In recent years, there has been a proliferation of theoretical graph models, e.g., preferential attachment and small-world models, motivated by real-world graphs such as the Internet topology. To address the natural question of which model is best for a particular data set, we propose a model selection criterion for graph models. Since each model is in fact a probability distribution over graphs, we suggest using Maximum Likelihood to compare graph models and select their parameters. Interestingly, for the case of graph models, computing likelihoods is a difficult algorithmic task. However, we design and implement MCMC algorithms for computing the maximum likelihood for four popular models: a power-law random graph model, a preferential attachment model, a small-world model, and a uniform random graph model. We hope that this novel use of ML will objectify comparisons between graph models.
{"title":"Graph model selection using maximum likelihood","authors":"Ivona Bezáková, A. Kalai, R. Santhanam","doi":"10.1145/1143844.1143858","DOIUrl":"https://doi.org/10.1145/1143844.1143858","url":null,"abstract":"In recent years, there has been a proliferation of theoretical graph models, e.g., preferential attachment and small-world models, motivated by real-world graphs such as the Internet topology. To address the natural question of which model is best for a particular data set, we propose a model selection criterion for graph models. Since each model is in fact a probability distribution over graphs, we suggest using Maximum Likelihood to compare graph models and select their parameters. Interestingly, for the case of graph models, computing likelihoods is a difficult algorithmic task. However, we design and implement MCMC algorithms for computing the maximum likelihood for four popular models: a power-law random graph model, a preferential attachment model, a small-world model, and a uniform random graph model. We hope that this novel use of ML will objectify comparisons between graph models.","PeriodicalId":124011,"journal":{"name":"Proceedings of the 23rd international conference on Machine learning","volume":"55 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2006-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125987582","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 report that mixtures of m multinomial logistic regression can be used to approximate a class of 'smooth' probability models for multiclass responses. With bounded second derivatives of log-odds, the approximation rate is O(m-2/s) in Hellinger distance or O(m-4/s) in Kullback-Leibler divergence. Here s = dim(x) is the dimension of the input space (or the number of predictors). With the availability of training data of size n, we also show that 'consistency' in multiclass regression and classification can be achieved, simultaneously for all classes, when posterior based inference is performed in a Bayesian framework. Loosely speaking, such 'consistency' refers to performance being often close to the best possible for large n. Consistency can be achieved either by taking m = mn, or by taking m to be uniformly distributed among {1, ...,mn} according to the prior, where 1 ≺ mn ≺ na in order as n grows, for some a ∈ (0, 1).
{"title":"A note on mixtures of experts for multiclass responses: approximation rate and Consistent Bayesian Inference","authors":"Yang Ge, Wenxin Jiang","doi":"10.1145/1143844.1143886","DOIUrl":"https://doi.org/10.1145/1143844.1143886","url":null,"abstract":"We report that mixtures of m multinomial logistic regression can be used to approximate a class of 'smooth' probability models for multiclass responses. With bounded second derivatives of log-odds, the approximation rate is O(m-2/s) in Hellinger distance or O(m-4/s) in Kullback-Leibler divergence. Here s = dim(x) is the dimension of the input space (or the number of predictors). With the availability of training data of size n, we also show that 'consistency' in multiclass regression and classification can be achieved, simultaneously for all classes, when posterior based inference is performed in a Bayesian framework. Loosely speaking, such 'consistency' refers to performance being often close to the best possible for large n. Consistency can be achieved either by taking m = mn, or by taking m to be uniformly distributed among {1, ...,mn} according to the prior, where 1 ≺ mn ≺ na in order as n grows, for some a ∈ (0, 1).","PeriodicalId":124011,"journal":{"name":"Proceedings of the 23rd international conference on Machine learning","volume":"47 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2006-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125059147","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}
Kernel-based learning algorithms, such as Support Vector Machines (SVMs) or Perceptron, often rely on sequential optimization where a few examples are added at each iteration. Updating the kernel matrix usually requires matrix-vector multiplications. We propose a new method based on transposition to speedup this computation on sparse data. Instead of dot-products over sparse feature vectors, our computation incrementally merges lists of training examples and minimizes access to the data. Caching and shrinking are also optimized for sparsity. On very large natural language tasks (tagging, translation, text classification) with sparse feature representations, a 20 to 80-fold speedup over LIBSVM is observed using the same SMO algorithm. Theory and experiments explain what type of sparsity structure is needed for this approach to work, and why its adaptation to Maxent sequential optimization is inefficient.
