arXiv subjects

Manfred Opper

Publications and source records attributed to Manfred Opper.

At least 37 records · Page 2Linked to original sources

arXiv2022

Deterministic particle flows for constraining stochastic nonlinear systems

Devising optimal interventions for constraining stochastic systems is a challenging endeavour that has to confront the interplay between randomness and nonlinearity. Existing methods for identifying the necessary dynamical adjustments resort either to space discretising solutions of ensuing partial differential equations, or to iterative stochastic path sampling schemes. Yet, both approaches become computationally demanding for increasing system dimension. Here, we propose a generally applicable and practically feasible non-iterative methodology for obtaining optimal dynamical interventions for diffusive nonlinear systems. We estimate the necessary controls from an interacting particle approximation to the logarithmic gradient of two forward probability flows evolved following deterministic particle dynamics. Applied to several biologically inspired models, we show that our method provides the necessary optimal controls in settings with terminal-, transient-, or generalised collective-state constraints and arbitrary system dynamics.

cond-mat.stat-mech↗

arXiv2022

Analysis of Random Sequential Message Passing Algorithms for Approximate Inference

We analyze the dynamics of a random sequential message passing algorithm for approximate inference with large Gaussian latent variable models in a student-teacher scenario. To model nontrivial dependencies between the latent variables, we assume random covariance matrices drawn from rotation invariant ensembles. Moreover, we consider a model mismatching setting, where the teacher model and the one used by the student may be different. By means of dynamical functional approach, we obtain exact dynamical mean-field equations characterizing the dynamics of the inference algorithm. We also derive a range of model parameters for which the sequential algorithm does not converge. The boundary of this parameter range coincides with the de Almeida Thouless (AT) stability condition of the replica symmetric ansatz for the static probabilistic model.

cs.LG↗

arXiv2021

Adaptive Inducing Points Selection For Gaussian Processes

Gaussian Processes (\textbf{GPs}) are flexible non-parametric models with strong probabilistic interpretation. While being a standard choice for performing inference on time series, GPs have few techniques to work in a streaming setting. \cite{bui2017streaming} developed an efficient variational approach to train online GPs by using sparsity techniques: The whole set of observations is approximated by a smaller set of inducing points (\textbf{IPs}) and moved around with new data. Both the number and the locations of the IPs will affect greatly the performance of the algorithm. In addition to optimizing their locations, we propose to adaptively add new points, based on the properties of the GP and the structure of the data.

stat.ML↗

arXiv2021

Nonlinear Hawkes Process with Gaussian Process Self Effects

Traditionally, Hawkes processes are used to model time--continuous point processes with history dependence. Here we propose an extended model where the self--effects are of both excitatory and inhibitory type and follow a Gaussian Process. Whereas previous work either relies on a less flexible parameterization of the model, or requires a large amount of data, our formulation allows for both a flexible model and learning when data are scarce. We continue the line of work of Bayesian inference for Hawkes processes, and our approach dispenses with the necessity of estimating a branching structure for the posterior, as we perform inference on an aggregated sum of Gaussian Processes. Efficient approximate Bayesian inference is achieved via data augmentation, and we describe a mean--field variational inference approach to learn the model parameters. To demonstrate the flexibility of the model we apply our methodology on data from three different domains and compare it to previously reported results.

stat.ML↗

arXiv2021

Exact solution to the random sequential dynamics of a message passing algorithm

We analyze the random sequential dynamics of a message passing algorithm for Ising models with random interactions in the large system limit. We derive exact results for the two-time correlation functions and the speed of convergence. The {\em de Almedia-Thouless} stability criterion of the static problem is found to be necessary and sufficient for the global convergence of the random sequential dynamics.

cond-mat.dis-nn↗

arXiv2020

A Mathematical Model of Local and Global Attention in Natural Scene Viewing

Understanding the decision process underlying gaze control is an important question in cognitive neuroscience with applications in diverse fields ranging from psychology to computer vision. The decision for choosing an upcoming saccade target can be framed as a selection process between two states: Should the observer further inspect the information near the current gaze position (local attention) or continue with exploration of other patches of the given scene (global attention)? Here we propose and investigate a mathematical model motivated by switching between these two attentional states during scene viewing. The model is derived from a minimal set of assumptions that generates realistic eye movement behavior. We implemented a Bayesian approach for model parameter inference based on the model's likelihood function. In order to simplify the inference, we applied data augmentation methods that allowed the use of conjugate priors and the construction of an efficient Gibbs sampler. This approach turned out to be numerically efficient and permitted fitting interindividual differences in saccade statistics. Thus, the main contribution of our modeling approach is two--fold; first, we propose a new model for saccade generation in scene viewing. Second, we demonstrate the use of novel methods from Bayesian inference in the field of scan path modeling.

