Neuroevolution has yet to scale up to complex reinforcement learning tasks that require large networks. Networks with many inputs (e.g. raw video) imply a very high dimensional search space if encoded directly. Indirect methods use a more compact genotype representation that is transformed into networks of potentially arbitrary size. In this paper, we present an indirect method where networks are encoded by a set of Fourier coefficients which are transformed into network weight matrices via an inverse Fourier-type transform. Because there often exist network solutions whose weight matrices contain regularity (i.e. adjacent weights are correlated), the number of coefficients required to represent these networks in the frequency domain is much smaller than the number of weights (in the same way that natural images can be compressed by ignore high-frequency components). This "compressed" encoding is compared to the direct approach where search is conducted in the weight space on the high-dimensional octopus arm task. The results show that representing networks in the frequency domain can reduce the search-space dimensionality by as much as two orders of magnitude, both accelerating convergence and yielding more general solutions.
@cite_6 @cite_8 used a simple set of program instructions to describe perceptron weight matrix showing that short programs can represent scalable solutions to problems that require a regular weight vector.
@cite_6 @cite_8
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Approximation of the permanent of a matrix with nonnegative entries is a well studied problem. The most successful approach to date for general matrices uses Markov chains to approximately sample from a distribution on weighted permutations, and Jerrum, Sinclair, and Vigoda developed such a method they proved runs in polynomial time in the input. The current bound on the running time of their method is O(n7(log n)4). Here we present a very different approach using sequential acceptance rejection, and show that for a class of dense problems this method has an O(n4 log n) expected running time. We introduce new Perceptron-based algorithms for the online multitask binary classification problem. Under suitable regularity conditions, our algorithms are shown to improve on their baselines by a factor proportional to the number of tasks. We achieve these improvements using various types of regularization that bias our algorithms towards specific notions of task relatedness. More specifically, similarity among tasks is either measured in terms of the geometric closeness of the task reference vectors or as a function of the dimension of their spanned subspace. In addition to adapting to the online setting a mix of known techniques, such as the multitask kernels of , our analysis also introduces a matrix-based multitask extension of the p-norm Perceptron, which is used to implement spectral co-regularization. Experiments on real-world data sets complement and support our theoretical findings. Given a large overcomplete dictionary of basis vectors, the goal is to simultaneously represent L>1 signal vectors using coefficient expansions marked by a common sparsity profile. This generalizes the standard sparse representation problem to the case where multiple responses exist that were putatively generated by the same small subset of features. Ideally, the associated sparse generating weights should be recovered, which can have physical significance in many applications (e.g., source localization). The generic solution to this problem is intractable and, therefore, approximate procedures are sought. Based on the concept of automatic relevance determination, this paper uses an empirical Bayesian prior to estimate a convenient posterior distribution over candidate basis vectors. This particular approximation enforces a common sparsity profile and consistently places its prominent posterior mass on the appropriate region of weight-space necessary for simultaneous sparse recovery. The resultant algorithm is then compared with multiple response extensions of matching pursuit, basis pursuit, FOCUSS, and Jeffreys prior-based Bayesian methods, finding that it often outperforms the others. Additional motivation for this particular choice of cost function is also provided, including the analysis of global and local minima and a variational derivation that highlights the similarities and differences between the proposed algorithm and previous approaches. The Perceptron algorithm, despite its simplicity, often performs well in online classification tasks. The Perceptron becomes especially effective when it is used in conjunction with kernel functions. However, a common difficulty encountered when implementing kernel-based online algorithms is the amount of memory required to store the online hypothesis, which may grow unboundedly as the algorithm progresses. Moreover, the running time of each online round grows linearly with the amount of memory used to store the hypothesis. In this paper, we present the Forgetron family of kernel-based online classification algorithms, which overcome this problem by restricting themselves to a predefined memory budget. We obtain different members of this family by modifying the kernel-based Perceptron in various ways. We also prove a unified mistake bound for all of the Forgetron algorithms. To our knowledge, this is the first online kernel-based learning paradigm which, on one hand, maintains a strict limit on the amount of memory it uses and, on the other hand, entertains a relative mistake bound. We conclude with experiments using real datasets, which underscore the merits of our approach. Recovering low-rank and sparse matrices from incomplete or corrupted