summaries: The authors introduce a variational inference method that leverages Annealed Importance Sampling (AIS) and Langevin/HMC MCMC. In particular HMC dynamics are used to target a sequence of intermediate bridging densities in order to generate approximate samples from the target posterior distribution. Importantly, since the method is uncorrected (i.e. does without a accept/reject step) the resulting lower bound is fully differentiable. This makes various hard-to-set parameters (e.g. temperatures for bridging densities) learnable. In experiments the authors demonstrate that their method, Uncorrected Hamiltonian Annealing (UHA), leads to tight lower bounds on the log evidence.  The paper proposed to use the uncorrected HMC kernel in an AIS-type variational scheme. This enables optimizing the ELBO via reparameterization gradients for many algorithm tuning parameters, including step size, momentum covariance, and annealing schedule. The method is shown to outperform competitors in most cases.  The paper proposes UHA, an original and clever modification of Hamiltonian AIS. The idea is to change the AIS ratio such that it becomes tunable, but can still be computed in closed form (ratios of momentum distributions). This is achieved by dropping the accept-reject step in the HMC update such that a single UHA step only involves (re)sampling momenta and leapfrog integration. This modification allows the authors to tune parameters such as the proposal distribution, the step size and damping coefficients (used in momentum resampling). Many interesting ideas for future extensions are proposed. Experiments on various models demonstrate that UHA obtains a tighter ELBO than other combinations of VI and HMC.   The paper propose to use uncorrected HMC kernels (i.e. HMC kernels without accept-reject steps) in AIS to make the estimate of log normalisation constant differentiable w.r.t. to various parameters (e.g. step-size for numerical integration). The author(s) name(s) the method UHA. The promise is that one can then using reparameterisation gradient to tune these parameters. Through experiments on inference tasks, the benefit of tuning parameters in UHA is demonstrated against other methods that use HMC in AIS. For experiments on VAE training, UHA with tuned parameters has shown better test ELBO than IW. This paper studies the annealed important sampling (AIS) using the Hamiltonian dynamics kernels in each interpolant. To circumvent the problem that the Hamiltonian dynamics involves the non-differentiable operations, this paper proposes to discard the accept-reject operations in the kernel, and derives the resulting density ratios in a simple form related to the momentum variables. The resulting approach, named Uncorrected Hamiltonian Annealing (UHA), is a fully differentiable AIS method. This paper validates the effectiveness of UHA through extensive experiments and highlights the importance of differentiability. 
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sections.text: Given an unnormalized target distribution we want to obtain approximate samples from it and a tight lower bound on its (log) normalization constant logZ. Annealed Importance Sampling (AIS) with Hamiltonian MCMC is a powerful method that can be used to do this. Its main drawback is that it uses non-differentiable transition kernels, which makes tuning its many parameters hard. We propose a framework to use an AIS-like procedure with Uncorrected Hamiltonian MCMC, called Uncorrected Hamiltonian Annealing. Our method leads to tight and differentiable lower bounds on logZ. We show empirically that our method yields better performances than other competing approaches, and that the ability to tune its parameters using reparameterization gradients may lead to large performance improvements.
sections.heading: 1 Introduction
sections.text: Variational Inference (VI) [4, 41, 45] is a method to do approximate inference on a target distribution p(z) = p̄(z)/Z that is only known up to the normalization constant Z. The basic insights are, first, that the evidence lower bound (ELBO) Eq(z)[log p̄(z)− log q(z)] lower-bounds logZ and, second, that maximizing the ELBO is equivalent to minimizing the KL-divergence from q to p. The simplest VI method chooses a parameterized family for q and optimizes its parameters to maximize the ELBO.
  A recent direction involves combining VI with Markov chain Monte Carlo (MCMC) [34, 43]. These methods can be seen as an instance of the auxiliary VI framework [2] – they create an augmented variational distribution that represents all intermediate random variables generated during the MCMC procedure. An augmented target distribution that attempts to capture the inverse MCMC dynamics is optimized jointly with this variational distribution. However, it has been observed that capturing inverse dynamics is challenging [43, §5.4] (further discussion in Section 4).
  Annealed Importance Sampling (AIS) [22, 27] is a powerful technique used to build augmented distributions without the need of learning inverse dynamics. While it was originally proposed to estimate expectations using importance sampling, it can be easily used to build lower bounds on normalization constants of intractable densities [18, 44]. AIS creates a sequence of densities that bridge from a tractable initial approximation q to the target p̄. Then, the augmented variational distribution is given by a sequence of MCMC kernels targeting each bridging density, while the augmented target uses the reversals of those kernels. It turns out that the ratio of these augmented distributions can be computed using only evaluations of the bridging densities. Combining Hamiltonian MCMC kernels with AIS has been observed to produce strong lower bounds [35, 44].
