Direct policy search has been widely applied in modern reinforcement learning and continuous control. However, the theoretical properties of direct policy search on nonsmooth robust control synthesis have not been fully understood. The optimal $\mathcal{H}_\infty$ control framework aims at designing a policy to minimize the closed-loop $\mathcal{H}_\infty$ norm, and is arguably the most fundamental robust control paradigm. In this work, we show that direct policy search is guaranteed to find the global solution of the robust $\mathcal{H}_\infty$ state-feedback control design problem. Notice that policy search for optimal $\mathcal{H}_\infty$ control leads to a constrained nonconvex nonsmooth optimization problem, where the nonconvex feasible set consists of all the policies stabilizing the closed-loop dynamics. We show that for this nonsmooth optimization problem, all Clarke stationary points are global minimum. Next, we identify the coerciveness of the closed-loop $\mathcal{H}_\infty$ objective function, and prove that all the sublevel sets of the resultant policy search problem are compact. Based on these properties, we show that Goldstein's subgradient method and its implementable variants can be guaranteed to stay in the nonconvex feasible set and eventually find the global optimal solution of the $\mathcal{H}_\infty$ state-feedback synthesis problem. Our work builds a new connection between nonconvex nonsmooth optimization theory and robust control, leading to an interesting global convergence result for direct policy search on optimal $\mathcal{H}_\infty$ synthesis.

Reinforcement Learning (RL) traditionally relies on scalar reward signals, limiting its ability to leverage the rich semantic knowledge often available in real-world tasks. In contrast, humans learn efficiently by combining numerical feedback with language, prior knowledge, and common sense. We introduce Prompted Policy Search (ProPS), a novel RL method that unifies numerical and linguistic reasoning within a single framework. Unlike prior work that augment existing RL components with language, ProPS places a large language model (LLM) at the center of the policy optimization loop-directly proposing policy updates based on both reward feedback and natural language input. We show that LLMs can perform numerical optimization in-context, and that incorporating semantic signals, such as goals, domain knowledge, and strategy hints can lead to more informed exploration and sample-efficient learning. ProPS is evaluated across fifteen Gymnasium tasks, spanning classic control, Atari games, and MuJoCo environments, and compared to seven widely-adopted RL algorithms (e.g., PPO, SAC, TRPO). It outperforms all baselines on eight out of fifteen tasks and demonstrates substantial gains when provided with domain knowledge. These results highlight the potential of unifying semantics and numerics for transparent, generalizable, and human-aligned RL.

We also provide a general convergence analysis to support our empirical findings. Although our analysis is similar to CPI's, it has a key difference: as long as MBOC succeeds, we can provide a larger policy improvement than CPI at each iteration.

We also provide a general convergence analysis to support our empirical findings. Although our analysis is similar to CPI's, it has a key difference: as long as MBOC succeeds, we can provide a larger policy improvement than CPI at each iteration.

We introduce the first direct policy search algorithm which provably converges to the globally optimal dynamic filter for the classical problem of predicting the outputs of a linear dynamical system, given noisy, partial observations. Despite the ubiquity of partial observability in practice, theoretical guarantees for direct policy search algorithms, one of the backbones of modern reinforcement learning, have proven difficult to achieve. This is primarily due to the degeneracies which arise when optimizing over filters that maintain an internal state. In this paper, we provide a new perspective on this challenging problem based on the notion of informativity, which intuitively requires that all components of a filter's internal state are representative of the true state of the underlying dynamical system. We show that informativity overcomes the aforementioned degeneracy. Specifically, we propose a regularizer which explicitly enforces informativity, and establish that gradient descent on this regularized objective - combined with a "reconditioning step" - converges to the globally optimal cost at a O (1 /T) rate.

However, the limited accessibility of data in FL still poses many challenges, such as insufficient training data and local data bias.