DOCSLIB.ORG

Gaussian Process Volatility Model

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Gaussian Process Volatility Model Gaussian Process Volatility Model Yue Wu [email protected] Jose´ Miguel Hernandez´ Lobato [email protected] Zoubin Ghahramani [email protected] University of Cambridge, Department of Engineering, Cambridge CB2 1PZ, UK Abstract 1986). The accurate prediction of time-changing vari- GARCH has further inspired a host of variants and ex- ances is an important task in the modeling of fi- tensions A review of many of these models can be found nancial data. Standard econometric models are in Hentschel(1995). Most of these GARCH variants at- often limited as they assume rigid functional re- tempt to address one or both limitations of GARCH: a) lationships for the variances. Moreover, function the assumption of a linear dependency between the cur- parameters are usually learned using maximum rent volatility and past volatilities, and b) the assumption likelihood, which can lead to overfitting. To ad- that positive and negative returns have symmetric effects dress these problems we introduce a novel model on volatility. Asymmetric effects are often observed, as for time-changing variances using Gaussian Pro- large negative returns send measures of volatility soaring, cesses. A Gaussian Process (GP) defines a distri- while large positive returns do not (Bekaert & Wu, 2000; bution over functions, which allows us to cap- Campbell & Hentschel, 1992). ture highly flexible functional relationships for the variances. In addition, we develop an on- Most solutions proposed in these GARCH variants involve: line algorithm to perform inference. The algo- a) introducing nonlinear functional relationships for the rithm has two main advantages. First, it takes a volatility, and b) adding asymmetric terms in the functional Bayesian approach, thereby avoiding overfitting. relationships. However, the GARCH variants do not funda- Second, it is much quicker than current offline in- mentally address the problem that the functional relation- ference procedures. Finally, our new model was ship of the volatility is unknown. In addition, these variants evaluated on financial data and showed signifi- can have a high number of parameters, which may lead to cant improvement in predictive performance over overfitting when learned using maximum likelihood. current standard models. More recently, volatility modeling has received attention within the machine learning community, with the develop- ment of copula processes (Wilson & Ghahramani, 2010) 1. Introduction and heteroscedastic Gaussian processes (Lazaro-Gredilla´ & Titsias, 2011). These models leverage the flexibility Time series of financial returns often exhibit heteroscedas- of Gaussian Processes (Rasmussen & Williams, 2006) to ticity, that is the standard deviation or volatility of the re- model the unknown relationship in the variances. How- arXiv:1402.3085v1 [stat.ME] 13 Feb 2014 turns is time-dependent. In particular, large returns (either ever, these models do not address the asymmetric effects of positive or negative) are often followed by returns that are positive and negative returns on volatility. also large in size. The result is that financial time series frequently display periods of low and high volatility. This In this paper we introduce a new non-parametric volatility phenomenon is known as volatility clustering (Cont, 2001). model, called the Gaussian Process Volatility Model (GP- Several univariate models have been proposed for capturing Vol). This new model is more flexible, as it is not lim- this property. The best known are the Autoregressive Con- ited by a fixed functional form. Instead a prior distribution ditional Heteroscedasticity model (ARCH) (Engle, 1982) is placed on possible functions using GPs, and the func- and its extension, the Generalised Autoregressive Con- tional relationship is learned from the data. Furthermore, ditional Heteroscedasticity model (GARCH) (Bollerslev, GP-Vol explicitly models the asymmetric effects on volatil- ity from positive and negative returns. Our new volatility st Proceedings of the 31 International Conference on Machine model is evaluated in a series of experiments on real fi- Learning, Beijing, China, 2014. JMLR: W&CP volume 32. Copy- right 2014 by the author(s). nancial returns, comparing it against popular econometric Gaussian Process Volatility Model models, namely GARCH, EGARCH (Nelson, 1991) and