  However, these bounds are sensitive to numerous parameters, such as the initial distribution, bridging schedule, and parameters of the MCMC kernels. It would be desirable to optimize these parameters
  35th Conference on Neural Information Processing Systems (NeurIPS 2021).
  to tighten the bound. Unfortunately, the presence of Metropolis-Hastings acceptance steps means that the the final estimator is non-differentiable, and thus reparameterization gradients cannot be used.
  In this work, we propose Uncorrected Hamiltonian Annealing (UHA), a differentiable alternative to Hamiltonian AIS. We define an augmented variational distribution using Hamiltonian MCMC kernels, but dropping the accept-reject steps. This is motivated by the fact that Hamiltonian dynamics sometimes have high acceptance rates. Since these uncorrected MCMC kernels do not exactly hold the bridging densities invariant, an augmented target distribution cannot be defined in terms of reversals. Instead, we define our augmented target by deriving an algorithm for the exact reversal of the original (corrected) MCMC kernel and dropping the accept-reject step. Surprisingly, this yields a very simple expression for the resulting lower bound.
  We use reparameterization gradients to tune various parameters involved in the lower bound produced by UHA, including the initial approximation q, parameters of the uncorrected MCMC kernel, and the bridging densities. Experimentally, tuning all these leads to large gains. For example, in several inference tasks we observe that tuning UHA with K = 64 bridging densities gives better results than traditional Hamiltonian AIS with K = 512.
  Finally, we use UHA to train VAEs [24, 31]. In this case we observe that using UHA leads to higher ELBOs. In addition, we observe that increasing the number of bridging densities with UHA consistently leads to better results, and that for a large enough number of bridging densities the variational gap (difference between ELBO and true log-likelihood) becomes small, and models with higher log-likelihood are obtained.
sections.heading: 2 Preliminaries
sections.text: Variational inference and augmentation. Suppose that p(z) = 1Z p̄(z) is some target density, where p̄ is unnormalized and Z = ∫ p̄(z)dz is the corresponding normalizer, and let
  ELBO(q(z), p̄(z)) = E q(z)
  log p̄(z)
  q(z) (1)
  be the "ELBO operator". Variational inference (VI) is based on the fact that for any q(z) we have [4]
  logZ = ELBO(q(z), p̄(z)) + KL(q(z)‖p(z)). (2)
  In VI, the parameters of q are tuned to maximize the "evidence lower bound" (ELBO). Since the KL-divergence is non-negative, this is always a lower bound on logZ. Also, maximizing the ELBO is equivalent to minimizing the KL-divergence from q to p.
  To get tighter bounds and better approximations recent work has made use of augmented distributions [2, 21]. Let z1:M = (z1, · · · , zM ) and suppose that p̄(z1:M ) = p̄(zM )p(z1:M−1|zM ) augments the original target density while preserving its normalization constant. Then, for any q(z1:M ) we have
  logZ = ELBO(q(z1:M ), p̄(z1:M )) + KL(q(z1:M )‖p(z1:M )). (3)
  The first term is called the "augmented" ELBO and again lower bounds logZ. By the chain rule of KL-divergence [12], the KL-divergence from q to p over z1:M upper-bounds the KL-divergence over zM . This justifies using the marginal of q over zM to approximate the original target distribution.
  Annealed Importance Sampling. A successful approach for creating augmented distributions is Annealed Importance Sampling (AIS) [27]. It creates an augmented proposal distribution q by applying a sequence of transition densities Tm(zm+1|zm), and an augmented target by defining transition densities Um(zm|zm+1). This gives the augmented densities
  q(z1:M ) = q(z1) M−1∏ m=1 Tm(zm+1|zm) and p̄(z1:M ) = p̄(zM ) M−1∏ m=1 Um(zm|zm+1). (4)
  Naively, the ratio of these densities is
  p̄(z1:M ) q(z1:M ) = p̄(zM ) q(z1) M−1∏ m=1 Um(zm|zm+1) Tm(zm+1|zm) . (5)
  To define the transitions Tm and Um, AIS creates a sequence of unnormalized densities π̄1, . . . , π̄M−1 that “bridge” from a starting distribution q to the target p̄, meaning that π̄1 is close to q and π̄M−1 is close to p̄. Then, for each intermediate distribution, Tm(zm+1|zm) is chosen to be a Markov kernel that holds πm invariant, and Um to be the reversal of Tm with respect to πm, defined as
  Um(zm|zm+1) = T (zm+1|zm) πm(zm)
  πm(zm+1) . (6)
  This choice produces a simplification so that eq. 5 becomes
  p̄(z1:M ) q(z1:M ) = p̄(zM ) q(z1) M−1∏ m=1 π̄m(zm) π̄m(zm+1) . (7)
  This can be easily evaluated without needing to evaluate the transition densities. The ratio from eq. 7 can be used to get an expression for the lower bound ELBO(q(z1:M ), p̄(z1:M )). Research has shown that the AIS augmentation may lead to extremely tight lower bounds [18, 17, 35, 44].
