Variational excess risk bound  for general state space models
Authors: Elisabeth Gassiat
Sylvain Le Corff
Venue: Accepted by TMLR ()

Abstract: In this paper, we consider variational autoencoders (VAE) for general state space models.  We consider a backward factorization of the variational distributions to analyze the excess risk associated with VAE. Such backward factorizations were recently proposed to perform online variational learning  and to obtain upper bounds on the variational estimation error. When independent trajectories of sequences are observed and under strong mixing assumptions on the state space model and on the variational distribution, we provide an oracle inequality explicit in the number of samples and in the length of the observation sequences. We then derive consequences of this theoretical result. In particular, when the data distribution is given by a state space model, we provide an upper bound for the Kullback-Leibler divergence between the data distribution and its estimator and between the variational posterior and the estimated state space posterior distributions. Under classical assumptions, we prove that our results can be applied to Gaussian backward kernels built with dense and recurrent neural networks.

Published in Transactions on Machine Learning Research (06/2024)
Variational excess risk bound for general state space models
Elisabeth Gassiat
Université Paris-Saclay, CNRS
Laboratoire de mathématiques d' d'Orsay 91405, Orsoy, France
Sylvain Le Corff
sylvain. le corffflarroommn--niirrrsste
LPSM
Sorbonne Université, UMR CNRS 8001 Paris, France
Reviewed on OpenReview; https: // openrevvee.......... net/ pet//foum?? id= 360X7A866
Abstract
In this paper, consider variational autoencoders (VAE) for general state space mod- els. We consider backward factorization of the variational distributions to analyze the excess risk associated with VAE. Such backward factorizations were recently proposed to perform online variational learning Campbell et al. (2021) and to obtain upper bounds on the variational estimation error Chagneux et al. (2022). When independent trajectories of sequences are observed and under strong mixing assumptions on the state space model and on the variational distribution, we provide an oracle incquality explicit in the number of samples and in the length of the observation sequences. We then derive consequences of this theoretical result. In particular, when the data distribution is given by state space model, we provide upper bound for the Kullback-Leibler divergence between the data distribution and its estimator and between the variational posterior and the est imattdd state space posterior distributions. Under classical assumptions, we prove that our results can be applied to Gaussian backward kernels built with dense and recurrent neural networks.
1 Introduction
Deep generative models have been increasingly used and analyzed for the past few years. In this setting, Variational antocncoders (VAEs) offer the possibility to simultancously model and train (i) the conditional distribut of the observation given latent. variables referred to as the decoder, and (ii) a variational approx- imation of the conditional distribution of the latent variable given the observation referred to the encoder.
They have been succossfully appliod in many contexts such image generation (Vahdat & Kautz 2020
text generation (Bowman et al. 2015), state estimation and image reconstruction (Cohen et al. 2022
Variational inference has been widely and satisfactorily used for many practical applications but its theoret- ical properties has been analyzed only very recently. Theoretical guarantees have been mostly proposed for variational inference procedures in settings where datasets are based on independent data and for mean-fiee field approximations. Tn Huggins et al. (2020), the authors provided variational error bounds, particular for the estimation of the posterior mean and covariance. In Chérief-Abbdellatif & Alquier (2018), the authors established the concentration of variational approximations of posterior distributions for mixtures of general laws using PAC-Bayesian theory. The PAC-Bayesian theory has also been used in Mbacke et al. (2023) where the authors controlled in particular the L2 reconstruction loss under the true data distribution for VAEs. In addition, Tang & Yang (2021) provided a theoretical analysis of the excess risk for Empirical Bayes Variational Autoencoders for both parametric and nonparametric settings. They derived a set of generic assumptions to obtain an oracle inequality explicit the number of samples and proposed an upper bound for the total variation distance between the true distribution of the observations and a variational approximation combining the empirical distribution of the dataset and the proposed VAE architecture.