g(xt) = θxt + λ jxtj : GJR-GARCH (Glosten et al., 1993). Overall, we found our proposed model has the best predictive performance. In ad- Asymmetry in the effects of positive and negative returns is dition, the functional relationship learned by our model is introduced through the function g(xt). Then if the coeffi- highly intuitive and automatically discovers the nonlinear cient θ is negative, negative returns will increase volatility. and asymmetric features that previous models attempt to Another popular GARCH extension with asymmetric effect capture. of returns is GJR-GARCH (Glosten et al., 1993): The second main contribution of the paper is the develop- q p r 2 X 2 X 2 X 2 ment of an online algorithm for learning GP-Vol. GP-Vol is σt = α0+ αjxt−j + βiσt−i + γkxt−kIt−k ; an instance of a Gaussian Process State Space Model (GP- j=1 i=1 k=1 SSM). Most previous work on GP-SSMs (Ko & Fox, 2009; (4) Deisenroth et al., 2009; Deisenroth & Mohamed, 2012) ( 0 ; if xt−k ≥ 0 have focused on developing approximation methods for It−k = : 1 ; x < 0 filtering and smoothing the hidden states in GP-SSM, as- if t−k suming known GP transition dynamics. Only very recently 2 The asymmetric effect is captured by γkx It−k, which has Frigola et al.(2013) addressed the problem of learning t−k is nonzero if xt−k < 0. both the hidden states and the transition dynamics by us- ing Particle Gibbs with ancestor sampling (PGAS) (Lind- sten et al., 2012). In this paper, we introduce a new online 3. Gaussian Process State Space Models algorithm for performing inference on GP-SSMs. Our al- GARCH, EGARCH and GJR-GARCH can be all repre- gorithm has similar predictive performance as PGAS on fi- sented as General State-Space or Hidden Markov models nancial datasets, but is much quicker as inference is online. (HMM) (Baum & Petrie, 1966; Doucet et al., 2001), with the unobserved dynamic variances being the hidden states. 2. Review of GARCH and GARCH variants Transition functions for the hidden states are fixed and as- sumed to be linear in these models. The linear assumption The standard heteroscedastic variance model for financial limits the flexibility of these models. data is GARCH. GARCH assumes a Gaussian observation model (1) and a linear transition function so that the time- More generally, a non-parametric approach can be taken 2 varying variance σt is linearly dependent on p previous where a Gaussian Process prior is placed on the transition variance values and q previous squared time series values: function, so that the functional form can be learned from data. This Gaussian Process state space model (GP-SSM) 2 xt ∼ N (0; σt ) ; (1) is a generalization of HMM. The two class of models dif- q p fer in two main ways. First, in HMM the transition func- 2 X 2 X 2 σt = α0 + αjxt−j + βiσt−i ; (2) tion has fixed functional form, while in GP-SSM it is rep- j=1 i=1 resented by a GP. Second, in GP-SSM the states do not have Markovian structure once the transition function is where xt are the values of the return time series being mod- eled. The GARCH(p,q) generative model is flexible and marginalized out, see Section5 for details. can produce a variety of clustering behavior of high and However, the flexibility of GP-SSMs comes at a cost. low volatility periods for different settings of the model co- Specifically, inference in GP-SSMs is complicated. Most efficients, α1; : : : ; αq and β1; : : : ; βp. previous work on GP-SSMs (Ko & Fox, 2009; Deisenroth While GARCH is flexible, it has several limitations. First, et al., 2009; Deisenroth & Mohamed, 2012) have focused 2 2 on developing approximation methods for filtering and a linear relationship between σt−p:t−1 and σt is assumed. Second, the effect of positive and negative returns is the smoothing the hidden states in GP-SSM assuming known 2 GP dynamics. A few papers considered learning the GP same due to the quadratic term xt−j. However, it is of- ten observed that large negative returns lead to sharp rises dynamics and the states, but for special cases of GP-SSMs. in volatility, while positive returns do not (Bekaert & Wu, For example, Turner et al.(2010) applied EM to obtain 2000; Campbell & Hentschel, 1992). maximum likelihood estimates for parametric systems that can be represented by GPs. Recently, Frigola et al.