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Bayesian neural networks (BNNs) are a rich and flexible class of models that have several advantages over standard feedforward networks, but are typically expensive to train on large-scale data. In this thesis, we explore the use of small-variance asymptotics-an approach to yielding fast algorithms from probabilistic models-on various Bayesian neural network models. We first demonstrate how small-variance asymptotics shows precise connections between standard neural networks and BNNs; for example, particular sampling algorithms for BNNs reduce to standard backpropagation in the small-variance limit. We then explore a more complex BNN where the number of hidden units is additionally treated as a random variable in the model. While standard sampling schemes would be too slow to be practical, our asymptotic approach yields a simple method for extending standard backpropagation to the case where the number of hidden units is not fixed. We show on several data sets that the resulting algorithm has benefits over backpropagation on networks with a fixed architecture. / 2019-01-02T00:00:00Z

In this thesis we concentrate on instances of the GLE which can be represented in a Markovian form in an extended phase space. We extend previous results on the geometric ergodicity of this class of GLEs using Lyapunov techniques, which allows us to conclude ergodicity for a large class of GLEs relevant to molecular dynamics applications. The main body of this thesis concerns the numerical discretisation of the GLE in the extended phase space representation. We generalise numerical discretisation schemes which have been previously proposed for the underdamped Langevin equation and which are based on a decomposition of the vector field into a Hamiltonian part and a linear SDE. Certain desirable properties regarding the accuracy of configurational averages of these schemes are inherited in the GLE context. We also rigorously prove geometric ergodicity on bounded domains by showing that a uniform minorisation condition and a uniform Lyapunov condition are satisfied for sufficiently small timestep size. We show that the discretisation schemes which we propose behave consistently in the white noise and overdamped limits, hence we provide a family of universal integrators for Langevin dynamics. Finally, we consider multiple-time stepping schemes making use of a decomposition of the fluctuation-dissipation term into a reversible and non-reversible part. These schemes are designed to efficiently integrate instances of the GLE whose Markovian representation involves a high number of auxiliary variables or a configuration dependent fluctuation-dissipation term. We also consider an application of dynamics based on the GLE in the context of large scale Bayesian inference as an extension of previously proposed adaptive thermostat methods. In these methods the gradient of the log posterior density is only evaluated on a subset (minibatch) of the whole dataset, which is randomly selected at each timestep. Incorporating a memory kernel in the adaptive thermostat formulation ensures that time-correlated gradient noise is dissipated in accordance with the fluctuation-dissipation theorem. This allows us to relax the requirement of using i.i.d. minibatches, and explore a variety of minibatch sampling approaches.

Orientador: Roberto E. Lagos Monaco / Banca: José Antonio Roversi / Banca: Bernardo Laks / Resumo: Neste trabalho estudamos o modelo do movimento Browniano de uma partícula carregada sob a ação de campos elétrico e magnético, externos e homogêneos, no formalismo de Langevin. Calculamos a energia cinética média através do teorema da flutuação-dissipação e obtivemos uma expressão para a temperatura efetiva das partículas Brownianas em função da temperatura do reservatório e dos campos externos. Esta temperatura efetiva mostrou-se sempre maior que a temperatura do reservatório, o que explica a expressão "portadores quentes". Estudamos essa temperatura efetiva no regime assintótico, ou seja, no estado estacionário atingido em tempos muito longos (quando comparado com o tempo de colisão) e a utilizamos para escrever as equações de transporte em semicondutores, denominadas equações de Shockley generalizadas sendo que incluem nesse caso também a ação do campo magnético. Uma aplicação direta e relevante foi a modelagem para o já conhecido efeito Gunn para portadores assumidos como Brownianos. A temperatura efetiva calculada por nós no regime transiente permitiu estudar também os efeitos do reservatório na relaxação da temperatura efetiva à temperatura terminal (de não equilíbrio e estacionária). Nossos resultados no que diz respeito ao efeito Gunn, embora seja o modelo mais simples de um portador Browniano, mostrou uma surpreendente concordância com resultados experimentais, sugerindo que modelos mais sofisticados devam incluir os elementos apresentados neste estudo / Abstract: We present a Brownian model for a charged particle in a field of forces, in particular, electric and magnetic external homogeneous fields, within the Langevin formalism. We compute the average kinetic energy via the fluctuation dissipation and obtain an expression for the Brownian particle's effective temperature. The latter is a function of the heat bath temperature and both external fields. This effective temperature is always greater than the heat bath temperature, therefore the expression "hot carriers". This effective temperature, in the asymptotic regime, the stationary state at long times (greater than the collision time), is used to write down the transport equations for semiconductors, namely the generalized Shockley equations, now incorporating the magnetic field effect. A direct and relevant application follows: a model for the well known Gunn effect, assuming a Brownian scheme. In the transient regime the computed effective temperature also allow us to probe some features of the heat bath, as the effective temperature relaxes to its terminal stationary value. As for our results in the Gunn effect model, the simplest of all in a Brownian scheme, we obtain a surprisingly good agreement with experimental data, suggesting that more involved models should include our minimal assumptions / Mestre