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In this paper, consider data sets consisting in independent copies of sequences with length T+1 having distribution Pp. We aim at extending the theoretical results on variational inference procedures in two directions. First, we set the focus on the use of VAËs for general state space models, i.e. settings where the decoding distribution Py of the observations depends on an unobserved Markoy chain. State space models are a ubiquitous class of latent variable models for sequential data, see for instance Marino et al.] (2018); Lin al. (2018); Krishnan et al. (2017). In addition, instead of using mean-field approximations, we consider variational encoding distributions Qo satisfying a backward factorization as proposed in [Campbel] et al. 2021); Chagneux et al. (2022). In Chagneux et al. (2022), the authors derived the first theoretical results providing upper bounds on the state decoding estimation error when using variational inference with backward factorization and no such results were proposed for state space models using a mean-field approx- imation. This factorization was used in Campbell et al. (2021) to define new online variationa estimation algorithms, where observations are processed mu-the-fly. In state space models, the true posterior distribu- tion of the latent states given the observations admits a backward Markovian dccomposition. Therefore this lactorization allows to introduce variational family which fits the data structure which is not the case of mean-field approximations.
Our results give the first (up to our knowledge) theorctical guarantees on the trained variational approxi- mation in this setting.
We provide assumptions on the decoding and variational encoding kernels under which we prove an oracle incquality for the risk explicit in particular in the number of samples and in the longth of the observation sequences, see Theorem 3.1 This result is established using an alternative formulation of (Tang & Yang 2021 Theorem 3) in our state space setting and with an explicit dependency on some constants to track all terms depending on the number of observations. The variance term has the usual rate 1/n up to log n terms in the sample size and grows as T in the length T of the sample sequences. This allows to understand when the procedure leads to a decoding distribution that approximates well the data distribution together with a coding distribution which approximates well the decoding stale distribution.
In particular, when data are generated from a general state space model, and when Pp belongs to the decoding family of distributions, we give an upper bound also explicit in the way the backward coding kernels approximate the backward decoding kernels, see Corollary
We analyse settings in which our results hold, in particular settings with Gaussian backward kernels based on Multi-Layer Perceptrons (MLPss) and on Recurrent Neural Networks (RNNs).
Our thcoretical results provide the first excess risk bounds in context of VAE for state space models. The proposed upper bound has the same behaviour in as the ones obtained in the classical statistical literature for parametric models, in which 1/n is the parametric rate and log factors come from the concentration of empirical measures. This behaviour was recovered in (Tang & Yang 2021 Theorem 3) in the variational learning context. Regarding theoretical results for VAEs in the state space models context, we are only aware of Chagneux et al. (2022) where the authors control the variational posterior error for the estimation of expectations of additive functionals with an upper bound having linear growth in T. In this context, we target more challenging objective and it is not surprising that the upper bound on the excess risk may have a larger than linear growth in T.
The paper is organised as follows. The general sctting and notations for state space models and variational learning are given in Section 2 Assumptions and theoretical results are proposed in Section 3 along with discussions on specific deep architectures used in practice. A discussion with insights for future works is given in Section 4 Detailed technical sctting ot the assumptions and detailed proofs of theoretical results are given in Appendices. Additional proofs highlight that when the state and observation spaces are compact our main results hold are given in Appendix G
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2 State space model and variational estimation
The observations are n sequences Yor, 1 i S n, taking values a measurable space (Y,)). We use awa as a short-hand notation for (auy...,, (v) for < u & v and any sequence (ae)ezo- In the following, we need consider quantities depending on observation sequences yo.t, which highlighted with super-indices all what follows.
2.1 Coding distribution with state space modeling
Throughout the paper, all quantities related to the coding distribution depend parameter E O θ C Rd To be used as coding distribution for the observations, we consider state-space model, i.c. bivariate discrete-lime process {(X,, 220 where (Xx)220 is a hidden Markov chain in a measurable space (X, The imitial distribution X of Xo has density { with respect to reference measure μ and for all f > 0, the conditional distribution of X(+1 given Xu depends only on X,, it is written MX),,,) and has density :.....). In such model, the observations {YrJuCeET are assumed to be independent conditionally on No.r and, for all 0 & 1 & T, the distribution of Y, given Xo.r depends on X, only, is written Go(X,, :); and has density y + (X,, with respect to a reference measure v.
Example 1. Consider nonlinear state space model defined by Xo ~ S in Rd and for all 0 Sstsss 1,
X = fo(,)) t6
where RR. - Rdd is parametric function and (E1)1SITT are i..d. with distribution NN E) and inde- pendent of Xo- In this setting, mo .) is the Gaussian density with mean fo (x()) and variance E. The observations are defined, for all 0 S t T,
h (x.) +1
where ho : R.d + R2 is parametric function and (ne)sSiST are i.i.d. with distribution. N(O, A) and inde- pendent of (Xar,, Su.T). In this selling, y + 36 (Xx) is the Gaussian densily with mean he (Xx) and voriance A.
In this context, the joint probability distribution Po of (Xn.r, Yo.r) has density with respect to ( s (((++) given, for all E O, zo. XT+1 and all your E YT+1, by
Po.orT (Fur,, Vorr) ((ro)o) (zr) II mo (xe-1, (#t)
and the coding distribution of sequence Yo.r is denoted by PÝ. The joint smoothing distribution, i.c. the conditional distribution of Xo.r given Yo.r = Vo.T; ; is given for every measurable function h by $ 8.0.Tyr (h) J x(dd0ogg (xo) II M1 (xx) (Ix)h(Zor))
S [x(dro)g) (zo) II' Mo(x;-1, dr, ) (xt)
The probability density of Th No: denote the the filtering distribution at time t, i.c. the conditional distribution of X, given Yo:- = yorr, with a similar convention for the probability densities. The joint smoothing distribution can also be written
9.0:TIT is denoted by No.r T- In the following, we use the notation T to
T-1
h gerr
8.0.T (
No-T (dxr)
II -o - (ET drr-
(1)
where B ---
-
is the backward kernel at time T t defined by T No:T - (mm
BBooT= P ( (TT d
with probability density with re- - (WT .). For all T, 0, your E YT+1, the loglikelihood of the observations
0,7
spect to a denoted by Mo:T - 1 is:
gon (8) log L (8)
where
L L (0) / pa 0 ( yorr ( )((doo.))
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Published in Transactions on Machine Learning Research (06/2024)
2.2 Variational learning with backward factorization
The joint smoothing distribution is usually intractable and focus in this paper variational learning to perform approximate maximum likelihood. All quantities related to the encoding distribution depend on a parameter E Q, & C 124 Following Campbell et al. (2021); Chagneux et al. (2022), we propose a backward variational formulation mimicking
Qua (dro.)) Q (drT) II
0 1----t-117- --r------- (XT- (xT-1,0T----))
t=0
where -- (resp. Q has probability density C 1 (resp. g with respect to the reference measure μs.
In this setting, the ELBO writes, for all 0 O yp E o,, and for a sequence of observations Yo.r,
ELBO (0, p (0) KT. Your Yorr 0.0:11P)
Here, (Q)) denotes the Cullback-Teibles divergence between probability distributions Q and P, that is KT. (QI) = = Te [og((((()/ Maximizing the empirical ELBO given by Ei= oTB (0, (0) is equivalent to minimizing the following loss function
1 (0, s, YoT),,
(0,
n
where
w0, p, Yotr) = log PD( 0.T + KL (xx0r ( (0,u:r11)
_ (0)
Define
(nrr,punr) argmingg@. (0,r(0,)))
(2)
Although this is not the foeus of this paper, note that in practice the estimators given by P2 cannot be computed explicitly, in particular when the decoder or the encoder parameterized by neural networks. In this case, common approach is to obtain approximate estimators using stochastic gradient descent-based approaches like ADAM Kingma & Ba 2015), see for instance Chagneux et al. (2022) and the references therein. In these settings, the gradient descent is performed using mini batches trajectories for each parameter update and the unknown expectations are approximated using Monte Carlo methods.
Such a procedure is a so-called M-estimation method in the statistical literature. The intuition is that with large data sets, that is when is large, the ELBO is close to the expected (under the unknown distribution of the data) value of w, and the estimated decoding and coding parameters are close to minimizing this expected value. An important body of work in the statistical community has been devoted to develop very general settings in which non asymptotic bounds on the risk of M-estimators, referred to as oracle incqualitics, can be given, see van de Geer (2000) as carly reference, or Wainwright (2019) and the references therein for more recent results. Moreover, oracle inequalities are obviously the only properly one can hope for such estimators, the other propertics being consequences the oracle incquality. In the following section, we thus first provide assumptions under which we obtain an oracle inequality for the coding and decoding distributions with parameters defined in (2) and then discuss consequences. Note that though (2) defines the parameters when the algorithm has reached the optimal value, could relax the definition by minimizing to some error term that would be added to the upper bound of the oracle inequality.
2.3 Examples
We can consider for instance generative models where the transition kernels and emission distributions are Gaussian and parameterized by neural networks.
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Gaussian backward kernels with dense networks.
For all x E X, x' + m²( is the Gaussian probability density function with mean po(x), and variance Ex(r) where (pO(z), Eo(x)) = MLP"(x) with MLP a dense Multi-layer network with input x and weights given by 0.
For all 1 2 S T, x EX, x' + No.r 12,t-11e(2, 2') is the Gaussian probability density function with mean yoor (x), and variance E (1((-1((()) where (n (x), - (z)) = MLP ('.t-1je
t-1jt (x) with MLP a dense Multi-layer network with input a and weights depending on sp.
[-1]t
Another example where the forward Markov kernels and the backward variational kernels are Gaussian is given by the Chaotic Recurrent Neural Network (CRNN) described Campbell et al. (2021) Zhao et al.
(0022 Chagneux et al. (0022).
Gaussian backward kernels with recurrent networks. A natural parameterization is also to recurrent neural network which updates internal state (st)120 from which the backward variational kernels and filtering density are built. For all t 2 0, define s, = (s where RNN is a recurrent neural network, and let x' + o and variance E where (me, Ex) MLP
1 x') be the Gaussian probability density function with mean 1
3
Main results
3.1 Assumptions
In this section, propose set of assumptions the kernel densities ma and 20. 4p,t)t+1 0 s 1, and
on the conditional densities 5%, under which we are able to prove an oracle inequality. The precise setting of those assumptions detailed in Appendix B We discuss in Section 3.3 how they can be applied specific architectures used in practice. Additional discussions on the assumptions are provided in Appendix where we prove that usual compact state space models are covered by theory.
As can be seen in the definition of the FL,BO, we shall need to control smoothing expectations both in the coding distribution and in the decoding distribution. In the state space model literature, Assumption H is usual for this purpose.
H1 There exist probability measures 77- and 77+ on (X, X') and constants 0 < a_ < <0+ < 00 such that a- (resp. 7+7+) is a uniform lower bound (resp. uniform upper bound) for and Mg. Similarly, there exist probability measures A_ and A on (X, X) such that for all yo: E YT+1, there exist goor and / 0 such that @ (resp. you X4) uniform lower bound (resp. uniform upper bound) for 2 and Quate+1
In the state space model literature, H2 is usual for the study of asymptotic properties of maximum likelihood estimators.
H2 For all E Y, infage f 9 (z)) (dx) = c-(x) > 0 and supace f (z(x))++ = c+(y) < co.
More assumptions are needed to manage the complexity of the models and to get a nonasymptotic control of the risk of the estimators. These controls are obtained with Assumptions H36
We constrain the kernels and the conditional densities to be Lipschitz in the parameters with Lipschitz coelficient depending on the variables.
H3 There exists M, G K t-1jp KT such that m²(x,x) is Lipschitz in 0 with Lipschitz cocfficient
a(x,x'), (x) is Lipschitz in 0 with Lipschitz coefficient G9(x), o 1p (x, x') is Lipschitz in yp with Lipschitz coefficient K K (a', x), your (z) is Lipschitz in y with Lipschitz coefficient K3 (z).
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We shall also need Lipschitz properties of the functions comparing logarithms of the backward kernels of the coding and decoding distributions. The precise technical setting of the assumption is given in Appendix B
H4 The difference between logarithms of the backward kernels of the coding and of the decoding dis- tributions satisfy Lipschitz properties with respect to the parameters. Moreover, the integral with respect to X of this difference is uniformly upper bounded.
We shall need to prove that w is a Lipschitz function of the parameters, and we need an upper bound on the -o-norm of the Lipschitz coefficient. For this purpose, consider moment assumptions. The precise technical setting of the assumption is given in Appendix B
H5 There exists which is an upper bound of several moments involving quantities defined in the previous assumptions.
The last assumption is used to get concentration properties, as usual in the statistical literature to get theoretical guarantees with finite samples. It involves Orliez norms, the precise definition of which given in Appendix
H6 There exists a, and B > 0 such that the Orlicz norm of order a, of several functions is upper bounded with B.
3.2 Oracle inequalities and consequences
Our main result. is an oracle inequality for the risk. The so-called variance term has the usual rale 1/n up to 1ogn terms in the sample size n. It is proved to grow much as 7 in the length T of the sample sequences. We assume that θ and & are compact spaces, and that the sum of their diameters is bounded by do- Thcorem 3.1. Assume that H hold. Then, there cvist constants co; C1 022 D which depend on a+, a-, Oo,, A, B and do only, such that with probability at least 1 - coexp(-ci (d. log (nga))
/ T (n,,T Sn.T, (o.
S inf { (1+ + c2(1 + 1 Dd,
log (oggd
n
where Er mingee,ped f (0, sp your (po (p((o))) and d, = dg + dp.
In Chagneux et al. (2022), under similar assumptions as the strong mixing assumptions of our paper, the authors obtain a linear growth in T" as an upper bound on the variational posterior error when the objective is compute expectations of additive functionals. In our setting we target a more challenging objective i.. obtaining excess risk bounds and full control on the divergence between the true data distribution and the estimated one (Corollary 3.2). Indeed, as an intermediate step, the proof of Theorem 3.1 requires similar bounds as the one in Chagneux et al. (2022). The sketch of proof we follow (see below) leads to a growth of order T We do not claim that our upper bounds is optimal but it is not surprising that it has a larger growth than the one of Chagneux et al. (2022). Finding another method to prove a bound of better order if possible is an open problem.
In our assumptions, many constants depend the state dimension and on the dimension of the parameter space. In this paper, decided to set the focus the dependency the number of samples and on T. A reason is that tracking the dependency on the dimension is challenging in state space models. Tracking the dependency on for mixing constants is a contribution on its own for general state spaces without assuming too restrictive conditions on the model. This is the focus future works.
Proof. To prove Theorem 3.1 we use Theorem +11 which an improved formulation of (Tang & Yang. 2021 Thcorem 3), provod Appendix C. which we track the dependency in n and T. Theorem C.1 provides an oracle inequality in which the upper bound sum of bias term and a variance term. The bias term Ey. The variance term the product of rate in the sample size and a complexity term. The rate