(2013) A more flexible and often cited GARCH extension is Ex- learned both the hidden states and the GP dynamics using ponential GARCH (EGARCH) (Nelson, 1991): PGAS (Lindsten et al., 2012). Unfortunately PGAS is a q p full MCMC inference method, and can be expensive com- 2 X X 2 log(σt ) = α0 + αjg(xt−j) + βi log(σt−i) ; (3) putationally. In this paper, we present an online Bayesian j=1 i=1 inference algorithm for learning the hidden states v1:T , the Gaussian Process Volatility Model unknown function f, and any hyper-parameters θ of the tion would be of the form: model. Our algorithm has similar predictive performance as PGAS, but is much quicker. vt = f(vt−1; vt−2; :::; vt−p; xt−1; xt−2; :::; xt−q) + : (8) 4. Gaussian Process Volatility Model 5. Bayesian Inference for GP-Vol We introduce now our new nonparametric volatility model, In the standard GP regression setting the inputs and tar- which is an instance of GP-SSM. We call our new model gets are observed, then the function f can be learned using the Gaussian Process Volatility Model (GP-Vol): exact inference. However, this is not the case in GP-Vol, 2 xt ∼ N (0; σt ) ; (5) where some inputs and all targets vt are unknown.
Load more

Recommended publications
  • Scalable Nonparametric Bayesian Inference on Point Processes with Gaussian Processes
    Scalable Nonparametric Bayesian Inference on Point Processes with Gaussian Processes Yves-Laurent Kom Samo [email protected] Stephen Roberts [email protected] Deparment of Engineering Science and Oxford-Man Institute, University of Oxford Abstract 2. Related Work In this paper we propose an efficient, scalable Non-parametric inference on point processes has been ex- non-parametric Gaussian process model for in- tensively studied in the literature. Rathbum & Cressie ference on Poisson point processes. Our model (1994) and Moeller et al. (1998) used a finite-dimensional does not resort to gridding the domain or to intro- piecewise constant log-Gaussian for the intensity function. ducing latent thinning points. Unlike competing Such approximations are limited in that the choice of the 3 models that scale as O(n ) over n data points, grid on which to represent the intensity function is arbitrary 2 our model has a complexity O(nk ) where k and one has to trade-off precision with computational com- n. We propose a MCMC sampler and show that plexity and numerical accuracy, with the complexity being the model obtained is faster, more accurate and cubic in the precision and exponential in the dimension of generates less correlated samples than competing the input space. Kottas (2006) and Kottas & Sanso (2007) approaches on both synthetic and real-life data. used a Dirichlet process mixture of Beta distributions as Finally, we show that our model easily handles prior for the normalised intensity function of a Poisson data sizes not considered thus far by alternate ap- process. Cunningham et al.
    [Show full text]
  • Deep Neural Networks As Gaussian Processes
    Published as a conference paper at ICLR 2018 DEEP NEURAL NETWORKS AS GAUSSIAN PROCESSES Jaehoon Lee∗y, Yasaman Bahri∗y, Roman Novak , Samuel S. Schoenholz, Jeffrey Pennington, Jascha Sohl-Dickstein Google Brain fjaehlee, yasamanb, romann, schsam, jpennin, [email protected] ABSTRACT It has long been known that a single-layer fully-connected neural network with an i.i.d. prior over its parameters is equivalent to a Gaussian process (GP), in the limit of infinite network width. This correspondence enables exact Bayesian inference for infinite width neural networks on regression tasks by means of evaluating the corresponding GP. Recently, kernel functions which mimic multi-layer random neural networks have been developed, but only outside of a Bayesian framework. As such, previous work has not identified that these kernels can be used as co- variance functions for GPs and allow fully Bayesian prediction with a deep neural network. In this work, we derive the exact equivalence between infinitely wide deep net- works and GPs. We further develop a computationally efficient pipeline to com- pute the covariance function for these GPs. We then use the resulting GPs to per- form Bayesian inference for wide deep neural networks on MNIST and CIFAR- 10. We observe that trained neural network accuracy approaches that of the corre- sponding GP with increasing layer width, and that the GP uncertainty is strongly correlated with trained network prediction error. We further find that test perfor- mance increases as finite-width trained networks are made wider and more similar to a GP, and thus that GP predictions typically outperform those of finite-width networks.
