Above are papers related to the following paper: [Paper_ID] Paper #68505. 	[Title] On Learning Conjunctions with Malicious Noise [Abstract] We show how to learn monomials in the presence of malicious noise, when the underlined distribution is a product distribution. We show that our results apply not only to product distributions but to a wide class of distributions. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #249858. 	[Title] 25 Learning in Hybrid Noise Environments Using Statistical Queries [Abstract] We consider formal models of learning from noisy data. Specifically, we focus on learning in the probability approximately correct model as defined by Valiant. Two of the most widely studied models of noise in this setting have been classification noise and malicious errors. However, a more realistic model combining the two types of noise has not been formalized. We define a learning environment based on a natural combination of these two noise models. We first show that hypothesis testing is possible in this model. We next describe a simple technique for learning in this model, and then describe a more powerful technique based on statistical query learning. We show that the noise tolerance of this improved technique is roughly optimal with respect to the desired learning accuracy and that it provides a smooth tradeoff between the tolerable amounts of the two types of noise. Finally, we show that statistical query simulation yields learning algorithms for other combinations of noise models, thus demonstrating that statistical query specification truly An important goal of research in machine learning is to determine which tasks can be automated, and for those which can, to determine their information and computation requirements. One way to answer these questions is through the development and investigation of formal models of machine learning which capture the task of learning under plausible assumptions. In this work, we consider the formal model of learning from examples called "probably approximately correct" (PAC) learning as defined by Valiant [Val84]. In this setting, a learner attempts to approximate an unknown target concept simply by viewing positive and negative examples of the concept. An adversary chooses, from some specified function class, a hidden f0; 1g-valued target function defined over some specified domain of examples and chooses a probability distribution over this domain. The goal of the learner is to output in both polynomial time and with high probability, an hypothesis which is "close" to the target function with respect to the distribution of examples. The learner gains information about the target function and distribution by interacting with an example oracle. At each request by the learner, this oracle draws an example randomly according to the hidden distribution, labels it according to the hidden target function, and returns the labelled example to the learner. A class of functions F is said to be PAC learnable if captures the generic fault tolerance of a learning algorithm. 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 #68495. 	[Title] PAC-Learning PROLOG clauses with or without errors [Abstract] In a nutshell we can describe a generic ILP problem as following: given a set E of (positive and negative) examples of a target predicate, and some background knowledge B about the world (usually a logic program including facts and auxiliary predicates), the task is to find a logic program H (our hypothesis) such that all positive examples can be deduced from B and H, while no negative example can. In this paper we review some of the results achieved in this area and discuss the techniques used. Moreover we prove the following new results: * Predicates described by non-recursive, local clauses of at most k literals are PAC-learnable under any distribution. This generalizes a previous result that was valid only for constrained clauses. * Predicates that are described by k non-recursive local clauses are PAC-learnable under any distribution. This generalizes a previous result that was non construc tive and valid only under some class of distributions. Finally we introduce what we believe is the first theoretical framework for learning Prolog clauses in the presence of errors. To this purpose we introduce a new noise model, that we call the fixed attribute noise model, for learning propositional concepts over the Boolean domain. This new noise model can be of its own interest. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #12182. 	[Title] How to Use Expert Advice (Extended Abstract) [Abstract] We analyze algorithms that predict a binary value by combining the predictions of several prediction strategies, called experts. Our analysis is for worst-case situations, i.e., we make no assumptions about the way the sequence of bits to be predicted is generated. We measure the performance of the algorithm by the difference between the expected number of mistakes it makes on the bit sequence and the expected number of mistakes made by the best expert on this sequence, where the expectation is taken with respect to the randomization in the predictions. We show that the minimum achievable difference is on the order of the square root of the number of mistakes of the best expert, and we give efficient algorithms that achieve this. Our upper and lower bounds have matching leading constants in most cases. We give implications of this result on the performance of batch learning algorithms in a PAC setting which improve on the best results currently known in this context. We also extend our analysis to the case in which log loss is used instead of the expected number of mistakes. 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 #164. 	