Generative Learning With Euler Particle Transport

1 INTRODUCTION . The ability to efficiently sample from complex distributions plays a key role in a variety of prediction and inference tasks in machine learning and statistics ( Salakhutdinov , 2015 ) . The long-standing methodology for learning an underlying distribution relies on an explicit statistical data model , which can be difficult to specify in many applications such as image analysis , computer vision and natural language processing . In contrast , implicit generative models do not assume a specific form of the data distribution , but rather learn a nonlinear map to transform a reference distribution to the target distribution . This modeling approach has been shown to achieve impressive performance in many machine learning tasks ( Reed et al. , 2016 ; Zhu et al. , 2017 ) . Generative adversarial networks ( GAN ) ( Goodfellow et al. , 2014 ) , variational auto-encoders ( VAE ) ( Kingma & Welling , 2014 ) and flow-based methods ( Rezende & Mohamed , 2015 ) are important representatives of implicit generative models . In this paper , we propose an Euler particle transport ( EPT ) approach for learning a generative model by integrating ideas from optimal transport , numerical ODE , density-ratio estimation and deep neural networks . We formulate the problem of generative learning as that of finding a nonlinear transform that pushes forward a reference to the target based on the quadratic Wasserstein distance . Since it is challenging to solve the resulting Monge-Ampère equation , we consider the continuity equation derived from the linearization of the Monge-Ampère equation , which is a gradient flows converging to the target distribution . We solve the Mckean-Vlasov equation associated with the gradient flow using the forward Euler method . The resulting EPT that pushes forward the reference distribution to the target distribution is a composition of a sequence of simple residual maps , which are computationally stable and easy to train . The residual maps are completely determined by the density ratios between the distributions at the current iterations and the target distribution . We estimate density ratios based on the Bregman divergence with a gradient regularizer using deep density-ratio fitting . We establish bounds on the approximation errors due to linearization of the Monge-Ampère equation , Euler discretization of the Mckean-Vlasov equation , and deep density-ratio estimation . Our result on the error rate for the proposed density-ratio estimators improves the minimax rate of nonparametric estimation via exploring the low-dimensional structure of the data and circumvents the “ curse of dimensionality ” . Experimental results on multi-mode synthetic data and comparisons with stateof-the-art GANs on benchmark data support our theoretical findings and demonstrate that EPT is computationally more stable and easier to train than GANs . Using simple ReLU ResNets without batch normalization and spectral normalization , we obtained results that are better than or comparable with those using GANs trained with such tricks . 2 EULER PARTICLE TRANSPORT . Let X ∈ Rm be a random vector with distribution ν , and let Z be a random vector with distribution µ . We assume that µ has a known and simple form . Our goal is to construct a transformation T such that T # µ = ν , where T # µ denotes the push-forward distribution of µ by T , that is , the distribution of T ( Z ) . Then we can sample from ν by first generating a Z ∼ µ and calculate T ( Z ) . In practice , ν is unknown and only a random sample { Xi } ni=1 i.i.d . ν is available . We must construct T based on the sample . There may exist multiple transports T with T # µ = ν . The optimal transport is the one that minimizes the quadratic Wasserstein distance between µ and ν defined by W2 ( µ , ν ) = { inf γ∈Γ ( µ , ν ) E ( Z , X ) ∼γ [ ‖Z −X‖22 ] } 1 2 , ( 1 ) where Γ ( µ , ν ) denotes the set of couplings of ( µ , ν ) ( Villani , 2008 ; Ambrosio