Deterministic Sampling in MPC: dsMPPI/dsCEM

• Deterministic sampling is a method that replaces stochastic sampling with precomputed low-discrepancy samples to uniformly cover the proposal distribution.
• It leverages offline LCD construction and runtime linear transformations to minimize variance and efficiently map samples into evolving control distributions.
• These techniques produce smoother control trajectories and improved sample efficiency, making them viable for real-time and embedded MPC applications.

Deterministic sampling refers to a class of sampling-based optimal control and trajectory optimization methods in which the stochastic exploration step—typically involving Monte Carlo draws from a distribution—is replaced entirely by a fixed set of optimally chosen, low-discrepancy samples. In modern model predictive control (MPC) frameworks such as Model Predictive Path Integral (MPPI) and the Cross-Entropy Method for MPC (CEM-MPC), deterministic sampling gives rise to algorithms such as deterministic-sampling CEM (dsCEM) and deterministic-sampling MPPI (dsMPPI). These approaches achieve variance reduction, improved sample efficiency, and substantially smoother control policies compared to their randomly sampled counterparts, especially in nonlinear, time-correlated, or low-sample regimes (Walker et al., 7 Jan 2026, Walker et al., 7 Oct 2025).

1. Foundations and Motivation

Standard sampling-based MPC approaches such as MPPI and CEM-MPC estimate optimal control actions by simulating $N$ trajectories through the application of i.i.d. Gaussian noise to a nominal control sequence. However, random sampling exhibits several limitations:

• Poor support coverage: Random samples tend to cluster and leave gaps, requiring large N for accurate expectation estimates ("low-discrepancy" is only achieved asymptotically).
• Chattering and non-smoothness: Lack of temporal correlation between samples induces jerky control signals, which are undesirable in physical systems.
• High computational burden: Large numbers of rollouts are necessary for reliable optimization, challenging real-time applications or embedded deployments (Walker et al., 7 Oct 2025).

Deterministic sampling addresses these limitations by precomputing a set of quadrature-like “optimal Dirac points”—called LCD samples (from Localized Cumulative Distributions)—which span the proposal distribution in a uniform, low-discrepancy manner. These samples are then systematically transformed at run-time into the current proposal distribution, enabling more reliable estimation and efficient use of computational resources (Walker et al., 7 Oct 2025, Walker et al., 7 Jan 2026).

2. Deterministic Sampling Construction and Transformation

The deterministic sample design process involves:

• LCD Construction: For a reference density $f(x)$ (typically $N(0,I)$), the localized cumulative distribution (LCD) is defined by:

$$F_f(m, b) = \int_{\mathbb{R}^d} K(x; m, b)f(x)dx, \qquad K(x; m, b) = \exp\left(-\frac{\|x-m\|^2}{2b^2}\right)$$

A finite set of $K$ sample points $\{\tilde u_i\}$ is chosen to minimize a Cramér–von Mises–type discrepancy between $f$ and its Dirac mixture approximation $$\hat f(x) = \frac{1}{K}\sum_{i=1}^K \delta(x - \tilde u_i)$$.

• Offline Optimization: These points are solved for by offline gradient-based minimization of the integrated squared difference between $F_f$ and $F_{\hat f}$, producing a deterministic, identity-covariance set with superior uniformity (Walker et al., 7 Oct 2025).
• Runtime Linear Transformation: The sample set is mapped on-the-fly into the current proposal $f(x)$0 by $f(x)$1. Here, $f(x)$2 is the Cholesky factor of $f(x)$3. This step ensures sample points populate the high-probability region of the current search space (Walker et al., 7 Oct 2025, Walker et al., 7 Jan 2026).
• Temporal Smoothness: To induce smooth time evolution in control sequences, a colored-noise prior is used by embedding a Toeplitz correlation matrix $f(x)$4 from a $f(x)$5 power spectrum; the sample transformation becomes $f(x)$6 with colored base samples (Walker et al., 7 Jan 2026).

3. Detailing dsCEM and dsMPPI Algorithms

Deterministic Cross-Entropy Method (dsCEM)

In dsCEM, the sample replacement occurs within the iterative CEM-MPC loop:

• Deterministic sample generation: $f(x)$7 for $f(x)$8
• Cost evaluation: $f(x)$9
• Elite selection and update: next mean and covariance $N(0,I)$0, $N(0,I)$1 are computed over the $N(0,I)$2 lowest-cost samples (elites). Weighted updates or uniform weights are standard.
• Momentum/adaptive schemes: Optional momentum averaging and two alternatives for covariance adaptation: fixed temporal correlation with adaptive marginal variances, or full covariance adaptation (Walker et al., 7 Oct 2025).

