Exact Learning Algorithm Research Articles

Bayesian Network Structure Learning Using Improved A* with Constraints from Potential Optimal Parent Sets

Learning the structure of a Bayesian network and considering the efficiency and accuracy of learning has always been a hot topic for researchers. This paper proposes two constraints to solve the problem that the A* algorithm, an exact learning algorithm, is not efficient enough to search larger networks. On the one hand, the parent–child set constraints reduce the number of potential optimal parent sets. On the other hand, the path constraints are obtained from the potential optimal parent sets to constrain the search process of the A* algorithm. Both constraints are proposed based on the potential optimal parent sets. Experiments show that the time efficiency of the A* algorithm can be significantly improved, and the ability of the A* algorithm to search larger Bayesian networks can be improved by the two constraints. In addition, compared with the globally optimal Bayesian network learning using integer linear programming (GOBNILP) algorithm and the max–min hill-climbing (MMHC) algorithm, which are state of the art, the A* algorithm enhanced by constraints still performs well in most cases.

Conjunctive Queries: Unique Characterizations and Exact Learnability

We answer the question of which conjunctive queries are uniquely characterized by polynomially many positive and negative examples and how to construct such examples efficiently. As a consequence, we obtain a new efficient exact learning algorithm for a class of conjunctive queries. At the core of our contributions lie two new polynomial-time algorithms for constructing frontiers in the homomorphism lattice of finite structures. We also discuss implications for the unique characterizability and learnability of schema mappings and of description logic concepts.

Learning Large DAGs by Combining Continuous Optimization and Feedback Arc Set Heuristics

Bayesian networks represent relations between variables using a directed acyclic graph (DAG). Learning the DAG is an NP-hard problem and exact learning algorithms are feasible only for small sets of variables. We propose two scalable heuristics for learning DAGs in the linear structural equation case. Our methods learn the DAG by alternating between unconstrained gradient descent-based step to optimize an objective function and solving a maximum acyclic subgraph problem to enforce acyclicity. Thanks to this decoupling, our methods scale up beyond thousands of variables.

Learning the structure of Bayesian networks with ancestral and/or heuristic partition

Developing efficient strategies for searching larger Bayesian networks in exact structure learning is an open challenge. In this study, ancestral and heuristic partition constraints are proposed to develop a series of exact learning algorithms, in which an ancestral partition is used to prune the order graph of a Bayesian network, and a heuristic partition is utilized to improve the tightness of the heuristic function. Algorithms for calculating these two types of constraints are established through thorough theoretical proof. Comparative experiments have been undertaken with state-of-the-art algorithms. It has been demonstrated that an algorithm improved with the proposed ancestral partition or combined ancestral and heuristic partition outperforms the algorithm in its original form, and it can have lower running time, fewer expanded states, and higher accuracy, as well as the ability to search larger networks within 100 nodes.

Causal constraint pruning for exact learning of Bayesian network structure

How to improve the efficiency of exact learning of the Bayesian network structure is a challenging issue. In this paper, four different causal constraints algorithms are added into score calculations to prune possible parent sets, improving state-of-the-art learning algorithms' efficiency. Experimental results indicate that exact learning algorithms can significantly improve the efficiency with only a slight loss of accuracy. Under causal constraints, these exact learning algorithms can prune about 70% possible parent sets and reduce about 60% running time while only losing no more than 2% accuracy on average. Additionally, with sufficient samples, exact learning algorithms with causal constraints can also obtain the optimal network. In general, adding max-min parents and children constraints has better results in terms of efficiency and accuracy among these four causal constraints algorithms.