  Hamiltonian Dynamics. Many MCMC methods used to sample from p(z) are based on Hamiltonian dynamics [3, 8, 29, 42]. The idea is to create an augmented distribution p(z, ρ) = p(z)S(ρ), where S(ρ) is a distribution over a momentum variable ρ (e.g. a Multivariate Gaussian). Then, one can define numerical integration schemes where z and ρ evolve while nearly holding p(z, ρ) constant. When corrected by a Metropolis-Hastings acceptance step, this can be made to exactly hold p(z, ρ) invariant. This is alternated with a scheme that resamples the momentum ρ while holding S(ρ) invariant. When Hamiltonian dynamics work well, z can quickly move around, suppressing random-walk behavior.
  There are a variety of different Hamiltonian MCMC methods, corresponding to different integration schemes, momentum distributions, and ways of resampling the momentum. For instance, HMC and Langevin dynamics use the leapfrog integrator, a Gaussian for the momentum variables and a full resampling of the momentum variables at each step [29, 42]. On the other hand, if the momentum variables are only partially resampled, the under-damped variants of HMC and Langevin dynamics are recovered [29]. It was observed that partial resampling may lead to improved perfomance [9].
  It is easy to integrate Hamiltonian dynamics into AIS. First, define an augmented target p̄(z, ρ) = p̄(z)S(ρ) and an augmented starting distribution q(z, ρ) = q(z)S(ρ). Then, create a series of augmented densities π̄1(z, ρ), . . . , π̄M−1(z, ρ) bridging the two as π̄m(z, ρ) = π̄m(z)S(ρ). Finally, define the forward transition Tm(zm+1, ρm+1|zm, ρm) to be an iteration of a Hamiltonian MCMC method that leaves πm(z, ρ) invariant. We will describe a single transition Tm as a sequence of three steps: (1) resample the momentum; (2) simulate Hamiltonian dynamics and apply an accept-reject step; and (3) negate the momentum. The precise process that defines the transition is shown in Alg. 1. Note that this algorithm is quite general, and compatible with HMC, Langevin dynamics and their underdamped variants (by selecting an appropriate integrator and resampling method).
  Algorithm 1 Corrected Tm(zm+1, ρm+1|zm, ρm) 1. Sample ρ′m from some s(ρ ′ m|ρm) that leaves S(ρ) invariant. Set z′m ← zm.
  2. Simulate Hamiltonian dynamics as (z′′m, ρ′′m)← Tm(z′m, ρ′m). Calculate an acceptance probability α = min (1, π̄m(z′′m, ρ′′m)/π̄m(z′m, ρ′m)). With probability α, set (z′′′m , ρ′′′m)← (z′′m, ρ′′m). Otherwise, set (z′′′m , ρ′′′m)← (z′m, ρ′m). 3. Reverse the momentum as (zm+1, ρm+1)← (z′′′m ,−ρ′′′m). return (zm+1, ρm+1)
  Representing Tm this way makes it easy to show it holds the density πm(z, ρ) invariant. The overall strategy is to show that each of the steps 1-3 holds πm invariant, and so does the composition of them [29, §3.2]. For steps 1 and 3 this is trivial, provided that S(ρ) = S(−ρ). For step 2, we require that the simulation Tm has unit Jacobian and satisfies T −1m = Tm. Then, Tm can be interpreted as a symmetric Metropolis-Hastings proposal, meaning the Metroplis-Hastings acceptance probability α is as given. A typical choice for Tm that satisfies these requirements is the leapfrog integrator with a momentum reversal at the end. (This reversal then gets "un-reversed" in step 3 for accepted moves.)
