Boolean satisfiability (SAT) as a canonical NP-complete decision problem is one of the most important problems in computer science. In practice, real-world SAT sentences are drawn from a distribution that may result in efficient algorithms for their solution. Such SAT instances are likely to have shared characteristics and substructures. This work approaches the exploration of a family of SAT solvers as a learning problem. In particular, we relate polynomial time solvability of a SAT subset to a notion of margin between sentences mapped by a feature function into a Hilbert space. Provided this mapping is based on polynomial time computable statistics of a sentence, we show that the existance of a margin between these data points implies the existance of a polynomial time solver for that SAT subset based on the Davis-Putnam-Logemann-Loveland algorithm. Furthermore, we show that a simple perceptron-style learning rule will find an optimal SAT solver with a bounded number of training updates. We derive a linear time computable set of features and show analytically that margins exist for important polynomial special cases of SAT. Empirical results show an order of magnitude improvement over a state-of-the-art SAT solver on a hardware verification task.

Linear chains and trees are basic building blocks in many applications of graphical models. Although exact inference in these models can be performed by dynamic programming, this computation can still be prohibitively expensive with non-trivial target variable domain sizes due to the quadratic dependence on this size. Standard message-passing algorithms for these problems are inefficient because they compute scores on hypotheses for which there is strong negative local evidence. For this reason there has been significant previous interest in beam search and its variants; however, these methods provide only approximate inference. This paper presents new efficient exact inference algorithms based on the combination of it column generation and pre-computed bounds on the model's cost structure. Improving worst-case performance is impossible. However, our method substantially speeds real-world, typical-case inference in chains and trees. Experiments show our method to be twice as fast as exact Viterbi for Wall Street Journal part-of-speech tagging and over thirteen times faster for a joint part-of-speed and named-entity-recognition task. Our algorithm is also extendable to new techniques for approximate inference, to faster two-best inference, and new opportunities for connections between inference and learning.

Some of the most compelling applications of online convex optimization, including online prediction and classification, are unconstrained: the natural feasible set is R^n. Existing algorithms fail to achieve sub-linear regret in this setting unless constraints on the comparator point x* are known in advance. We present an algorithm that, without such prior knowledge, offers near-optimal regret bounds with respect to _any_ choice of x*. In particular, regret with respect to x* = 0 is _constant_. We then prove lower bounds showing that our algorithm's guarantees are optimal in this setting up to constant factors.

Latent linear dynamical systems with generalised-linear observation models arise in a variety of applications, for example when modelling the spiking activity of populations of neurons. Here, we show how spectral learning methods for linear systems with Gaussian observations (usually called subspace identification in this context) can be extended to estimate the parameters of dynamical system models observed through non-Gaussian noise models. We use this approach to obtain estimates of parameters for a dynamical model of neural population data, where the observed spike-counts are Poisson-distributed with log-rates determined by the latent dynamical process, possibly driven by external inputs. We show that the extended system identification algorithm is consistent and accurately recovers the correct parameters on large simulated data sets with much smaller computational cost than approximate expectation-maximisation (EM) due to the non-iterative nature of subspace identification. Even on smaller data sets, it provides an effective initialization for EM, leading to more robust performance and faster convergence. These benefits are shown to extend to real neural data.

For modeling data matrices, this paper introduces Probabilistic Co-Subspace Addition (PCSA) model by simultaneously capturing the dependent structures among both rows and columns. Briefly, PCSA assumes that each entry of a matrix is generated by the additive combination of the linear mappings of two features, which distribute in the row-wise and column-wise latent subspaces. Consequently, it captures the dependencies among entries intricately, and is able to model the non-Gaussian and heteroscedastic density. Variational inference is proposed on PCSA for approximate Bayesian learning, where the updating for posteriors is formulated into the problem of solving Sylvester equations. Furthermore, PCSA is extended to tackling and filling missing values, to adapting its sparseness, and to modelling tensor data. In comparison with several state-of-art approaches, experiments demonstrate the effectiveness and efficiency of Bayesian (sparse) PCSA on modeling matrix (tensor) data and filling missing values.

Learning the number of clusters is a key problem in data clustering. We present dip-means, a novel robust incremental method to learn the number of data clusters that may be used as a wrapper around any iterative clustering algorithm of the k-means family. In contrast to many popular methods which make assumptions about the underlying cluster distributions, dip-means only assumes a fundamental cluster property: each cluster to admit a unimodal distribution. The proposed algorithm considers each cluster member as a ''viewer'' and applies a univariate statistic hypothesis test for unimodality (dip-test) on the distribution of the distances between the viewer and the cluster members. Two important advantages are: i) the unimodality test is applied on univariate distance vectors, ii) it can be directly applied with kernel-based methods, since only the pairwise distances are involved in the computations. Experimental results on artificial and real datasets indicate the effectiveness of our method and its superiority over analogous approaches.

We present a novel multilabel/ranking algorithm working in partial information settings. The algorithm is based on 2nd-order descent methods, and relies on upper-confidence bounds to trade-off exploration and exploitation. We analyze this algorithm in a partial adversarial setting, where covariates can be adversarial, but multilabel probabilities are ruled by (generalized) linear models. We show $O(T^{1/2}\log T)$ regret bounds, which improve in several ways on the existing results. We test the effectiveness of our upper-confidence scheme by contrasting against full-information baselines on real-world multilabel datasets, often obtaining comparable performance.

In this paper we dicuss a novel framework for multiclass learning, defined by a suitable coding/decoding strategy, namely the simplex coding, that allows to generalize to multiple classes a relaxation approach commonly used in binary classification. In this framework a relaxation error analysis can be developed avoiding constraints on the considered hypotheses class. Moreover, we show that in this setting it is possible to derive the first provably consistent regularized methods with training/tuning complexity which is {\em independent} to the number of classes. Tools from convex analysis are introduced that can be used beyond the scope of this paper.

In this work we study how the stimulus distribution influences the optimal coding of an individual neuron. Closed-form solutions to the optimal sigmoidal tuning curve are provided for a neuron obeying Poisson statistics under a given stimulus distribution.

We present a method for approximate inference for a broad class of non-conjugate probabilistic models. In particular, for the family of generalized linear model target densities we describe a rich class of variational approximating densities which can be best fit to the target by minimizing the Kullback-Leibler divergence. Our approach is based on using the Fourier representation which we show results in efficient and scalable inference.