances (noise), building learning models and evaluating them. It integrates a set of state-of-the-art inference algorithms, a set of general noise handling algorithms, and abundant functions for model training and evaluation. CEKA is written in Java with core classes being compatible with the well-known machine learning tool WEKA, which makes the utilization of the functions in WEKA much easier.","PeriodicalId":14794,"journal":{"name":"J. Mach. Learn. Res.","volume":"10 1","pages":"2853-2858"},"PeriodicalIF":0.0,"publicationDate":"2015-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85227326","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Bayesian network classifiers are a powerful machine learning tool. In order to evaluate the expressive power of these models, we compute families of polynomials that sign-represent decision functions induced by Bayesian network classifiers. We prove that those families are linear combinations of products of Lagrange basis polynomials. In absence of V-structures in the predictor sub-graph, we are also able to prove that this family of polynomials does indeed characterize the specific classifier considered. We then use this representation to bound the number of decision functions representable by Bayesian network classifiers with a given structure.
{"title":"Decision boundary for discrete Bayesian network classifiers","authors":"Gherardo Varando, C. Bielza, P. Larrañaga","doi":"10.5555/2789272.2912086","DOIUrl":"https://doi.org/10.5555/2789272.2912086","url":null,"abstract":"Bayesian network classifiers are a powerful machine learning tool. In order to evaluate the expressive power of these models, we compute families of polynomials that sign-represent decision functions induced by Bayesian network classifiers. We prove that those families are linear combinations of products of Lagrange basis polynomials. In absence of V-structures in the predictor sub-graph, we are also able to prove that this family of polynomials does indeed characterize the specific classifier considered. We then use this representation to bound the number of decision functions representable by Bayesian network classifiers with a given structure.","PeriodicalId":14794,"journal":{"name":"J. Mach. Learn. Res.","volume":"62 1","pages":"2725-2749"},"PeriodicalIF":0.0,"publicationDate":"2015-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77806302","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this work we address the Ev-SVM model proposed by Perez-Cruz et al. as an extension of the traditional v support vector classification model (v-SVM). Through an enhancement of the range of admissible values for the regularization parameter v, the Ev-SVM has been shown to be able to produce a wider variety of decision functions, giving rise to a better adaptability to the data. However, while a clear and intuitive geometric interpretation can be given for the v-SVM model as a nearest-point problem in reduced convex hulls (RCH-NPP), no previous work has been made in developing such intuition for the Ev-SVM model. In this paper we show how Ev-SVM can be reformulated as a geometrical problem that generalizes RCH-NPP, providing new insights into this model. Under this novel point of view, we propose the RapMinos algorithm, able to solve Ev-SVM more efficiently than the current methods. Furthermore, we show how RapMinos is able to address the Ev-SVM model for any choice of regularization norm lp ≥1 seamlessly, which further extends the SVM model flexibility beyond the usual Ev-SVM models.
{"title":"Geometric intuition and algorithms for Ev-SVM","authors":"Á. Jiménez, A. Takeda, J. Lázaro","doi":"10.5555/2789272.2789283","DOIUrl":"https://doi.org/10.5555/2789272.2789283","url":null,"abstract":"In this work we address the Ev-SVM model proposed by Perez-Cruz et al. as an extension of the traditional v support vector classification model (v-SVM). Through an enhancement of the range of admissible values for the regularization parameter v, the Ev-SVM has been shown to be able to produce a wider variety of decision functions, giving rise to a better adaptability to the data. However, while a clear and intuitive geometric interpretation can be given for the v-SVM model as a nearest-point problem in reduced convex hulls (RCH-NPP), no previous work has been made in developing such intuition for the Ev-SVM model. In this paper we show how Ev-SVM can be reformulated as a geometrical problem that generalizes RCH-NPP, providing new insights into this model. Under this novel point of view, we propose the RapMinos algorithm, able to solve Ev-SVM more efficiently than the current methods. Furthermore, we show how RapMinos is able to address the Ev-SVM model for any choice of regularization norm lp ≥1 seamlessly, which further extends the SVM model flexibility beyond the usual Ev-SVM models.","PeriodicalId":14794,"journal":{"name":"J. Mach. Learn. Res.","volume":"29 1","pages":"323-369"},"PeriodicalIF":0.0,"publicationDate":"2015-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81136110","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Majid Alkaee Taleghan, Thomas G. Dietterich, Mark Crowley, K. Hall, H. Albers
In a simulator-defined MDP, the Markovian dynamics and rewards are provided in the form of a simulator from which samples can be drawn. This paper studies MDP planning algorithms that attempt to minimize the number of simulator calls before terminating and outputting a policy that is approximately optimal with high probability. The paper introduces two heuristics for efficient exploration and an improved confidence interval that enables earlier termination with probabilistic guarantees. We prove that the heuristics and the confidence interval are sound and produce with high probability an approximately optimal policy in polynomial time. Experiments on two benchmark problems and two instances of an invasive species management problem show that the improved confidence intervals and the new search heuristics yield reductions of between 8% and 47% in the number of simulator calls required to reach near-optimal policies.
