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Guidance with Spherical Gaussian Constraint for Conditional Diffusion
Feb 05, 2024
Recent advances in diffusion models attempt to handle conditional generative tasks by utilizing a differentiable loss function for guidance without the need for additional training. While these methods achieved certain success, they often compromise on sample quality and require small guidance step sizes, leading to longer sampling processes. This paper reveals that the fundamental issue lies in the manifold deviation during the sampling process when loss guidance is employed. We theoretically show the existence of manifold deviation by establishing a certain lower bound for the estimation error of the loss guidance. To mitigate this problem, we propose Diffusion with Spherical Gaussian constraint (DSG), drawing inspiration from the concentration phenomenon in high-dimensional Gaussian distributions. DSG effectively constrains the guidance step within the intermediate data manifold through optimization and enables the use of larger guidance steps. Furthermore, we present a closed-form solution for DSG denoising with the Spherical Gaussian constraint. Notably, DSG can seamlessly integrate as a plugin module within existing training-free conditional diffusion methods. Implementing DSG merely involves a few lines of additional code with almost no extra computational overhead, yet it leads to significant performance improvements. Comprehensive experimental results in various conditional generation tasks validate the superiority and adaptability of DSG in terms of both sample quality and time efficiency.
Two Sides of The Same Coin: Bridging Deep Equilibrium Models and Neural ODEs via Homotopy Continuation
Oct 14, 2023Deep Equilibrium Models (DEQs) and Neural Ordinary Differential Equations (Neural ODEs) are two branches of implicit models that have achieved remarkable success owing to their superior performance and low memory consumption. While both are implicit models, DEQs and Neural ODEs are derived from different mathematical formulations. Inspired by homotopy continuation, we establish a connection between these two models and illustrate that they are actually two sides of the same coin. Homotopy continuation is a classical method of solving nonlinear equations based on a corresponding ODE. Given this connection, we proposed a new implicit model called HomoODE that inherits the property of high accuracy from DEQs and the property of stability from Neural ODEs. Unlike DEQs, which explicitly solve an equilibrium-point-finding problem via Newton's methods in the forward pass, HomoODE solves the equilibrium-point-finding problem implicitly using a modified Neural ODE via homotopy continuation. Further, we developed an acceleration method for HomoODE with a shared learnable initial point. It is worth noting that our model also provides a better understanding of why Augmented Neural ODEs work as long as the augmented part is regarded as the equilibrium point to find. Comprehensive experiments with several image classification tasks demonstrate that HomoODE surpasses existing implicit models in terms of both accuracy and memory consumption.
Reduced Policy Optimization for Continuous Control with Hard Constraints
Oct 14, 2023Recent advances in constrained reinforcement learning (RL) have endowed reinforcement learning with certain safety guarantees. However, deploying existing constrained RL algorithms in continuous control tasks with general hard constraints remains challenging, particularly in those situations with non-convex hard constraints. Inspired by the generalized reduced gradient (GRG) algorithm, a classical constrained optimization technique, we propose a reduced policy optimization (RPO) algorithm that combines RL with GRG to address general hard constraints. RPO partitions actions into basic actions and nonbasic actions following the GRG method and outputs the basic actions via a policy network. Subsequently, RPO calculates the nonbasic actions by solving equations based on equality constraints using the obtained basic actions. The policy network is then updated by implicitly differentiating nonbasic actions with respect to basic actions. Additionally, we introduce an action projection procedure based on the reduced gradient and apply a modified Lagrangian relaxation technique to ensure inequality constraints are satisfied. To the best of our knowledge, RPO is the first attempt that introduces GRG to RL as a way of efficiently handling both equality and inequality hard constraints. It is worth noting that there is currently a lack of RL environments with complex hard constraints, which motivates us to develop three new benchmarks: two robotics manipulation tasks and a smart grid operation control task. With these benchmarks, RPO achieves better performance than previous constrained RL algorithms in terms of both cumulative reward and constraint violation. We believe RPO, along with the new benchmarks, will open up new opportunities for applying RL to real-world problems with complex constraints.