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in is the usual one, and the complexity term is itself product of the constants 01 and D involved in the assumptions of Theorem C together with the dimensional constant d,. The main purpose of the remaining of the proof is to understand how and D depend on T, the length of the sequences. This requires to provide assumptions that can be verified in specific models such as the models described in Scctions 2.3 and 3.3 and to control precisely behaviours of terms involving smoothing (coding and decoding) distributions which is known to be challenging. The detailed proof is given in the Appendix D We show that ai may be upper bounded with CT for some constant C' 0, and that D may be upper bounded with DT, resulting in product of order 73.
Note that
I ( Pn,y, , (o:r))po(oorr))(nm KL
(Pb
PY
8n,5 ) + Er, KL
(orrr
Yoir Pu.r, 0:T
D or O:TIT)
If the upper bound in Theorem 3.1 small, then the distribution PD of the observations well approximated by the decoding observational distribution PY and the decoding distribution of the latent state distribution
b,.z
given data & Yor is also in average well approximated by the coding distribution Q Yor PurT
In the same way,
Er mingee KL (Po Ep,, KL (o Yoer (
In case the data follows a state space distribution given by some decoding distribution, that is if there exists H* E e such that Pp = Pg., the oracle inequality Theorem 3.1 becomes, by taking 0 = 8" to upper bound Er,
( pX
(Qrrr Pn. .urr
Katr Yar O.I.T: S (1 min KL (o
Yoir
Yorr 6.:TI
KL
P 8n.5
+ Epy.
KL KL
+ c2 + 1 Dd.73
(3)
for any 7 >>0 O. the following corollary, we assume that the coding backward kernels are chosen such that they are good approximations the backward decoding kernels in Kullback-Teibler divergence.
H7 There exists e > O, such that for all E there exists yp E + such that for all your E Y1+11
KL.
Se
and for all 1 TT
KL (Q Bo.t-1je SE.
Corollary Assume there eaists 6* E θ such that PD Pp. Assume moreover E7 Then under the same with probability at least.
assumptions as in Theorem 3.1 for the constants co, Cr, C2 D in Theorem 3.1 1 -ceep(l-c (o, log mlnna)) for any 7
KL (p) p PX PT.T +Epy KL
(Q)a Yor 0:T
o Yor
Pn, 0:TTIT (L+N)[++)))tt))))>)) Dd.
log d. (l)og)))
P..T.:.T
When the data distribution is given by state space model, Corollary provides upper bound for the Lullbeck-Liibler divergence between the data distribution and its estimator and between the variational postcrior and the estimated state space posterior distributions. This result sheds additional light on the quality of variational reconstruction in state space models with respect to (Chagneux el al., 2022 Proposition 3). In (Chagneux et al. 2022 Proposition 3), the authors provided upper bounds on the error between conditional expectations state functionals under the true posterior distribution and under its variational approximation. In both settings, designing coding backward kernels that are good approximations of the true backward decoding kernels is enough to obtain quantitative controls the reconstruction error.

