Parallel Variational Monte Carlo (ICML 2026)

Background

State-space models (SSMs) provide a general framework for describing time-evolving systems, in which an unobserved latent state evolves according to Markovian dynamics and observations are generated conditionally on that state. Owing to this flexibility, SSMs are applied across domains as varied as target tracking, option pricing, ecology, meteorology, and neuroscience.

Training becomes considerably more difficult once the transition and observation models are parameterised by neural networks, yielding a deep state space model (DSSM). Two largely distinct paradigms have emerged for this purpose. Auto-encoding DSSMs frame the SSM as a variational auto-encoder (VAE) and optimise a variational lower bound; this permits fully parallel, scalable training but yields loose bounds and is poorly suited to supervised or semi-supervised objectives. Differentiable sequential Monte Carlo (DSMC) instead backpropagates through a particle filter, naturally supporting supervised losses and providing tighter bounds, but its resampling step imposes an inherently sequential forward pass that scales poorly on modern hardware.

This leaves an open question: whether the parallel efficiency of the auto-encoding approach can be combined with the tighter bounds and supervised training afforded by sequential Monte Carlo.

Parallel Variational Monte Carlo

We propose parallel variational Monte Carlo (PVMC), a training method that bridges the two paradigms and inherits the strengths of each. As with auto-encoding DSSMs, PVMC dispenses with the sequential proposal mechanism of particle filtering, enabling efficient parallel execution of the forward pass. As with DSMC, it constructs an importance-weighted approximation to the posterior over latent states, thereby supporting supervised training and the learning of meaningful latent representations.

In contrast to DSMC, rather than targeting the online filtering posterior, the state at time $t$ conditioned on observations up to $t$, PVMC targets the marginal smoothing posterior, the distribution of the latent state at each time-step conditioned on the entire observation sequence. Particles are sampled independently at every time-step and subsequently reweighted to approximate this smoothing distribution, accounting for every trajectory that can be formed through the set of proposed particles.

Our contributions are threefold:

• We introduce PVMC, the first end-to-end differentiable particle smoother with unbiased gradient updates and a statistically consistent posterior approximation.
• We derive a new evidence lower bound (ELBO) for training generative DSSMs that is tighter than standard single-sample or importance-weighted objectives.
• We develop a hardware-aware recursion for the importance weights, based on parallel prefix and suffix scans, achieving a 10× speed-up over the fastest competing DSMC approach and up to 100× over unbiased DSMC approaches in our experiments on an NVIDIA RTX 4090 GPU.

Experiments Results

We evaluate PVMC in three settings: a linear-Gaussian system for which exact smoothing is available, a nonlinear supervised state-estimation benchmark, and a real-world generative modelling task on financial time series.

Linear-Gaussian — state estimation

In this setting the exact marginal posterior is available via the Rauch-Tung-Striebel (RTS) smoother, allowing us to verify that PVMC’s forward pass behaves as a Bayesian smoother. We compare against the Kalman filter, the classical two-filter smoother (TFS), and the parallel-in-time smoother d-SMC, and evaluate PVMC using both an analytic Kalman proposal and a learned neural proposal.

PVMC attains a substantially lower mean error relative to the exact posterior means than the baselines, while improving on the runtime of the sequential smoothing baseline TFS. The learned proposal matches the analytic one across all metrics, indicating that PVMC’s ELBO gradients provide an effective learning signal.

Prey-predator model — state estimation

We next consider a nonlinear, supervised setting: a stochastic Lotka-Volterra (prey-predator) model in which the observation process is known but the dynamics are parameterised by a learned neural network. We compare against differentiable particle filter baselines (Stop-gradient, Soft, Diffusion), the mixture density particle smoother (MDPS), and a P-VAE ablation that employs PVMC’s importance sampler but trains with a standard VAE objective.

PVMC achieves the best overall results across all metrics — state-estimation error, filtering error, and distributional discrepancy — while training in under two minutes and converging reliably across all 20 seeds. The DSMC baselines are both slower and markedly less stable, frequently failing to converge.

Financial time series — generative modelling

Finally, we evaluate PVMC as a generative model on the daily close price of the S&P 500 Index (SPX), a setting in which no ground-truth latent state is assumed to exist. Here the latent variable is interpreted as a low-dimensional Markov summary of the state of the economy. We assess whether generated trajectories reproduce key stylised facts of the returns: volatility clustering, measured via the autocorrelation of absolute and squared daily returns, and distributional shape, measured via skewness and kurtosis. Baselines include the time-causal VAE (TC-VAE), the deep Markov model (DMM), a soft-resampling DPF, and the P-VAE ablation.

PVMC most consistently captures the short-term autocorrelation structure of SPX returns and produces skewness and kurtosis distributions closest to those observed in the real data. Neither DMM nor Soft-DPF captures the correlation structure, indicating a failure to learn meaningful temporal dynamics.

Conclusion

We introduced parallel variational Monte Carlo (PVMC), a training paradigm for deep state space models that combines parallel scalability with principled posterior inference. By targeting the marginal smoothing posterior through importance weighting rather than resampling, PVMC remains fully parallel while providing a statistically consistent smoother with unbiased gradient estimates. Across both simulated and real-world tasks, it robustly trains DSSMs for generative and discriminative objectives, achieving state-of-the-art accuracy while attaining an approximately 10× speed-up over the fastest baseline on the supervised state-estimation task.

BibTeX

@inproceedings{brady2026pvmc,
   title     = {Efficient Learning of Deep State Space Models via Importance Smoothing},
   author    = {Brady, John-Joseph and N\"usken, Nikolas and Li, Yunpeng},
   booktitle = {Proceedings of the International Conference on Machine Learning ({ICML})},
   year      = {2026},
}