What is the optimal way to approximate a high-dimensional diffusion process by one in which the coordinates are independent? This paper presents a construction, called the \emph{independent projection}, which is optimal for two natural criteria. First, when the original diffusion is reversible with invariant measure $\rho_*$, the independent projection serves as the Wasserstein gradient flow for the relative entropy $H(\cdot\,|\,\rho_*)$ constrained to the space of product measures. This is related to recent Langevin-based sampling schemes proposed in the statistical literature on mean field variational inference. In addition, we provide both qualitative and quantitative results on the long-time convergence of the independent projection, with quantitative results in the log-concave case derived via a new variant of the logarithmic Sobolev inequality. Second, among all processes with independent coordinates, the independent projection is shown to exhibit the slowest growth rate of path-space entropy relative to the original diffusion. This sheds new light on the classical McKean-Vlasov equation and recent variants proposed for non-exchangeable systems, which can be viewed as special cases of the independent projection.

Multi-task reinforcement learning could enable robots to scale across a wide variety of manipulation tasks in homes and workplaces. However, generalizing from one task to another and mitigating negative task interference still remains a challenge. Addressing this challenge by successfully sharing information across tasks will depend on how well the structure underlying the tasks is captured. In this work, we introduce our new architecture, Projected Task-Specific Layers (PTSL), that leverages a common policy with dense task-specific corrections through task-specific layers to better express shared and variable task information. We then show that our model outperforms the state of the art on the MT10 and MT50 benchmarks of Meta-World consisting of 10 and 50 goal-conditioned tasks for a Sawyer arm.