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Proof. The result follows from equation 77 and the fact that for any E θ and yp E d, for any your,
KT. (Q%
No.r
20 4.0:T
Σ
KL p,t-
BB.t-ite) + KT. (
(Q%17 pp.
T
3.3 Applications
In this soction, we consider generative models where the transition kernels and emission distributions are Gaussian in various classical settings. We show that under weak assumptions on these models, some as- sumptions of main results hold. Establishing that all assumptions are satisfied in general settings, i.e. without very specific assumptions on the architectures, is more challenging problem.
We prove in Appendix G that H holds in particular for compact state spaces. We also prove that the functions Mer7 are upper-bounded explicitly, and that dleer and 620.6-1 are lower and upper-bounder ex- 1,,,pp
0,t-1|e
plicitly. This allows to obtain explicit constants in H4 Providing additional comments on the assumptions requires assumptions on the observation space or on the dependency of the variational distributions on the observations. When the observation space compact we can also obtain a uniform control with respect to the observations of these upper bounds which crucial check and H6
Gaussian backward kernels with dense networks. transition kernels and emission distributions are Gaussian and parameterized by dense networks following
We consider a generative model where the
Scction 2.3
For all x X, x' + mo(x,x') is the Gaussian probability density function with mean po(x;, and variance Eo(r) where (Ho(x), En(x)) MLP*(x) with MLP a dense Multi-layer network with input x and weights given by 0. In this case, if the output layer of MLP is such that is bounded and I 2n (z) < E (i.e. (x) - E and Σ - Mn (xx are positive semi-definite matrices) for all x E X, then there exist constants c, 2 such that for all x, x' E X,
cexp (-Xx x) ma( 2) Texp (-Aa(z))
where X is the smallest eigenvalue of Σ and X is the largest eigenvalue of E and where
a<x) = = ((1|=) - M)
1 a(x)=
2= 1
+ (||= - ||z]|sm + (M m (mlgg))
with m infaex,eee [1]) and M suprex,oce [[) This implies that H holds. In order to check F3 if we assume also that for all x E X, + p0(x) and + Eg (x) are continuously differentiable and that θ compact then there exists M such that for all 0, E θ and x, x' X,
|mo(x, ma (x,xx))) s M(2,2')100 0'112
c We can check H4 for log fft- as other items can be verified following the same steps. Assuming that M (z, is Gaussian probability density with mean (o (x) and variance -0,4-1 (z). Wo. -1 Under similar regularity assumptions on the networks providing M8 (x) and - (g (x),, when
is compact, H holds.
For all 1 s x' - go.rr g - (x, (x,2") x') is the Gaussian probability density function with mean " ()]=)1((()) and variance y - (x) where (1 (z), E (x)=))(())) (x)) = MLP 1-11 (x) with MLP t-1le dense Multi-layer network with input and weights depending s. In this case, is s00r -t11
the output layer of MLP is such that your is bounded and W0.7 I to t-1lt
(z)) (=))= 2-11/
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(i.e. (E (- (2))- -
i/t and 21-1|t - (=x))-1(((2))) (z)) are positive semi-definite matrices) for all
a E X, then there exist constants M0u7 1|t 1-1 such that for all a, x' E X,
N0-7 exp 1/-
1 Tx) S -p i|t (e', S e-jj exp (p)x))
where 1-ilee is the smallest eigenvalue of 1-1 and 1/t is the largest eigenvalue of E where
2/-1 and
B(x) 1 [||=| M[e1 ) 10 ([|=]] 10m0-mmm---- m z-1ee
2 || | "oor
+(M 1-1lt m-jpp 22
vour m S (
with mi-ije inf (1]=1))())) and Mi-ije supzex 11 (z)]l- Similar assumptions can be used for your using dense neural networks with bounded output. Under similar regularity assumptions on 9p.7 p,e-11e and E 1/c-1/c than for μe, and variance Es, we may prove that H3 holds when & is compact.
Gaussian backward kernels with recurrent networks. A natural parameterization is also to use recurrent neural network which updates an internal state (s1)120 from which the backward variational kernels and filtering density are built. For all & 2 0, define 8; = (N- where RNN is a recurrent Na.r (x, x') be the Gaussian probability density function with mean f0:r neural network, and let x' + 4
and variance - where (i) (s) If the network MLP is bounded similarly as in the dense neural network case, then the backward variational kernels satisfy HD
Functional autoregressive models. The discussion neural networks also indicates that the as- sumptions can be verified for some classical statistical models. Assume for instance that X = R and that for all 0 E O, x E X, x' + +ma(zx²') is the Gaussian probability density function with mean fo(x), and variance (z). Then, H[ holds for mg when -00 < infzex,0ee fo(x) S supzex,eEe folx) < oo and -o < infzex,eee (x) I supzEx,ace (x) < 20.
Assume that at each time t 2 0, Y, hx(())) + 6,, where (5ehe2 are independent Gaussian random variables. Assume also that hx()) MLP (X,) where MLP" is a dense neural network with bounded output layer, then H2 holds. Assume that for all E X, 0 he(x) is continuously differentiable and that O is compact, for all y E Y, there exists G such that for all f, 0' e O and E X,
Gaussian emission densities.
(x)(x) g (z) S (*())) 8'112.
which means that H8 holds for the emission distributions.
4 Discussion
In this paper, we used backward decomposition of variational posterior distributions to propose the first theoretical results for variational autoencoders (VAE) applied to general state space models. Under strong mixing assumptions on the state space model and on the variational distribution, we provide in particular an oracle inequality and an upper bound for the Kullback-Leibler divergence between the data distribution and its estimator.
Although these results are the first theoretical guarantees for VAE in the context of state space models, we believe that this is the first step to solve challenging open problems. First, in order to cover wider variety of applications, weakening the strong mixing assumptions, for instance using local Docblin assumptions, would be very interesting although it is a challenge when analyzing the stability of smoothing distributions. Another research direction is to understand how our results can be extended in settings where the observations are processed online, i.c. cases where the parameters are updated when new observations are reccived but never stored. To the best of our knowledge, online variational estimation has recently been explored with methodologies but without any theoretical guarantees.
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Andrew Campbell, Yuyang Shi, Thomas Rainforth, and Arnaud Doucct. Online variational filtering and parameter learning. Advances in Newral Information Processing Systems, 34, 2021.
Olivier Cappé, Eric Moulines, and Tobias Ryden. Inference in Hidden Markov Models (Springer Series in Statistics). Springer-Veriie Verlag, Berlin, Heidelberg, 2005. ISBN 0387402888 0887402640.
Mathis Chagneux, Elisabeth Gassiat, Pierre Gloaguen, and Sylvain Le Corff. Additive smoothing error in backward variational inference for general state-space models. arXiv preprint arXiv:2206.000 2022.
Badr-Eddine
and Pierre Alquier. Consistency variational Bayes inference for estimation and model selection in mixtures. Electronic Journal of Statistics, 12(2):2995 3035, 2018, doi: 10.1214/
18- 18-EJS1475. URL https: org/ 10. 1014/11111111 E8JJ1447
Max Cohen, Guillaume Quispe, Sylvain Le Corff, Charles Ollion, and Eric Moulines. Diffusion bridges vector quantized variational autoencoders. In Kamalika Chaudhuri, Stefanie Jegelka, Le Song, Csaba Szepesvari, Gang Niu, and Sivan Sabato (eds.), Proceedings of the 39th International Conference on Machine Learning, volume 162 of Proceedings of Machine Learning Research, pp. 4141-4156. PMLR, 17-23 Jul 2022. URL https: https:////cceeedng...o.m mlr presss/v62/ c2bpe/2ccb.b..... html.
Yohamn De Castro, Elisabeth Gassiat, and Sylvain Le Corff. Consistent estimation of the filtering and marginal smoothing distributions in nonparametric hidden markov models. IEEE Transactions on Infor- mation Theory, 6388):4758-4777 2017.
Randal Douc, Fric Moulines, and David Stoffer. Nonlinear time series: Theory, methods and applications with ezamples. CRC press, 2014.
Pierre Gloaguen, Sylvain Le Corff, and Jimmyy Olsson. A pseudo-marginal sequential Monte Carlo online smoothing algorithm. Bernoulli, 28(4):2606 - 2633, 2022. doi: 10.3150/21-BEJ1431 URL hRttps/////oo org/10. 25511003150/21-B1111
Jonathan Huggins, Mikolaj Kasprzak, Trevor Campbell, and Tamara Broderick. Validated variational in- ference via practical posterior error bounds. In International Conference Artificial Intelligence and Statistics, pp. 1792-1802. PMLR, 2020.
Diederik Kingma and Jimmy Ba. Adam: A method for stochastic optimization. In International Conference on Learning Representations (ICLR), San Diega, CA, USA, 2015.
Rahul Krishnan, Uri Shalit, and David Sontag. Structured inference networks for nonlinear state space models. In AAAI Conference on Artificial Intelligence, volume 31, 2017.
Wu Lin, Mohammad Emtiyaz Khan, and Nicolas Hubacher. Variational message passing with structured inference networks. In International Conference Learning Represenlations, 2018.
Joseph Marino, Milan Cvitkovic, and Yisong Yue. A general method for amorlizing variational filtering. In Advances in neurol information processing systems (NeurIPS), volume 31, 2018
Sokhna Diarra Mbacke, Florence Clerc, and Pascal Germain. Statistical guarantees for variational autoen- coders using pac-bayesian theory. Advances in Neural Information Processing Systems, 2023.
Jimmy Olsson, Olivier Cappé, Randal Douc, and Éric Moulines. Sequential Monte Carlo smoothing with application to parameter estimation in nonlinear state space models. Bernoulli, 14(1):155 179, 2008. doi: 10.3150/07-BEJ61160 URL //L/
Rong Tang and Yun Yang. On empirical bayes variational autoencoder: An excess risk bound. In Proceedings of Machine Learning Research, Conference Learning Theory COLT 2021, 2021.
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Published in Transactions on Machine Learning Research (06/2024)
Arash Vahdat and Jan Kautz. NVAE: A deep hierarchical variational autoencoder. Advonces in neural information processing systems, 33:19667-19677 2020.
Sara de Geer. Empirical processes in M-estimation. Cambridge University Pross, 2000.
Martin J. Wainwright. High-dimensiooaal statistics, volume 48 of Cambridge Series in Statistical and Prob- abilistic Mathemalics. Cambridge University Press, Cambridge, 2019. ISBN 978-1-108-49802-3 doi: A non-asymptotic viewpoint.
10.1017/97811000 198627777. URL http:s///aai..org//00000///11111
Yuan Zhao, Josue Nassar, lan Jordan, Mónica Bugallo, and Il Memming Park. Streaming variational monte carlo. IEEE transactions on pattern analysis and machine intelligence, (0(1):1150-116)) 2022.
A Notations.
In the following, for all measures À and 7 (X, and all transition kernels K consider the follow- ing nolations. For all measurable sets A c X x X, À & v(A) [1(x, (dd))))(dd) and X @ K(A) (= ('))(dr))((())) dæ'), for all measurable sets B c X, XK(B) J (dzr)]n(r') (x, dr'), and for all real-valued measurable functions h on (X, X(h) = M(d))/(x). For all measurable functions h1, h2, we write h @ h2 xx) + hı (z())/((z').
For all a > 0, define on 12, the function taa : x exp(x*)- 11
For all real-valued random variables X, define the Orlicz norm of order by
| = infAso { (((()))) 1
For all probability measures P and Q defined the same probability space, |P Q1iiv will denote the total variation norm between P and Q.
B Assumptions
In this section, we give the precise setting of the assumptions.
H1 There exist probability measures and n on (') and constants 0 < a < <0 < oo such that for all E O, x E X, all measurable set A,
(A) S ((A) S (A4)+(A)
and
a_) S Mo(a, A) < (4)+(A)
There exist probability measures A_ and d on (X. such that for all your E YT+1, there exist gler and yoor such that for all yp $, l 0, E X, all measurable set. A,
Horr X (A) &
Q 4,1 (z, A) < @o X (A).
In addition, for all 4 all you E yT+1 and all measurable set A,
yon. x_ (A) HoeT (A) S yuur X4 (A).
H2 For all Y, inface f (z (dx) (v) 0 and supace f (x) c4(v) <
We consider also the following notation suPace nh and infoe = g"
11

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Published in Transactions on Machine Learning Research (06/2024)
H3 There exists M such that for all 0,0' θ and x, x, o' E X,
Imo(z,x') - mo (2 2')) (x,,2') 112
For all 1 1st t S T, yo:r: there exists K Ki-jj such that for all p, o E & and x, x' E X,
pp.t- A (I, - g 11 (2. x¹) S Ki (z', x)|l - 4'112.
your
In addition, there exists KYo such that for all 4: q' E o and Z ZE X
(I)
No (w)) ky (+)]lo 42112.
For all y E Y, there exists G such that for all 0, 0' E θ and x E EX,
(x)(x) - 5 (x)) s (* 112
Define, for S 1,
h ho _ (T) = log p0r 1/t (Te, 11-1) - log 0,,0-111 1/1 (ze, 22-1)
(4)
and, by convention, 1 (1T-11 2T) = log 200 T 1|T (2T,, (0-1, - log b T
8 1|T (2T, 11) + log p (xT)
log o0 (IT).
H4 For all your E YT+1 and all STSTT
|/>((lee)
(cc:)) :>/
sup
yoor 1 (z.;
= =0g yo. < <00,
and for all 0, o E O p. E 1stST
/ A+ @ (dzzrr)) |log 2 (2, - log 9 1- 1/1 (z, 22){
S
c1,
No.T 1|:p - 421122 ,
f *+ a+ (ddrdd))) boog b ( (x)) log 1
1 (3r, 2²))
110 -0112
2,
/ (+(( log no-T (x) log To
l r (3x)
<
Mor 3.7
||p - 9'112
i X+ (dx) log oyor (x) log " (z)
Nor
||0 0
where A is defined in [
H5 There exists A such that the following inequalities are satisfied.
E
Your Yo.r
27
S A, E
Your Yo.r
)1 SA,
27
0,7
for all 0 O
m2 (Y,)²
(( (our) 27
(2 )9
< A, E
S A,,
for all 151ST,
(w) (oNorryaaat-a yot-)) 21 )
7 2 (M @ gir-1 V29
E
< A, E
AA,
c_ (Y, /cc (Y,)²
12