{"title":"Fast transpose methods for kernel learning on sparse data","authors":"P. Haffner","doi":"10.1145/1143844.1143893","DOIUrl":"https://doi.org/10.1145/1143844.1143893","url":null,"abstract":"Kernel-based learning algorithms, such as Support Vector Machines (SVMs) or Perceptron, often rely on sequential optimization where a few examples are added at each iteration. Updating the kernel matrix usually requires matrix-vector multiplications. We propose a new method based on transposition to speedup this computation on sparse data. Instead of dot-products over sparse feature vectors, our computation incrementally merges lists of training examples and minimizes access to the data. Caching and shrinking are also optimized for sparsity. On very large natural language tasks (tagging, translation, text classification) with sparse feature representations, a 20 to 80-fold speedup over LIBSVM is observed using the same SMO algorithm. Theory and experiments explain what type of sparsity structure is needed for this approach to work, and why its adaptation to Maxent sequential optimization is inefficient.","PeriodicalId":124011,"journal":{"name":"Proceedings of the 23rd international conference on Machine learning","volume":"67 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2006-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127110138","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}
Michael Fink, S. Shalev-Shwartz, Y. Singer, S. Ullman
We describe a general framework for online multiclass learning based on the notion of hypothesis sharing. In our framework sets of classes are associated with hypotheses. Thus, all classes within a given set share the same hypothesis. This framework includes as special cases commonly used constructions for multiclass categorization such as allocating a unique hypothesis for each class and allocating a single common hypothesis for all classes. We generalize the multiclass Perceptron to our framework and derive a unifying mistake bound analysis. Our construction naturally extends to settings where the number of classes is not known in advance but, rather, is revealed along the online learning process. We demonstrate the merits of our approach by comparing it to previous methods on both synthetic and natural datasets.
{"title":"Online multiclass learning by interclass hypothesis sharing","authors":"Michael Fink, S. Shalev-Shwartz, Y. Singer, S. Ullman","doi":"10.1145/1143844.1143884","DOIUrl":"https://doi.org/10.1145/1143844.1143884","url":null,"abstract":"We describe a general framework for online multiclass learning based on the notion of hypothesis sharing. In our framework sets of classes are associated with hypotheses. Thus, all classes within a given set share the same hypothesis. This framework includes as special cases commonly used constructions for multiclass categorization such as allocating a unique hypothesis for each class and allocating a single common hypothesis for all classes. We generalize the multiclass Perceptron to our framework and derive a unifying mistake bound analysis. Our construction naturally extends to settings where the number of classes is not known in advance but, rather, is revealed along the online learning process. We demonstrate the merits of our approach by comparing it to previous methods on both synthetic and natural datasets.","PeriodicalId":124011,"journal":{"name":"Proceedings of the 23rd international conference on Machine learning","volume":"30 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2006-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127185170","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 a series of new feature weighting algorithms, all stemming from a new interpretation of RELIEF as an online algorithm that solves a convex optimization problem with a margin-based objective function. The new interpretation explains the simplicity and effectiveness of RELIEF, and enables us to identify some of its weaknesses. We offer an analytic solution to mitigate these problems. We extend the newly proposed algorithm to handle multiclass problems by using a new multiclass margin definition. To reduce computational costs, an online learning algorithm is also developed. Convergence theorems of the proposed algorithms are presented. Some experiments based on the UCI and microarray datasets are performed to demonstrate the effectiveness of the proposed algorithms.