    [Show full text]
  • Gaussian Process Dynamical Models for Human Motion
    IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 30, NO. 2, FEBRUARY 2008 283 Gaussian Process Dynamical Models for Human Motion Jack M. Wang, David J. Fleet, Senior Member, IEEE, and Aaron Hertzmann, Member, IEEE Abstract—We introduce Gaussian process dynamical models (GPDMs) for nonlinear time series analysis, with applications to learning models of human pose and motion from high-dimensional motion capture data. A GPDM is a latent variable model. It comprises a low- dimensional latent space with associated dynamics, as well as a map from the latent space to an observation space. We marginalize out the model parameters in closed form by using Gaussian process priors for both the dynamical and the observation mappings. This results in a nonparametric model for dynamical systems that accounts for uncertainty in the model. We demonstrate the approach and compare four learning algorithms on human motion capture data, in which each pose is 50-dimensional. Despite the use of small data sets, the GPDM learns an effective representation of the nonlinear dynamics in these spaces. Index Terms—Machine learning, motion, tracking, animation, stochastic processes, time series analysis. Ç 1INTRODUCTION OOD statistical models for human motion are important models such as hidden Markov model (HMM) and linear Gfor many applications in vision and graphics, notably dynamical systems (LDS) are efficient and easily learned visual tracking, activity recognition, and computer anima- but limited in their expressiveness for complex motions. tion. It is well known in computer vision that the estimation More expressive models such as switching linear dynamical of 3D human motion from a monocular video sequence is systems (SLDS) and nonlinear dynamical systems (NLDS), highly ambiguous.
    [Show full text]
  • Modelling Multi-Object Activity by Gaussian Processes 1
    LOY et al.: MODELLING MULTI-OBJECT ACTIVITY BY GAUSSIAN PROCESSES 1 Modelling Multi-object Activity by Gaussian Processes Chen Change Loy School of EECS [email protected] Queen Mary University of London Tao Xiang E1 4NS London, UK [email protected] Shaogang Gong [email protected] Abstract We present a new approach for activity modelling and anomaly detection based on non-parametric Gaussian Process (GP) models. Specifically, GP regression models are formulated to learn non-linear relationships between multi-object activity patterns ob- served from semantically decomposed regions in complex scenes. Predictive distribu- tions are inferred from the regression models to compare with the actual observations for real-time anomaly detection. The use of a flexible, non-parametric model alleviates the difficult problem of selecting appropriate model complexity encountered in parametric models such as Dynamic Bayesian Networks (DBNs). Crucially, our GP models need fewer parameters; they are thus less likely to overfit given sparse data. In addition, our approach is robust to the inevitable noise in activity representation as noise is modelled explicitly in the GP models. Experimental results on a public traffic scene show that our models outperform DBNs in terms of anomaly sensitivity, noise robustness, and flexibil- ity in modelling complex activity. 1 Introduction Activity modelling and automatic anomaly detection in video have received increasing at- tention due to the recent large-scale deployments of surveillance cameras. These tasks are non-trivial because complex activity patterns in a busy public space involve multiple objects interacting with each other over space and time, whilst anomalies are often rare, ambigu- ous and can be easily confused with noise caused by low image quality, unstable lighting condition and occlusion.