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Published in Transactions on Machine Learning Research (06/2024)
T
27
yor
Σ
x 8 1 (K 2 (o S
where for all yorrs p(yarr) = 1- yuur for all 0 ST,
C+ (Y))PuGG)))
c- (xx))cc__Y SA
and for all 0 SSTST, all SSST.
c (Y)()) & p(M (ann)+nuMMMMMM1110 27
E
S S..
c- (- (ooa-(())) c- (V,))
(V,)c-
H6 There exists a, and B > 0 such that for all T 1,
po (lo.))ll S BT and
(0Xor) supo
Σ
T
A 8 A+
)
S BT,
and for all 0 OSIST,
log c+ (Xi) V co S B.
C An oracle inequality adapted from Tang & Yang (2021)
We propose an alternative formulation of Theorem 3 in Tang & Yang 2021 in which we provide the precise behavior of the constant the variance term. To avoid introducing too many new notations, we formulato the results of Tang & Yang (2021) choosing the observation to be Yo.T, the latent variables to be Xor in our setting.
Condition A. Vorr YT+1,
There exist a and function b such that for all O, O' E O, x o o o
I (0, - (0', sp, )l s ((vor)))(( pp) (')l2,
with E(b a1.
Assumption A.
There exist ax > 0 and D > such that
supe {
L Your (0)
Yor ) 0,T
{108
+ KL. (2
DD.
(5)
PD (YoT)
Thcorem C.1. Assume that ④ and are compact spaces and that the sum of their diameter is upper bounded by do- Assume moreover that Condition A and Assumption A hold. Then, there exist constants co. Cı, which depend do, a1 and as,, and universal constant cz, such that with probability at least coexpl-ci(dd log mllaa.),
/ To (0n,T, Pn, Vo. worr)po(vo:))d(mmo & inl { (1 (1+ (1+ c2(1 02(++ a1 Dd. (logg
where Ex mingee, (00, yo. (
and d. = do + dgp
Theorem 3), in which we track the dependencies of the constants with respect to 01. In (Tang Yang 2021 Lemma 14), a multiplicative term vai is required on the r.h.s. of the incquality. Then on page 24 third line the inequality needs again /a, on the r.h.s., and the end of the proof follows by multiplying 5, by Vaj. We obtain that in (Tang & Yang 2021 Theorem 3), their constant 62 proportional to d1-
Proof. We follow the proof of (Tang & Yang, 2021
13

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Published in Transactions on Machine Learning Research (06/2024)
D Proof of Theorem 3.1
First, Assumption A of Theorem 3.1 holds with D DT for some positive constant D depending on B.
This is a consequence of the first point in H6 Proposition E.4 and Proposition E.5
We now prove that Condition A Theorem 3.1 holds i.e. that for all 0, 0', 4,, 4', Vo.T.
100,0' ;4,,p' 4 yo:r) x (0, 4,, Vo.r) w00, ,p', yorr)) ((yo:T)]((0,))) (p) (0', 42)l2
with S CCT for some C >>0 0. Write, for all 0, f1, 22, yo:r:
log P:2 (Nor) Dn.r
Moor
v0 (0,41. (1.1))
toir E
0,0:1 (Xo:T)
Note that
A 0', 4p,4' yo.r) $ 1
poo:T (0) pluer (8')l + Vo. -
- (0', 42)1
_0..
Write
Ego (0,4) EVOIT (0', ,'') s (0, sp, vo.T) + : v0.)).
where
A (0 pp (9. -0 (0, 00,0090 sp) (0,
A2 0', 11, v0 HO:T) = 10 (0, 10 (p) - -. (0', 1p 4/)l
Therefore,
A00,0', p. yorr)
1 10 (0)
eqp (0') A1 (1 40 vp', yorr) (2 0', 90, sp Vorr)
By Proposition
E.1
Proposition
E.2 and Proposition
E.3 get that for all 0, 0", sp, sp,, and all yorr.
A0 0', 9, v', M)rT) S (310))) + (x())))) |0 - 0'112 + (x2(V0r)) + ka 110 - 4'112,
where
Ki (1(:T))) = 0_c_ (yo)
(G
+ Σ of (vr) t=1
c+ (L + 7+ @ M(M 9t *ggh)
+ n (G
(6)
(_c((1r-1)
with M 9tt 2 ld (x, x0) M(x, D' (z)g") (x'), and for all t,
40² 1
L2(Yo))=
Σ g'
(ys)s (0-c))))) 7 8 (M g" 1 0 u(Gu))
1
(7)
0²
==
with E = 1 - 0-/0+-
T
T
N2(2O.T) (8
2
yorr
Σ
d
)((307))
s/s+
(8)
T
k3(yo:r) = O
gle
youT 1.1
+ vo.r
(9)
t=1
and
<a(yo:T) gy0.r
gun.r
20st 2
1
(10)
in which e. 200. 2,t 3.T and d0e are defined in Using H5 it is easy to prove that Bbs, (Vo.r)
E[Ke(Vorr))], ()ss())) and BiK, (Vo:r)? are upper bounded by 2 for constant that depends only a+, a- and A, and Theorem 3.1 follows.
14

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Published in Transactions on Machine Learning Research (06/2024)
E
Additional proofs
Proposition E.1. Assume that hold. For all 0, 0' E + and all your E Y
g (0) f7 (6)) S K1 (1(oor)) 1- 11122
where
k1(Yo:))= (G
O-C- (y)) E -oc( { ( n 8 MMMM- gum
T a+
+ 17 (G
O-C- (y1-1)
with M. gue-1 30 (x, x²) Ma, x' jiut (x) ( (x"), where Tu-1 defined in Lemma
F.2
Proof. For all 8, 8' θ and all your E YT+1, with the convention po(volv-1) = po(()))
C (0) eT (6') = Σ
(log po po (ell log po (Velvore-1)
For all / O,
Po (V-1B000-1)= =
dxc))² (xx)
Note first that
po(2ol30:011) 2 c
so that
1 (0) -00 (o(()) Po (yo))
po Po(velNoa-1) - po - (uel3oa-1)
C-c- (20)
C- (1))
For t == 0, using that (yVo f x(d2z)) 9 " (zo), Assumptions and H3 yield
(po po (yo) ++7+(Gu)) 1l
In addition,
Po (e(VlBo0e-1) - per (ye (ollb)-e1)) =
( 10:0--1
(dxr-1) (dtre-
Mh (det- Mo Mo( 1, dxx)g (xx)
0,2-1
+ ( (M dxx))
Ma (Iz-1, drx)) 96 (Ie)+
1 - (dx (2 dee) (g (xx) g6 (xx))
Using Lemma Assumptions H and
we gel.
(Po (o
po (ye (0ot-11) s (20))(0) 0 $ 0000111
+
0+
/ 11t 8 (C) + (G+)+ 110 -0'122
0_c_ (vh-)
The proof is completed by using Lemma
Proposition E.2. Assume that ! hold. Then,
(1 4, s No.T) & (2(U0.T)]l - -
where
9..0:11 (Xo:r)
2 (Xo:r)
A (i s4 sp, Jo:r) =
E
log
E wour Baggr
log
@ 00.0.T T (x.T)
No.T
20.0-TI (No:T)
with p(yO:T)
g and
T
T
R2 (2(Uo:T)
(D)or))
Vo.T
Σ s=t-1
d 8 (K
alo+1 (x330o)))))0000)=" )((30)))()
t=1
15