  Since Tm holds πm invariant, we can define Um as the reversal of Tm wrt πm. Then, eq. 7 becomes
  p̄(z1:M , ρ1:M ) q(z1:M , ρ1:M ) = p̄(zM , ρM ) q(z1, ρ1) M−1∏ m=1 π̄m(zm, ρm) π̄m(zm+1, ρm+1) . (8)
  Using this ratio we get an expression for the lower bound ELBO(q(z1:M , ρ1:M ), p̄(z1:M , ρ1:M )) obtained with Hamiltonian AIS. While this method has been observed to yield strong lower bounds on logZ [35, 44] (see also Section 5.2), its performance depends on many parameters: initial distribution q(z), momentum distribution S, momentum resampling scheme, simulator Tm, and bridging densities. We would like to tune these parameters by maximizing the ELBO using reparameterization-based estimators. However, due to the accept-reject step required by the Hamiltonian MCMC transition, the resulting bound is not differentiable, and thus reparameterization gradients are not available.
sections.heading: 3 Uncorrected Hamiltonian Annealing
sections.text: The contribution of this paper is the development of uncorrected Hamiltonian Annealing (UHA). This method is similar to Hamiltonian AIS (eq. 8), but yields a differentiable lower bound. The main idea is simple. For any transitions Tm and Um, by the same logic as in eq. 5, we can define the ratio
  p̄(z1:M , ρ1:M ) q(z1:M , ρ1:M ) = p̄(zM , ρM ) q(z1, ρ1) M−1∏ m=1 Um(zm, ρm|zm+1, ρm+1) Tm(zm+1, ρm+1|zm, ρm) . (9)
  Hamiltonian AIS defines Tm as a Hamiltonian MCMC kernel that holds πm invariant, and Um as the reversal of Tm with respect to πm. While this leads to a nice simplification, there is no requirement that these choices be made. We can use any transitions as long as the ratio Um/Tm is tractable.
  We propose to use the "uncorrected" versions of the transitions Tm and Um used by Hamiltonian AIS, obtained by dropping the accept-reject steps. To get an expression for the uncorrected Um we first derive the reversal Um used by Hamiltonian AIS (Alg. 2). These uncorrected transitions are no longer reversible with respect to the bridging densities πm(z, ρ), and thus we cannot use the simplification used by AIS to get eq. 8. Despite this, we show that the ratio Um/Tm for the uncorrected transitions can still be easily computed (Thm. 2). This produces a differentiable estimator, meaning the parameters can be tuned by stochastic gradient methods designed to maximize the ELBO.
  We start by deriving the process that defines the transition Um used by Hamiltonian AIS. This is shown in Alg. 2. It can be observed that Um follows the same three steps of Tm (resample momentum, Hamiltonian simulation with accept-reject, momentum negation), but in reverse order.
  Algorithm 2 Corrected Um(zm, ρm|zm+1, ρm+1) 1. Set (z′′′m , ρ′′′m)← (zm+1,−ρm+1). 2. Simulate Hamiltonian dynamics as (z′′m, ρ′′m)← Tm(z′′′m , ρ′′′m).
  Calculate an acceptance probability α = min (1, π̄m(z′′m, ρ′′m)/π̄m(z′′′m , ρ′′′m)). With probability α, set (z′m, ρ′m)← (z′′m, ρ′′m). Otherwise, set (z′m, ρ′m)← (z′′′m , ρ′′′m).
  3. Sample ρm from srev(ρm|ρ′m), the reversal of s(ρ′m|ρm) with respect to S(ρm). Set zm ← z′m. return (zm, ρm)
  Lemma 1. The corrected Um (Alg. 2) is the reversal of the corrected Tm (Alg. 1) with respect to πm.
  (Proof Sketch). First, we claim the general result that if T1, T2 and T3 have reversals U1, U2 and U3, respectively, then the composition T = T1 ◦T2 ◦T3 has reversal U = U3 ◦U2 ◦U1 (all reversals with respect to same density π). Then, we apply this to the corrected Tm and Um: Tm is the composition of three steps that hold πm invariant. Thus, its reversal Um is given by the composition of the reversals of those steps, applied in reversed order. A full proof is in Appendix F.
  We now define the "uncorrected" transitions used by UHA, shown in Algs. 3 and 4. These are just the transitions used by Hamiltonian AIS but without the accept-reject steps. (If Hamiltonian dynamics are simulated exactly, the acceptance rate is one and the uncorrected and corrected transitions are equivalent.) We emphasize that, for the "uncorrected" transitions, Tm does not exactly hold πm invariant and Um is not the reversal of Tm. Thus, their ratio does not give a simple expression in terms of π̄m as in eq. 8. Nevertheless, the following result shows that their ratio has a simple form. Theorem 2. Let Tm and Um be the uncorrected transitions defined in Algs. 3 and 4, and let the dynamics simulator Tm(z, ρ) be volume preserving and self inverting. Then,
  Um(zm, ρm|zm+1, ρm+1) Tm(zm+1, ρm+1|zm, ρm) = S(ρm) S(ρ′m) , (10)
  Algorithm 3 Uncorrected Tm(zm+1, ρm+1|zm, ρm) 1. Sample ρ′m from some s(ρ ′ m|ρm) that leaves S(ρ) invariant. Set z′m ← zm.