Tese (doutorado)—Universidade de Brasília, Departamento de Matemática, 2011. / Submitted by Albânia Cézar de Melo (albania@bce.unb.br) on 2011-11-25T12:40:14Z No. of bitstreams: 1 2011_FabianoFortunatoTeixeiraSantos.pdf: 351242 bytes, checksum: 25b73c3f71893e33ffed49df66bb44f2 (MD5) / Approved for entry into archive by Leila Fernandes (leilabiblio@yahoo.com.br) on 2011-12-14T11:53:08Z (GMT) No. of bitstreams: 1 2011_FabianoFortunatoTeixeiraSantos.pdf: 351242 bytes, checksum: 25b73c3f71893e33ffed49df66bb44f2 (MD5) / Made available in DSpace on 2011-12-14T11:53:08Z (GMT). No. of bitstreams: 1 2011_FabianoFortunatoTeixeiraSantos.pdf: 351242 bytes, checksum: 25b73c3f71893e33ffed49df66bb44f2 (MD5) / Para a equação de Langevin generalizada (ELG) governada por um ruído de cauda pesada, determinamos duas classes de soluções. Neste caso, ao contrário da equação de Langevin clássica, o cálculo de Itô não pode ser aplicado para obter soluções em média quadrática. Nossa abordagem baseia-se nas propriedades da transformada de Laplace para processos estáveis e na identificação da ELG como uma equação de Volterra estocástica. Para o índice de estabilidade 1 < α ≤ 2 mostramos que a conjectura de A. V. Medino [24], é realmente uma classe de soluções em probabilidade. Além disso, mostramos que algumas séries de Fourier-Stieltjes aleatórias convergem para a solução da ELG e discutimos o papel do índice de estabilidade no modo de convergência. ______________________________________________________________________________ ABSTRACT / For the Generalized Langevin Equation (GLE) driven by heavy-tailed noise we derive several classes of solutions. In this case, unlike the classical Langevin Equation case, the Ito’s calculus cannot be applied to obtain mean square solutions. Our approach relies on the properties of Laplace transforms for stable processes and on the identification of GLE as Volterra stochastic integro-differential equation. For stability index 1 < α ≤ 2 we show that Medino’s conjecture [24] is indeed a class of solutions in probability. Moreover, making use of random Fourier-Stieltjes series we exhibit approximating series that converge to the solution and discuss the role of stability index in the convergence mode.

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[pt] As flutuações de preços e de outras grandezas observáveis nos mercados financeiros apresentam comportamentos não triviais, tais como longas correlações temporais, não gaussianidade ou leis de escala, cuja origem não é ainda bem compreendida. Neste trabalho investigamos possíveis mecanismos determinísticos e estocásticos responsáveis pelas distribuições de probabilidade anômalas observadas para os índices de mercado e para os volumes de ações comercializadas. No primeiro caso, consideramos a expansão de Kramers-Moyal como ponto de partida para descrever a evolução das densidades de probabilidade. Para a modelagem dos volumes negociados, consideramos misturas estatísticas que surgem das flutuações em escalas longas dos parâmetros internos que descrevem a dinâmica em escalas mais curtas. Este estudo provê uma demonstração consistente, a partir de análise empírica de séries temporais reais, de como funções de densidade de probabilidade com caudas em lei de potência podem emergir através de mecanismos diversos, tais como processos estocásticos com flutuações aditivo-multiplicativas, ou como resultado de misturas estatísticas. / [en] The fluctuations of prices and other observables in financial markets have non-trivial behaviors, such as long temporal correlations, non-Gaussianity or scaling laws, whose origin is not well understood so far. In this work we have investigated possible deterministic and stochastic mechanisms responsible for the anomalous probability distributions observed for market indexes and volumes of traded shares. In the first case, we consider the Kramers-Moyal expansion as a starting point to describe the evolution of probability densities. For the modelling of trading volumes, we consider the mixed statistics that emerges from the long-scale fluctuations of inner parameters that describe the dynamics on shorter scales. This study provides a consistent demonstration, from empirical analysis of real time series, on how probability density functions with power laws tails may emerge through various mechanisms, such as stochastic processes with additive-multiplicative fluctuations or as a result of mixed statistics.

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