Ergodic Inference: Accelerate Convergence by Optimisation

1 INTRODUCTION . Statistical inference methods in machine learning are dominated by two approaches : simulation and optimisation . Markov chain Monte Carlo ( MCMC ) is a well-known simulation-based method , which promises asymptotically unbiased samples from arbitrary distributions at the cost of expensive Markov simulations . Variational inference ( VI ) is a well-known method using optimisation , which fits a parametric approximation to the target distribution . VI is biased but offers a computationally efficient generation of approximate samples . There is a recent trend of hybrid methods of MCMC and VI to achieve a better balance between computational efficiency and bias . Hybrid methods often use MCMC or VI as an algorithmic component of the other . In particular , Salimans et al . ( 2015 ) proposed a promising modified VI method that reduces approximation bias by using MCMC transition kernels . Another technique reduces the computational complexity of MCMC by initialising the Markov simulation from a pretrained variational approximation ( Hoffman , 2017 ; Han et al. , 2017 ) . Levy et al . ( 2018 ) proposed to improve MCMC using flexible non-linear transformations given by neural networks and gradientbased auto-tuning strategies . In this work , we propose a novel hybrid method , called ergodic inference ( EI ) . EI improves over both MCMC and VI by tuning the hyper-parameters of a flexible finite-step MCMC chain so that its last state sampling distribution converges fast to a target distribution . EI optimises a tractable objective function which only requires to evaluate the logarithm of the unnormalized target density . Furthermore , unlike in traditional MCMC methods , the samples generated by EI from the last state of the MCMC chain are independent and have no correlations . EI offers an appealing option to balance computational complexity vs. bias on popular benchmarks in machine learning . Compared with previous hybrid methods , EI has following advantages : • EI ’ s hyperparameter tuning produces sampling distributions with lower approximation bias . • The bias is guaranteed to decrease as the length of the MCMC chain increases . • By stopping gradient computations , EI has less computational cost than related baselines . We also state some disadvantages of our method : • The initial state distribution in EI ’ s MCMC chain has to have higher entropy than the target . • The computational complexity per simulated sample of EI is in general higher than in VI . 2 BACKGROUND . 2.1 MONTE CARLO STATISTICAL INFERENCE . Monte Carlo ( MC ) statistical inference approximates expectations under a given distribution using simulated samples . Given a target distribution π , MC estimations of an expectation Eπ [ f ( x ) ] are defined as empirical average of the evaluation of f on samples from π . To generate samples from π , we assume that the unnormalized density function π∗ ( x ) can be easily computed . In a Bayesian setting we typically work with π∗ ( x|y ) given by the product of the prior p ( x ) and the likelihood p ( y|x ) , where y denotes observed variables and x denotes the model parameters specifying p ( y|x ) . 2.2 MARKOV CHAIN MONTE CARLO . Markov chain Monte Carlo ( MCMC ) casts inference as simulation of ergodic Markov chains that converge to the target π . The MCMC kernel M ( x′|x ) is characterised by the detailed balance ( DB ) property : π ( x ) M ( x′|x ) = π ( x′ ) M ( x|x′ ) . Given an unnormalised target density π∗ , an MCMC kernel can be constructed in three steps : first , sample an auxiliary random variable r from an auxiliary distribution qφ1 with parameters φ1 ; second , create a new candidate sample as ( x ′ , r′ ) = fφ2 ( xt−1 , r ) , where fφ2 is a deterministic function with parameters φ2 ; finally , accept the proposal as xt = x ′ with probability pMH = min { 0 , π∗ ( x′ ) qφ1 ( r′ ) / [ π∗ ( xt−1 ) qφ1 ( r ) ] } , otherwise duplicate the previous sample as xt = xt−1 . The last step is well known in the literature as the Metropolis-Hastings ( M-H ) correction step ( Robert & Casella , 2005 ) and it results in MCMC kernels that satisfy the DB condition . In the following , we denote the joint MCMC parameters ( φ1 , φ2 ) by φ . If fφ2 does not preserve volume , then it requires a Jacobian correction factor in the ratio in pMH . Hamiltonian Monte Carlo ( HMC ) is a successful MCMC method which has drawn great attention . A few recent works based on this method are Salimans et al . ( 2015 ) ; Hoffman ( 2017 ) ; Levy et al . ( 2018 ) . In HMC , the auxiliary distribution qφ1 is often chosen to be Gaussian with zero-mean and a constant diagonal covariance matrix specified by φ1 . The most common fφ2 in HMC is a numeric integrator called the leapfrog algorithm , which simulates Hamiltonian dynamics defined by log π∗ ( Neal , 2010 ) . The leapfrog integrator requires the gradient of log π∗ and a step size parameter given by φ2 . Given any initial state x0 , MCMC can generate asymptotically unbiased samples x1 : n. For this , MCMC iteratively simulates the next sample xt through the application of the MCMC transition kernel to the previous sample xt−1 . It is well known in the literature that MCMC is computationally demanding ( MacKay , 2002 ; Bishop , 2006 ) . In particular , it is often necessary to run sufficiently long burn-in MCMC simulations to reduce simulation bias . Another drawback of MCMC is sample correlation , which increases the variance of the MC estimator ( Neal , 2010 ) . To avoid strong sample correlation , the common practice in MCMC to tune hyper-parameters manually using sample quality metrics like effective sample size , ( Hoffman , 2017 ; Robert & Casella , 2005 ) , which has been developed into automated gradient-based tuning strategies in recent work ( Levy et al. , 2018 ) . 