observations is an important problem in statistics, machine learning, computer vision, as well as signal and image processing. In theory, this problem can be solved by the natural convex joint mixed relaxations (i.e., l 1 -norm and trace norm) under certain conditions. However, all current provable algorithms suffer from superlinear per-iteration cost, which severely limits their applicability to large-scale problems. In this paper, we propose a scalable, provable and structured robust bilinear factorization (RBF) method to recover low-rank and sparse matrices from missing and grossly corrupted data, i.e., robust matrix completion (RMC), or incomplete and grossly corrupted measurements, i.e., compressive principal component pursuit (CPCP). Specifically, we first present two small-scale matrix trace norm regularized bilinear factorization models for RMC and CPCP problems, in which repetitively calculating SVD of a large-scale matrix is replaced by updating two much smaller factor matrices. Then, we apply the alternating direction method of multipliers (ADMM) to efficiently solve the RMC problems. Finally, we provide the convergence analysis of our algorithm, and extend it to address general CPCP problems. Experimental results verified both the efficiency and effectiveness of our method compared with the state-of-the-art methods. Weight decay was proposed to reduce overfitting which often appears in the generalization tasks of artificial neural nets. Here weight decay is applied to a well defined model system based on a single layer perceptron, which exhibits strong overfitting. Since we know for this system the optimal nonoverfitting solution we can compare the effect of the weight decay with this solution. A strategy to find the optimal weight decay strength, which leads to the optimal solution for any number of examples, is proposed. An integrated model for ranking scientific publications together with authors and journals recently presented in [Bini, Del Corso, Romani, ETNA 2008] is closely analyzed. The model, which relies on certain adjacency matrices H,K and F obtained from the relations of citation, authorship and publication, provides the ranking by means of the Perron vector of a stochastic matrix obtained by combining H,K and F. Some perturbation theorems concerning the Perron vector previously introduced by the authors are extended to more general cases and a counterexample to a property previously addressed by the authors is presented. The theoretical results confirm the consistency and effectiveness of our model. Some paradigmatic examples are reported together with some results obtained on a real set of data. The perceptron algorithm, developed mainly in the machine learning literature, is a simple greedy method for finding a feasible solution to a linear program (alternatively, for learning a threshold function). In spite of its exponential worst-case complexity, it is often quite useful, in part due to its noise-tolerance and also its overall simplicity. In this paper, we show that a randomized version of the perceptron algorithm along with periodic rescaling runs in polynomial-time. The resulting algorithm for linear programming has an elementary description and analysis. It is widely believed that the back-propagation algorithm in neural networks, for tasks such as pattern classification, overcomes the limitations of the perceptron. The authors construct several counterexamples to this belief. They also construct linearly separable examples which have a unique minimum which fails to separate two families of vectors, and a simple example with four two-dimensional vectors in a single-layer network showing local minima with a large basin of attraction. Thus, back-propagation is guaranteed to fail in the first example, and likely to fail in the second example. It is shown that even multilayered (hidden-layer) networks can also fail in this way to classify linearly separable problems. Since the authors' examples are all linearly separable, the perceptron would correctly classify them. The results disprove the presumption, made in recent years, that, barring local minima, back-propagation will find the best set of weights for a given problem. > In this paper we apply multilabel classification algorithms to the EUR-Lex database of legal documents of the European Union. For this document collection, we studied three different multilabel classification problems, the largest being the categorization into the EUROVOC concept hierarchy with almost 4000 classes. We evaluated three algorithms: (i) the binary relevance approach which independently trains one classifier per label; (ii) the multiclass multilabel perceptron algorithm, which respects dependencies between the base classifiers; and (iii) the multilabel pairwise perceptron algorithm, which trains one classifier for each pair of labels. All algorithms use the simple but very efficient perceptron algorithm as the underlying classifier, which makes them very suitable for large-scale multilabel classification problems. The main challenge we had to face was that the almost 8,000,000 perceptrons that had to be trained in the pairwise setting could no longer be stored in memory. We solve this problem by resorting to the dual representation of the perceptron, which makes the pairwise approach feasible for problems of this size. The results on the EUR-Lex database confirm the good predictive performance of the pairwise approach and demonstrates the feasibility of this approach for large-scale tasks. Some integrated models for ranking scientific publications together with authors and journals are presented and analyzed. The models rely on certain adjacency matrices obtained from the relationships between citations, authors