Probability Distributions.- Linear Models for Regression.- Linear Models for Classification.- Neural Networks.- Kernel Methods.- Sparse Kernel Machines.- Graphical Models.- Mixture Models and EM.- Approximate Inference.- Sampling Methods.- Continuous Latent Variables.- Sequential Data.- Combining Models.

Pattern Recognition and Machine Learning

The formalism of probabilistic graphical models provides a unifying framework for capturing complex dependencies among random variables, and building large-scale multivariate statistical models. Graphical models have become a focus of research in many statistical, computational and mathematical fields, including bioinformatics, communication theory, statistical physics, combinatorial optimization, signal and image processing, information retrieval and statistical machine learning. Many problems that arise in specific instances — including the key problems of computing marginals and modes of probability distributions — are best studied in the general setting. Working with exponential family representations, and exploiting the conjugate duality between the cumulant function and the entropy for exponential families, we develop general variational representations of the problems of computing likelihoods, marginal probabilities and most probable configurations. We describe how a wide variety of algorithms — among them sum-product, cluster variational methods, expectation-propagation, mean field methods, max-product and linear programming relaxation, as well as conic programming relaxations — can all be understood in terms of exact or approximate forms of these variational representations. The variational approach provides a complementary alternative to Markov chain Monte Carlo as a general source of approximation methods for inference in large-scale statistical models.

/pdf/graphical-models-exponential-families-and-variational-3i6dwlfta1.pdf

Graphical Models, Exponential Families, and Variational Inference

One of the core problems of modern statistics is to approximate difficult-to-compute probability densities. This problem is especially important in Bayesian statistics, which frames all inference about unknown quantities as a calculation involving the posterior density. In this article, we review variational inference (VI), a method from machine learning that approximates probability densities through optimization. VI has been used in many applications and tends to be faster than classical methods, such as Markov chain Monte Carlo sampling. The idea behind VI is to first posit a family of densities and then to find a member of that family which is close to the target density. Closeness is measured by Kullback–Leibler divergence. We review the ideas behind mean-field variational inference, discuss the special case of VI applied to exponential family models, present a full example with a Bayesian mixture of Gaussians, and derive a variant that uses stochastic optimization to scale up to massive data...

Variational Inference: A Review for Statisticians

When a parameter space has a certain underlying structure, the ordinary gradient of a function does not represent its steepest direction, but the natural gradient does. Information geometry is used for calculating the natural gradients in the parameter space of perceptrons, the space of matrices (for blind source separation), and the space of linear dynamical systems (for blind source deconvolution). The dynamical behavior of natural gradient online learning is analyzed and is proved to be Fisher efficient, implying that it has asymptotically the same performance as the optimal batch estimation of parameters. This suggests that the plateau phenomenon, which appears in the backpropagation learning algorithm of multilayer perceptrons, might disappear or might not be so serious when the natural gradient is used. An adaptive method of updating the learning rate is proposed and analyzed.

Natural gradient works efficiently in learning

In this article, we review probabilistic topic models: graphical models that can be used to summarize a large collection of documents with a smaller number of distributions over words. Those distributions are called "topics" because, when fit to data, they capture the salient themes that run through the collection. We describe both finite-dimensional parametric topic models and their Bayesian nonparametric counterparts, which are based on the hierarchical Dirichlet process (HDP). We discuss two extensions of topic models to time-series data-one that lets the topics slowly change over time and one that lets the assumed prevalence of the topics change. Finally, we illustrate the application of topic models to nontext data, summarizing some recent research results in image analysis.

/pdf/probabilistic-topic-models-4cq6hpiogr.pdf

Probabilistic Topic Models

We consider the problem of learning the optimal action-value function in the discounted-reward Markov decision processes (MDPs). We prove a new PAC bound on the sample-complexity of model-based value iteration algorithm in the presence of the generative model, which indicates that for an MDP with N state-action pairs and the discount factor γ ∈ [0, 1) only O(N log(N/δ)/(1 - γ)3e2)) samples are required to find an e-optimal estimation of the action-value function with the probability 1 - δ. We also prove a matching lower bound of Θ(N log(N/δ)/((1 - γ)3e2)) on the sample complexity of estimating the optimal action-value function by every RL algorithm. To the best of our knowledge, this is the first matching result on the sample complexity of estimating the optimal (action-) value function in which the upper bound matches the lower bound of RL in terms of N, e, δ and 1/(1-γ). Also, both our lower bound and our upper bound significantly improve on the state-of-the-art in terms of 1/(1 - γ).

