Diversifying Parallel Ergodic Search: A Signature Kernel Evolution Strategy

Effective robotic exploration in continuous domains requires planning trajectories that maximize coverage over a predefined region. A recent development, Stein Variational Ergodic Search (SVES), proposed parallel ergodic exploration (a key approach within the field of robotic exploration), via Stein variational inference that computes a set of candidate trajectories approximating the posterior distribution over the solution space trajectories. While this approach leverages GPU parallelism well, the trajectories in the set might not be distinct enough, leading to a suboptimal set. In this paper, we propose two key methods to diversify the solution set of this approach. First, we leverage the signature kernel within the SVES framework, introducing a pathwise, sequence-sensitive interaction that preserves the Markovian structure of the trajectories and naturally spreads paths across distinct regions of the search space. Second, we propose a derivative-free evolution-strategy interpretation of SVES that exploits batched, GPU-friendly fitness evaluations and can be paired with approximate gradients whenever analytic gradients of the kernel are unavailable or computationally intractable. The resulting method both retains SVES's advantages while diversifying the solution set and extending its reach to black-box objectives. Across planar forest search, 3D quadrotor coverage, and model-predictive control benchmarks, our approach consistently reduces ergodic cost and produces markedly richer trajectory sets than SVES without significant extra tuning effort.

Prior work has observed that the non-convex nature of the ergodic objective can yield multiple locally optimal trajectories under different initializations [9]. Yet explicitly reasoning over-and sampling from-a distribution of such trajectories is computationally prohibitive. Moreover, there is no assurance that distinct initial conditions will avoid collapsing onto the same ergodic solution, a critical shortcoming in online exploration scenarios where mode collapse can incur catastrophic performance failures.

Stein variational inference methods show promise in providing the necessary tools to approximate distributions of trajectories in a computationally tractable manner [10]. Motivated by these strengths, [2] proposes Stein variational ergodic search (SVES), a formulation of ergodic exploration as a Stein variational inference problem: by applying Stein variational gradient descent to the space of robot trajectories. However, SVES neither fully exploits the inherent Markovian structure of trajectories nor avoids a strong dependence on gradient information. Moreover, although SVES alludes to promoting diversity among trajectories, it does not concretely define this notion or provide what "diversity" entails.

Contributions: We address the challenges mentioned above by "diversifying" Stein Variational Ergodic Search (SVES). Our approach is built on two complementary pillars. First, we leverage the signature kernel similar to [11], enabling us to encode the intrinsic sequential (or Markovian) and geometric properties of trajectories. The signature kernel naturally discriminates between subtly different paths, thus promoting diversity. Second, we incorporate Stein Variational Covariance Matrix Adaptation Evolution Strategy (SV-CMA-ES) [12] and Simultaneous Perturbation Stochastic Approximation (SPSA) [13] into SVES, drawing on recent progress in Stein Variational Evolution Strategies, this approach not only eliminates the need for gradients-often costly to compute-by harnessing GPU parallelism, making it viable for real-time applications, but also enables the use of complex non-differentiable or black-box loss functions and constraints. Later we show empirically that this approach leads to better optimization results and a more diverse set of solutions.