{"title":"Iterative RELIEF for feature weighting","authors":"Yijun Sun, Jian Li","doi":"10.1145/1143844.1143959","DOIUrl":"https://doi.org/10.1145/1143844.1143959","url":null,"abstract":"We propose a series of new feature weighting algorithms, all stemming from a new interpretation of RELIEF as an online algorithm that solves a convex optimization problem with a margin-based objective function. The new interpretation explains the simplicity and effectiveness of RELIEF, and enables us to identify some of its weaknesses. We offer an analytic solution to mitigate these problems. We extend the newly proposed algorithm to handle multiclass problems by using a new multiclass margin definition. To reduce computational costs, an online learning algorithm is also developed. Convergence theorems of the proposed algorithms are presented. Some experiments based on the UCI and microarray datasets are performed to demonstrate the effectiveness of the proposed algorithms.","PeriodicalId":124011,"journal":{"name":"Proceedings of the 23rd international conference on Machine learning","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2006-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129250373","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}
O. Banerjee, L. Ghaoui, A. d’Aspremont, G. Natsoulis
We consider the problem of fitting a large-scale covariance matrix to multivariate Gaussian data in such a way that the inverse is sparse, thus providing model selection. Beginning with a dense empirical covariance matrix, we solve a maximum likelihood problem with an l1-norm penalty term added to encourage sparsity in the inverse. For models with tens of nodes, the resulting problem can be solved using standard interior-point algorithms for convex optimization, but these methods scale poorly with problem size. We present two new algorithms aimed at solving problems with a thousand nodes. The first, based on Nesterov's first-order algorithm, yields a rigorous complexity estimate for the problem, with a much better dependence on problem size than interior-point methods. Our second algorithm uses block coordinate descent, updating row/columns of the covariance matrix sequentially. Experiments with genomic data show that our method is able to uncover biologically interpretable connections among genes.
{"title":"Convex optimization techniques for fitting sparse Gaussian graphical models","authors":"O. Banerjee, L. Ghaoui, A. d’Aspremont, G. Natsoulis","doi":"10.1145/1143844.1143856","DOIUrl":"https://doi.org/10.1145/1143844.1143856","url":null,"abstract":"We consider the problem of fitting a large-scale covariance matrix to multivariate Gaussian data in such a way that the inverse is sparse, thus providing model selection. Beginning with a dense empirical covariance matrix, we solve a maximum likelihood problem with an l1-norm penalty term added to encourage sparsity in the inverse. For models with tens of nodes, the resulting problem can be solved using standard interior-point algorithms for convex optimization, but these methods scale poorly with problem size. We present two new algorithms aimed at solving problems with a thousand nodes. The first, based on Nesterov's first-order algorithm, yields a rigorous complexity estimate for the problem, with a much better dependence on problem size than interior-point methods. Our second algorithm uses block coordinate descent, updating row/columns of the covariance matrix sequentially. Experiments with genomic data show that our method is able to uncover biologically interpretable connections among genes.","PeriodicalId":124011,"journal":{"name":"Proceedings of the 23rd international conference on Machine learning","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2006-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129935821","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}
Alex Graves, Santiago Fernández, F. Gomez, J. Schmidhuber
Many real-world sequence learning tasks require the prediction of sequences of labels from noisy, unsegmented input data. In speech recognition, for example, an acoustic signal is transcribed into words or sub-word units. Recurrent neural networks (RNNs) are powerful sequence learners that would seem well suited to such tasks. However, because they require pre-segmented training data, and post-processing to transform their outputs into label sequences, their applicability has so far been limited. This paper presents a novel method for training RNNs to label unsegmented sequences directly, thereby solving both problems. An experiment on the TIMIT speech corpus demonstrates its advantages over both a baseline HMM and a hybrid HMM-RNN.
{"title":"Connectionist temporal classification: labelling unsegmented sequence data with recurrent neural networks","authors":"Alex Graves, Santiago Fernández, F. Gomez, J. Schmidhuber","doi":"10.1145/1143844.1143891","DOIUrl":"https://doi.org/10.1145/1143844.1143891","url":null,"abstract":"Many real-world sequence learning tasks require the prediction of sequences of labels from noisy, unsegmented input data. In speech recognition, for example, an acoustic signal is transcribed into words or sub-word units. Recurrent neural networks (RNNs) are powerful sequence learners that would seem well suited to such tasks. However, because they require pre-segmented training data, and post-processing to transform their outputs into label sequences, their applicability has so far been limited. This paper presents a novel method for training RNNs to label unsegmented sequences directly, thereby solving both problems. An experiment on the TIMIT speech corpus demonstrates its advantages over both a baseline HMM and a hybrid HMM-RNN.","PeriodicalId":124011,"journal":{"name":"Proceedings of the 23rd international conference on Machine learning","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2006-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130011220","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 address the issue of the learnability of concept classes under three classification noise models in the probably approximately correct framework. After introducing the Class-Conditional Classification Noise (CCCN) model, we investigate the problem of the learnability of concept classes under this particular setting and we show that concept classes that are learnable under the well-known uniform classification noise (CN) setting are also CCCN-learnable, which gives CN = CCCN. We then use this result to prove the equality between the set of concept classes that are CN-learnable and the set of concept classes that are learnable in the Constant Partition Classification Noise (CPCN) setting, or, in other words, we show that CN = CPCN.
