Daniel Frisch

Markus Walker, Daniel Frisch, Uwe D. Hanebeck

Cross-entropy method model predictive control (CEM--MPC) is a powerful gradient-free technique for nonlinear optimal control, but its performance is often limited by the reliance on random sampling. This conventional approach can lead to inefficient exploration of the solution space and non-smooth control inputs, requiring a large number of samples to achieve satisfactory results. To address these limitations, we propose deterministic sampling CEM (dsCEM), a novel framework that replaces the random sampling step with deterministic samples derived from localized cumulative distributions (LCDs). Our approach introduces modular schemes to generate and adapt these sample sets, incorporating temporal correlations to ensure smooth control trajectories. This method can be used as a drop-in replacement for the sampling step in existing CEM-based controllers. Experimental evaluations on two nonlinear control tasks demonstrate that dsCEM consistently outperforms state-of-the-art iCEM in terms of cumulative cost and control input smoothness, particularly in the critical low-sample regime.

Markus Walker, Marcel Reith-Braun, Daniel Frisch et al.

Sampling-based model predictive control (MPC) algorithms, such as model predictive path integral (MPPI), enable approximate, gradient-free solutions to optimal control problems by drawing samples from a proposal distribution, evaluating their trajectory costs, and updating the proposal parameters accordingly. However, these approaches typically rely on heuristics for adjusting hyperparameters, such as temperature or momentum, or manual tuning. We propose a trust region formulation for sampling-based MPC that constrains updates of the proposal distribution via a principled Kullback--Leibler (KL) divergence bound and, optionally, an entropy lower bound. This replaces heuristic hyperparameter adaptation with values that are optimal w.r.t. the underlying Lagrangian. We further improve sample efficiency and convergence by combining the trust region update with deterministic localized cumulative distribution (LCD)-based sampling. Experiments on two benchmark environments demonstrate that the proposed trust region update achieves faster convergence and better sample efficiency in low-sample and low-iteration regimes, especially when paired with deterministic LCD-based sampling.

Andrew Millard, Joshua Murphy, Daniel Frisch et al.

Markov chain Monte Carlo (MCMC) methods are a powerful but computationally expensive way of performing non-parametric Bayesian inference. MCMC proposals which utilise gradients, such as Hamiltonian Monte Carlo (HMC), can better explore the parameter space of interest if the additional hyper-parameters are chosen well. The No-U-Turn Sampler (NUTS) is a variant of HMC which is extremely effective at selecting these hyper-parameters but is slow to run and is not suited to GPU architectures. An alternative to NUTS, Change in the Estimator of the Expected Square HMC (ChEES-HMC) was shown not only to run faster than NUTS on GPU but also sample from posteriors more efficiently. Sequential Monte Carlo (SMC) samplers are another sampling method which instead output weighted samples from the posterior. They are very amenable to parallelisation and therefore being run on GPUs while having additional flexibility in their choice of proposal over MCMC. We incorporate (ChEEs-HMC) as a proposal into SMC samplers and demonstrate competitive but faster performance than NUTS on a number of tasks.

Daniel Frisch, Uwe D. Hanebeck

We consider estimating the parameters of a Gaussian mixture density with a given number of components best representing a given set of weighted samples. We adopt a density interpretation of the samples by viewing them as a discrete Dirac mixture density over a continuous domain with weighted components. Hence, Gaussian mixture fitting is viewed as density re-approximation. In order to speed up computation, an expectation-maximization method is proposed that properly considers not only the sample locations, but also the corresponding weights. It is shown that methods from literature do not treat the weights correctly, resulting in wrong estimates. This is demonstrated with simple counterexamples. The proposed method works in any number of dimensions with the same computational load as standard Gaussian mixture estimators for unweighted samples.