Predictive Dynamic Sampling in Control & Inference

• Predictive dynamic sampling is defined as a method that combines forward prediction, adaptive control, and dynamic updates to optimize performance in time-varying environments.
• It employs techniques from model predictive control, such as parallel Monte Carlo rollouts and adaptive covariance adjustment, to refine control strategies in real time.
• Applications span nonlinear MPC, adaptive signal processing, and probabilistic inference, leveraging GPU-accelerated simulations to ensure efficient and robust performance.

Predictive dynamic sampling encompasses algorithms and architectures that employ forward prediction, adaptive or parallelized sampling, and dynamic incorporation of new information to optimize control, inference, or signal reconstruction in time-varying systems. This paradigm arises wherever the underlying process, system, or probabilistic model evolves, or where sampling itself must be targeted to maximize informational or practical efficiency. Techniques span nonlinear model predictive control (MPC) via path integral methods, probabilistic inference in dynamic graphical models, bandit learning in non-stationary environments, analog-to-digital conversion for sparse signals, and adaptive experimental design for spatiotemporal fields.

1. Predictive Dynamic Sampling in Model Predictive Control

At the core of sampling-based MPC, predictive dynamic sampling methods repeatedly generate entire sequences of control inputs—rather than single-step proposals—by drawing samples around a nominal controller and simulating their outcomes forward over a finite horizon. In the Model Predictive Path Integral (MPPI) framework, each control $v_t$ at time $t$ is sampled from a Gaussian centered on a nominal $\bar u_t$, forming $K$ sampled input trajectories $U^k = \bar U + \mathcal E^k$ where $\bar U = [\bar u_0,...,\bar u_{T-1}]$ and $\mathcal E^k$ is a noise sequence (Pezzato et al., 2023). All samples are simulated in parallel, e.g., using GPU-accelerated engines like IsaacGym, which obviate the need for analytic models of robot or environment dynamics.

The cost functional for each trajectory $\tau^k$ is computed, commonly in discounted form,

$$S(\tau^k) = \sum_{t=0}^{T-1}\gamma^t\,C(x_{t}^k, v_{t}^k),\quad \gamma\in(0,1]$$

where $C$ is user-defined. The importance of each sample is weighted via a path-integral exponential,

$$w^k = \frac{1}{\eta}\exp\left(-\frac{1}{\lambda}\,S(\tau^k)\right),\quad \eta = \sum_{j=1}^K\exp\left(-\frac{1}{\lambda}S(\tau^j)\right)$$

with temperature parameter $\lambda$. The new nominal controller $\bar U$ is updated as the expectation over weights:

$\bar U \gets \sum_{k=1}^K w^k U^k$

and only the first control $\bar u_0$ is executed; horizon shifting and reseeding close the sampling loop.

Covariance-variable importance sampling extends classical MPPI by allowing dynamic adaptation of both the mean and covariance of the trajectory sampler, greatly accelerating convergence in ill-conditioned or highly nonlinear systems (Williams et al., 2015). Empirical results demonstrate that predictive dynamic sampling matches or exceeds the cost performance and computational efficiency of optimization-based controllers (DDP, fmincon), especially as GPU parallelization scales to thousands of rollouts at real-time rates.

2. Generalizations: Multimodal and Biased Predictive Sampling

Classic implementations of MPPI employ a unimodal Gaussian sampler around the previous plan. However, arbitrary and multimodal sampling distributions can be constructed by fusing outputs from multiple ancillary controllers—classical, learned, or hand-designed (Trevisan et al., 2024). The Biased-MPPI framework uses an importance weighting design that cancels out the requirement to compute the likelihood ratio $p(V^k)/g(V^k)$, where $g(V)$ is the sampler and $p(V)$ the uncontrolled process distribution, by incorporating a shifted cost $$\widetilde S(V) = S(V)+\lambda\log\frac{p(V)}{g(V)}$$; under the introduced bias, importance weights depend only on the path cost:

$w_k = \exp(-S(V^k)/\lambda)$

Multimodal sampling improves robustness against local minima and variance-driven failure in high-dimensional optimization. Each ancillary branch provides a full candidate trajectory, and the final control sequence is derived by weighted averaging over all sampled paths, including deterministic and stochastic branches. Empirical evidence confirms substantial reductions in collisions, violations, and faster convergence compared to unimodal proposals in decentralized multi-agent and real-world robotic tasks.

