Above are papers related to the following paper: [Paper_ID] Paper #1130929. 	[Title] A model f w depending on a set of parameters w is used to estimate [Abstract] We have introduced earlier the use of regularisation in the learning procedure. It should now be understood that regularisation is most often a necessity to increase the quality of the results. Even when the unregularised solution is acceptable, it is likely that some regularisation will produce an improvement in performance. There does not exist any method giving directly the best value for the regularisation parameter ~, even in the linear case. The topic of this chapter is thus to propose some methods to estimate the best value. The best ~ being the one that leads to the smallest generalisation error, the methods presented and compared here propose estimators of the generalisation error. This estimation can then be used to approximate the best regularisation level. In sections 3.2 to 3.4 we present validation-based techniques. They estimate the generalisation error on the basis of some extra data. In sections 3.6 to 3.9, we deal with algebraic estimates of this error, that do not use any extra data, but rely on a number of assumptions. The contribution of this chapter is to present all these techniques and analyse them on the same ground. We also present some short derivations clarifying the links between different estimators of generalisation error, as well as a comparison between them. During the course of this chapter, the error will be the quadratic difference. For the validation-based methods, it is possible to consider any kind of error without modification of the method. On the other hand, the algebraic estimates are specific to the quadratic cost. Adapting them to another cost function would require to derive new expressions for the estimators. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #45188. 	[Title] A Bound on the Error of Cross Validation Using the Approximation and Estimation Rates, with [Abstract] We give an analysis of the generalization error of cross validation in terms of two natural measures of the difficulty of the problem under consideration: the approximation rate (the accuracy to which the target function can be ideally approximated as a function of the number of hypothesis parameters), and the estimation rate (the deviation between the training and generalization errors as a function of the number of hypothesis parameters). The approximation rate captures the complexity of the target function with respect to the hypothesis model, and the estimation rate captures the extent to which the hypothesis model suffers from overfitting. Using these two measures, we give a rigorous and general bound on the error of cross validation. The bound clearly shows the tradeoffs involved with making fl the fraction of data saved for testing too large or too small. By optimizing the bound with respect to fl, we then argue (through a combination of formal analysis, plotting, and controlled experimentation) that the following qualitative properties of cross validation behavior should be quite robust to significant changes in the underlying model selection problem: Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #45189. 	[Title] An Experimental and Theoretical Comparison of Model Selection Methods on simple model selection problems, the [Abstract] We investigate the problem of model selection in the setting of supervised learning of boolean functions from independent random examples. More precisely, we compare methods for finding a balance between the complexity of the hypothesis chosen and its observed error on a random training sample of limited size, when the goal is that of minimizing the resulting generalization error. We undertake a detailed comparison of three well-known model selection methods | a variation of Vapnik's Guaranteed Risk Minimization (GRM), an instance of Rissanen's Minimum Description Length Principle (MDL), and cross validation (CV). We introduce a general class of model selection methods (called penalty-based methods) that includes both GRM and MDL, and provide general methods for analyzing such rules. We provide both controlled experimental evidence and formal theorems to support the following conclusions: * The class of penalty-based methods is fundamentally handicapped in the sense that there exist two types of model selection problems for which every penalty-based method must incur large generalization error on at least one, while CV enjoys small generalization error Despite the inescapable incomparability of model selection methods under certain circumstances, we conclude with a discussion of our belief that the balance of the evidence provides specific reasons to prefer CV to other methods, unless one is in possession of detailed problem-specific information. on both. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #89547. 	[Title] Rigorous Learning Curve Bounds from Statistical Mechanics [Abstract] In this paper we introduce and investigate a mathematically rigorous theory of learning curves that is based on ideas from statistical mechanics. The advantage of our theory over the well-established Vapnik-Chervonenkis theory is that our bounds can be considerably tighter in many cases, and are also more reflective of the true behavior (functional form) of learning curves. This behavior can often exhibit dramatic properties such as phase transitions, as well as power law asymptotics not explained by the VC theory. The disadvantages of our theory are that its application requires knowledge of the input distribution, and it is limited so far to finite cardinality function classes. We illustrate our results with many concrete examples of learning curve bounds derived from our theory. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #350362. 	[Title] Improving Regressors using Boosting Techniques [Abstract] In the regression context, boosting and bagging are techniques to build a committee of regressors that may be superior to a single regressor. We use regression trees as fundamental building blocks in bagging committee machines and boosting committee machines. Performance is analyzed on three non-linear functions and the Boston housing database. In all cases, boosting is at least equivalent, and in most cases better than bagging in terms of prediction error. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #976284. 	[Title] Algorithmic Stability and Sanity-Check Bounds for Leave-One-Out Cross-Validation [Abstract] In this paper we prove sanity-check bounds for the error of the leave-one-out cross-validation estimate of the generalization error: that is, bounds showing that the worst-case error of this estimate is not much worse than that of the training error estimate. The name sanity-check refers to the fact that although we often expect the leave-one-out estimate to perform considerably better than the training error estimate, we are here only seeking assurance that its performance will not be considerably worse. Perhaps surprisingly, such assurance has been given only for rather limited cases in the prior literature on cross-validation. Any nontrivial bound on the error of leave-one-out must rely on some notion of algorithmic stability. Previous bounds relied on the rather strong notion of hypothesis stability, whose application was primarily limited to nearest-neighbor and other local algorithms. Here we introduce the new and weaker notion of error stability, and apply it to obtain sanity-check bounds for leave-one-out for other classes of learning algorithms, including training error minimization procedures and Bayesian algorithms. We also provide lower bounds demonstrating the necessity of error stability for proving bounds on the error of the leave-one-out estimate, and the fact that for training error minimization algorithms, in the worst case such bounds must still depend on the Vapnik-Chervonenkis dimension of the hypothesis class. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #192734. 	[Title] Preventing "Overfitting" of Cross-Validation Data [Abstract] Suppose that, for a learning task, we have to select one hypothesis out of a set of hypotheses (that may, for example, have been generated by multiple applications of a randomized learning algorithm). A common approach is to evaluate each hypothesis in the set on some previously unseen cross-validation data, and then to select the hypothesis that had the lowest cross-validation error. But when the cross-validation data is partially corrupted such as by noise, and if the set of hypotheses we are selecting from is large, then "folklore" also warns about "overfitting" the cross-validation data [Klockars and Sax, 1986, Tukey, 1949, Tukey, 1953]. In this paper, we explain how this "overfitting" really occurs, and show the surprising result that it can be overcome by selecting a hypothesis with a higher cross-validation error, over others with lower cross-validation errors. We give reasons for not selecting the hypothesis with the lowest cross-validation error, and propose a new algorithm, LOOCVCV, that uses a computa-tionally efficient form of leave-one-out cross-validation to select such a hypothesis. Finally, we present experimental results for one domain, that show LOOCVCV consistently beating picking the hypothesis with the lowest cross-validation error, even when using reasonably large cross-validation sets. Please summarise all the given information.