  2. Simulate Hamiltonian dynamics as (z′′m, ρ′′m)← Tm(z′m, ρ′m). 3. Reverse the momentum as (zm+1, ρm+1)← (z′′m,−ρ′′m). return (zm+1, ρm+1)
  Algorithm 4 Uncorrected Um(zm, ρm|zm+1, ρm+1) 1. Set (z′′m, ρ′′m)← (zm+1,−ρm+1). 2. Simulate Hamiltonian dynamics as (z′m, ρ′m)← Tm(z′′m, ρ′′m). 3. Sample ρm from srev(ρm|ρ′m), the reversal of s(ρ′m|ρm) with respect to S(ρm). Set zm ← z′m. return (zm, ρm)
  where ρ′m is the second component of Tm(zm+1,−ρm+1). (That is, ρ′m from Algs. 3 and 4.) (Proof Sketch.) We consider variants of Algs. 3 and 4 in which each time z is assigned we add Gaussian noise with some variance aI . We then derive the densities for Tm and Um using the rule for transformation of densities under invertible mappings, using that Tm is self-inverting and volume preserving. Taking the ratio gives eq. 10. Since this is true for arbitrary a, we take the stated result as the limit as a→ 0. A full proof is in Appendix G.
  As an immediately corollary of eq. 9 and Theorem 2 we get that for UHA
  p̄(z1:M , ρ1:M ) q(z1:M , ρ1:M ) = p̄(zM ) q(z1) M−1∏ m=1 S(ρm+1) S(ρ′m) . (11)
  This ratio can be used to get an expression for the lower bound ELBO(q(z1:M , ρ1:M ), p̄(z1:M , ρ1:M )) obtained with UHA. As mentioned in Section 2, the parameters of the augmented distributions are tuned to maximize the ELBO, equivalent to minimizing the KL-divergence from q to p̄. While computing this ELBO exactly is typically intractable, an unbiased estimate can be obtained using a sample from q(z1:M , ρ1:M ) as shown in Alg. 5. If sampling is done using reparameterization, then unbiased reparameterization gradients may be used together with stochastic optimization algorithms to optimize the lower bound. In contrast, the variational lower bound obtained with Hamiltonian AIS (see Alg. 6 in Appendix A) does not allow the computation of unbiased reparameterization gradients.
  Algorithm 5 Generating the (differentiable) uncorrected Hamiltonian annealing variational bound. Sample z1 ∼ q and ρ1 ∼ S. Initialize estimator as L ← − log q(z1). for m = 1, 2, · · · ,M − 1 do
  Run uncorrected Tm (Alg. 3) on input (zm, ρm), storing ρ′m and the output (zm+1, ρm+1). Update estimator as L ← L+ log (S(ρm+1)/S(ρ′m)).
  Update estimator as L ← L+ log p̄(zM ). return R
sections.heading: 3.1 Algorithm Details
sections.text: Simulation of dynamics. We use the leapfrog operator with step-size to simulate Hamiltonian dynamics. This has unit Jacobian and satisfies Tm = T −1m (if the momentum is negated after the simulation), which are the properties required for eq. 11 to be correct (see Theorem 2).
  Momentum distribution and resampling. We set the momentum distribution S(ρ) = N (ρ|0,Σ) to be a Gaussian with mean zero and covariance Σ. The resampling distribution s(ρ′|ρ) must hold this distribution invariant. As is common we use s(ρ′|ρ) = N (ρ′|ηρ, (1− η2)Σ), where η ∈ [0, 1) is the damping coefficient. If η = 0, the momentum is completely replaced with a new sample from S in each iteration (used by HMC and Langevin dynamics [29, 42]). For larger η, the momentum becomes correlated between iterations, which may help suppress random walk behavior and encourage faster mixing [9] (used by the underdamped variants of HMC and Langevin dynamics [29]).
  Bridging densities. We set π̄m(z, ρ) = q(z, ρ)1−βm p̄(z, ρ)βm , where βm ∈ [0, 1] and βm < βm+1.
