Understanding and certifying the generalization performance of machine learning algorithms - i.e. obtaining theoretical estimates of the test error from the training error - is a central theme of statistical learning theory. Among the many complexity measures used to derive such guarantees, Rademacher complexity yields sharp, data-dependent bounds that apply well beyond classical VC-dimension theory. In this study, we formalize the generalization error bound by Rademacher complexity in Lean 4, building on measure-theoretic probability theory available in the Mathlib library. Our development provides a mechanically-checked pipeline from the definitions of empirical and expected Rademacher complexity, through a formal symmetrization argument and a bounded-differences analysis, to high-probability uniform deviation bounds via a formally proved McDiarmid inequality. A key technical contribution is a reusable mechanism for lifting results from countable hypothesis classes (where measurability of suprema is straightforward in Mathlib) to separable topological index sets via a reduction to a countable dense subset. As worked applications of the abstract theorem, we mechanize standard empirical Rademacher bounds for linear predictors under 𝓁₂ and 𝓁₁ regularizations, and we also formalize a Dudley-type entropy integral bound based on covering numbers and a chaining construction.

We propose an efficient framework using Dynnikov coordinates for homotopy-aware multi-agent path planning in planar domains that may contain obstacles. We developed a method for generating multiple homotopically distinct solutions for the multi-agent path planning problem in planar domains by combining our framework with revised prioritized planning and proved its completeness under specific assumptions. Experimentally, we demonstrated that our method is significantly faster than a method without Dynnikov coordinates. We also confirmed experimentally that homotopy-aware planning contributes to avoiding locally optimal solutions when searching for low-cost trajectories for a swarm of agents in a continuous environment.

Designing a safe policy for uncertain environments is crucial in real-world control systems. However, this challenge remains inadequately addressed within the Markov decision process (MDP) framework. This paper presents the first algorithm guaranteed to identify a near-optimal policy in a robust constrained MDP (RCMDP), where an optimal policy minimizes cumulative cost while satisfying constraints in the worst-case scenario across a set of environments. We first prove that the conventional policy gradient approach to the Lagrangian max-min formulation can become trapped in suboptimal solutions. This occurs when its inner minimization encounters a sum of conflicting gradients from the objective and constraint functions. To address this, we leverage the epigraph form of the RCMDP problem, which resolves the conflict by selecting a single gradient from either the objective or the constraints. Building on the epigraph form, we propose a bisection search algorithm with a policy gradient subroutine and prove that it identifies an $\varepsilon$-optimal policy in an RCMDP with $\widetilde{\mathcal{O}}(\varepsilon^{-4})$ robust policy evaluations.

We study the reinforcement learning (RL) problem in a constrained Markov decision process (CMDP), where an agent explores the environment to maximize the expected cumulative reward while satisfying a single constraint on the expected total utility value in every episode. While this problem is well understood in the tabular setting, theoretical results for function approximation remain scarce. This paper closes the gap by proposing an RL algorithm for linear CMDPs that achieves $\widetilde{\mathcal{O}}(\sqrt{K})$ regret with an episode-wise zero-violation guarantee. Furthermore, our method is computationally efficient, scaling polynomially with problem-dependent parameters while remaining independent of the state space size. Our results significantly improve upon recent linear CMDP algorithms, which either violate the constraint or incur exponential computational costs.

To find the shortest paths for all pairs on manifolds with infinitesimally defined metrics, we introduce a framework to generate them by predicting midpoints recursively. To learn midpoint prediction, we propose an actor-critic approach. We prove the soundness of our approach and show experimentally that the proposed method outperforms existing methods on several planning tasks, including path planning for agents with complex kinematics and motion planning for multi-degree-of-freedom robot arms.

The human brain’s plasticity allows for the integration of artificial body parts into the human body. Leveraging this, embodied systems realize intuitive interactions with the environment. We introduce a novel concept: embodied swarm robots. Swarm robots constitute a collective of robots working in harmony to achieve a common objective, in our case, serving as functional body parts. Embodied swarm robots can dynamically alter their shape, density, and the correspondences between body parts and individual robots. We contribute an investigation of the influence on embodiment of swarm robot-specific factors derived from these characteristics, focusing on a hand. Our paper is the first to examine these factors through virtual reality (VR) and real-world robot studies to provide essential design considerations and applications of embodied swarm robots. Through quantitative and qualitative analysis, we identified a system configuration to achieve the embodiment of swarm robots.

Multi-agent path planning (MAPP) is the problem of planning collision-free trajectories from start to goal locations for a team of agents. This work explores a relatively unexplored setting of MAPP where streams of agents have to go through the starts and goals with high throughput. We tackle this problem by formulating a new variant of MAPP called periodic MAPP in which the timing of agent appearances is periodic. The objective with periodic MAPP is to find a periodic plan, a set of collision-free trajectories that the agent streams can use repeatedly over periods, with periods that are as small as possible. To meet this objective, we propose a solution method that is based on constraint relaxation and optimization. We show that the periodic plans once found can be used for a more practical case in which agents in a stream can appear at random times. We confirm the effectiveness of our method compared with baseline methods in terms of throughput in several scenarios that abstract autonomous intersection management tasks.