    [Show full text]
  • Financial Time Series Volatility Analysis Using Gaussian Process State-Space Models
    Financial Time Series Volatility Analysis Using Gaussian Process State-Space Models by Jianan Han Bachelor of Engineering, Hebei Normal University, China, 2010 A thesis presented to Ryerson University in partial fulfillment of the requirements for the degree of Master of Applied Science in the Program of Electrical and Computer Engineering Toronto, Ontario, Canada, 2015 c Jianan Han 2015 AUTHOR'S DECLARATION FOR ELECTRONIC SUBMISSION OF A THESIS I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I authorize Ryerson University to lend this thesis to other institutions or individuals for the purpose of scholarly research. I further authorize Ryerson University to reproduce this thesis by photocopying or by other means, in total or in part, at the request of other institutions or individuals for the purpose of scholarly research. I understand that my dissertation may be made electronically available to the public. ii Financial Time Series Volatility Analysis Using Gaussian Process State-Space Models Master of Applied Science 2015 Jianan Han Electrical and Computer Engineering Ryerson University Abstract In this thesis, we propose a novel nonparametric modeling framework for financial time series data analysis, and we apply the framework to the problem of time varying volatility modeling. Existing parametric models have a rigid transition function form and they often have over-fitting problems when model parameters are estimated using maximum likelihood methods. These drawbacks effect the models' forecast performance. To solve this problem, we take Bayesian nonparametric modeling approach.
    [Show full text]
  • Gaussian-Random-Process.Pdf
    The Gaussian Random Process Perhaps the most important continuous state-space random process in communications systems in the Gaussian random process, which, we shall see is very similar to, and shares many properties with the jointly Gaussian random variable that we studied previously (see lecture notes and chapter-4). X(t); t 2 T is a Gaussian r.p., if, for any positive integer n, any choice of coefficients ak; 1 k n; and any choice ≤ ≤ of sample time tk ; 1 k n; the random variable given by the following weighted sum of random variables is Gaussian:2 T ≤ ≤ X(t) = a1X(t1) + a2X(t2) + ::: + anX(tn) using vector notation we can express this as follows: X(t) = [X(t1);X(t2);:::;X(tn)] which, according to our study of jointly Gaussian r.v.s is an n-dimensional Gaussian r.v.. Hence, its pdf is known since we know the pdf of a jointly Gaussian random variable. For the random process, however, there is also the nasty little parameter t to worry about!! The best way to see the connection to the Gaussian random variable and understand the pdf of a random process is by example: Example: Determining the Distribution of a Gaussian Process Consider a continuous time random variable X(t); t with continuous state-space (in this case amplitude) defined by the following expression: 2 R X(t) = Y1 + tY2 where, Y1 and Y2 are independent Gaussian distributed random variables with zero mean and variance 2 2 σ : Y1;Y2 N(0; σ ). The problem is to find the one and two dimensional probability density functions of the random! process X(t): The one-dimensional distribution of a random process, also known as the univariate distribution is denoted by the notation FX;1(u; t) and defined as: P r X(t) u .
    [Show full text]
  • Gaussian Markov Processes
    C. E. Rasmussen & C. K. I. Williams, Gaussian Processes for Machine Learning, the MIT Press, 2006, ISBN 026218253X. c 2006 Massachusetts Institute of Technology. www.GaussianProcess.org/gpml Appendix B Gaussian Markov Processes Particularly when the index set for a stochastic process is one-dimensional such as the real line or its discretization onto the integer lattice, it is very interesting to investigate the properties of Gaussian Markov processes (GMPs). In this Appendix we use X(t) to define a stochastic process with continuous time pa- rameter t. In the discrete time case the process is denoted ...,X−1,X0,X1,... etc. We assume that the process has zero mean and is, unless otherwise stated, stationary. A discrete-time autoregressive (AR) process of order p can be written as AR process p X Xt = akXt−k + b0Zt, (B.1) k=1 where Zt ∼ N (0, 1) and all Zt’s are i.i.d. Notice the order-p Markov property that given the history Xt−1,Xt−2,..., Xt depends only on the previous p X’s. This relationship can be conveniently expressed as a graphical model; part of an AR(2) process is illustrated in Figure B.1. The name autoregressive stems from the fact that Xt is predicted from the p previous X’s through a regression equation. If one stores the current X and the p − 1 previous values as a state vector, then the AR(p) scalar process can be written equivalently as a vector AR(1) process. Figure B.1: Graphical model illustrating an AR(2) process.