{"title":"PAC optimal MDP planning with application to invasive species management","authors":"Majid Alkaee Taleghan, Thomas G. Dietterich, Mark Crowley, K. Hall, H. Albers","doi":"10.5555/2789272.2912119","DOIUrl":"https://doi.org/10.5555/2789272.2912119","url":null,"abstract":"In a simulator-defined MDP, the Markovian dynamics and rewards are provided in the form of a simulator from which samples can be drawn. This paper studies MDP planning algorithms that attempt to minimize the number of simulator calls before terminating and outputting a policy that is approximately optimal with high probability. The paper introduces two heuristics for efficient exploration and an improved confidence interval that enables earlier termination with probabilistic guarantees. We prove that the heuristics and the confidence interval are sound and produce with high probability an approximately optimal policy in polynomial time. Experiments on two benchmark problems and two instances of an invasive species management problem show that the improved confidence intervals and the new search heuristics yield reductions of between 8% and 47% in the number of simulator calls required to reach near-optimal policies.","PeriodicalId":14794,"journal":{"name":"J. Mach. Learn. Res.","volume":"31 1","pages":"3877-3903"},"PeriodicalIF":0.0,"publicationDate":"2015-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79032748","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper describes a new paradigm of machine learning, in which Intelligent Teacher is involved. During training stage, Intelligent Teacher provides Student with information that contains, along with classification of each example, additional privileged information (for example, explanation) of this example. The paper describes two mechanisms that can be used for significantly accelerating the speed of Student's learning using privileged information: (1) correction of Student's concepts of similarity between examples, and (2) direct Teacher-Student knowledge transfer.
{"title":"Learning using privileged information: similarity control and knowledge transfer","authors":"V. Vapnik, R. Izmailov","doi":"10.5555/2789272.2886814","DOIUrl":"https://doi.org/10.5555/2789272.2886814","url":null,"abstract":"This paper describes a new paradigm of machine learning, in which Intelligent Teacher is involved. During training stage, Intelligent Teacher provides Student with information that contains, along with classification of each example, additional privileged information (for example, explanation) of this example. The paper describes two mechanisms that can be used for significantly accelerating the speed of Student's learning using privileged information: (1) correction of Student's concepts of similarity between examples, and (2) direct Teacher-Student knowledge transfer.","PeriodicalId":14794,"journal":{"name":"J. Mach. Learn. Res.","volume":"7 1","pages":"2023-2049"},"PeriodicalIF":0.0,"publicationDate":"2015-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76458118","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the problem of predicting online the labeling of a graph. We consider a novel setting for this problem in which, in addition to observing vertices and labels on the graph, we also observe a sequence of just vertices on a second graph. A latent labeling of the second graph selects one of K labelings to be active on the first graph. We propose a polynomial time algorithm for online prediction in this setting and derive a mistake bound for the algorithm. The bound is controlled by the geometric cut of the observed and latent labelings, as well as the resistance diameters of the graphs. When specialized to multitask prediction and online switching problems the bound gives new and sharper results under certain conditions.
