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PAC-Bayesian Reconstruction Guarantees for Time Series Variational Autoencoders
Chloé Hashimoto-Cullen, Ghislain Agoua, Benjamin Guedj, Sylvain Le Corff
TL;DR
VAEs are widely used for sequential data, but their theoretical guarantees largely focus on i.i.d. settings. This paper develops PAC-Bayesian reconstruction guarantees for VAEs with Markovian latent structure, showing that the bounds can remain stable with trajectory length while identifying assumptions and their scope.
Problem
Theoretical guarantees for VAEs applied to time series remain limited, especially beyond i.i.d. settings.
Method
The paper extends PAC-Bayesian reconstruction analysis to sequential VAEs with structured, Markovian variational distributions.
Results
The resulting generalisation bound can remain stable with respect to trajectory length.
Takeaways & Limitations
The framework provides a theoretically grounded first step for understanding VAE-based time-series models and informing sequential model design.
Takeaways & Limitations
The analysis relies on assumptions including control of variational bias, sub-gamma reconstruction losses, and a decoder fixed independently of observations.
Abstract
from arXiv · showhide
Forecasting time series accurately is critical for applications with complex data ranging from energy systems to healthcare and finance. Among current state of the art models, generative latent variable models are increasingly implemented; yet principled generalisation guarantees for modern latent variable models remain limited. In particular, while Variational AutoEncoders are widely used for sequential data, their theoretical analysis is largely restricted to i.i.d. settings. In this work, we develop a PAC-Bayesian framework for latent variables models applied to time series. Building on reconstruction-based bounds, we extend PAC-Bayesian guarantees to Markovian latent structures, capturing temporal dependencies through a sequential generative process. These guarantees do not grow with the length of the trajectory. Our bounds depend on assumptions which are common in the literature; we provide an example framework where they would be verified to show that they are not as restrictive as they may seem.
1 Introduction
This work addresses the limited theoretical guarantees for VAEs on dependent time-series data by developing PAC-Bayesian reconstruction guarantees for sequential models with structured latent distributions. It extends the analysis to Markovian dependencies and illustrates the assumptions through a lightweight discrete-latent construction.
- VAEs are widely used for time-series applications, but few theoretical guarantees exist for this model family.
- Existing PAC-Bayesian reconstruction guarantees for VAEs do not cover time series.
- Under suitable mixing and regularity assumptions, the resulting bounds remain controlled as trajectory length increases.
- The paper develops PAC-Bayesian guarantees for sequential VAEs with structured variational distributions and Markovian dependencies.
- A lightweight discrete-latent instantiation makes the Markovian structure and mixing assumptions explicit as a proof of concept.
2 Background and Related Works
The background introduces VAEs, PAC-Bayesian risk bounds, and variational representations for dependent data. It positions the paper as addressing the open problem of PAC-Bayesian reconstruction and sampling guarantees for VAEs in state-space models.
- A VAE uses a decoder to model observations from latent variables and an encoder to approximate the unavailable posterior latent distribution.
- The standard VAE objective combines reconstruction loss with a β-weighted KL divergence regularisation term.
- PAC-Bayesian theory bounds the gap between theoretical and empirical risks for a randomly selected posterior.
- PAC-Bayesian guarantees for VAEs in state-space models remain largely open, motivating bounds for dependent data.
3 PAC-Bayesian Bounds on Reconstructing Time Series
The section develops PAC-Bayesian reconstruction guarantees for sequential VAEs with structured, including Markovian, variational distributions. Under boundedness, smoothness, mixing, and stability assumptions, the resulting bounds retain statistical rates while remaining controlled as trajectory length increases.
- Bound assumptions: Sub-gamma reconstruction-loss assumptions extend the framework beyond bounded losses and bounded observation spaces.This supports squared reconstruction and negative log-likelihood losses on non-compact spaces under a uniform sub-gamma condition.
- Rates: Under Kn = O(1), the upper-bound rate is O(n^-1/2) for fixed confidence and O((log n/n)^1/2) when ηn = n^-α.More generally, Kn = O(n^κ) gives rate O(n^(κ−1)/2) when κ ∈ (0, 1).
- Markovian guarantees: Theorem 3 provides a generalisation bound for Markovian variational distributions whose three terms include KL divergences and a structural stability contribution.The bound controls the bias between variational expectations across observation sequences under the strong-mixing assumption H1.
- Markovian guarantees: The 1/T reconstruction-loss normalisation offsets linear growth, yielding bounds that remain stable as trajectory length increases.This time-uniform behavior relies on the stated structural assumptions for the variational distribution and latent process.