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Published in Transactions on Machine Learning Research (06/2024)
Proof. For all pp 4, 0t,0stsT 1, define
2 p.,0 Ap (To:T)
2..7 (IT)
II u=T
&, 1/4 (Ta, Ta Ia-1) II
Lg 10',u-1/u (Tus Iu-1)
a=t
f+2 II
UpT
(ITT)
p
(Fus Fu-
II um++1
ap' 1|u (T Fn-
u=T
with the convention II= ap.a 1---1/u (In Iu- and I1 ap ia
(Tus fu- 1, and for t T
N 2/p (M0:r)
4g.r (xT)
II
pp 29'u-1/o 1/1 (Tis Fu- Ip (ppp11aa
(XT)
II
Lo 1/u (Tus Ju-
o=T
u=T
Therefore,
A p0, (o:T)
E vo-T
(Xe (Xe- 1, x,)]
E Bopor 200000
h (X (Xe-1:
E your 2. aa12
hoorr
(x
where
S S T, are defined equation
Note first that if > 8 + 1, then
E ggoor 1117
(1.00p (X,-1, X.) = 0 so that
T
T
A1 s,, 9/, No.r)
=
Σ
10. (x Xx) (, s=1-1
E
(=1
For all t S s + 1, write for all measurable set A,
P.9 (A) (A)= / [A(())) Fo-I (#T)u(dZT)
II
2 Tp 1/u
(Fur Fu- (u-
==T
No:r , (A) /
La(xx)
N0. (oT)u(deT)
II u=
p 1/l (Tiu Dn () 1) 1s (1o41, xss)addx.)
Therefore,
B F
Norr (X, (X.-1, =
yo.T P4,8
Q II
Jou - 1/1
Q -
MorT
Using 1 the backward variational kernels satisfy a Doeblin condition, see Doue et al. so that
2014 Section 6.1.3),
E M Sppe"
1 (X, (X,,1, x.)) s 2
Q p 1-1--10
where for all measurable functions S. osc(s) sup (fxx) f(x'))- By H1 and
H4
osc (Q Toir 11r 1100
Vo $
/
- 1e-- 21 ( 1 Tour (x )u
yoir
//
< 20y0rr
(x ب (d
< 20 uour.
16

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Published in Transactions on Machine Learning Research (06/2024)
Noting that by H3
[++
1u -u al S p..T
2 II
8.0- 1/5 s 11:p 4'1l2 S (0 @ (k 11p - 121122
K T0
=
concludes the proof.
Proposition E.3. Assume that 1 hold. Then,
(2 8² ,, sp, yorr) I ks((or)) 1lve 40112 + nia (a((yur) 110
8 1l12
where
youT log
0..0:r ((o:T)((o:T)) 90,0:T T (Xo:T)
Tett Uo-T N0.7 Per 00:TT T (Xor))
(Xo:T)
02,0,'', (o:r)
R vo-r 9gl.0:7 a..
R vo: o0er
log
with
k3(yO:T) Shoer
fuuer Σ
Mor + 3,T
and K4(yo:T) Duur
plueT
Muse1 + c/,6
2,6
and where oue,, 22
NoeT 3,T
and ch are defined in 44
Proof. By definition,
42,0:r (Xorr) your Pon-Ty (Xo:T)
No.r
log 1p0r0:r .0:2 (oor)
voer
A2(0, 4,, .u:r)
E Tour 10e7
log
E, Vo-r
Vo.T or 0'00: (Xo:r)
( (XorT)
No T Tet 00:T (Xo.T)
18 yo-r .o.r
log
log
1 yo.r 110:TIT (Xo:T)
No.T & W.:T (XO.T)
vo:r 20.0T
1 (X X,) - 1,8' up" (X Xo)
1120r "1,8ip
Ma.
where h0 1st T, defined in equation
For t T and all Ft-1: E X,
(,8.p (x)-1;
your (Iz_1, xx) s log 9 ( xt-1) log p - (Ie, xc-1))
log - (t,, log 1 Tt-1)
Using
and H
E po-r 9 oar
log - ( x(-1) - log 2- (Ie, 201)
(0 / 2+ @ ) (dldd') log your
1og 1/1 (x, log g (z, (2,
< (020:ry2 114p - 4'1l2
Similarly,
[og 12001 1 (12,, x4-1) - log uo.c-1 1 (xt, Fz .1-1)
X4 80 X+ (doda') log gooe - 1 (x,,
log b f (2r, a²)
< (g (
y01-1 110 0'112
For T,, remains bound Egr which is forward by using H and
log No.7 (XT) log No:T (XT)) + log 6 (XT) log
Mo:7 (XTr
o.r
17

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Published in Transactions on Machine Learning Research (06/2024)
Proposition E.4. Assume that 2 and HD hold. Then, there erists such that
log T. (0)
sup AE
SCT
Vo.
Proof. For all 0 E θ and all your Y with the convention po(voly-11 po(u)),
T log po (v
log T.7.
No:T (0) = - (0)
As pa(())= f x(drro)g (2o), by H1 (10) S pe())) S 04c4(30). For all > 0,
/
Po (vo1
1b 1 (dx;=1)Ma (2) (xx))
so that by 2 0-c (ye) $ (oollBoo & C+C+ (3r). Using the second point in H6 and the triangular inequality concludes the proof.
Proposition E.5. Assume that 1 and HC hold. Then, there erists B >> 0 such that
pu KL (0
P0.T
Ya.r
BT
Proof. For all 0 O, p E b, yo.r E YT+1,
0.77 (Xo:T)
Ho. KT.
100.7 (O.T
B.,, yorr
log
0,0:T T (Xorr)
Must
your 4p.00
N (Xe1,,
where Vo.r 1,
1 S T, are defined in equation
By for all SIST,
E. No:T [N:.0.7p (x X²)
S (100 )2 A+ (1220rp)
which concludes the proof by 16
F
Technical results
Lemma F.1. Assume that 1 and 2 hold. For all O, all 2 0, all you E YT+1, positive measurable function h,
n (18' h))
0+ (9% h))
S T (h)
C- (V()
Proof. At time O, we have ( x x(dro) (zo) so that by F12
a_ 11 (46 h)
T2 (h) - 04 (10 h)
O-c (yu)
Similarly,
2 (dxx) xx 9 (Ire) / D (.
so that by and H2
a a- < 0,5
0 0+ 0+1 (g c_ (3:)
(h) -
04c(
18

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Published in Transactions on Machine Learning Research (06/2024)
Lemma F.2. Assume that H
22 and H
hold. Then, for all 8, 0' E t 1,
Le( 11
bv
where
Lz(00:) = 40
{ {
2 p (g @ MM + 7 (Gbb)
c- (va)
lo
==0
with = 1--1--1- 0-/0+.
101-0-10
Proof. The proof follows the same lines as the proof of (De Castro et. al. 2017 Proposition 2.1); which was in the setting a discrete state space. For t > O, note that $0 (dre) 36 (ax) f
dxx)//o, (Vooe) where co,+(Vor))
J (ar (.))) 0,t-1
dzz)).
Consider the forward kernel at time t defined, for all E O, all Y1 Y, E Rd and probability mea- sure by
J mola' 22)9 (x)' (x) J mo(a', "ma("',""))) x²)9 (dx))dd))
Therefore, D. = Fg T 0.1-
and for all 0, 8' E,,
T
Vo.
Hh 0t
F F Moie-t 01 8,1-
Fill T Mo:r-1
0,t
1,s (Veit) + FV T Tunee-
F 0
T
where
(Us:)) = Fot
F 10, 00.
20:2
Fb ... F 20
0,0+1
with the convention F Φy 0 Lo Consider also the backward function ( and the forward smooth- ing kernel FVoa defined by s|t,0
gM++sit (xx) = f M1( dr.- nh (I++1) ... Mo Ma(a,-1,d (z.) Peeelt
gle+ (x, (2z))m((ss-11 2.) (x.)
(x
I 3 3 (arr""((MMMMa) (z) dz) (x)
Following for instance
CCpppé
2005
Chapter 4), we can write for all probability measure 7,
10.417 = To.ele Doa-ti p 0,0
F 0, 0,
where Vo. & 1i Therefore,
T Tae
T Most
(0 oole 0,0+1le" Flott o F Foo+1ie"
Fite
M0.-1 0
Fdt F T 11
0
where ole x 08o+in Fy,
a 00 and fo. x gBet1 -. Note that by H1 for all measurable sets
I (dr.s) (3 (z)gg (x)
(x A)
a+ I n4 (dx))) 3 (x)9e Vo (x)
that
100.0 F.c. 0,s+1/e" ... 1.-11 30.000000 E2t
p 0,41
55
70.00r. -0.00.
0.
1/s
19