q-bio.NC↗

arXiv2020

Interacting particle solutions of Fokker-Planck equations through gradient-log-density estimation

Fokker-Planck equations are extensively employed in various scientific fields as they characterise the behaviour of stochastic systems at the level of probability density functions. Although broadly used, they allow for analytical treatment only in limited settings, and often is inevitable to resort to numerical solutions. Here, we develop a computational approach for simulating the time evolution of Fokker-Planck solutions in terms of a mean field limit of an interacting particle system. The interactions between particles are determined by the gradient of the logarithm of the particle density, approximated here by a novel statistical estimator. The performance of our method shows promising results, with more accurate and less fluctuating statistics compared to direct stochastic simulations of comparable particle number. Taken together, our framework allows for effortless and reliable particle-based simulations of Fokker-Planck equations in low and moderate dimensions. The proposed gradient-log-density estimator is also of independent interest, for example, in the context of optimal control.

cond-mat.stat-mech↗

arXiv2020

GP-ETAS: Semiparametric Bayesian inference for the spatio-temporal Epidemic Type Aftershock Sequence model

The spatio-temporal Epidemic Type Aftershock Sequence (ETAS) model is widely used to describe the self-exciting nature of earthquake occurrences. While traditional inference methods provide only point estimates of the model parameters, we aim at a full Bayesian treatment of model inference, allowing naturally to incorporate prior knowledge and uncertainty quantification of the resulting estimates. Therefore, we introduce a highly flexible, non-parametric representation for the spatially varying ETAS background intensity through a Gaussian process (GP) prior. Combined with classical triggering functions this results in a new model formulation, namely the GP-ETAS model. We enable tractable and efficient Gibbs sampling by deriving an augmented form of the GP-ETAS inference problem. This novel sampling approach allows us to assess the posterior model variables conditioned on observed earthquake catalogues, i.e., the spatial background intensity and the parameters of the triggering function. Empirical results on two synthetic data sets indicate that GP-ETAS outperforms standard models and thus demonstrate the predictive power for observed earthquake catalogues including uncertainty quantification for the estimated parameters. Finally, a case study for the l'Aquila region, Italy, with the devastating event on 6 April 2009, is presented.

stat.AP↗

arXiv2020

A Dynamical Mean-Field Theory for Learning in Restricted Boltzmann Machines

We define a message-passing algorithm for computing magnetizations in Restricted Boltzmann machines, which are Ising models on bipartite graphs introduced as neural network models for probability distributions over spin configurations. To model nontrivial statistical dependencies between the spins' couplings, we assume that the rectangular coupling matrix is drawn from an arbitrary bi-rotation invariant random matrix ensemble. Using the dynamical functional method of statistical mechanics we exactly analyze the dynamics of the algorithm in the large system limit. We prove the global convergence of the algorithm under a stability criterion and compute asymptotic convergence rates showing excellent agreement with numerical simulations.

cs.LG↗

arXiv2020

Automated Augmented Conjugate Inference for Non-conjugate Gaussian Process Models

We propose automated augmented conjugate inference, a new inference method for non-conjugate Gaussian processes (GP) models. Our method automatically constructs an auxiliary variable augmentation that renders the GP model conditionally conjugate. Building on the conjugate structure of the augmented model, we develop two inference methods. First, a fast and scalable stochastic variational inference method that uses efficient block coordinate ascent updates, which are computed in closed form. Second, an asymptotically correct Gibbs sampler that is useful for small datasets. Our experiments show that our method are up two orders of magnitude faster and more robust than existing state-of-the-art black-box methods.

stat.ML↗

arXiv2020

Understanding the dynamics of message passing algorithms: a free probability heuristics

We use freeness assumptions of random matrix theory to analyze the dynamical behavior of inference algorithms for probabilistic models with dense coupling matrices in the limit of large systems. For a toy Ising model, we are able to recover previous results such as the property of vanishing effective memories and the analytical convergence rate of the algorithm.

cond-mat.stat-mech↗

arXiv2020

Analysis of Bayesian Inference Algorithms by the Dynamical Functional Approach

We analyze the dynamics of an algorithm for approximate inference with large Gaussian latent variable models in a student-teacher scenario. To model nontrivial dependencies between the latent variables, we assume random covariance matrices drawn from rotation invariant ensembles. For the case of perfect data-model matching, the knowledge of static order parameters derived from the replica method allows us to obtain efficient algorithmic updates in terms of matrix-vector multiplications with a fixed matrix. Using the dynamical functional approach, we obtain an exact effective stochastic process in the thermodynamic limit for a single node. From this, we obtain closed-form expressions for the rate of the convergence. Analytical results are excellent agreement with simulations of single instances of large models.