An exact quantum algorithm for testing 3-junta in Boolean functions with one uncomplemented product

This paper modifies Chen’s algorithm, which is the first exact quantum algorithm for testing 2-junta, and proposes an exact quantum learning algorithm for finding dependent variables of the Boolean function $$ f: \left\{ {0, 1} \right\}^{n} \to \left\{ {0, 1} \right\} $$ with one uncomplemented product of three variables. Typically, the dependent variables are obtained by evaluating the function $$ 2n $$ times in the worst case. However, our proposed quantum algorithm only requires $$ O\left( {\log_{2} n} \right) $$ function operations in the worst case. In addition, the average number to perform the function is evaluated. Our algorithm requires an average of $$ 7.23 $$ function operations at the most when $$ n \ge 16 $$ . We also show that our algorithm cannot solve $$ k $$ -junta problem with one uncomplemented product if $$ 4 \le k < n/2 $$ .

Learning With End-Users in Distribution Grids: Topology and Parameter Estimation

Efficient operation of distribution grids in the smart-grid era is hindered by the limited presence of real-time nodal and line meters. In particular, this prevents the easy estimation of grid topology and associated line parameters that are necessary for control and optimization efforts in the grid. This article studies the problems of topology and parameter estimation in radial balanced distribution grids where measurements are restricted to only the leaf nodes and all intermediate nodes are unobserved/hidden. To this end, we propose two exact learning algorithms that use balanced voltage and injection measured only at the end users. The first algorithm requires time-stamped voltage samples, statistics of nodal power injections, and permissible line impedances to recover the true topology. The second and improved algorithm requires only time-stamped voltage and complex power samples to recover both the true topology and impedances without any additional input (e.g., number of grid nodes, statistics of injections at hidden nodes, and permissible line impedances). We prove the correctness of both learning algorithms for grids where unobserved buses/nodes have a degree greater than three and discuss extensions to regimes where that assumption doesn't hold. Further, we present computational and, more important, the sample complexity of our proposed algorithm for joint topology and impedance estimation. We illustrate the performance of the designed algorithms through numerical experiments on the IEEE and custom power distribution models.

Genomic alterations associated with the progression from castration-sensitive to castration-resistant metastatic prostate cancer based on machine learning analysis of cell-free DNA genomic profile.

e17596 Background: Metastatic castration-sensitive prostate cancer (mCSPC) eventually progresses to metastatic castration-resistant prostate cancer (mCRPC), which has few treatment options and carries a poor prognosis. We hypothesize that there are specific genomic alterations (GAs) associated with the progression from mCSPC to mCRPC. Methods: Patients (Pts) with mCSPC and mCRPC undergoing next-generation sequencing of cell-free DNA by a CLIA certified lab (G360, Guardant Health Inc., Redwood City, CA) as a part of routine care were retrospectively identified. Principal components analysis, an unsupervised ML algorithm, was used for data exploration and visualization. A combination of feature selection and supervised machine learning classification algorithms were used to identify genes associated with mCRPC. Gene Ontology enrichment analysis was used to identify pathways enriched for mCRPC-associated GAs. Patterns of mCRPC-associated GAs at a gene- and pathway-level were identified by Bayesian networks fitted using an exact structure learning algorithm. Results: 154 Pts with mCSPC and 187 Pts with mCRPC were included. A set of 16 GAs that robustly distinguished mCRPC from mCSPC (PPV = 94%, specificity = 91%) using supervised machine learning algorithms. These GAs, primarily amplifications, corresponded to AR, MAPK signaling, PI3K signaling, G1/S cell cycle, and receptor tyrosine kinases (RTKs). Positive statistical dependencies were observed between genes in these pathways. At a pathway-level, the presence of G1/S GAs in mCRPC samples increased the likelihood of harboring GAs in RTK, MAPK, and PI3K signaling. Limitations: The retrospective nature of our study means that unknown exposures could act as confounding variables, however this is representative of real-world clinical settings. Although the strength of this study is inclusion of clinically annotated patient samples, the limitation is that patients with mCSPC and mCRPC were unmatched. Conclusions: These results provide evidence that progression from mCSPC to mCRPC is associated with stereotyped concomitant gain-of-function in the RTK, PI3K, MAPK, and G1/S pathways in addition to AR. Upon external validation, these hypothesis generating data may warrant further investigation into combinatorial therapies that target these pathways.