  Computing gradients. We set the initial distribution q(z1) to be a Gaussian, and perform all sampling operations in Alg. 5 using reparameterization [24, 31, 39]. Thus, the whole procedure is differentiable and reparameterization-based gradients may be used to tune parameters by maximizing the ELBO. These parameters include the initial distribution q(z1), the covariance Σ of the momentum distribution, the step-size of the integrator, the damping coefficient η of the momentum resampling distribution, and the parameters of the bridging densities (including β), among others. As observed in Section 5.2.1 tuning all of these parameters may lead to considerable performance improvements.
sections.heading: 4 Related Work
sections.text: UHA and slight variations have been proposed in concurrent work by Thin et al. [38], who use uncorrected Langevin dynamics together with the uncorrected reversal to build variational lower bounds, and by Zhang et al. [46], who proposed UHA with under-damped Langevin dynamics together with a convergence analysis for linear regression models.
  There are three other lines of work that produce differentiable variational bounds integrating Monte Carlo methods. One is Hamiltonian VI (HVI) [34, 43]. It uses eq. 9 to build a lower bound on logZ, with Tm set to an uncorrected Hamiltonian transition (like UHA but without bridging densities) and Um set to conditional Gaussians parameterized by learnable functions. Typically, a single transition is used, and the parameters of the transitions are learned by maximizing the resulting ELBO.1
  A second method is given by Hamiltonian VAE (HVAE) [7], based on Hamiltonian Importance sampling [28]. They augment the variational distribution with momentum variables, and use the leapfrog integrator to simulate Hamiltonian dynamics (a deterministic invertible transformation with unit Jacobian) with a tempering scheme as a target-informed flow [30, 37].
  The third method is Importance Weighting (IW) [6, 13, 15]. Here, the idea is that ELBO(q(z), p̄(z)) ≤ E log 1K ∑ k p̄(zk)/q(zk), and that the latter bound can be optimized, rather than the traditional ELBO. More generally, other Monte-Carlo estimators can be used [16].
  Some work defines novel contrastive-divergence-like objectives in terms of the final iteration of an MCMC chain [32, 26]. These do not provide an ELBO-like variational bound. While in some cases the initial distribution can be optimized to minimize the objective [32], gradients do not flow through the MCMC chains, meaning MCMC parameters cannot be optimized by gradient methods.
  For latent variable models, Hoffman [19] suggested to run a few MCMC steps after sampling from the variational distribution before computing gradients with respect to the model parameters, which is expected to "debias" the gradient estimator to be closer to the true likelihood gradient. The variational distribution is simultaneously learned to optimize a standard ELBO. (AIS can also be used [14].)
sections.heading: 5 Experiments and Results
sections.text: This section presents results using UHA for Bayesian inference problems on several models of varying dimensionality and for VAE training. We compare against Hamiltonian AIS, IW, HVI and HVAE. We report the performance of each method for different values of K, the number of likelihood evaluations required to build the lower bound (e.g. number of samples used for IW, number of bridging densities plus one for UHA). Note that, for a fixed K, all methods have the same oracle complexity (i.e. number of target/target’s gradient evaluation), and that for K = 1 they all reduce to plain VI.
  For UHA and Hamiltonian AIS we use under-damped Langevin dynamics, that is, we perform just one leapfrog step per transition and partially resample momentum. We implement all algorithms using Jax [5].
  1The formulation of HVI allows the use of more than one transition. However, this leads to an increased number of reverse models that must be learned, and thus not typically used in practice. Indeed, experiments by Salimans et al. [34] use only one HMC step while varying the number of leapfrog integration steps, and results from Wolf et al. [43] show that increasing the number of transitions may actually yield worse bounds (they conjecture that this is due to the difficulty of learning inverse dynamics.).
sections.heading: 5.1 Toy example
sections.text: This section compares results obtained with UHA and IW when the target is set to a factorized Studentt with mean zero, scale one, and three degrees of freedom. We tested three different dimensionalities: 20, 200 and 500. In all cases we have logZ = 0, so we can exactly analyze the tightness of the bounds obtained by the methods. We set the initial approximation to be a mean-field Gaussian, and optimize the objective using Adam [23] with a step-size of 0.001 for 5000 steps. For UHA we tune the initial approximation q(z), the integrator’s step-size and the damping coefficient η.
  We ran UHA for K ∈ {4, 16, 64, 128} and IW for K ∈ {128, 1024}. Table 1 shows the results for the three dimensionalities considered. It can be observed that UHA performs significantly better than IW as the dimensionality increases; for the target with dimension 500, UHA with K = 16 yields better bounds than IW with K = 1024. On the other hand, the methods perform similarly for the low dimensional target. Finally, in this case both methods have similar time costs. For instance, for K = 128 UHA takes 14.2 seconds to optimize and IW takes 13.9.
sections.heading: 5.2 Inference tasks
sections.text: This section shows results using UHA for Bayesian inference tasks. For this set of experiments, for UHA we tune the initial distribution q(z), the integrator’s step-size and the damping coefficient η. We include detailed results tuning more parameters in Section 5.2.1.
