Action Manifold Smoothing: A Lipschitz Pathway Perspective on High-Dimensional Reinforcement Learning

High-dimensional continuous control remains challenging in deep reinforcement learning, where algorithms like TD3 and SAC often collapse. We propose a unifying \textbf{Lipschitz Pathway} framework that decomposes instability into four amplification stages, namely action parameterization ($L_1$), dynamics sensitivity ($L_2$), Q-network curvature ($L_3$), and temporal-difference (TD) target stability ($L_4$), where errors compound multiplicatively along the learning pipeline. Our analysis identifies a \textit{discrete-continuous mismatch} as the root cause: value functions trained from sparse point samples must generalize over continuous manifolds, leading to multiplicative error amplification along the pathway. To address this, we introduce \textbf{Action Manifold Smoothing (AMS)}, which replaces point-wise TD targets with orthogonally-sampled neighborhood averages, jointly regularizing $L_3$ (via implicit Laplacian smoothing) and $L_4$ (via local manifold supervision). We further characterize when Lipschitz-constrained Q-networks and geometric action priors are beneficial based on task structure. Empirically, AMS enables both TD3 and SAC to achieve over 400 reward on the 38-D Dog Run task within 1M steps, where baselines fail. These results validate the Lipschitz pathway as a principled framework for diagnosing and solving stability bottlenecks in high-dimensional control.

Modern AI agents that learn to walk, run, or manipulate objects often collapse when the task becomes physically complex—imagine teaching a robot dog with 38 joints to run. These failures are usually treated as bugs to patch, but we found a deeper reason: the AI is trying to draw a smooth, continuous map of "what every action is worth," using only a handful of scattered data points. In high dimensions, this map is full of false peaks and cliffs, which mislead the agent and cause learning to derail. Our solution, Action Manifold Smoothing, gently averages the agent's value estimates over a small neighborhood of nearby actions, rather than trusting a single noisy point. This is analogous to a surveyor estimating ground elevation from several nearby measurements instead of one footstep. With this simple change, two standard learning algorithms that previously failed completely on a simulated 38-joint robot dog now succeed within a single day of training. The result suggests that respecting the geometry of how value changes with action— rather than adding more tricks—is the key to scaling reinforcement learning to high-dimensional physical control.