2.3 MC ESTIMATION USING VARIATIONAL INFERENCE . Variational inference ( VI ) is a popular alternative to MCMC for generating approximate samples from π . Unlike MCMC reducing sample bias by long burn-in simulation , VI casts the sample bias reduction as an optimisation problem , where a parametric approximate sampling distribution P is fit to the target π . In particular , VI optimises the evidence lower bound ( ELBO ) given by LELBO ( P ‖ π∗ ) = Ep [ log π∗ ( x ) ] +H ( P ) , ( 1 ) whereH ( P ) = −Ep [ log p ( x ) ] , also known as the entropy , must be tractable to compute . LELBO ( PT ‖ π∗ ) is a lower bound on the log normalising constant logZ = log ∫ π∗ ( x ) dx . This bound is tight when P = π. therefore , the approximation bias in VI can be defined as the gap between LELBO ( P ‖ π∗ ) and logZ , that is , ∆bias ( P ) = logZ − LELBO ( P ‖ π ) = DKL ( P ‖ π ) ≥ 0 , ( 2 ) where DKL ( P ‖ π ) denotes the Kullback-Leibler ( KL ) divergence . Variational approximations often belong to simple parametric families like the multivariate Gaussian distribution with diagonal covariance matrix . This results in computationally efficient algorithms for bias reduction and sample generation , but may also produce highly biased samples in cases of over-simplified approximation that ignores correlation . Designing variational approximation to achieve low bias under the constraint of tractable entropy and efficient sampling procedures is possible using flexible distributions parameterised by neural networks ( NNs ) ( Rezende & Mohamed , 2015 ; Kingma et al. , 2016 ) . However , how to design such NNs for VI is still a research challenge . 2.4 HYBRID METHODS AND VARIANTS OF MCMC AND VI . The balance between computational efficiency and bias is a challenge at the heart of all inference methods . MCMC represents a family of simulation-based methods that guarantee low-bias samples at cost of expensive simulations ; VI represents a family of optimisation-based methods that generate high-bias samples at a low computational cost . Many recent works seek a better balance between efficiency and bias by combining MCMC and VI . Salimans et al . ( 2015 ) proposed to reduce variational bias by optimising an ELBO specified in terms of the tractable joint density of short MCMC chains . The idea seems initially promising , but the proposed ELBO becomes looser and looser as the chain grows longer . Caterini et al . ( 2018 ) construct an alternative ELBO for HMC that still has problems since the auxiliary momentum variables are sampled only once at the beginning of the chain , which reduces the empirical performance of HMC . Inspired by contrastive divergence , Ruiz & Titsias ( 2019 ) proposed a novel variational objective function to optimise variational parameters by adding additional term that minimise the KL between a MCMC distribution and variational approximation to reduce variational bias . Hoffman ( 2017 ) and Han et al . ( 2017 ) proposed to replace expensive burn-in simulations in MCMC with samples from pre-trained variational approximations . This approach is effective at finding good initial proposal distributions . However , it does not offer a solution for tuning HMC parameters ( Hoffman , 2017 ) , which are critical for good empirical performance . Another line of research has focused on improving inference using flexible distributions , which are transformed from simple parametric distributions by non-linear non-volume preserving ( NVP ) functions . Levy et al . ( 2018 ) proposed to tune NVP parameterised MCMC w.r.t . a variant of the expected squared jumped distance ( ESJD ) loss proposed by Pasarica & Gelman ( 2010 ) . Song et al . ( 2017 ) proposed a similar auto-tuning for NVP parameterised MCMC using an adversarial loss . 3 ERGODIC INFERENCE . Ergodic inference ( EI ) is motivated by the well-known convergence of MCMC chains ( Robert & Casella , 2005 ) : MCMC chains converge in terms of the total variation ( TV ) distance between the marginal distribution of the MCMC chain and the target π . Inspired by the convergence property of MCMC chains , we define an ergodic approximation PT to π with T MCMC steps as following . Given a parametric distribution P0 with tractable density p0 ( x0 ; φ0 ) parameterlized by φ0 and an MCMC kernel M ( x′|x ; φ ) constructed using the unnormalised target density π∗ and with MCMC hyperparameter φ , an ergodic approximation of π is the marginal distribution of the final state of an T -step MCMC chain initialized from P0 : pT ( xT ; φ0 , φ ) = ∫ T∏ t=1 M ( xt|xt−1 ; φ ) p0 ( x0 ; φ0 ) dx0 : T−1 . ( 3 ) We call φ0 and φ the ergodic parameters of PT . Well known in MCMC literature like ( Robert & Casella , 2005 ; Murray & Salakhutdinov , 2008 ) , the ergodic approximation pT converges to π after every MCMC transition and with sufficiently long chain pT is guaranteed to be arbitrarily close to π with arbitrary φ and φ0 . It is important to clarify that ergodic approximation is different from the modified variational methods like ( Ruiz & Titsias , 2019 ) which only optimise the variational parameters φ0 , but the optimisation objective functions involve MCMC similation . In the following section , we show how EI can tune the ergodic parameters to minimise the bias of PT as an approximation to the target π with finite T .

Summary: The paper presents a new hybrid method to unify MCMC and VI. The key idea is to interpret a ﬁnite-length MCMC/HMC chain as a parametric procedure, whose parameters can be optimized via a VI-motivated objective. Specifically, the authors propose to modify the well-known ELBO (which is now non-trivial due to the intractable entropy) to form a new constrained and tractable objective. The presented techniques are tested on synthetic datasets and with the experiments of a VAE on MNIST.