[Title] Learning polynomials with queries: The highly noisy case  task for the case when F [Abstract] Given a function f mapping n-variate inputs from a finite field F into F , we consider the task of reconstructing a list of all n-variate degree d polynomials which agree with f on a tiny but non-negligible fraction, ffi, of the input space. We give a randomized algorithm for solving this task which accesses f as a black box and runs in time polynomial in 1 d=jF j). For the special case when d = 1, we solve this problem for all * def jF j &gt; 0. In this case the running time of our algorithm is bounded by a polynomial in 1 * ; n and exponential in d. Our algorithm generalizes a previously known algorithm, due to Goldreich and Levin, that solves this Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #12197. 	[Title] Pac Learning, Noise, and Geometry [Abstract] This paper describes the probably approximately correct model of concept learning, paying special attention to the case where instances are points in Euclidean n-space. The problem of learning from noisy training data is also studied. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #12198. 	[Title] Exact Identification of Read-once Formulas Using Fixed Points of Amplification Functions [Abstract] In this paper we describe a new technique for exactly identifying certain classes of read-once Boolean formulas. The method is based on sampling the input-output behavior of the target formula on a probability distribution that is determined by the fixed point of the formula's amplification function (defined as the probability that a 1 is output by the formula when each input bit is 1 independently with probability p). By performing various statistical tests on easily sampled variants of the fixed-point distribution, we are able to efficiently infer all structural information about any logarithmic-depth formula (with high probability). We apply our results to prove the existence of short universal identification sequences for large classes of formulas. We also describe extensions of our algorithms to handle high rates of noise, and to learn formulas of unbounded depth in Valiant's model with respect to specific distributions. Most of this research was carried out while all three authors were at MIT Laboratory for Computer Science with support provided by ARO Grant DAAL03-86-K-0171, DARPA Contract N00014-89-J-1988, NSF Grant CCR-88914428, and a grant from the Siemens Corporation. R. Schapire received additional support from AFOSR Grant 89-0506 while at Harvard University. S. Goldman is currently supported in part by a G.E. Foundation Junior Faculty Grant and NSF Grant CCR-9110108. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #9515. 	[Title] Tracking Drifting Concepts By Minimizing Disagreements [Abstract] In this paper we consider the problem of tracking a subset of a domain (called the target) which changes gradually over time. A single (unknown) probability distribution over the domain is used to generate random examples for the learning algorithm and measure the speed at which the target changes. Clearly, the more rapidly the target moves, the harder it is for the algorithm to maintain a good approximation of the target. Therefore we evaluate algorithms based on how much movement of the target can be tolerated between examples while predicting with accuracy *. Furthermore, the complexity of the class H of possible targets, as measured by d, its VC-dimension, also effects the difficulty of tracking the target concept. We show that if the problem of minimizing the number of disagreements with a sample from among concepts in a class H can be approximated to within a factor k, then there is a simple tracking algorithm for H which can achieve a probability * of making a mistake if the target movement rate is at most a constant times * 2 =(k(d + k) ln 1 * ), where d is the Vapnik-Chervonenkis dimension of H. Also, we show that if H is properly PAC-learnable, then there is an efficient (randomized) algorithm that with high probability approximately minimizes disagreements to within a factor of 7d + 1, yielding an efficient tracking algorithm for H which tolerates drift rates up to a constant times * 2 =(d 2 ln 1 In addition, we prove complementary results for the classes of halfspaces and axis-aligned hy- perrectangles showing that the maximum rate of drift that any algorithm (even with unlimited computational power) can tolerate is a constant times * 2 =d. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #12210. 	[Title] Efficient Distribution-free Learning of Probabilistic Concepts [Abstract] In this paper we investigate a new formal model of machine learning in which the concept (boolean function) to be learned may exhibit uncertain or probabilistic behavior|thus, the same input may sometimes be classified as a positive example and sometimes as a negative example. Such probabilistic concepts (or p-concepts) may arise in situations such as weather prediction, where the measured variables and their accuracy are insufficient to determine the outcome with certainty. We adopt from the Valiant model of learning [27] the demands that learning algorithms be efficient and general in the sense that they perform well for a wide class of p-concepts and for any distribution over the domain. In addition to giving many efficient algorithms for learning natural classes of p-concepts, we study and develop in detail an underlying theory of learning p-concepts. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #10169. 	[Title] Learning Boolean Concepts in the Presence of Many Irrelevant Features [Abstract]  Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #124734. 	