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Published in Transactions on Machine Learning Research (06/2024)
with 1 - 0-/ot. This yields
yau 0,t
2 o'
c N0,00, - -0,00,011llo +
Fot 00
F. 2 20:1-1
0², 0-
For all bounded measurable functions h,
J Po,ate Pr 0,8 8¹ .- (1r)))())))))))
p. T Ma.
00 (3r, (o.)8000 ( (1s))())))))dda)))
hoo .te(h))))) - v0,00
(a/((b) -r
13 (Irs) F. t 0' -- (d)r))
I 1 (x,)): ( (5s))((darr))
O.alt
0,,00,1 (h) + 11.0' (h).
where
Boole (Irs) Fd 4 Vo.s
.0- (xx)) F 2o 00 (Ix)) ( h(zs)p(dz.)
guo. 0,00, (h)
J B (1 P
p 1 b0 (I (dd)))
S 1ggttr (Ie) 1 (((r))
Po (xx))
Fyg W
s- (I.) F0
- (Ix) u(dx.))
(h)
0.0'
S 0 (I') 2 (I.) (d(dx))
10 (xx) F 20 (I, (((z))
Note that for all x, E X, by HI
a- / n (dd.s)))))) 11 (I.-1) ... mo( (xe))) (z))p(dzs+22))) 00 (x*))
$ s / 1+(xx4+1)9) geti (x++1)))))))))))) ... (1)(===1,40)99 (#:)p(dz+42.)
so that
|3
0, (h) + 50 0,0022 (h) < 2//////c || 1: 20001
4 20
6'
bv inf 2
gy (2s)
8,
$ 2
| |//loc 0.5 0
-F
$ --1
00 1l
For all bounded measurable function h,
Fø Mo F. 4 20:0011 h
S R1 Rs
where
(mo(x', x)gg (x) - mo (a', (.,.))ga 2) (z)) 2 (dx' (dz))((z)
R1
S (mo('', x²)g"" (a") 2 0f 9-1 (dx' u((d"))
ma (a', 99 (2") x (dx' (dz')/(zz) (4(2")))))))))))))) jd(z"") I mai (x', xd' g9 (I") D - (ddx))((xx""))
(molaz (x', x² )0" (z") mg (a', x²))9"7 (z")) D 1 (da') (((a")
Ra
I ma (m' x² (x") (0 (dx"))(ddz)
By 113 and Lemma
0+ R < d_c_ (va)
o_ c_ (v.11) n @ (3 3. MM + (G'%) 110 0011 17
The same upper bound can be obtained for R2
f mer (x', )% (x" D (dx' (dx')h(z ((({=))
||h|| oo
f mmg (x', g" (x" (b")) (dx'))((d""
This yields
Fgg &
No. lov 5 0-c (y/)
20
1/ (g @ 9th M) + n- (G 110- 0'
0 c (
(y--1)
which concludes the proof.
20

---

Published in Transactions on Machine Learning Research (06/2024)
G Checking assumptions
In this section, we provide additional assumptions the state space model and on the variational family to support that our assumptions can be verified.
A1 There cxist constants a- < <04 < oo such that for all x e X,
0_ S 5(2) a
and for all 0 e O x, x' EX,
S mm( (I, x') S 0+
For all your E Yr+1, there exist yho > and tut > 0 such that for all E 2, t 0, all x, E X,
your I your (I,, x²) S y00.
In addition, for all E 4, all your E YT+1 and all x E EX,
fuer S d (x) S plu.
Assumption AT is known as a strong miiingniiingggoooo assumption and allows to verify H[ quantitative bounds on approximation of joint smoothing distributions, see for instance Olsson et al. (200S) Gloaguen et al. (2022p. It typically requires the state space to be compact. In scttings where the bacwkard variartional kernels are Gaussian and obtained with neural networks which uniformly bounded with respect to the time index and the observations, 000rr and yaarr do not depend the observations.
is classical to obtain
A2 For all E Y, infece J (z) c_ (y) 0 and supace J 9 (=) c4 (v) < 00.
Lemma Lemma G.2 and Proposition G.3 allow to obtain explicit constants in H We prove that the functions N0 are upper-bounded explicitly, and that daor and 1 1 are lower and upper-bounded 1,t-1
explicitly, in particular with respect to the obscrvation sequence.
When the observation space is compact we can also obtain a uniform control with respect to the observations of these quantities which is crucial to check 15 and 6
Lemma G.1. Assume that 1 and hold. For all 0 E O, all 1 2 O, all your, lt.
96 (II)
0 -+9 (21) σ T_ c_ (111
yo0 (2,) <
0 +.c4(1))
Proof. At time 0, we have (x0) x ((ro) (Ho) so that by A A
96 (xt)
0+ (ar) c_ (vo)
I 22 <
T+ c. (16)
Similarly,
(x:) x g (xx) / 2 (dx;- (z,-1,
Mont 0
so that by AT and A
σ 0- M1 (xr)
T+ 1 (Ix))
< 0,t (ax) -
0 C+ (Vr)
σ c_ (1bb
21

---

Published in Transactions on Machine Learning Research (06/2024)
Lemma G.2. Assume that 1 and A2 hold. For all 8, all 1 st S T, all yo.r, 21-1; Dt.
2 90 (z1-1)
I 0 0,1-111 (Tt, x2-1) v oigg - (It-1)
1uou-1
o c+(
o² c- (i
and for 1 1S1ST-1
1130-7 1081pl0000 $ $llog 10 (yoo)_( V log 0+ (VO.T)
c2c (yt-1)gtt-111 (x)-1) c2 (Vt-1)
of 2(c (
+ sup xe-1EX
log
log c (yt-1)
and
8-9 (xT) 0+c+
v log -
T18,pplo S $log200
20 v log 2000
+ sup
log
xTeX
(VT)
o (Vr-1)g 1)g (xT-1) v log
(
sup -X
log
of c+ (yr-1)
c²c (YT
where h,,0,p SSIS,, are defined in equalion
Proof. By Lemma
-² o²10 pe-1 (Ix-1)
o2 Mt-1 (81-1)
I 20:t-1 (0-))))000)
0+ C+ (Vo-1)
σ_ (y1-1)
Since
Uh. ( (Z xto)
b 0,1-1/t [t-1)
J p (It-1, Te ) (cdx
we get
o2 c_ (11-1)g8 3/-1 (x
22 (1)-1)) 11-1 (Ig-1) < 1 (1it xt-1)
2 c+ 02(4(()-1)
0²c (2c-(yt-11)
Now by equation 4. for 1 s 11,
11 (II-1, x4)) = log gorr 1,1--11 (Ie: (Ee: x,-1)
log Mout
(2,, xe-1) so that
111-113
10:r 11 (x1-1; < llg log 4 (Vo.T))
o2 σ2c (11 1)99 y0-1 (x v log 22c+(86-199) 11-1 (xC-1)
+ log
o7 (fc+(11-1 (yt-1)
0²c-(1t-11
which concludes the proof. In addition, using that
hEOT 1.0,p (ET-1, IT) = log vou - 1/ (IT, IT- log b 11 (IT, xT + log 2 (ØT) - log 6: (IT)
16 (IT,
yields
T- ur (PT)
0+gh UT (AT)
Ao (XT-1, 12) < log 20_ 20-00000000 (VorT) )l V log 20 (14:z) )1 + log 0+C (Br)
v log
T_c_ o_c-VyT
o (UT (0r-1)90
1990 yr
(xT
oict (yT 11)00 yr ((T--1)
+ log
V
log
o. c+ (JT
0² c_ (VT-1)
22

---

Published in Transactions on Machine Learning Research (06/2024)
Proposition G.3. Assume that A A2 and H hold. Then HA holds. More precisely, for all your YT+1 and all 0 S S
/ M(dr)
sup EE
1 (Ix, :))
= =iter < 00,
where u. spe |1 mo is given in Lemma G.2 For all 0, E E 9 g Φ 1S1ST,
/ A M(dadz') [log 1pe (2r,2') - log your
your
Co' e- 1 (Ir, x')
S
Wo.r 1,t
11 - 4'112
/ A(dzdx)) log 08.-11 111-111(22, (2, x') - log b
o', 1- (I,x')
No:t-1
S
2,t
-
|0- 10-0'11 8'112
/ A(dx) loog 1
NorT (x) l vo.T (x)
q,'
- 5 110 4'1l2
/ >(dx) log & (x) - log &8 No.T (x)
S 2 10 - -01l12 :
No.
where yoer = (JODT)-1X > @ (KCTTT)) y0u-- = (0m)))-1((3)) ) and ct.7 = 20 --_ (UT ((( infrex gor (z)).
inf 9 (x)), 3,
yorr
Proof. For all p,pee E Φ, TSts,,
Ap.t-11e 11e (I, x²) - 200 -p-e
your
1/t (2, x¹)
log 1 (I, x') log H0.7
1-111 111 (2, (2, 5
2 9p.t-11r 1/r x²) A 0 - 1/c (I. x²)
x²)
so that by AD and HB
vour log 9p.t-11e (I, - log 2 2 x²) S (0
K ( (a', x)1 - 4'1l,
an we can choose ynorr = (D - 8 (K) Sitilarryy for all 42 40 E $
< (x) 20 (x))
Bo-T log o. (x)) - log
- (x)
your (x) ^ 0 (z))
so that by A anddd H3
llg 2 (xx) - log y0-r (x) S 3 K7 (x))((x>>>>>>>>)>>))))) - 10'l
and we can choose
uo- = ( ((Koo.)) For all 0,0' C,t
b
0 1/e (I., x8)
b f f 1 (Ir, x3)
1
log 6 gu (I, x') - log 1 (2, -
- 1|t (I,
^ 1 b/ 1
-e- i10 (I.
By Lemma G.2
1/r (2, x3) S 0f c+ (16-2)
log bg 1
i1e (1) x²)- - log 20 1
-² gt: (x(-1) - 11 (2,, - Most
Then, noting that og.=1 (2, write
0 (x mm x) (z) where co(x) ʃ 0. (x') (mo) pd)ddd)),
(0 (x') - (a'=
1,80.4 - _1, (1,) x²) - - 11 (Ir, .²) 5
co(x)
- mm (1' x))
( (x'))mrraa (x))
Mont
cpr (x) (xr(x) co(x)
co(x)
+
cx(z)
23