{"title":"CN = CPCN","authors":"L. Ralaivola, François Denis, C. Magnan","doi":"10.1145/1143844.1143935","DOIUrl":"https://doi.org/10.1145/1143844.1143935","url":null,"abstract":"We address the issue of the learnability of concept classes under three classification noise models in the probably approximately correct framework. After introducing the Class-Conditional Classification Noise (CCCN) model, we investigate the problem of the learnability of concept classes under this particular setting and we show that concept classes that are learnable under the well-known uniform classification noise (CN) setting are also CCCN-learnable, which gives CN = CCCN. We then use this result to prove the equality between the set of concept classes that are CN-learnable and the set of concept classes that are learnable in the Constant Partition Classification Noise (CPCN) setting, or, in other words, we show that CN = CPCN.","PeriodicalId":124011,"journal":{"name":"Proceedings of the 23rd international conference on Machine learning","volume":"169 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2006-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129754286","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}
Ashesh Jain, Michael Hu, Nathan D. Ratliff, Drew Bagnell, Martin A Zinkevich
Imitation learning of sequential, goal-directed behavior by standard supervised techniques is often difficult. We frame learning such behaviors as a maximum margin structured prediction problem over a space of policies. In this approach, we learn mappings from features to cost so an optimal policy in an MDP with these cost mimics the expert's behavior. Further, we demonstrate a simple, provably efficient approach to structured maximum margin learning, based on the subgradient method, that leverages existing fast algorithms for inference. Although the technique is general, it is particularly relevant in problems where A* and dynamic programming approaches make learning policies tractable in problems beyond the limitations of a QP formulation. We demonstrate our approach applied to route planning for outdoor mobile robots, where the behavior a designer wishes a planner to execute is often clear, while specifying cost functions that engender this behavior is a much more difficult task.
{"title":"Maximum margin planning","authors":"Ashesh Jain, Michael Hu, Nathan D. Ratliff, Drew Bagnell, Martin A Zinkevich","doi":"10.1145/1143844.1143936","DOIUrl":"https://doi.org/10.1145/1143844.1143936","url":null,"abstract":"Imitation learning of sequential, goal-directed behavior by standard supervised techniques is often difficult. We frame learning such behaviors as a maximum margin structured prediction problem over a space of policies. In this approach, we learn mappings from features to cost so an optimal policy in an MDP with these cost mimics the expert's behavior. Further, we demonstrate a simple, provably efficient approach to structured maximum margin learning, based on the subgradient method, that leverages existing fast algorithms for inference. Although the technique is general, it is particularly relevant in problems where A* and dynamic programming approaches make learning policies tractable in problems beyond the limitations of a QP formulation. We demonstrate our approach applied to route planning for outdoor mobile robots, where the behavior a designer wishes a planner to execute is often clear, while specifying cost functions that engender this behavior is a much more difficult task.","PeriodicalId":124011,"journal":{"name":"Proceedings of the 23rd international conference on Machine learning","volume":"38 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2006-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126471238","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 show that several important Bayesian bounds studied in machine learning, both in the batch as well as the online setting, arise by an application of a simple compression lemma. In particular, we derive (i) PAC-Bayesian bounds in the batch setting, (ii) Bayesian log-loss bounds and (iii) Bayesian bounded-loss bounds in the online setting using the compression lemma. Although every setting has different semantics for prior, posterior and loss, we show that the core bound argument is the same. The paper simplifies our understanding of several important and apparently disparate results, as well as brings to light a powerful tool for developing similar arguments for other methods.
{"title":"On Bayesian bounds","authors":"A. Banerjee","doi":"10.1145/1143844.1143855","DOIUrl":"https://doi.org/10.1145/1143844.1143855","url":null,"abstract":"We show that several important Bayesian bounds studied in machine learning, both in the batch as well as the online setting, arise by an application of a simple compression lemma. In particular, we derive (i) PAC-Bayesian bounds in the batch setting, (ii) Bayesian log-loss bounds and (iii) Bayesian bounded-loss bounds in the online setting using the compression lemma. Although every setting has different semantics for prior, posterior and loss, we show that the core bound argument is the same. The paper simplifies our understanding of several important and apparently disparate results, as well as brings to light a powerful tool for developing similar arguments for other methods.","PeriodicalId":124011,"journal":{"name":"Proceedings of the 23rd international conference on Machine learning","volume":"93 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2006-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129222547","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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