Deterministic Sampling MPPI (dsMPPI)

dsMPPI integrates these deterministic samples into the path integral (MPPI) framework using exponential soft weights rather than the CEM hard-threshold elite set:

• Sample transformation: $N(0,I)$3, $N(0,I)$4
• Trajectory simulation and cost: Each rollout is propagated, computing $N(0,I)$5
• Exponential weighting: $N(0,I)$6
• Soft averaging for parameter updates: The mean and diagonal variance are updated by exponentially weighted averages.
• Momentum smoothing: Update $N(0,I)$7 and $N(0,I)$8 via $N(0,I)$9-weighted smoothing to prevent premature collapse and improve exploration/exploitation trade-offs.
• Adaptive temperature: $$F_f(m, b) = \int_{\mathbb{R}^d} K(x; m, b)f(x)dx, \qquad K(x; m, b) = \exp\left(-\frac{\|x-m\|^2}{2b^2}\right)$$0 is adapted based on an effective sample size metric ($$F_f(m, b) = \int_{\mathbb{R}^d} K(x; m, b)f(x)dx, \qquad K(x; m, b) = \exp\left(-\frac{\|x-m\|^2}{2b^2}\right)$$1) so as to keep importance ratios well-behaved (Walker et al., 7 Jan 2026).

Both dsCEM and dsMPPI support alternative variation schemes, such as multi-iteration subsetting (drawing a subset from a larger pool across iterations) and coordinate permutation to increase effective exploration without reintroducing randomness (Walker et al., 7 Jan 2026).

4. Theoretical Insights and Computational Complexity

Deterministic sampling for MPC confers distinct theoretical benefits:

• Variance reduction: Deterministic LCD samples uniformly tile the underlying distribution, minimizing quadrature error relative to i.i.d. sampling. For a fixed computational budget, deterministic methods achieve lower estimation variance and hence require fewer trajectories for equivalent accuracy (Walker et al., 7 Oct 2025, Walker et al., 7 Jan 2026).
• Convergence and bias-variance tradeoff: Momentum-smoothing ($$F_f(m, b) = \int_{\mathbb{R}^d} K(x; m, b)f(x)dx, \qquad K(x; m, b) = \exp\left(-\frac{\|x-m\|^2}{2b^2}\right)$$2) and adaptive temperature tuning ($$F_f(m, b) = \int_{\mathbb{R}^d} K(x; m, b)f(x)dx, \qquad K(x; m, b) = \exp\left(-\frac{\|x-m\|^2}{2b^2}\right)$$3) regulate the update magnitude and ensure stability. dsCEM provides exact moment matching for quadratic costs and linear dynamics (Walker et al., 7 Jan 2026).
• Complexity: Both dsCEM and dsMPPI maintain $$F_f(m, b) = \int_{\mathbb{R}^d} K(x; m, b)f(x)dx, \qquad K(x; m, b) = \exp\left(-\frac{\|x-m\|^2}{2b^2}\right)$$4 total runtime per control step, with batch trajectory simulation being predominant. Deterministic sample generation and linear-algebraic updates are negligible relative to forward simulation costs (Walker et al., 7 Jan 2026, Walker et al., 7 Oct 2025).

5. Key Hyperparameters and Tuning Practices

Critical hyperparameters and empirically effective ranges for dsCEM and dsMPPI include (Walker et al., 7 Jan 2026):