  Models. We consider four models: Brownian motion (d = 32), which models a Brownian Motion process with a Gaussian observation model; Convection Lorenz bridge (d = 90), which models a nonlinear dynamical system for atmospheric convection; and Logistic regression with the a1a (d = 120) and madelon (d = 500) datasets. The first two obtained from the “Inference gym” [36].
  Baselines. We compare UHA against IW, HVAE, a simple variant of HVI, and Hamiltonian AIS (HAIS). For all methods which rely on HMC (i.e. all except IW) we use a singe integration step-size common to all dimensions and fix the momentum distribution to a standard Gaussian. For HVI we learn the initial distribution q(z), integration step-size and the reverse dynamics Um (set to a factorized Gaussian with mean and variance given by affine functions), and for HVAE we learn q(z), and the tempering scheme (we use the quadratic scheme parameterized by a single parameter).
  Training details. We set q(z) to be a mean-field Gaussian initialized to a maximizer of the ELBO, and tune the parameters of each method by running Adam for 5000 steps. We repeat all simulations for different step-sizes in {10−3, 10−4, 10−5}, and select the best one for each method. Since Hamiltonian AIS’ parameters cannot be tuned by gradient descent, we find a good pair ( , η) by grid search. We consider η ∈ {0.5, 0.9, 0.99} and three values of that correspond to three different rejection rates: 0.05, 0.25 and 0.5. We tested all 9 possible combinations and selected the best one.
  Results are shown in Fig. 1. Our method yields better lower bounds than all other competing approaches for all models considered, and that increasing the number of bridging densities consistently leads to better results. The next best performing method is Hamiltonian AIS. IW also shows a good performance for the lower dimensional model Brownian motion. However, for models of higher dimensionality IW leads to bounds that are several nats worse than the ones achieved by UHA. Finally, HVI and HVAE yield bounds that are much worse than those achieved by the other three methods, and do not appear to improve consistently for larger K. For HVAE, these results are consistent with
  the ones in the original paper [7, §4], in that higher K may sometimes hurt performance. For HVI, we believe this is related to the use of just one HMC step and suboptimal inverse dynamics.
  Optimization times for Plain VI, IW and UHA (the latter two with K = 32) are 2.4, 3.4 and 4.4 seconds for the Brownian motion dataset, 2.5, 6.8 and 6.9 seconds for Lorenz convection, 2.8, 8.3 and 19.9 seconds for Logistic regression (A1A), and 4.6, 16.6 and 121.2 seconds for Logistic regression (Madelon). While IW and UHA have the same oracle complexity for the same K, we see that the difference between their time cost depends on the specific model under consideration. All other methods that use HMC have essentially the same time cost as UHA.
sections.heading: 5.2.1 Tuning More Parameters with UHA
sections.text: A basic version of UHA involves fitting a variational distribution using plain VI, and then tuning the integration step-size and the damping coefficient η. However, more parameters could be tuned:
  • Moment distribution cov Σ: We propose to learn a diagonal matrix instead of using the identity. • Bridging densities’ coefficients βm: Typically βm = m/M . We propose to learn the sequence β,
  with the restrictions β0 = 0, βM = 1, βm < βm+1 and βm ∈ [0, 1]. • Initial distribution q(z): Instead of fixing q(z) to be a maximizer of the typical ELBO, we propose
  to learn it to maximize the augmented ELBO obtained using UHA. • Integrator’s step-size : Instead of learning a unique step-size , we propose to learn a step-size
  that is a function of β, i.e. (β). In our experiments we use an affine function. • Bridging densities parameters ψ: Instead of setting them-th bridging density to be q1−βmpβm , we
  propose to set it to q1−βmψ(βm) p βm , where qψ(βm) is a mean-field Gaussian with a mean and diagonal covariance specified as affine functions of β.
  We consider the four models described previously and compare three methods: UHA tuning all parameters described above, UHA tuning only the pair ( , η), and Hamiltonian AIS with parameters ( , η) obtained by grid-search. We perform the comparison forK ranging from 2 to 512. (ForK ≥ 64 we tune the UHA’s parameters using K = 64 and extrapolate them as explained in Appendix D.)
  Results are shown in Fig. 2. It can be observed that tuning all parameters with UHA leads to significantly better lower bounds than those obtained by Hamiltonian AIS (or UHA tuning only and η). Indeed, for the Logistic regression models, UHA tuning all parameters for K = 64 leads to results comparable to the ones obtained by Hamiltonian AIS with K = 512.