et al. , 2008 ) . Suppose that µ and ν have densities q and p with respect to the Lesbeque measure , respectively . Then the optimal transport map T such that T # µ = ν is characterized by the Monge-Ampère equation ( Brenier , 1991 ; McCann , 1995 ; Santambrogio , 2015 ) . Specifically , the minimization problem in ( 1 ) admits a unique solution γ = ( 1 , T ) # µ with T = ∇Ψ , µ-a.e. , where 1 is the identity map and ∇Ψ is the gradient of the potential function Ψ : Rm → R. This function is convex and satisfies the Monge-Ampère equation det ( ∇2Ψ ( z ) ) = q ( z ) p ( ∇Ψ ( z ) ) , z ∈ Rm . ( 2 ) Therefore , to find the optimal transport T , it suffices to solve ( 2 ) for Ψ . However , it is challenging to solve this degenerate elliptic equation due to its highly nonlinear nature . Below we describe the proposed EPT method for obtaining an approximate solution of the MongeAmpère equation ( 2 ) . It consists of the following steps : ( a ) linearizing ( 2 ) via residual maps , ( b ) determining the velocity fields governing the stochastic McKean-Vlasov equation resulting from the linearization , ( c ) calculating the forward Euler particle transport map and , ( d ) training the EPT map by estimating the velocity fields from data . Since velocity fields are completely determined by density ratios , this step amounts to nonparametric density ratio estimation . We also provide bounds on the errors due to linearization , discretization and estimation . Mathematical details and proofs are given in the appendix . Linearization via residual map A basic approach to addressing the difficulty due to nonlinearity is linearization . We use a linearization method based on the residual map Tt , Φt = ∇Ψ = 1+ t∇Φt , t ≥ 0 , ( 3 ) where Φt : Rm → R1 is a function to be chosen such that the law of Tt , Φt ( Z ) approaches ν as t increases ( Villani , 2008 ) . We give the specific form of Φt below , see Theorem B.1 in the appendix for details . This linearization scheme leads to the stochastic process Xt : Rm → Rm satisfying the McKeanVlasov equation d dt Xt ( x ) = vt ( Xt ( x ) ) , t ≥ 0 , with X0 ∼ µ , µ- a.e . x ∈ Rm , ( 4 ) where vt is the velocity vector field of Xt . In addition , we have vt = ∇Φt . Thus vt also determines the residual map ( 3 ) . The details of the derivation are given in Theorems B.2 and B.1 . in the appendix . Therefore , estimating the residual map ( 3 ) is equivalent to estimating vt . The movement of Xt along t is completely governed by vt , given the initial value . We choose a vt to decrease the discrepancy between the distribution of Xt , say µt , at time t and the target ν with respect to a properly chosen measure . An equivalent formulation of ( 4 ) is through the gradient flow { µt } t≥0 with { vt } t≥0 as its velocity fields , see Proposition B.1 in the appendix . Computationally it is more convenient to work with ( 4 ) . Determining velocity field The basic intuition is that we should move in the direction that decreases the differences between µt and the target ν . We use an energy functional L [ µt ] to measure such differences . An important energy functional L [ µt ] is the f -divergence ( Ali & Silvey , 1966 ) , L [ µt ] = Df ( µt‖ν ) = ∫ Rm p ( x ) f ( qt ( x ) p ( x ) ) dx , ( 5 ) where qt is the density of µt , p is the density of ν and f : R+ → R is assumed to be a twicedifferentiable convex function with f ( 1 ) = 0 . We choose Φt such that L [ µt ] is minimized . We show in Theorem B.1 in the appendix that Φt ( x ) = −f ′ ( rt ( x ) ) and vt ( x ) = ∇Φt ( x ) . Therefore , vt ( x ) = −f ′′ ( rt ( x ) ) ∇rt ( x ) , where rt ( x ) = qt ( x ) p ( x ) , x ∈ Rm . For example , if we use the χ2-divergence with f ( c ) = ( c−1 ) 2/2 , then vt ( x ) = ∇rt ( x ) is simply the gradient of the density ratio . Other types of velocity fields can be obtained by using different energy functionals such as the Lebesgue norm of the density difference , i.e. , L [ µt ] = ∫ Rm |qt ( x ) −p ( x ) | 2dx , see Section B.2 for details . The forward