Parameter	Typical Range	Significance
$$F_f(m, b) = \int_{\mathbb{R}^d} K(x; m, b)f(x)dx, \qquad K(x; m, b) = \exp\left(-\frac{\|x-m\|^2}{2b^2}\right)$$5	50–300	# deterministic samples per iteration
$$F_f(m, b) = \int_{\mathbb{R}^d} K(x; m, b)f(x)dx, \qquad K(x; m, b) = \exp\left(-\frac{\|x-m\|^2}{2b^2}\right)$$6	10–50	Horizon length (problem dependent)
$$F_f(m, b) = \int_{\mathbb{R}^d} K(x; m, b)f(x)dx, \qquad K(x; m, b) = \exp\left(-\frac{\|x-m\|^2}{2b^2}\right)$$7	2–5 (usually 3)	CEM/MPPI inner iterations
$$F_f(m, b) = \int_{\mathbb{R}^d} K(x; m, b)f(x)dx, \qquad K(x; m, b) = \exp\left(-\frac{\|x-m\|^2}{2b^2}\right)$$8	$$F_f(m, b) = \int_{\mathbb{R}^d} K(x; m, b)f(x)dx, \qquad K(x; m, b) = \exp\left(-\frac{\|x-m\|^2}{2b^2}\right)$$9cost	Inverse temperature; adapt via $K$0
$K$1	0.9–0.99	Momentum for mean/covariance updates
$K$2	0.5–1.5	Colored-noise exponent for smoothness
$K$3	1–5	Buffer size (tracking best rollouts)

A frequent guideline: start with $K$4, $K$5, $K$6, and tune $K$7 so that exponential weights are neither too diffuse nor collapsed, monitoring effective sample size $K$8. For smoothing, set $K$9 (Walker et al., 7 Jan 2026).

6. Empirical Performance and Comparison

Empirical evaluation on canonical nonlinear control tasks (e.g., cart-pole swing-up, truck backer-upper) demonstrates:

• Smoother trajectories: dsMPPI yields ≈30% smoother controls (quantified by cumulative $\{\tilde u_i\}$0) compared to dsCEM and ≈60% compared to classic MPPI. On the cart-pole, smoothness improves from $\{\tilde u_i\}$1 (random MPPI) to $\{\tilde u_i\}$2 (dsMPPI) (Walker et al., 7 Jan 2026).
• Sample efficiency: dsCEM achieves comparable or lower cumulative cost than iCEM using $\{\tilde u_i\}$3; e.g., on the mountain car, dsCEM achieves 40% lower cost and 50% smoother inputs at $\{\tilde u_i\}$4 (Walker et al., 7 Oct 2025).
• Computational parity: These deterministic-sampling methods do not incur extra asymptotic online costs relative to their random-sampling equivalents (Walker et al., 7 Jan 2026, Walker et al., 7 Oct 2025).

Method	Cart-pole Cost	Smoothness	Truck Cost	Smoothness
MPPI	245.3 ± 12.1	1.12e5 ± .10e5	162.8 ± 8.4	0.48e5 ± .05e5
Iterative MPPI	198.7 ± 9.3	0.83e5 ± .08e5	140.2 ± 6.7	0.39e5 ± .04e5
dsCEM	185.5 ± 8.7	0.61e5 ± .05e5	131.9 ± 5.4	0.31e5 ± .03e5
dsMPPI	188.2 ± 9.0	0.42e5 ± .03e5	133.4 ± 5.9	0.28e5 ± .02e5

Key findings: dsMPPI and dsCEM match or outperform random-sampling variants both in control cost and input smoothness, with the largest gains in demanding low-sample regimes (Walker et al., 7 Jan 2026, Walker et al., 7 Oct 2025).

7. Extensions, Generalizations, and Limitations

The deterministic-sampling principle is generic and can be transferred between MPPI and CEM approaches, as well as other stochastic sampling-based optimizers. LCD design can also incorporate task-specific priors (e.g. anisotropic variances, stronger colored-noise correlations) (Walker et al., 7 Oct 2025, Walker et al., 7 Jan 2026).

Known limitations include:

• Scalability of LCD construction: The offline computation of the optimal Dirac mixture becomes challenging for high-dimensional spaces, and transforming LCD samples using non-isotropic covariances may degrade optimality.
• Extension to other sampling schemes: In the context of direct policy optimization, deterministic sigma-point collocation achieves exact recovery of the LQR solution for linear–quadratic–Gaussian systems and reduces variance for mildly nonlinear cases (Howell et al., 2020).

A plausible implication is that direct deterministic-sampling frameworks may further benefit from adaptive Dirac-point generation and online adaptation when operating in high-dimensional or time-varying uncertainty regimes.

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References:

• (Walker et al., 7 Jan 2026) Smooth Sampling-Based Model Predictive Control Using Deterministic Samples
• (Walker et al., 7 Oct 2025) Sample-Efficient and Smooth Cross-Entropy Method Model Predictive Control Using Deterministic Samples
• (Howell et al., 2020) Direct Policy Optimization using Deterministic Sampling and Collocation