and publications, which together give a suitable irreducible stochastic matrix whose Perron vector provides the ranking. Some perturbation theorems concerning the Perron vectors of nonnegative irreducible matrices are proved. These theoretical results provide a validation of the consistency and effectiveness of our models. Several examples are reported together with some results obtained on a real set of data. This paper considers the problem of recovering an unknown sparse p×p matrix X from an m×m matrix Y=AXB T , where A and B are known m×p matrices with m≪p. The main result shows that there exist constructions of the sketching matrices A and B so that even if X has O(p) nonzeros, it can be recovered exactly and efficiently using a convex program as long as these nonzeros are not concentrated in any single row column of X. Furthermore, it suffices for the size of Y (the sketch dimension) to scale as m = O(√(# nonzeros in X) × log p). The results also show that the recovery is robust and stable in the sense that if X is equal to a sparse matrix plus a perturbation, then the convex program we propose produces an approximation with accuracy proportional to the size of the perturbation. Unlike traditional results on sparse recovery, where the sensing matrix produces independent measurements, our sensing operator is highly constrained (it assumes a tensor product structure). Therefore, proving recovery guarantees require nonstandard techniques. Indeed, our approach relies on a novel result concerning tensor products of bipartite graphs, which may be of independent interest. This problem is motivated by the following application, among others. Consider a p×n data matrix D, consisting of n observations of p variables. Assume that the correlation matrix X:=DD T is (approximately) sparse in the sense that each of the p variables is significantly correlated with only a few others. Our results show that these significant correlations can be detected even if we have access to only a sketch of the data S=AD with A ∈ R m×p . We investigate the problem of reconstructing a high-dimensional nonnegative sparse vector from lower-dimensional linear measurements. While much work has focused on dense measurement matrices, sparse measurement schemes can be more efficient both with respect to signal sensing as well as reconstruction complexity. Known constructions use the adjacency matrices of expander graphs, which often lead to recovery algorithms which are much more efficient than minimization. However, prior constructions of sparse measurement matrices rely on expander graphs with very high expansion coefficients which make the construction of such graphs difficult and the size of the recoverable sets very small. In this paper, we introduce sparse measurement matrices for the recovery of nonnegative vectors, using perturbations of the adjacency matrices of expander graphs requiring much smaller expansion coefficients, hereby referred to as minimal expanders. We show that when minimization is used as the reconstruction method, these constructions allow the recovery of signals that are almost three orders of magnitude larger compared to the existing theoretical results for sparse measurement matrices. We provide for the first time tight upper bounds for the so called weak and strong recovery thresholds when minimization is used. We further show that the success of optimization is equivalent to the existence of a “unique” vector in the set of solutions to the linear equations, which enables alternative algorithms for minimization. We further show that the defined minimal expansion property is necessary for all measurement matrices for compressive sensing, (even when the non-negativity assumption is removed) therefore implying that our construction is tight. We finally present a novel recovery algorithm that exploits expansion and is much more computationally efficient compared to minimization. Abstract— A linear space-time block code (STBC) is a vector space spanned by its defining weight matrices over the real number field. We define a Quadratic Form (QF), called the Hurwitz-Radon QF (HRQF), on this vector space and give a QF interpretation of the ML decoding complexity of a STBC. It is shown that the ML decoding complexity is only a function of the weight matrices defining the code and their ordering, and not of the channel realization (even though the equivalent channel when sphere decoding is used depends on the channel realization) or the number of receive antennas. It is shown that the ML decoding complexity is completely captured into a single matrix obtained from the HRQF. Also, given a set of weight matrices, an algorithm to obtain the best ordering of them leading to the least ML decoding complexity is presented. The well known classes of low ML decoding complexity codes (multi-group decodable codes, fast decodable codes and fast group decodable codes) are presented in the framework of HRQF. The authors consider the problem of learning a linear threshold function (a halfspace in n dimensions, also called a "perceptron"). Methods for solving this problem generally fall into two categories. In the absence of noise, this problem can be formulated as a linear program and solved in polynomial time with the ellipsoid algorithm (or interior point methods). On the other hand, simple greedy algorithms such as the perceptron algorithm seem to work well in practice and can be made noise tolerant; but, their running time depends on a separation parameter (which quantifies the amount of "wiggle room" available) and can be exponential in the description length of the input. They show how simple greedy methods can be used to find weak hypotheses (hypotheses that classify noticeably