Exact Learning Algorithms, Betting Games, and Circuit Lower Bounds

This article extends and improves the work of Fortnow and Klivans [2009], who showed that if a circuit class C has an efficient learning algorithm in Angluin’s model of exact learning via equivalence and membership queries [Angluin 1988], then we have the lower bound EXP NP not C. We use entirely different techniques involving betting games [Buhrman et al. 2001] to remove the NP oracle and improve the lower bound to EXP not C. This shows that it is even more difficult to design a learning algorithm for C than the results of Fortnow and Klivans [2009] indicated. We also investigate the connection between betting games and natural proofs, and as a corollary the existence of strong pseudorandom generators. Our results also yield further evidence that the class of Boolean circuits has no efficient exact learning algorithm. This is because our separation is strong in that it yields a natural proof [Razborov and Rudich 1997] against the class. From this we conclude that an exact learning algorithm for Boolean circuits would imply that strong pseudorandom generators do not exist, which contradicts widely believed conjectures from cryptography. As a corollary we obtain that if strong pseudorandom generators exist, then there is no exact learning algorithm for Boolean circuits.

Efficiently Testing Sparse GF(2) Polynomials

We give the first algorithm that is both query-efficient and time-efficient for testing whether an unknown function f:{0,1}n→{−1,1} is an s-sparse GF(2) polynomial versus ε-far from every such polynomial. Our algorithm makes poly(s,1/ε) black-box queries to f and runs in time n⋅poly(s,1/ε). The only previous algorithm for this testing problem (Diakonikolas et al. in Proceedings of the 48th Annual Symposium on Foundations of Computer Science, FOCS, pp. 549–558, 2007) used poly(s,1/ε) queries, but had running time exponential in s and super-polynomial in 1/ε.Our approach significantly extends the “testing by implicit learning” methodology of Diakonikolas et al. (Proceedings of the 48th Annual Symposium on Foundations of Computer Science, FOCS, pp. 549–558, 2007). The learning component of that earlier work was a brute-force exhaustive search over a concept class to find a hypothesis consistent with a sample of random examples. In this work, the learning component is a sophisticated exact learning algorithm for sparse GF(2) polynomials due to Schapire and Sellie (J. Comput. Syst. Sci. 52(2):201–213, 1996). A crucial element of this work, which enables us to simulate the membership queries required by Schapire and Sellie (J. Comput. Syst. Sci. 52(2):201–213, 1996), is an analysis establishing new properties of how sparse GF(2) polynomials simplify under certain restrictions of “low-influence” sets of variables.

Efficient learning algorithms yield circuit lower bounds

We describe a new approach for understanding the difficulty of designing efficient learning algorithms. We prove that the existence of an efficient learning algorithm for a circuit class C in Angluin's model of exact learning from membership and equivalence queries or in Valiant's PAC model yields a lower bound against C. More specifically, we prove that any subexponential time, deterministic exact learning algorithm for C (from membership and equivalence queries) implies the existence of a function f in EXP NP such that f ∉ C . If C is PAC learnable with membership queries under the uniform distribution or exact learnable in randomized polynomial-time, we prove that there exists a function f ∈ BPEXP (the exponential time analog of BPP ) such that f ∉ C . For C equal to polynomial-size, depth-two threshold circuits (i.e., neural networks with a polynomial number of hidden nodes), our result shows that efficient learning algorithms for this class would solve one of the most challenging open problems in computational complexity theory: proving the existence of a function in EXP NP or BPEXP that cannot be computed by circuits from C. We are not aware of any representation-independent hardness results for learning depth-2, polynomial-size neural networks with respect to the uniform distribution. Our approach uses the framework of the breakthrough result due to Kabanets and Impagliazzo showing that derandomizing BPP yields non-trivial circuit lower bounds.