[Title] On Learning Visual Concepts and DNF Formulae [Abstract] We consider the problem of learning DNF formulae in the mistake-bound and the PAC models. We develop a new approach, which is called polynomial explainability, that is shown to be useful for learning some new subclasses of DNF (and CNF) formulae that were not known to be learnable before. Unlike previous learnability results for DNF (and CNF) formulae, these subclasses are not limited in the number of terms or in the number of variables per term; yet, they contain the subclasses of k-DNF and k-term-DNF (and the corresponding classes of CNF) as special cases. We apply our DNF results to the problem of learning visual concepts and obtain learning algorithms for several natural subclasses of visual concepts that appear to have no natural boolean counterpart. On the other hand, we show that learning some other natural subclasses of visual concepts is as hard as learning the class of all DNF formulae. We also consider the robustness of these results under various types of noise. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #12350. 	[Title] Cryptographic Limitations on Learning Boolean Formulae and Finite Automata [Abstract] In this paper we prove the intractability of learning several classes of Boolean functions in the distribution-free model (also called the Probably Approximately Correct or PAC model) of learning from examples. These results are representation independent, in that they hold regardless of the syntactic form in which the learner chooses to represent its hypotheses. Our methods reduce the problems of cracking a number of well-known public-key cryptosys- tems to the learning problems. We prove that a polynomial-time learning algorithm for Boolean formulae, deterministic finite automata or constant-depth threshold circuits would have dramatic consequences for cryptography and number theory: in particular, such an algorithm could be used to break the RSA cryptosystem, factor Blum integers (composite numbers equivalent to 3 modulo 4), and detect quadratic residues. The results hold even if the learning algorithm is only required to obtain a slight advantage in prediction over random guessing. The techniques used demonstrate an interesting duality between learning and cryptography. We also apply our results to obtain strong intractability results for approximating a gener - alization of graph coloring. fl This research was conducted while the author was at Harvard University and supported by an A.T.& T. Bell Laboratories scholarship. y Supported by grants ONR-N00014-85-K-0445, NSF-DCR-8606366 and NSF-CCR-89-02500, DAAL03-86-K-0171, DARPA AFOSR 89-0506, and by SERC. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #157761. 	[Title] On the Complexity of Function Learning [Abstract] The majority of results in computational learning theory are concerned with concept learning, i.e. with the special case of function learning for classes of functions with range f0; 1g. Much less is known about the theory of learning functions with a larger range such as IN or IR. In particular relatively few results exist about the general structure of common models for function learning, and there are only very few nontrivial function classes for which positive learning results have been exhibited in any of these models. We introduce in this paper the notion of a binary branching adversary tree for function learning, which allows us to give a somewhat surprising equivalent characterization of the optimal learning cost for learning a class of real-valued functions (in terms of a max-min definition which does not involve any "learning" model). Another general structural result of this paper relates the cost for learning a union of function classes to the learning costs for the individual function classes. Furthermore, we exhibit an efficient learning algorithm for learning convex piecewise linear functions from IR d into IR. Previously, the class of linear functions from IR d into IR was the only class of functions with multi-dimensional domain that was known to be learnable within the rigorous framework of a formal model for on-line learning. Finally we give a sufficient condition for an arbitrary class F of functions from IR into IR that allows us to learn the class of all functions that can be written as the pointwise maximum of k functions from F . This allows us to exhibit a number of further nontrivial classes of functions from IR into IR for which there exist efficient learning algorithms. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #10177. 	[Title] Learning in the Presence of Malicious Errors [Abstract] In this paper we study an extension of the distribution-free model of learning introduced by Valiant [23] (also known as the probably approximately correct or PAC model) that allows the presence of malicious errors in the examples given to a learning algorithm. Such errors are generated by an adversary with unbounded computational power and access to the entire history of the learning algorithm's computation. Thus, we study a worst-case model of errors. Our results include general methods for bounding the rate of error tolerable by any learning algorithm, efficient algorithms tolerating nontrivial rates of malicious errors, and equivalences between problems of learning with errors and standard combinatorial optimization problems. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #4804. 	