    [Show full text]
  • FULLY BAYESIAN FIELD SLAM USING GAUSSIAN MARKOV RANDOM FIELDS Huan N
    Asian Journal of Control, Vol. 18, No. 5, pp. 1–14, September 2016 Published online in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.1002/asjc.1237 FULLY BAYESIAN FIELD SLAM USING GAUSSIAN MARKOV RANDOM FIELDS Huan N. Do, Mahdi Jadaliha, Mehmet Temel, and Jongeun Choi ABSTRACT This paper presents a fully Bayesian way to solve the simultaneous localization and spatial prediction problem using a Gaussian Markov random field (GMRF) model. The objective is to simultaneously localize robotic sensors and predict a spatial field of interest using sequentially collected noisy observations by robotic sensors. The set of observations consists of the observed noisy positions of robotic sensing vehicles and noisy measurements of a spatial field. To be flexible, the spatial field of interest is modeled by a GMRF with uncertain hyperparameters. We derive an approximate Bayesian solution to the problem of computing the predictive inferences of the GMRF and the localization, taking into account observations, uncertain hyperparameters, measurement noise, kinematics of robotic sensors, and uncertain localization. The effectiveness of the proposed algorithm is illustrated by simulation results as well as by experiment results. The experiment results successfully show the flexibility and adaptability of our fully Bayesian approach ina data-driven fashion. Key Words: Vision-based localization, spatial modeling, simultaneous localization and mapping (SLAM), Gaussian process regression, Gaussian Markov random field. I. INTRODUCTION However, there are a number of inapplicable situa- tions. For example, underwater autonomous gliders for Simultaneous localization and mapping (SLAM) ocean sampling cannot find usual geometrical models addresses the problem of a robot exploring an unknown from measurements of environmental variables such as environment under localization uncertainty [1].
    [Show full text]
  • Mean Field Methods for Classification with Gaussian Processes
    Mean field methods for classification with Gaussian processes Manfred Opper Neural Computing Research Group Division of Electronic Engineering and Computer Science Aston University Birmingham B4 7ET, UK. opperm~aston.ac.uk Ole Winther Theoretical Physics II, Lund University, S6lvegatan 14 A S-223 62 Lund, Sweden CONNECT, The Niels Bohr Institute, University of Copenhagen Blegdamsvej 17, 2100 Copenhagen 0, Denmark winther~thep.lu.se Abstract We discuss the application of TAP mean field methods known from the Statistical Mechanics of disordered systems to Bayesian classifi­ cation models with Gaussian processes. In contrast to previous ap­ proaches, no knowledge about the distribution of inputs is needed. Simulation results for the Sonar data set are given. 1 Modeling with Gaussian Processes Bayesian models which are based on Gaussian prior distributions on function spaces are promising non-parametric statistical tools. They have been recently introduced into the Neural Computation community (Neal 1996, Williams & Rasmussen 1996, Mackay 1997). To give their basic definition, we assume that the likelihood of the output or target variable T for a given input s E RN can be written in the form p(Tlh(s)) where h : RN --+ R is a priori assumed to be a Gaussian random field. If we assume fields with zero prior mean, the statistics of h is entirely defined by the second order correlations C(s, S') == E[h(s)h(S')], where E denotes expectations 310 M Opper and 0. Winther with respect to the prior. Interesting examples are C(s, s') (1) C(s, s') (2) The choice (1) can be motivated as a limit of a two-layered neural network with infinitely many hidden units with factorizable input-hidden weight priors (Williams 1997).