- Bound assumptions: Proposition 1 requires neither encoder Lipschitz continuity nor a bounded observation space beyond boundedness of the loss itself.Its concentration dependence on sample size is similar to the earlier PAC-Bayesian contribution cited in the section.
- Instantiation: The discrete-latent instantiation makes Markovian and mixing assumptions explicit and illustrates how the bounds can guide structured variational representations.It uses a structured discrete variational distribution based on predictions from a pretrained forecasting model and a finite reference set of trajectories, as a proof of concept rather than a new algorithmic contribution.
4 Conclusion
The paper introduces PAC-Bayesian reconstruction bounds for VAEs with Markovian latent structure, extending guarantees beyond i.i.d. data. The resulting bound can remain stable as trajectory length increases, while the framework's assumptions impose practical limitations.
- The paper introduces a PAC-Bayesian reconstruction bound for VAEs with Markovian latent structure.
- The analysis provides theoretical guarantees for sequential latent-variable models beyond the i.i.d. setting.
- The generalisation bound can remain stable with respect to trajectory length, clarifying the relationship between temporal depth and sample size.
- The framework relies on assumptions including controlled variational-distribution bias and sub-gamma reconstruction losses.
- The work is presented as a first step toward integrating generalisation guarantees into sequential deep-learning-system design.
Broader Impact Statement
The authors describe the work as advancing theoretical understanding of a broadly used machine-learning model. They do not identify specific impacts for explicit discussion.
- The work advances theoretical understanding of a broadly used machine-learning model.
- The authors state that the findings may have broad impacts but do not explicitly specify them.
A Proofs of Theoretical Results
The appendix reworks a prior PAC-Bayesian lemma for time series and proves it using independence, exponential inequalities, change of measure, and logarithmic manipulation. The proof concludes by isolating the KL terms and normalizing by a positive parameter.
- The appendix reworks Mbacke et al.'s Lemma B.1 in the context of time series.
- The lemma applies to losses, priors, variational distributions, and positive λ for i.i.d. sequences sampled from the data-generating distribution.
- Independence of the samples enables the exponential expectation argument used in the proof.
- The Donsker-Varadhan change of measure converts the expectation into a bound involving the variational distribution.
- Taking logarithms, using linearity of expectation, moving KL divergences to the upper bound, and dividing by λ completes the proof.
A.1 Proof of Proposition 1
The proof of Proposition 1 uses bounded reconstruction losses and independent observations to obtain a high-probability inequality. It combines Hoeffding's lemma with Markov's inequality and a preceding bound on Γ_n(S).
- The reconstruction loss ℓθ(z_i^1:T, x_i^1:T) takes values in [0,b], enabling Hoeffding's lemma.
- Independence of the observations is used as a proof step.
- Markov's inequality yields the target statement with probability at least 1−η_n.
- The proof applies the preceding upper bound on Γ_n(S) to conclude the proposition.
A.2 Proof of Proposition 3
The proof controls variational-approximation bias for additive state functionals and derives a trajectory-length-dependent bound under H1 and H2. It uses uniform minorisation of backward kernels to establish contraction properties and bound reconstruction-loss expectations.
- Proposition 6: Under H1, the approximation error grows at most linearly with the number of observations.
- Proposition 6: Proposition 6 bounds the variational-approximation error for additive state functionals under H1 and H2.The result compares expectations across observation sequences.
- Backward-kernel contraction: Uniform minorisation of the backward kernels yields a Dobrushin-coefficient contraction used in the proof.The proof invokes standard results for uniformly minorised Markov chains.
- Assumptions: The assumptions impose bounds on functions and kernel quantities, including M, C, and lower and upper σϕ terms.These quantities appear in the stated corollary and define ρϕ through the ratio of σϕ terms.
- Reconstruction-loss bound: The proof applies the contraction result to bound the expected reconstruction loss by a factor involving C, M, and ρϕ.The displayed bound contains the factor 1/(1 − ρϕ) together with C and M.
A.3 Proof of Proposition 4
The proof of Proposition 4 combines independence, H3, Markov’s inequality, and the Donsker–Varadhan change-of-measure inequality. It then uses the preceding upper bound on Γn(S) to conclude the result.
- Proof of Proposition 4: Independence of the observations and H3 are applied with α = λn/n.
- Proof of Proposition 4: Markov’s inequality establishes an event holding with probability at least 1 −ηn.
- Proof of Proposition 4: On this event, the proof applies the Donsker–Varadhan change-of-measure inequality.
- Proof of Proposition 4: The preceding upper bound on Γn(S) completes the proof.