/pdf/on-the-sample-complexity-of-reinforcement-learning-with-a-yzejsumz4s.pdf

On the Sample Complexity of Reinforcement Learning with a Generative Model

We formulate tempo tracking in a Bayesian framework where a tempo tracker is modeled as a stochastic dynamical system. The tempo is modeled as a hidden state variable of the system and is estimated by a Kalman filter. The Kalman filter operates on a Tempogram, a wavelet-like multiscale expansion of a real performance. An important advantage of our approach is that it is possible to formulate both off-line or real-time algorithms. The simulation results on a systematically collected set of MIDI piano performances of Yesterday and Michelle by the Beatles shows accurate tracking of approximately 90% of the beats.

https://www.cmpe.boun.edu.tr/~cemgil/papers/cemgil-tt.pdf

On tempo tracking: Tempogram Representation and Kalman filtering

We reformulate a class of non-linear stochastic optimal control problems introduced by Todorov (2007) as a Kullback-Leibler (KL) minimization problem. As a result, the optimal control computation reduces to an inference computation and approximate inference methods can be applied to efficiently compute approximate optimal controls. We show how this KL control theory contains the path integral control method as a special case. We provide an example of a block stacking task and a multi-agent cooperative game where we demonstrate how approximate inference can be successfully applied to instances that are too complex for exact computation. We discuss the relation of the KL control approach to other inference approaches to control.

/pdf/optimal-control-as-a-graphical-model-inference-problem-489ku87qbe.pdf

Optimal control as a graphical model inference problem

Loopy and generalized belief propagation are popular algorithms for approximate inference in Markov random fields and Bayesian networks. Fixed points of these algorithms correspond to extrema of the Bethe and Kikuchi free energy (Yedidia et al., 2001). However, belief propagation does not always converge, which motivates approaches that explicitly minimize the Kikuchi/Bethe free energy, such as CCCP (Yuille, 2002) and UPS (Teh and Welling, 2002). Here we describe a class of algorithms that solves this typically non-convex constrained minimization problem through a sequence of convex constrained minimizations of upper bounds on the Kikuchi free energy. Intuitively one would expect tighter bounds to lead to faster algorithms, which is indeed convincingly demonstrated in our simulations. Several ideas are applied to obtain tight convex bounds that yield dramatic speed-ups over CCCP.

/pdf/approximate-inference-and-constrained-optimization-1b4rmfgi6y.pdf

Approximate inference and constrained optimization

We present a probabilistic generative model for timing deviations in expressive music performance. The structure of the proposed model is equivalent to a switching state space model. The switch variables correspond to discrete note locations as in a musical score. The continuous hidden variables denote the tempo. We formulate two well known music recognition problems, namely tempo tracking and automatic transcription (rhythm quantization) as filtering and maximum a posteriori (MAP) state estimation tasks. Exact computation of posterior features such as the MAP state is intractable in this model class, so we introduce Monte Carlo methods for integration and optimization. We compare Markov Chain Monte Carlo (MCMC) methods (such as Gibbs sampling, simulated annealing and iterative improvement) and sequential Monte Carlo methods (particle filters). Our simulation results suggest better results with sequential methods. The methods can be applied in both online and batch scenarios such as tempo tracking and transcription and are thus potentially useful in a number of music applications such as adaptive automatic accompaniment, score typesetting and music information retrieval.

/pdf/monte-carlo-methods-for-tempo-tracking-and-rhythm-3oyovbr25t.pdf

Bert Kappen

Papers

On the Sample Complexity of Reinforcement Learning with a Generative Model

On tempo tracking: Tempogram Representation and Kalman filtering

Optimal control as a graphical model inference problem

Approximate inference and constrained optimization

Monte Carlo methods for tempo tracking and rhythm quantization