    [Show full text]
  • When Gaussian Process Meets Big Data
    IEEE 1 When Gaussian Process Meets Big Data: A Review of Scalable GPs Haitao Liu, Yew-Soon Ong, Fellow, IEEE, Xiaobo Shen, and Jianfei Cai, Senior Member, IEEE Abstract—The vast quantity of information brought by big GP paradigms. It is worth noting that this review mainly data as well as the evolving computer hardware encourages suc- focuses on scalable GPs for large-scale regression but not on cess stories in the machine learning community. In the meanwhile, all forms of GPs or other machine learning models. it poses challenges for the Gaussian process (GP) regression, a d n well-known non-parametric and interpretable Bayesian model, Given n training points X = xi R i=1 and their { ∈ n } which suffers from cubic complexity to data size. To improve observations y = yi = y(xi) R , GP seeks to { ∈ }i=1 the scalability while retaining desirable prediction quality, a infer the latent function f : Rd R in the function space variety of scalable GPs have been presented. But they have not (m(x), k(x, x′)) defined by the7→ mean m(.) and the kernel yet been comprehensively reviewed and analyzed in order to be kGP(., .). The most prominent weakness of standard GP is that well understood by both academia and industry. The review of it suffers from a cubic time complexity (n3) because of scalable GPs in the GP community is timely and important due O to the explosion of data size. To this end, this paper is devoted the inversion and determinant of the n n kernel matrix × to the review on state-of-the-art scalable GPs involving two main Knn = k(X, X).
    [Show full text]
  • Stochastic Optimal Control Using Gaussian Process Regression Over Probability Distributions
    Stochastic Optimal Control Using Gaussian Process Regression over Probability Distributions Jana Mayer1, Maxim Dolgov2, Tobias Stickling1, Selim Özgen1, Florian Rosenthal1, and Uwe D. Hanebeck1 Abstract— In this paper, we address optimal control of non- In this paper, we will refer to the term value function without linear stochastic systems under motion and measurement uncer- loss of generality. tainty with finite control input and measurement spaces. Such A solution to a stochastic control problem can be found problems can be formalized as partially-observable Markov decision processes where the goal is to find policies via dy- using the dynamic programming (DP) algorithm that is based namic programming that map the information available to the on Bellman’s principle of optimality [1]. This algorithm relies controller to control inputs while optimizing a performance on two key principles. First, it assumes the existence of criterion. However, they suffer from intractability in scenarios sufficient statistics in form of a probability distribution that with continuous state spaces and partial observability which encompasses all the information up to the moment of policy makes approximations necessary. Point-based value iteration methods are a class of global approximate methods that regress evaluation. And second, it computes the policy by starting at the value function given the values at a set of reference points. the end of the planning horizon and going backwards in time In this paper, we present a novel point-based value iteration to the current time step and at every time step choosing a approach for continuous state spaces that uses Gaussian pro- control input that minimize the cumulative costs maintained cesses defined over probability distribution for the regression.
    [Show full text]
  • The Variational Gaussian Process
    Published as a conference paper at ICLR 2016 THE VARIATIONAL GAUSSIAN PROCESS Dustin Tran Harvard University [email protected] Rajesh Ranganath Princeton University [email protected] David M. Blei Columbia University [email protected] ABSTRACT Variational inference is a powerful tool for approximate inference, and it has been recently applied for representation learning with deep generative models. We de- velop the variational Gaussian process (VGP), a Bayesian nonparametric varia- tional family, which adapts its shape to match complex posterior distributions. The VGP generates approximate posterior samples by generating latent inputs and warping them through random non-linear mappings; the distribution over random mappings is learned during inference, enabling the transformed outputs to adapt to varying complexity. We prove a universal approximation theorem for the VGP, demonstrating its representative power for learning any model. For inference we present a variational objective inspired by auto-encoders and perform black box inference over a wide class of models. The VGP achieves new state-of-the-art re- sults for unsupervised learning, inferring models such as the deep latent Gaussian model and the recently proposed DRAW. 1 INTRODUCTION Variational inference is a powerful tool for approximate posterior inference. The idea is to posit a family of distributions over the latent variables and then find the member of that family closest to the posterior. Originally developed in the 1990s (Hinton & Van Camp, 1993; Waterhouse et al., 1996; Jordan et al., 1999), variational inference has enjoyed renewed interest around developing scalable optimization for large datasets (Hoffman et al., 2013), deriving generic strategies for easily fitting many models (Ranganath et al., 2014), and applying neural networks as a flexible parametric family of approximations (Kingma & Welling, 2014; Rezende et al., 2014).
    [Show full text]