3. Algorithmic Structure and Implementation

Predictive dynamic sampling in MPC is typified by the following recurrent algorithmic protocol:

1. State Observation and Environment Reset: Observe current state, initialize or reset simulation environments.
2. Sampling: Generate $K$ candidate control trajectories via Gaussian or multimodal samplers, possibly incorporating multiple ancillary controllers.
3. Parallel Rollout: Forward simulate each trajectory for the planning horizon, collect states and costs.
4. Weight Computation: Evaluate path-integral-based weights for each trajectory.
5. Controller Update: Aggregate samples via weighted averaging to update $\bar U$.
6. Execution and Horizon Shift: Implement the first control, shift the control sequence, and repeat.

Hyperparameters critical to performance include planning horizon $T$, total samples $K$, sampling covariance $\Sigma$, temperature $\lambda$, discount factor $\gamma$, and normalization bounds for weight sums. Real-world implementations leverage GPU-parallelizable simulations for efficient large-scale sampling (Pezzato et al., 2023).

4. Extension to Probabilistic Inference and Dynamic Graphical Models

Predictive dynamic sampling generalizes beyond control to probabilistic inference in time-varying graphical models (Feng et al., 2019, Feng et al., 2018). In dynamic Markov random fields (MRFs), as the model structure (graph, potentials) changes incrementally, predictive dynamic samplers maintain and update Monte Carlo sample pools efficiently. Algorithmic Lipschitz conditions ensure that small changes in model induce only local recalibration of sample trajectories, leading to incremental update costs proportional to the perturbation size rather than to global model dimensions.

For discrete graphical models, parallel Las Vegas sampling algorithms dynamically resample only affected variable subsets, leveraging a conditional Gibbs equilibrium property to ensure phase invariance and exactitude. This approach achieves $O(|D|)$ cost per update, where $|D|$ is the update size, versus $O(nN(n))$ cost for full resampling in static methods.

5. Applications in Signal Processing and Experimental Design

In analog front-end systems, dynamic predictive sampling underpins non-uniform, event-driven ADC architectures that rely on forward digital prediction windows to compress data and save power. By updating digital threshold windows based on previous samples, SAR quantization is selectively triggered only on unpredictable signal events (Tang et al., 2022). This yields substantial data compression and energy savings, especially for sparse physiological signals such as ECG.

For adaptive robotic sampling in spatiotemporal fields, predictive dynamic sampling couples learned fluid models (e.g., Neural ODEs) with reinforcement learning-based path planning. Predictions inform sampling locations to maximize information gain, and subsequent samples correct model drift and maintain bounded error across prolonged monitoring horizons (Manjanna et al., 2023).

6. Optimization, Multi-Agent Systems, and Scalability

Distributed predictive dynamic sampling algorithms, including multi-agent MPC, rely on consensus ADMM protocols and sampling-based policy optimization that scale efficiently by parallelizing local updates and information exchange between agents (Wang et al., 2022). Unified frameworks based on variational optimization, inference, and stochastic search provide convergence and sample complexity guarantees, supporting non-Gaussian policies, mixture models, and stein-variational updates. These methods have been demonstrated at scale for vehicular and aerial swarm scenarios, enabling near-constant runtime and per-agent cost across hundreds of agents.

7. Theoretical Guarantees, Limitations, and Practical Considerations

Predictive dynamic sampling carries recursive feasibility and Lyapunov stability in suboptimal MPC settings by preserving feasibility through cost-decreasing trajectory swaps and warm-starts (Bobiti et al., 2017). In stochastic control, adaptive sampling-based chance constraint approximation allows for less conservative but safe behavior as parametric uncertainty shrinks online (Teutsch et al., 2024).

Limitations arise in regions of strong coupling or low-temperature phases (for probabilistic graphical models), non-sparse or rapidly changing signals (for ADCs), and environments characterized by challenging local minima or high nonlinearity. Proper hyperparameter tuning, selection of informative ancillary controllers, and maintenance of feasible sets or control-invariant regions are necessary to guarantee performance and safety.

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Key Research References:

• Sampling-based Model Predictive Control with IsaacGym (Pezzato et al., 2023)
• Covariance-variable importance sampling for MPPI (Williams et al., 2015)
• Biased-MPPI for arbitrary sampling and controller fusion (Trevisan et al., 2024)
• Dynamic inference in probabilistic models (Feng et al., 2019, Feng et al., 2018)
• Dynamic predictive ADCs (Tang et al., 2022)
• Distributed sampling-based MPC (Wang et al., 2022)
• Adaptive stochastic predictive control (Teutsch et al., 2024)