cs.LG↗

arXiv2019

Tightening Bounds for Variational Inference by Revisiting Perturbation Theory

Variational inference has become one of the most widely used methods in latent variable modeling. In its basic form, variational inference employs a fully factorized variational distribution and minimizes its KL divergence to the posterior. As the minimization can only be carried out approximately, this approximation induces a bias. In this paper, we revisit perturbation theory as a powerful way of improving the variational approximation. Perturbation theory relies on a form of Taylor expansion of the log marginal likelihood, vaguely in terms of the log ratio of the true posterior and its variational approximation. While first order terms give the classical variational bound, higher-order terms yield corrections that tighten it. However, traditional perturbation theory does not provide a lower bound, making it inapt for stochastic optimization. In this paper, we present a similar yet alternative way of deriving corrections to the ELBO that resemble perturbation theory, but that result in a valid bound. We show in experiments on Gaussian Processes and Variational Autoencoders that the new bounds are more mass covering, and that the resulting posterior covariances are closer to the true posterior and lead to higher likelihoods on held-out data.

stat.ML↗

arXiv2019

Multi-Class Gaussian Process Classification Made Conjugate: Efficient Inference via Data Augmentation

We propose a new scalable multi-class Gaussian process classification approach building on a novel modified softmax likelihood function. The new likelihood has two benefits: it leads to well-calibrated uncertainty estimates and allows for an efficient latent variable augmentation. The augmented model has the advantage that it is conditionally conjugate leading to a fast variational inference method via block coordinate ascent updates. Previous approaches suffered from a trade-off between uncertainty calibration and speed. Our experiments show that our method leads to well-calibrated uncertainty estimates and competitive predictive performance while being up to two orders faster than the state of the art.

stat.ML↗

arXiv2019

Efficient Bayesian Inference of Sigmoidal Gaussian Cox Processes

We present an approximate Bayesian inference approach for estimating the intensity of an inhomogeneous Poisson process, where the intensity function is modelled using a Gaussian process (GP) prior via a sigmoid link function. Augmenting the model using a latent marked Poisson process and Pólya--Gamma random variables we obtain a representation of the likelihood which is conjugate to the GP prior. We estimate the posterior using a variational free--form mean field optimisation together with the framework of sparse GPs. Furthermore, as alternative approximation we suggest a sparse Laplace's method for the posterior, for which an efficient expectation--maximisation algorithm is derived to find the posterior's mode. Both algorithms compare well against exact inference obtained by a Markov Chain Monte Carlo sampler and standard variational Gauss approach solving the same model, while being one order of magnitude faster. Furthermore, the performance and speed of our method is competitive with that of another recently proposed Poisson process model based on a quadratic link function, while not being limited to GPs with squared exponential kernels and rectangular domains.

stat.ML↗

arXiv2019

Memory-free dynamics for the TAP equations of Ising models with arbitrary rotation invariant ensembles of random coupling matrices

We propose an iterative algorithm for solving the Thouless-Anderson-Palmer (TAP) equations of Ising models with arbitrary rotation invariant (random) coupling matrices. In the thermodynamic limit, we prove by means of the dynamical functional method that the proposed algorithm converges when the so-called de Almeida Thouless (AT) criterion is fulfilled. Moreover, we give exact analytical expressions for the rate of the convergence.

cond-mat.dis-nn↗

arXiv2019

Optimal decoding of dynamic stimuli encoded by heterogeneous populations of spiking neurons - a closed form approximation

Neural decoding may be formulated as dynamic state estimation (filtering) based on point process observations, a generally intractable problem. Numerical sampling techniques are often practically useful for the decoding of real neural data. However, they are less useful as theoretical tools for modeling and understanding sensory neural systems, since they lead to limited conceptual insight about optimal encoding and decoding strategies. We consider sensory neural populations characterized by a distribution over neuron parameters. We develop an analytically tractable Bayesian approximation to optimal filtering based on the observation of spiking activity, that greatly facilitates the analysis of optimal encoding in situations deviating from common assumptions of uniform coding. Continuous distributions are used to approximate large populations with few parameters, resulting in a filter whose complexity does not grow with the population size, and allowing optimization of population parameters rather than individual tuning functions. Numerical comparison with particle filtering demonstrates the quality of the approximation. The analytic framework leads to insights which are difficult to obtain from numerical algorithms, and is consistent with biological observations about the distribution of sensory cells' preferred stimuli.

q-bio.NC↗

arXiv2018

Efficient Gaussian Process Classification Using Polya-Gamma Data Augmentation

We propose a scalable stochastic variational approach to GP classification building on Polya-Gamma data augmentation and inducing points. Unlike former approaches, we obtain closed-form updates based on natural gradients that lead to efficient optimization. We evaluate the algorithm on real-world datasets containing up to 11 million data points and demonstrate that it is up to two orders of magnitude faster than the state-of-the-art while being competitive in terms of prediction performance.

stat.ML↗