[论文解读] Maximum Mean Discrepancy Gradient Flow

Michael Arbel, Anna Korba+2 more|arXiv (Cornell University)|Jun 11, 2019

Generative Adversarial Networks and Image Synthesis参考文献 49被引用 37

一句话总结

Introduces a Wasserstein gradient flow on the MMD, analyzes convergence to the global optimum, and proposes a noise-regularized particle algorithm for practical implementation.

ABSTRACT

We construct a Wasserstein gradient flow of the maximum mean discrepancy (MMD) and study its convergence properties. The MMD is an integral probability metric defined for a reproducing kernel Hilbert space (RKHS), and serves as a metric on probability measures for a sufficiently rich RKHS. We obtain conditions for convergence of the gradient flow towards a global optimum, that can be related to particle transport when optimizing neural networks. We also propose a way to regularize this MMD flow, based on an injection of noise in the gradient. This algorithmic fix comes with theoretical and empirical evidence. The practical implementation of the flow is straightforward, since both the MMD and its gradient have simple closed-form expressions, which can be easily estimated with samples.

研究动机与目标

• Motivate and construct a gradient flow on the space of probability measures endowed with the Wasserstein metric using the MMD as the objective.
• Derive and analyze the continuous-time and discrete-time (Euler) gradient flows of the MMD towards the target distribution.
• Investigate conditions for convergence to the global optimum and identify barriers to non-convex settings.
• Propose a regularization strategy by injecting noise into the gradient to improve convergence in practice and provide theoretical justification.

提出的方法

• Define the MMD between a fixed target distribution µ and a variable ν in a reproducing kernel Hilbert space, and express F(ν)=1/2 MMD^2(µ,ν).
• Formulate a gradient flow on P2(X) via the continuity equation with velocity Vt=−∇fµ,νt, yielding ∂tνt = div(νt ∇fµ,νt).
• Show that F(νt) decreases along the flow with dF(νt)/dt = −∫ ||∇fµ,νt(x)||^2 dνt(x).
• Provide a forward-Euler discretization νn+1=(I−γ∇fµ,νn)#νn) and establish conditions under which F(νn) decreases.
• Introduce a noisy gradient update Xn+1 = Xn − γ ∇fµ,νn(Xn + βnUn) as a regularized scheme.
• Present a practical particle-based algorithm updating Xi n+1 = Xi n − γ ∇fˆµ,ˆνn(Xi n + βnUi n) using samples from µ and νn and analyze its convergence.

实验结果

研究问题

• RQ1 Under what conditions does the MMD Wasserstein gradient flow converge to the global optimum?
• RQ2 How can the non-convexity of F be mitigated to ensure convergence in practice?
• RQ3 What regularization (via noise) best promotes global convergence without altering the true optimum?
• RQ4 How does a particle-based sampling implementation approximate the population flow and what are the convergence guarantees?

主要发现

• The MMD gradient flow in W2 is well-defined, with a Lyapunov decrease of F along the flow.
• A discrete forward-Euler scheme yields decreasing F provided the step-size γ is small enough (γ ≤ 2/3L).
• F is not generally displacement convex; instead it is Λ-displacement convex, leading to nontrivial convergence analysis and potential barriers.
• A regularization by injecting noise into the gradient (noisy update) can guarantee convergence to the global minimum under suitable conditions on the noise schedule.
• The proposed particle-based algorithm has polynomial-time per-iteration complexity and converges to the population flow as sample sizes increase, with quantified propagation of chaos results.
• Empirical evidence shows that noise-injected MMD flow can outperform plain MMD and KSD in training regression-like networks on synthetic tasks.