{"title":"Predicting a switching sequence of graph labelings","authors":"M. Herbster, Stephen Pasteris, M. Pontil","doi":"10.5555/2789272.2886813","DOIUrl":"https://doi.org/10.5555/2789272.2886813","url":null,"abstract":"We study the problem of predicting online the labeling of a graph. We consider a novel setting for this problem in which, in addition to observing vertices and labels on the graph, we also observe a sequence of just vertices on a second graph. A latent labeling of the second graph selects one of K labelings to be active on the first graph. We propose a polynomial time algorithm for online prediction in this setting and derive a mistake bound for the algorithm. The bound is controlled by the geometric cut of the observed and latent labelings, as well as the resistance diameters of the graphs. When specialized to multitask prediction and online switching problems the bound gives new and sharper results under certain conditions.","PeriodicalId":14794,"journal":{"name":"J. Mach. Learn. Res.","volume":"95 1","pages":"2003-2022"},"PeriodicalIF":0.0,"publicationDate":"2015-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79547396","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We propose an algorithm for two-class classification problems when the training data are imbalanced. This means the number of training instances in one of the classes is so low that the conventional classification algorithms become ineffective in detecting the minority class. We present a modification of the kernel Fisher discriminant analysis such that the imbalanced nature of the problem is explicitly addressed in the new algorithm formulation. The new algorithm exploits the properties of the existing minority points to learn the effects of other minority data points, had they actually existed. The algorithm proceeds iteratively by employing the learned properties and conditional sampling in such a way that it generates sufficient artificial data points for the minority set, thus enhancing the detection probability of the minority class. Implementing the proposed method on a number of simulated and real data sets, we show that our proposed method performs competitively compared to a set of alternative state-of-the-art imbalanced classification algorithms.
{"title":"Absent data generating classifier for imbalanced class sizes","authors":"Arash Pourhabib, B. Mallick, Yu Ding","doi":"10.5555/2789272.2912085","DOIUrl":"https://doi.org/10.5555/2789272.2912085","url":null,"abstract":"We propose an algorithm for two-class classification problems when the training data are imbalanced. This means the number of training instances in one of the classes is so low that the conventional classification algorithms become ineffective in detecting the minority class. We present a modification of the kernel Fisher discriminant analysis such that the imbalanced nature of the problem is explicitly addressed in the new algorithm formulation. The new algorithm exploits the properties of the existing minority points to learn the effects of other minority data points, had they actually existed. The algorithm proceeds iteratively by employing the learned properties and conditional sampling in such a way that it generates sufficient artificial data points for the minority set, thus enhancing the detection probability of the minority class. Implementing the proposed method on a number of simulated and real data sets, we show that our proposed method performs competitively compared to a set of alternative state-of-the-art imbalanced classification algorithms.","PeriodicalId":14794,"journal":{"name":"J. Mach. Learn. Res.","volume":"100 1","pages":"2695-2724"},"PeriodicalIF":0.0,"publicationDate":"2015-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85908770","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Gregory Plumb, D. Pachauri, R. Kondor, Vikas Singh
SnFFT is an easy to use software library written in the Julia language to facilitate Fourier analysis on the symmetric group (set of permutations) of degree n, denoted Sn and make it more easily deployable within statistical machine learning algorithms. Our implementation internally creates the irreducible matrix representations of Sn, and efficiently computes fast Fourier transforms (FFTs) and inverse fast Fourier transforms (iFFTs). Advanced users can achieve scalability and promising practical performance by exploiting various other forms of sparsity. Further, the library also supports the partial inverse Fourier transforms which utilizes the smoothness properties of functions by maintaining only the first few Fourier coefficients. Out of the box, SnFFT currently offers two non-trivial operations for functions defined on Sn, namely convolution and correlation. While the potential applicability of SnFFT is fairly broad, as an example, we show how it can be used for clustering ranked data, where each ranking is modeled as a distribution on Sn.