[Title] Toward Efficient Agnostic Learning [Abstract] In this paper we initiate an investigation of generalizations of the Probably Approximately Correct (PAC) learning model that attempt to significantly weaken the target function assumptions. The ultimate goal in this direction is informally termed agnostic learning, in which we make virtually no assumptions on the target function. The name derives from the fact that as designers of learning algorithms, we give up the belief that Nature (as represented by the target function) has a simple or succinct explanation. We give a number of positive and negative results that provide an initial outline of the possibilities for agnostic learning. Our results include hardness results for the most obvious generalization of the PAC model to an agnostic setting, an efficient and general agnostic learning method based on dynamic programming, relationships between loss functions for agnostic learning, and an algorithm for a learning problem that involves hidden variables. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #15429. 	[Title] A General Lower Bound on the Number of Examples Needed for Learning [Abstract] We prove a lower bound of ( 1 * ln 1 ffi + VCdim(C) * ) on the number of random examples required for distribution-free learning of a concept class C, where VCdim(C) is the Vapnik-Chervonenkis dimension and * and ffi are the accuracy and confidence parameters. This improves the previous best lower bound of ( 1 * ln 1 ffi + VCdim(C)), and comes close to the known general upper bound of O( 1 ffi + VCdim(C) * ln 1 * ) for consistent algorithms. We show that for many interesting concept classes, including kCNF and kDNF, our bound is actually tight to within a constant factor. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #77515. 	[Title] Prediction, Learning, Uniform Convergence, and Scale-sensitive Dimensions [Abstract] We present a new general-purpose algorithm for learning classes of [0; 1]-valued functions in a generalization of the prediction model, and prove a general upper bound on the expected absolute error of this algorithm in terms of a scale-sensitive generalization of the Vapnik dimension proposed by Alon, Ben-David, Cesa-Bianchi and Haussler. We give lower bounds implying that our upper bounds cannot be improved by more than a constant factor in general. We apply this result, together with techniques due to Haussler and to Benedek and Itai, to obtain new upper bounds on packing numbers in terms of this scale-sensitive notion of dimension. Using a different technique, we obtain new bounds on packing numbers in terms of Kearns and Schapire's fat-shattering function. We show how to apply both packing bounds to obtain improved general bounds on the sample complexity of agnostic learning. For each * &gt; 0, we establish weaker sufficient and stronger necessary conditions for a class of [0; 1]-valued functions to be agnostically learnable to within *, and to be an *-uniform Glivenko-Cantelli class. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #25805. 	[Title] Weakly Learning DNF and Characterizing Statistical Query Learning Using Fourier Analysis [Abstract] We present new results, both positive and negative, on the well-studied problem of learning disjunctive normal form (DNF) expressions. We first prove that an algorithm due to Kushilevitz and Mansour [16] can be used to weakly learn DNF using membership queries in polynomial time, with respect to the uniform distribution on the inputs. This is the first positive result for learning unrestricted DNF expressions in polynomial time in any nontrivial formal model of learning. It provides a sharp contrast with the results of Kharitonov [15], who proved that AC 0 is not efficiently learnable in the same model (given certain plausible cryptographic assumptions). We also present efficient learning algorithms in various models for the read-k and SAT-k subclasses of DNF. For our negative results, we turn our attention to the recently introduced statistical query model of learning [11]. This model is a restricted version of the popular Probably Approximately Correct (PAC) model [23], and practically every class known to be efficiently learnable in the PAC model is in fact learnable in the statistical query model [11]. Here we give a general characterization of the complexity of statistical query learning in terms of the number of uncorrelated functions in the concept class. This is a distribution-dependent quantity yielding upper and lower bounds on the number of statistical queries required for learning on any input distribution. As a corollary, we obtain that DNF expressions and decision trees are not even weakly learnable with fl This research is sponsored in part by the Wright Laboratory, Aeronautical Systems Center, Air Force Materiel Command, USAF, and the Advanced Research Projects Agency (ARPA) under grant number F33615-93-1-1330. Support also is sponsored by the National Science Foundation under Grant No. CC-9119319. Blum also supported in part by NSF National Young Investigator grant CCR-9357793. Views and conclusions contained in this document are those of the authors and should not be interpreted as necessarily representing official policies or endorsements, either expressed or implied, of Wright Laboratory or the United States Government, or NSF. respect to the uniform input distribution in polynomial time in the statistical query model. This result is information-theoretic and therefore does not rely on any unproven assumptions. It demonstrates that no simple modification of the existing algorithms in the computational learning theory literature for learning various restricted forms of DNF and decision trees from passive random examples (and also several algorithms proposed in the experimental machine learning communities, such as the ID3 algorithm for decision trees [22] and its variants) will solve the general problem. The unifying tool for all of our results is the Fourier analysis of a finite class of boolean functions on the hypercube. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #1108175. 	[Title] Cognitive Computation (Extended Abstract) [Abstract] Cognitive computation is discussed as a discipline that links together neurobiology, cognitive psychology and artificial intelligence. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #989397. 	[Title] PAC Learning Intersections of Halfspaces with Membership Queries (Extended Abstract) [Abstract]  Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #27606. 	[Title] Statistical Queries and Faulty PAC Oracles [Abstract] In this paper we study learning in the PAC model of Valiant [18] in which the example oracle used for learning may be faulty in one of two ways: either by misclassifying the example or by distorting the distribution of examples. We first consider models in which examples are misclassified. Kearns [12] recently showed that efficient learning in a new model using statistical queries is a sufficient condition for PAC learning with classification noise. We show that efficient learning with statistical queries is sufficient for learning in the PAC model with malicious error rate proportional to the required statistical query accuracy. One application of this result is a new lower bound for tolerable malicious error in learning monomials of k literals. This is the first such bound which is independent of the number of irrelevant attributes n. We also use the statistical query model to give sufficient conditions for using distribution specific algorithms on distributions outside their prescribed domains. A corollary of this result expands the class of distributions on which we can weakly learn monotone Boolean formulae. We also consider new models of learning in which examples are not chosen according to the distribution on which the learner will be tested. We examine three variations of distribution noise and give necessary and sufficient conditions for polynomial time learning with such noise. We show containments and separations between the various models of faulty oracles. Finally, we examine hypothesis boosting algorithms in the context of learning with distribution noise, and show that Schapire's result regarding the strength of weak learnabil-ity [17] is in some sense tight in requiring the weak learner to be nearly distribution free. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #67415. 	[Title] On the Learnability of Discrete Distributions (extended abstract) [Abstract]  Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #102884. 	[Title] Robust Trainability of Single Neurons [Abstract]  Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #1153262. 	[Title] Rationality and Intelligence [Abstract] The long-term goal of our field is the creation and understanding of intelligence. Productive research in AI, both practical and theoretical, benefits from a notion of intelligence that is precise enough to allow the cumulative development of robust systems and general results. The concept of rational agency has long been considered a leading candidate to fulfill this role. This paper outlines a gradual evolution in the formal conception of rationality that brings it closer to our informal conception of intelligence and simultaneously reduces the gap between theory and practice. Some directions for future research are indicated. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #1112574. 	[Title] Learning Switching Concepts [Abstract] We consider learning in situations where the function used to classify examples may switch back and forth between a small number of different concepts during the course of learning. We examine several models for such situations: oblivious models in which switches are made independent of the selection of examples, and more adversarial models in which a single adversary controls both the concept switches and example selection. We show relationships between the more benign models and the p-concepts of Kearns and Schapire, and present polynomial-time algorithms for learning switches between two k-DNF formulas. For the most adversarial model, we present a model of success patterned after the popular competitive analysis used in studying on-line algorithms. We describe a randomized query algorithm for such adversarial switches between two monotone disjunctions that is "1-competitive" in that the total number of mistakes plus queries is with high probability bounded by the number of switches plus some fixed polynomial in n (the number of variables). We also use notions described here to provide sufficient conditions under which learning a p-concept class "with a decision rule" implies being able to learn the class "with a model of probability." Please summarise all the given information.

Paper #68505 has the title `On Learning Conjunctions with Malicious Noise  
` and the abstract `We show how to learn monomials in the presence of malicious noise, when the underlined distribution is a product distribution. We show that our results apply not only to product distributions but to a wide class of distributions. 
`.
Question: Which category should Paper #68505 be classified as ?
You can select one from ['Neural_Networks', 'Case_Based', 'Theory', 'Reinforcement_Learning', 'Probabilistic_Methods', 'Rule_Learning', 'Genetic_Algorithms'].


Paper #68505 has the title `On Learning Conjunctions with Malicious Noise  
` and the abstract `We show how to learn monomials in the presence of malicious noise, when the underlined distribution is a product distribution. We show that our results apply not only to product distributions but to a wide class of distributions. 
`.
Question: Which category should Paper #68505 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: Theory