more than half of the examples) in polynomial time, without dependence on any separation parameter. This results in a polynomial-time algorithm for learning linear threshold functions in the PAC model in the presence of random classification noise. The algorithm is based on a new method for removing outliers in data. Specifically, for any set S of points in R sup n , each given to b bits of precision, they show that one can remove only a small fraction of S so that in the remaining set T, for every vector v, max sub x spl epsiv T (v spl middot x) sup 2 spl les poly(n,b)|T| sup -1 spl Sigma sub x spl epsiv T (v spl middot x) sup 2 . After removing these outliers, they are able to show that a modified version of the perceptron learning algorithm works in polynomial time, even in the presence of random classification noise. A traditional counterexample to a linear-time safety property shows the values of all signals at all times prior to the error. However, some signals may not be critical to causing the failure. A succinct explanation may help human understanding as well as speed up algorithms that have to analyze many such traces. In Bounded Model Checking (BMC), a counterexample is constructed from a satisfying assignment to a Boolean formula, typically in CNF. Modern SAT solvers usually assign values to all variables when the input formula is satisfiable. Deriving minimal satisfying assignments from such complete assignments does not lead to concise explanations of counterexamples because of how CNF formulae are derived from the models. Hence, we formulate the extraction of a succinct counterexample as the problem of finding a minimal assignment that, together with the Boolean formula describing the model, implies an objective. We present a two-stage algorithm for this problem, such that the result of each stage contributes to identify the “interesting” events that cause the failure. We demonstrate the effectiveness of our approach with an example and with experimental results. A novel logic program like language, weight constraint rules, is developed for answer set programming purposes. It generalizes normal logic programs by allowing weight constraints in place of literals to represent, e.g., cardinality and resource constraints and by providing optimization capabilities. A declarative semantics is developed which extends the stable model semantics of normal programs. The computational complexity of the language is shown to be similar to that of normal programs under the stable model semantics. A simple embedding of general weight constraint rules to a small subclass of the language called basic constraint rules is devised. An implementation of the language, the SMODELS system, is developed based on this embedding. It uses a two level architecture consisting of a front-end and a kernel language implementation. The front-end allows restricted use of variables and functions and compiles general weight constraint rules to basic constraint rules. A major part of the work is the development of an efficient search procedure for computing stable models for this kernel language. The procedure is compared with and empirically tested against satisfiability checkers and an implementation of the stable model semantics. It offers a competitive implementation of the stable model semantics for normal programs and attractive performance for problems where the new types of rules provide a compact representation. Abstract The sparse matrix–vector product is an important computational kernel that runs ineffectively on many computers with super-scalar RISC processors. In this paper we analyse the performance of the sparse matrix–vector product with symmetric matrices originating from the FEM and describe techniques that lead to a fast implementation. It is shown how these optimisations can be incorporated into an efficient parallel implementation using message-passing. We conduct numerical experiments on many different machines and show that our optimisations speed up the sparse matrix–vector multiplication substantially. Sparse matrix-vector multiplication is an important computational kernel in scientific applications. However, it performs poorly on modern processors because of a low compute-to-memory ratio and its irregular memory access patterns. This paper discusses the implementations of sparse matrix-vector algorithm using OpenMP to execute iterative methods on the Dawning S4800A1. Two storage formats (CSR and BCSR) for sparse matrices and three scheduling schemes (static, dynamic and guided) provided by the standard OpenMP are evaluated. We also compared these three schemes with non-zero scheduling, where each thread is assigned approximately the same number of non-zero elements. Experimental data shows that, the non-zero scheduling can provide the best performance in most cases. The current implementation provides satisfactory scalability for most of matrices. However, we only get a limited speedup for some large matrices that contain millions of non-zero elements. This paper shows that discriminative reranking with an averaged perceptron model yields substantial improvements in realization quality with CCG. The paper confirms the utility of including language model log probabilities as features in the model, which prior work on discriminative training with log linear models for HPSG realization had called into question. The perceptron model allows the combination of multiple n-gram models to be optimized and then augmented with both syntactic features and discriminative n-gram features. The full model yields a state-of-the-art BLEU score of 0.8506 on Section 23 of the CCGbank, to our knowledge the best score reported to date using a reversible, corpus-engineered grammar.