Paper #1130929 has the title `A model f w depending on a set of parameters w is used to estimate
` and the abstract `We have introduced earlier the use of regularisation in the learning procedure. It should now be understood that regularisation is most often a necessity to increase the quality of the results. Even when the unregularised solution is acceptable, it is likely that some regularisation will produce an improvement in performance. There does not exist any method giving directly the best value for the regularisation parameter ~, even in the linear case. The topic of this chapter is thus to propose some methods to estimate the best value. The best ~ being the one that leads to the smallest generalisation error, the methods presented and compared here propose estimators of the generalisation error. This estimation can then be used to approximate the best regularisation level. In sections 3.2 to 3.4 we present validation-based techniques. They estimate the generalisation error on the basis of some extra data. In sections 3.6 to 3.9, we deal with algebraic estimates of this error, that do not use any extra data, but rely on a number of assumptions. The contribution of this chapter is to present all these techniques and analyse them on the same ground. We also present some short derivations clarifying the links between different estimators of generalisation error, as well as a comparison between them. During the course of this chapter, the error will be the quadratic difference. For the validation-based methods, it is possible to consider any kind of error without modification of the method. On the other hand, the algebraic estimates are specific to the quadratic cost. Adapting them to another cost function would require to derive new expressions for the estimators. 
`.
Question: Which category should Paper #1130929 be classified as ?
You can select one from ['Neural_Networks', 'Case_Based', 'Theory', 'Reinforcement_Learning', 'Probabilistic_Methods', 'Rule_Learning', 'Genetic_Algorithms'].


Paper #1130929 has the title `A model f w depending on a set of parameters w is used to estimate
` and the abstract `We have introduced earlier the use of regularisation in the learning procedure. It should now be understood that regularisation is most often a necessity to increase the quality of the results. Even when the unregularised solution is acceptable, it is likely that some regularisation will produce an improvement in performance. There does not exist any method giving directly the best value for the regularisation parameter ~, even in the linear case. The topic of this chapter is thus to propose some methods to estimate the best value. The best ~ being the one that leads to the smallest generalisation error, the methods presented and compared here propose estimators of the generalisation error. This estimation can then be used to approximate the best regularisation level. In sections 3.2 to 3.4 we present validation-based techniques. They estimate the generalisation error on the basis of some extra data. In sections 3.6 to 3.9, we deal with algebraic estimates of this error, that do not use any extra data, but rely on a number of assumptions. The contribution of this chapter is to present all these techniques and analyse them on the same ground. We also present some short derivations clarifying the links between different estimators of generalisation error, as well as a comparison between them. During the course of this chapter, the error will be the quadratic difference. For the validation-based methods, it is possible to consider any kind of error without modification of the method. On the other hand, the algebraic estimates are specific to the quadratic cost. Adapting them to another cost function would require to derive new expressions for the estimators. 
`.
Question: Which category should Paper #1130929 be classified as ?
You can select one from ['Neural_Networks', 'Case_Based', 'Theory', 'Reinforcement_Learning', 'Probabilistic_Methods', 'Rule_Learning', 'Genetic_Algorithms'].

This paper should be classified into: Neural_Networks