Euler method Numerically , we need to discretize the McKean-Vlasov equation ( 4 ) . Let s > 0 be a small step size . We use the forward Euler method defined iteratively by : Tk = 1+ svk , ( 6 ) Xk+1 = Tk ( Xk ) , ( 7 ) µk+1 = ( Tk ) # µk , ( 8 ) where X0 ∼ µ , µ0 = µ , vk is the velocity field at the kth step , k = 0 , 1 , ... , K for some large K. The particle process { Xk } k≥0 is a discretized version of the continuous process { Xt } t≥0 in ( 4 ) . The final transport map is the composition of a sequence of simple residual maps T0 , T1 . . . , TK , i.e. , T = TK ◦ TK−1 · · · ◦ T0 . This updating scheme is based on the forward Euler method for solving equation ( 4 ) . This is the reason we refer to the proposed method as Euler particle transport ( EPT ) . Training EPT When the target ν is unknown and only a random sample is available , it is natural to learn ν by first estimating the discrete velocity fields vk at the sample level and then plugging the estimator of vk in ( 6 ) . For example , if we use the f -divergence as the energy functional , estimating vk ( x ) = −f ′′ ( rk ( x ) ) ∇rk ( x ) boils down to estimating the density ratios rk ( x ) = qk ( x ) /p ( x ) dynamically at each iteration k. Nonparametric density-ratio estimation using Bregman divergences and gradient regularizer are discussed in Section 4 below . Let v̂k be the estimated velocity fields at the kth iteration . The kth estimated residual map is T̂k = 1+ sv̂k . Finally , the trained map is T̂ = T̂K ◦ T̂K−1 ◦ · · · ◦ T̂0 . ( 9 ) Theoretical guarantees We establish the following bound on the approximation error due to the linearization of the Monge-Ampère equation under appropriate conditions : W2 ( µt , ν ) = O ( e−λt ) , ( 10 ) for some λ > 0 , see Proposition B.1 in the appendix . Therefore , µt converges to ν exponentially fast as t→∞ . For an integer K ≥ 1 and a small s > 0 , let { µst : t ∈ [ ks , ( k + 1 ) s ) , k = 0 , . . . , K } be a piecewise constant interpolation between µks and µ ( k+1 ) s , k = 0 , 1 , . . . , K. Under the assumption that the velocity fields vt are Lipschitz continuous with respect to ( x , µt ) , the discretization error of µst can be bounded in a finite time interval [ 0 , T ) as follows : sup t∈ [ 0 , T ) W2 ( µt , µst ) = O ( s ) . ( 11 ) The proof of ( 11 ) is given in Proposition B.2 in the appendix . The error bounds ( 10 ) and ( 11 ) imply that the distributions of the particles Xk generated by the EPT map defined in ( 7 ) with a small s and a sufficiently large k converges to the target ν at the rate of discretization size s. When training the EPT map , we use the deep neural networks to estimate the density ratios ( density differences ) with samples . In Theorem 4.1 , we provide an estimation error bound that improves the minimax rate of deep nonparametric estimation via exploring the low-dimensional structure of data and circumvents the “ curse of dimensionality. ” Thus this result is of independent interest in nonparametric estimation using deep neural networks .

Summary: This paper tackles generative modeling (sampling, in particular) via finding the push forward functions T (equivalently, the velocity fields v) that iteratively moves particles from a reference distribution toward the target data distribution. The velocity fields are solved by minimizing the f-divergence between the particle density at iteration k and the target data density, which is shown to be in the form of gradient of density ratio. Based on this intuition, the training stage becomes estimating the density ratio via neural networks, for each iteration k=1,...,K. However, estimating the density ratio can be quite difficult when two densities have little overlapping support. Thus, the author proposes to add gradient regularizer to the density ratio estimating function. The experiment on real-world computer vision benchmarks demonstrate reasonable sampling quality, and the FID score on CIFAR-10 is comparable to some GAN baselines in the generative modeling literature.