[Paper] Stochastic Deep Learning: A Probabilistic Framework for Modeling Uncertainty in Structured Temporal Data

Source: arXiv - 2601.05227v1

Overview

James Rice’s paper introduces Stochastic Latent Differential Inference (SLDI), a framework that fuses stochastic differential equations (SDEs) with variational autoencoders (VAEs). By embedding an Itô‑style SDE directly in the latent space, the method delivers continuous‑time uncertainty estimates for structured temporal data—think irregularly sampled sensor streams, high‑frequency finance ticks, or event‑based logs.

Key Contributions

• Latent‑space SDE integration – couples drift and diffusion neural nets with a VAE encoder/decoder, enabling data‑driven continuous‑time dynamics.
• Co‑parameterized adjoint network – learns the backward (gradient) dynamics alongside the forward SDE, forming a coupled forward‑backward system.
• Pathwise‑regularized adjoint loss – a novel regularizer that stabilizes training by controlling variance in stochastic gradient flows.
• Theoretical bridge – unifies variational inference, continuous‑time generative modeling, and control‑theoretic optimization under a rigorous stochastic‑calculus lens.
• Irregular‑sampling handling – naturally accommodates non‑uniform time steps without resorting to ad‑hoc interpolation.

Methodology

1. Base VAE – an encoder maps a sequence (or set of observations) to a latent distribution; a decoder reconstructs the original data.

2. Latent SDE layer – the latent variable evolves according to

   [ d\mathbf{z}t = f\theta(\mathbf{z}t, t),dt + g\phi(\mathbf{z}_t, t),dW_t, ]

   where (f_\theta) (drift) and (g_\phi) (diffusion) are small neural nets, and (W_t) is a standard Wiener process.

3. Adjoint network – a second neural net (\psi) learns the adjoint state (\lambda_t) that satisfies the backward stochastic differential equation needed for gradient computation.

4. Training objective – the ELBO (Evidence Lower Bound) is augmented with a pathwise‑regularized adjoint loss that penalizes high variance in (\lambda_t) along sampled trajectories.

5. Optimization – stochastic gradient descent is applied to the combined parameters ((\theta, \phi, \psi)) using reparameterization tricks for both the latent distribution and the SDE noise.

The whole pipeline can be implemented with modern autodiff libraries (e.g., PyTorch’s torchsde or TensorFlow Probability), requiring only a few extra lines beyond a standard VAE.

Results & Findings

• Training stability improves markedly: the variance of gradient estimates drops by ~30 % thanks to the adjoint regularizer.
• Continuous‑time interpolation: SLDI can query the latent state at any timestamp, outperforming discrete RNN baselines on out‑of‑sample forecasting.
• Uncertainty quality: calibration plots show that predicted confidence intervals contain the true values at the nominal rates (e.g., 95 % intervals cover ~94 % of test points).

Practical Implications

• Irregular data pipelines – developers no longer need to resample or impute missing timestamps; SLDI works directly on the raw time stamps.
• Risk‑aware forecasting – finance or IoT platforms can attach mathematically sound confidence bands to predictions, enabling better automated decision making (e.g., trigger alerts only when uncertainty is low).
• Model‑based control – the learned drift/diffusion networks can serve as differentiable simulators for reinforcement‑learning agents that must plan under stochastic dynamics.
• Scalable deployment – because the SDE is solved with adaptive solvers, inference cost scales with the effective temporal resolution rather than the raw number of observations, making it suitable for edge devices handling bursty sensor streams.

Limitations & Future Work

• Computational overhead – solving SDEs (especially with adaptive step sizes) adds ~2–3× runtime compared to a vanilla VAE; hardware‑accelerated SDE solvers are still emerging.
• Model interpretability – while drift/diffusion nets are expressive, extracting human‑readable dynamics (e.g., physical parameters) remains non‑trivial.
• Scalability to very high‑dimensional latents – the adjoint network grows with latent size, potentially limiting ultra‑large latent spaces.

Future directions suggested by the author include:

1. Coupling SLDI with graph neural networks for spatio‑temporal graphs.
2. Exploring symplectic SDE integrators to preserve physical invariants.
3. Extending the framework to handle multi‑modal data (e.g., video + audio) where each modality follows its own stochastic clock.

Authors

• James Rice

Paper Information

• arXiv ID: 2601.05227v1
• Categories: stat.ML, cs.LG, econ.EM, math.ST
• Published: January 8, 2026
• PDF: Download PDF