Applying dynamic Bayesian networks to perturbed gene expression data.

BackgroundA central goal of molecular biology is to understand the regulatory mechanisms of gene transcription and protein synthesis. Because of their solid basis in statistics, allowing to deal with the stochastic aspects of gene expressions and noisy measurements in a natural way, Bayesian networks appear attractive in the field of inferring gene interactions structure from microarray experiments data. However, the basic formalism has some disadvantages, e.g. it is sometimes hard to distinguish between the origin and the target of an interaction. Two kinds of microarray experiments yield data particularly rich in information regarding the direction of interactions: time series and perturbation experiments. In order to correctly handle them, the basic formalism must be modified. For example, dynamic Bayesian networks (DBN) apply to time series microarray data. To our knowledge the DBN technique has not been applied in the context of perturbation experiments.ResultsWe extend the framework of dynamic Bayesian networks in order to incorporate perturbations. Moreover, an exact algorithm for inferring an optimal network is proposed and a discretization method specialized for time series data from perturbation experiments is introduced. We apply our procedure to realistic simulations data. The results are compared with those obtained by standard DBN learning techniques. Moreover, the advantages of using exact learning algorithm instead of heuristic methods are analyzed.ConclusionWe show that the quality of inferred networks dramatically improves when using data from perturbation experiments. We also conclude that the exact algorithm should be used when it is possible, i.e. when considered set of genes is small enough.

The monotone theory for the PAC-model

In this paper we extend the Monotone Theory to the PAC-learning Model with membership queries. Using this extention we show that a DNF formula that has at least one “1/ poly -heavy” clause in one of its CNF representation (a clause that is not satisfied with probability 1/ poly ( n , s ) where n is the number of variables and s is the number of terms in f ) with respect to a distribution D is weakly learnable under this distribution. So DNF that are not weakly learnable under the distribution D do not have any “1/ poly -heavy” clauses in any of their CNF representations. A DNF f is called τ -CDNF if there is τ ′ > τ and a CNF representation of f that contains poly ( n , s ) clauses that τ ′ -approximates f according to a distribution D . We show that the class of all τ -CDNF is weakly ( τ + ϵ )-PAC-learnable with membership queries under the distribution D . We then show how to change our algorithm to a parallel algorithm that runs in polylogarithmic time with a polynomial number of processors. In particular, decision trees are (strongly) PAC-learnable with membership queries under any distribution in parallel in polylogarithmic time with a polynomial number of processors. Finally we show that no efficient parallel exact learning algorithm exists for decision trees.

The Query Complexity of Finding Local Minima in the Lattice

In this paper we study the query complexity of finding local minimum points of a boolean function. This task occurs frequently in exact learning algorithms for many natural classes, such as monotone DNF, O(log n)-term DNF, unate DNF, and decision trees. On the negative side, we prove that any (possibly randomized) algorithm that produces a local minimum of a function f chosen from a sufficiently “rich” concept class, using a membership oracle for f, must ask Ω(n2) membership queries in the worst case. In particular, this lower bound applies to the class of decision trees. A simple algorithm is known that achieves this lower bound. On the positive side, we show that for the class O(log n)-term DNF finding local minimum points requires only Θ(n log n) membership queries (and more generally Θ(tn) membership queries for t-term DNF with t≤n). This efficient procedure improves the time and query complexity of known learning algorithms for the class O(log n)-term DNF.

Deterministic Generative Models for Fast Feature Discovery

We propose a vector quantisation method which does not only provide a compact description of data vectors in terms codebook vectors, but also gives an explanation of codebook vectors as binary combinations of elementary features. This corresponds to the intuitive notion that, in the real world, patterns can be usefully thought of as being constructed by compositions from simpler features. The model can be understood as a generative model, in which the codebook vector is generated by a hidden binary state vector. The model is non-probabilistic in the sense that it assigns each data vector to a single codebook vector. We describe exact and approximate learning algorithms for learning deterministic feature representations. In contrast to probabilistic models, the deterministic approach allows the use of message propagation algorithms within the learning scheme. These are compared with standard mean-field/Gibbs sampling learning. We show that Generative Vector Quantisation gives a good performance in large scale real world tasks like image compression and handwritten digit analysis with up to 400 data dimensions.