---

Published in Transactions on Machine Learning Research (06/2024)
By A
/ A & Mdzdr')
(00,t-((x')
gu (x' (c0))o(22)
00 10 (x (m)))
2 xnx g (())1(x)
o+
0,1-1
$
8r ,-
Pb:c-1
and by Lemma G.2
we choose Mois-1
22
20 Le-1 //(0////0/0000 inf znx g (x))..
For all 0,0' E O
2o:T (x) 0 (x)
log di (2:) log og
0,
5
P8T (x) p (x)
By Lemma G.1
0 04C+ (2T)
log 0,. (x) log 13007 (xx)
S
100T P0.
(x) - 00 (
gur (x)
Therefore,
f A(dx) log P0.r (x)) log du $2 T_ inf gor (I))
C4C+ (UT)
and by Lemma G.2 we can choose 00.
C4T c (UT) (T( infaex ((T(x))) (x)).
If the observation space is compact, under standard regularity assumptions, all upper bounds can obtained uniformly with respect to the observations. Therefore, HB holds as soon as the integrals under μ. n- @ p and X4 & X4 are finite.
24


Reviews:
[{"review_id": "yeNqEqMBOB", "reviewer": "Reviewer_78cA", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "## Strengths\n- The paper provides some theoretical guarantees for VAEs for state space models.\n- The results are novel and seem to be sound.\n\n## Weaknesses\n- The draft is not very accessible to non-theoretical audience. It requires a good understand of the literature to understand and appreciate the paper.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper studies VAEs for state space models with a backward factorization proposed by Campbell et al. (2021).\nAn oracle inequality for the risk in terms of the number of samples and the length of the observation sequences is derived under strong mixing assumptions and with independent sequence trajectories.\nWhen data is generated by a state space model, an upper bound for the KL divergence between data and its estimator under the variational distribution is derived; the bound is in terms of how the backward coding kernels approximate the backward decoding kernels.\nGaussian backward kernels with dense and recurrent neural networks are used to show that these theoretical results hold in practice.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "The presentation can be made more accessible for non-theoretical audience. \n- Perhaps adding a paragraph motivating (1) why/how these results are useful and (2) how they could guide algorithm design.\n- Following about, a dedicated related work section to explain what kind of theoretical results are obtained by previous work in similar/related setups can be useful. Following this, missing gaps (like what this work focuses on) can be identified.\n\n## Misc\n- `\\citep` should be used for the two references in the title\n- I didn't know find where $M_\\theta$ (in H1) is first defined.", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "zy69Agw8UB", "reviewer": "Reviewer_BwgR", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n\nThe paper provides a concrete mathematical foundation for learning VAEs, which is a problem of great practical interest. Due to the complexity of the setup and analysis, I was not able to verify the proofs, but they seem careful and sophisticated.\n\nWeaknesses:\n\nThe paper suffers from excessive notational complexity and abstract presentation, which greatly hinders accessibility and broader comprehension among readers not deeply versed in variational inference. I would like to note that part of TMLR’s acceptance criteria is that the work should be of interest to TMLR’s audience, and a prerequisite for this is for the content of the paper to be understandable.\n\nDespite the mathematical rigor, the practical applicability of the derived oracle inequalities is not immediately clear, particularly given the abundance of assumptions required for the analysis. For example, the analysis hides constants involving density ratio bounds, and such constants could be considered to scale exponentially in the dimension in many situations. The paper contains little discussion of how to interpret the bounds.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper tackles the problem of providing oracle inequalities for variational autoencoders (VAEs). Notably, the paper offers an oracle inequality under strong mixing assumptions for both the state space model and variational distribution.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "Notation: it seems that $\\Phi^{y_{0:t}}$ is used both for the distribution of $X_{0:T}$ given $Y_{0:T} = y_{0:T}$ (when $t=T$) and for the distribution of $X_t$ given $Y_{0:t} = y_{0:t}$, which is an overload of notation; could you please clarify?\n\nThe manuscript would greatly benefit from a simplification of notation and a more intuitive presentation of the assumptions. Instead of tackling the general case first, is it possible to strip the model down to the simplest possible while retaining features of the problem (e.g., the functional autoregressive model discussed in the examples section) and present the assumptions and result tailored for this particular example before moving to the full results?\n\nA section early in the paper providing context and foundational understanding of variational inference, especially for readers unfamiliar with the field, would be highly valuable. For example, my understanding is that the $Q$ kernels are taken in applications to be parametrized by neural networks. Could you clarify how $(\\widehat \\theta_{n,T}, \\widehat\\varphi_{n,T})$ is computed in practice? If the optimization is intractable, please clarify that this paper addresses the idealized setting in which the optimization can be carried out exactly.\n\nInclude a technical overview section that outlines the high-level ideas of the proofs, distinguishing novel contributions from established techniques. Currently my impression is that the analysis rests on standard methods.\n\nThe paper should discuss the expected scaling of constants involved in the assumptions, particularly in light of their potential exponential scaling with ambient dimension.\n\nTypos:\n- Pg. 2, \"using a mean-field approximations\" -> \"using a mean-field approximation\"\n- Pg. 2, \"given for all\" -> \"given for every\"\n- Pg. 3, \"ELBO is closed\" -> \"ELBO is close\"\n- Pg. 4, \"close to minimize\" -> \"close to minimizing\"", "broader_impact_concerns": "None.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "9gSWCBC8RE", "reviewer": "Reviewer_qS78", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n- The paper offers a novel application of the VAE excess risk bound to state space models offering some theoretical backing to these techniques.\n- The paper thoroughly examines the assumptions required to derive the theorem and discuss their applicability in common state space modeling scenarios.\n- The claims made by the paper clearly match the derived technical results and contributions.\n\nWeaknesses (requested change number in brackets):\n- The writing could be improved by making some notations clearer and introducing definitions where they are needed. (1,2,3,4)\n- There could be more discussion around the specific implications of the result in the state space modeling setting. (6)\n- It is not clear how the use of the backward factorization links to the stated result. (5)", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper applies the result of Tang & Yang 2021 to the case of variational inference within state space models. The result bounds the sum of the KL divergence between the data distribution over observations and the model distribution for the observations plus the KL divergence between the variational posterior and true posterior. The bound is based on the error introduced when optimizing the ELBO based on a finite number of observations.\nThe authors find that the bound has a similar form as Tang & Yang but depends on the cube of the length of the sequence.\nThe paper then considers common parameterizations for state space models and considers whether the assumptions of the theorem are satisfied for these parameterizations.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "None of these concerns are critical for my recommendation of acceptance.\n\n1) I assume zeta refers to the initial distribution but this was not defined.\n2) It is not ideal to have m refer to both the transition distribution and the single sample ELBO, it would be good to change notation for one of these quantities.\n3) In the proof of theorem 3.1, Condition A of the theorem is referred to but is referred to but is not actually explicitly stated until the Appendix. I think at least a reference to the appendix is needed here or putting condition A in the main text.\n4) P^Y is only mentioned in the intro but I should be given a proper definition as it is used later on without any explicit definition given.\n5) It should be made clear what role the backward factorization plays in the derivation as this is mentioned during the motivation of the work. Is the theorem not possible to derive with a forward factorization?\n6) Some discussion of the derived result would be nice, e.g. the bound depends on the cube of the length of the observed sequence. However, in the work of Chagneux et al. (2022), the error seems to depend linearly on the length of the sequence. Is there any insight to be gained from considering why there is this discrepancy?", "broader_impact_concerns": "I have no concerns of the ethical impacts of the work.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: After the revisions, all reviewers were in favor of the paper being accepted, an opinion that concur with myself.  The general consensus was that even though the entirely theoretical and somewhat inaccessible nature of the work will reduce how widespread the scope of its appeal will be, it is still sound and potentially useful work that will be of interest to some in the TMLR audience.

As there are no notable outstanding concerns, I recommend accepting the paper "as is".  However, I would encourage the authors to try and make some further edits to improve readability and notational overhead for the camera-ready if possible.
Decision: Accept (poster)