  To verify what parameters lead to larger performance improvements, we tested UHA with K = 64 tuning different subsets of { , η,Σ, β, q(z), (β), ψ(β)}. Fig. 3 shows the results. It can be observed
  that tuning the bridging parameters β and the initial approximation q(z) leads to the largest gains in performance, and that tuning all parameters always outperforms tuning smaller subsets of parameters. We show a more thorough analysis, including more subsets and values of K in Appendix B.
  Finally, Appendix E shows results comparing UHA (tuning several parameters) against HMC, mean field VI and IW in terms of the approximation accuracy achieved on a logistic regression model with a fixed computational budget.
sections.heading: 5.3 VAE training
sections.text: Our method can be used to train latent variable models, such as Variational Auto-encoders (VAE) [24, 31]. In this case the initial approximation q(z|x) and the model p(x, z) are parameterized by two neural networks (encoder and decoder), whose parameters are trained by maximizing the ELBO. UHA can be used to train VAEs by augmenting these two distributions as described in Section 3.
  Datasets. We use three datasets: mnist [25] (numbers 1-9), emnist-letters [11] (letters A-Z), and kmnist [10] (cursive Kuzushiji). All consist on greyscale images of 28×28 pixels. In all cases we use stochastic binarization [33] and a training set of 50000 samples, a validation set of 10000 samples, and a test set of 10000 samples. All datasets are available in tensorflow-datasets [1].
  Baselines. We compare against Importance Weighted Auto-encoders [6] and plain VAE training [24].
  Architecture details. We set q(z|x) to a diagonal Gaussian, p(z) to a standard Normal, and p(x|z) to a Bernoulli. We consider two architectures for the encoder and decoder: (1) Feed forward networks with one hidden layer of size 450 and Relu non-linearities, with a latent space dimensionality of 64; (2) Architecture used by Burda et al. [6], feed forward networks with two hidden layers of size 200 with tanh non-linearities, with a latent space dimensionality of 50.
  Training details. In all cases the encoder and decoder are initialized to parameters that maximize the ELBO. For IW we tune the encoder and decoder parameters (using the doubly-reparameterized estimator [40]), and for UHA we tune the integration step-size , damping coefficient η, bridging parameters β, momentum covariance Σ (diagonal), and the decoder parameters. Following Caterini et al. [7] we constrain ∈ (0, 0.05) to avoid unstable behavior of the leapfrog discretization. We use Adam with a step-size of 10−4 to train for 100 epochs and use the validation set for early stopping. We repeated all simulations for three different random seeds. In all cases the standard deviation of the results was less than 0.1 nats (not shown in tables).
  All methods achieved better results using the architecture with one hidden layer. These results are shown in Tables 2 and 3. The first one shows the ELBO on the test set achieved for different values of K, and the second one the log-likelihood on the test set estimated with AIS [44]. It can be observed that UHA leads to higher ELBOs, higher log-likelihoods, and smaller variational gaps (difference between ELBO and log-likelihood) than IW for all datasets, with the difference between both methods’ performance increasing for increasing K. Notably, for K = 64, the variational gap for UHA becomes quite small, ranging from 0.8 to 1.4 nats depending on the dataset.
  Results for the architecture from Burda et al. [6] (two hidden layers) are shown in Tables 4 and 5 (Appendix C). Again, we observe that UHA consistently leads to higher ELBOs and the best test log-likelihood was consistently achieved by UHA with K = 64. However, for smaller K, IW sometimes had better log-likelihoods than UHA (despite worse ELBOs).
sections.heading: 6 Discussion
sections.text: Since UHA yields a differentiable lower bound, one could tune other parameters not considered in this work. For instance, a different momentum distribution per bridging density could be used, that is, π̄m(z, ρ) = π̄m(z)Sm(ρ). We believe additions such as this may yield further gains. Also, our method can be used to get tight and differentiable upper bounds on logZ using the reversed AIS procedure described by Grosse et al. [18].
  Finally, removing accept-reject steps might sometimes lead to instabilities during optimization if the step-size becomes large. We observed this effect when training VAEs on some datasets for the larger values of K. We solved this by constraining the range of (previously done by Caterini et al. [7]). While this simple solution works well, we believe that other approaches (e.g. regularization, automatic adaptation) could work even better. We leave the study of such alternatives for future work.
  Acknowledgments and Disclosure of Funding
  This material is based upon work supported in part by the National Science Foundation under Grant No. 1908577.
title: MCMC Variational Inference via Uncorrected Hamiltonian Annealing