JNN, a randomized algorithm for training multilayer networks in polynomial time

From an analytical approach of the multilayer network architecture, we deduce a polynomial-time algorithm for learning from examples. We call it JNN, for “Jacobian Neural Network”. Although this learning algorithm is a randomized algorithm, it gives a correct network with probability 1. The JNN learning algorithm is defined for a wide variety of multilayer networks, computing real output vectors, from real input vectors, through one or several hidden layers, with low assumptions on the activation functions of the hidden units. Starting from an exact learning algorithm, for a given database, we propose a regularization technique which improves the performance on applications, as can be verified on several benchmark problems. Moreover, the JNN algorithm does not require a priori statements on the network architecture, since the number of hidden units, for a one-hidden-layer network, is computed by learning. Finally, we show that a modular approach allows to learn with a reduced number of weights.

Learning Matrix Functions over Rings

Let R be a commutative Artinian ring with identity and let X be a finite subset of R . We present an exact learning algorithm with a polynomial query complexity for the class of functions representable as f(x) = Π i=1 n A i (x i ), where, for each 1 ≤ i ≤ n , A i is a mapping A i : X → R mi× mi+1 and m 1 = m n+1 = 1 . We show that the above algorithm implies the following results: 1. Multivariate polynomials over a finite commutative ring with identity are learnable using equivalence and substitution queries. 2. Bounded degree multivariate polynomials over Z n can be interpolated using substitution queries. 3. The class of constant depth circuits that consist of bounded fan-in MOD gates, where the modulus are prime powers of some fixed prime, is learnable using equivalence and substitution queries. Our approach uses a decision tree representation for the hypothesis class which takes advantage of linear dependencies. This paper generalizes the learning algorithm for automata over fields given in [BBB+].

Learning Boxes in High Dimension

We present exact learning algorithms that learn several classes of (discrete) boxes in {0,...,l-1} n . In particular we learn: (1) The class of unions of O(log n) boxes in time poly(n,log l) (solving an open problem of [16] and [12]; in [3] this class is shown to be learnable in time poly(n,l) ). (2) The class of unions of disjoint boxes in time poly(n,t,log l) , where t is the number of boxes. (Previously this was known only in the case where all boxes are disjoint in one of the dimensions; in [3] this class is shown to be learnable in time poly(n,t,l) .) In particular our algorithm learns the class of decision trees over n variables, that take values in {0,...,l-1} , with comparison nodes in time poly(n,t,log l) , where t is the number of leaves (this was an open problem in [9] which was shown in [4] to be learnable in time poly(n,t,l) ). (3) The class of unions of O(1) -degenerate boxes (that is, boxes that depend only on O(1) variables) in time poly(n,t, log l) (generalizing the learnability of O(1) -DNF and of boxes in O(1) dimensions). The algorithm for this class uses only equivalence queries and it can also be used to learn the class of unions of O(1) boxes (from equivalence queries only).

An exact learning algorithm for autoassociative neural networks with binary couplings

Exact solutions for the learning problem of autoassociative networks with binary couplings are determined by a new method. The use of a branch-and-bound algorithm leads to a substantial saving of computational time compared with complete enumeration. As a result, fully connected networks with up to 40 neurons could be investigated. The network capacity is found to be close to 0.83.

A subexponential exact learning algorithm for DNF using equivalence queries

We present a 2 O ̃ (√n) time exact learning algorithm for polynomial size DNF using equivalence queries only. In particular, DNF is PAC-learnable in subexponential time under any distribution. This is the first subexponential time PAC-learning algorithm for DNF under any distribution.