{"title":"SnFFT: a Julia toolkit for Fourier analysis of functions over permutations","authors":"Gregory Plumb, D. Pachauri, R. Kondor, Vikas Singh","doi":"10.5555/2789272.2912109","DOIUrl":"https://doi.org/10.5555/2789272.2912109","url":null,"abstract":"SnFFT is an easy to use software library written in the Julia language to facilitate Fourier analysis on the symmetric group (set of permutations) of degree n, denoted Sn and make it more easily deployable within statistical machine learning algorithms. Our implementation internally creates the irreducible matrix representations of Sn, and efficiently computes fast Fourier transforms (FFTs) and inverse fast Fourier transforms (iFFTs). Advanced users can achieve scalability and promising practical performance by exploiting various other forms of sparsity. Further, the library also supports the partial inverse Fourier transforms which utilizes the smoothness properties of functions by maintaining only the first few Fourier coefficients. Out of the box, SnFFT currently offers two non-trivial operations for functions defined on Sn, namely convolution and correlation. While the potential applicability of SnFFT is fairly broad, as an example, we show how it can be used for clustering ranked data, where each ranking is modeled as a distribution on Sn.","PeriodicalId":14794,"journal":{"name":"J. Mach. Learn. Res.","volume":"7 1","pages":"3469-3473"},"PeriodicalIF":0.0,"publicationDate":"2015-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81568737","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Blackwell's theory of approachability provides fundamental results for repeated games with vector-valued payoffs, which have been usefully applied in the theory of learning in games, and in devising online learning algorithms in the adversarial setup. A target set S is approachable by a player (the agent) in such a game if he can ensure that the average payoff vector converges to S, no matter what the opponent does. Blackwell provided two equivalent conditions for a convex set to be approachable. Standard approachability algorithms rely on the primal condition, which is a geometric separation condition, and essentially require to compute at each stage a projection direction from a certain point to S. Here we introduce an approachability algorithm that relies on Blackwell's dual condition, which requires the agent to have a feasible response to each mixed action of the opponent, namely a mixed action such that the expected payoff vector belongs to S. Thus, rather than projections, the proposed algorithm relies on computing the response to a certain action of the opponent at each stage. We demonstrate the utility of the proposed approach by applying it to certain generalizations of the classical regret minimization problem, which incorporate side constraints, reward-to-cost criteria, and so-called global cost functions. In these extensions, computation of the projection is generally complex while the response is readily obtainable.
{"title":"Response-based approachability with applications to generalized no-regret problems","authors":"A. Bernstein, N. Shimkin","doi":"10.5555/2789272.2831138","DOIUrl":"https://doi.org/10.5555/2789272.2831138","url":null,"abstract":"Blackwell's theory of approachability provides fundamental results for repeated games with vector-valued payoffs, which have been usefully applied in the theory of learning in games, and in devising online learning algorithms in the adversarial setup. A target set S is approachable by a player (the agent) in such a game if he can ensure that the average payoff vector converges to S, no matter what the opponent does. Blackwell provided two equivalent conditions for a convex set to be approachable. Standard approachability algorithms rely on the primal condition, which is a geometric separation condition, and essentially require to compute at each stage a projection direction from a certain point to S. Here we introduce an approachability algorithm that relies on Blackwell's dual condition, which requires the agent to have a feasible response to each mixed action of the opponent, namely a mixed action such that the expected payoff vector belongs to S. Thus, rather than projections, the proposed algorithm relies on computing the response to a certain action of the opponent at each stage. We demonstrate the utility of the proposed approach by applying it to certain generalizations of the classical regret minimization problem, which incorporate side constraints, reward-to-cost criteria, and so-called global cost functions. In these extensions, computation of the projection is generally complex while the response is readily obtainable.","PeriodicalId":14794,"journal":{"name":"J. Mach. Learn. Res.","volume":"198 1","pages":"747-773"},"PeriodicalIF":0.0,"publicationDate":"2015-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76894823","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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