Score-based generative modeling secretly minimizes the Wasserstein distance
Anonymous Author(s)
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Abstract
Score-based generative models are shown to achieve remarkable empirical performances in various applications such as image generation and audio synthesis.
However, a theoretical understanding of score-based diffusion models is still incomplete.
Recently, Song et al.
showed that the training objective of score-basedgenerative models is equivalent to minimizing the Kullback-Leibler divergenceof the generated distribution from the data distribution.
In this work, we showthat score-based models also minimize the Wasserstein distance between them.
Specifically, we prove that the Wasserstein distance is upper bounded by the squareroot of the objective function up to multiplicative constants and a fixed constantoffset.
Our proof is based on a novel application of the theory of optimal transport,which can be of independent interest to the society.
Our numerical experimentssupport our findings.
By analyzing our upper bounds, we provide a few techniquesto obtain tighter upper bounds
Introduction
The idea of gradually perturbing the data distribution to define the forward process and learningthe reverse process has recently demonstrated remarkable performances in generative modeling.
Score-based generative models [33, 35, 36] and diffusion probabilistic models [29, 17] have beenused in numerous applications, including image generation [25, 12], and audio synthesis [9, 21].
Both model families have the following two processes.
The given data distribution is incrementallyperturbed by adding small noise in the forward process.
This forward process can be understood asthe standard diffusion process [37].
The reverse process requires the so-called score function [2], thegradient of the log probability density function.
To approximate the score function, it is common tominimize a weighted combination of score matching losses.
Once the score function is approximated,one can generate samples from randomly generated noise by applying the learned reverse process.
Note that score-based models do not explicitly minimize any sort of similarity between the modeldistribution (the distribution of the generated samples) and the data distribution.
Therefore, itis unclear whether or not scored-based generative models can minimize divergence (or distance)between the model distribution and the data distribution.
In very recent work [31], the authors showthat optimizing the score matching loss is indeed equivalent to minimizing the upper bound on the
Kullback–Leibler divergence (KL divergence).
Therefore, it is expected that the model distribution issimilar (in the sense of KL divergence) to the data distribution if the score matching loss is smallenough.
A natural question follows – do score-based models minimize any other divergence or distancethan the KL divergence?
At first glance, this seems unlikely.
For instance, Generative Adversarial
Networks (GANs) [16] were shown to minimize different divergences depending on the choiceof the loss function.
The original GAN loss function turns out to capture the Jensen–Shannon divergence [16], while the W-GAN loss function captures the Wasserstein distance [3].
We refer thereaders to [10] for more details.
On the other hand, a unique aspect of score-based models is that the score function is defined in thesame space where data lies in.
In stark contrast, all the other popular generative models such as GANs[16], VAEs [19], and Normalizing Flows [20] use loss functions that only care about likelihoods anddo not directly operate in the knowledge of the underlying data space.
In this work, we answer this question by revealing the following property of score-based modeling:
Score-based modeling minimizes the Wasserstein distance.
We note that this is quite surprising as there is no direct relationship between the KL divergence andthe Wasserstein distance, in general.
Therefore, this finding is not implied by the aforementionedfinding.
[Figure]
Using the theory of optimal transport, we investigate the evolution of data distribution based on
Partial differential equations (PDEs).
This allows us to estimate the Wasserstein distance betweenthe given data distribution and the distribution of the generated samples.
Our work theoreticallyguarantees that the score-based generative models approximate data distributions in the space ofprobability measures equipped with the Wasserstein distance.
Our main contributions and findingsare summarized as follows:
• (Theorem 1 & Corollary 1) Using the theory of optimal transport, we prove that the Wasser-
stein distance between the data distribution and the model distribution is bounded from
above.
The upper bound is given as the square root of the score-matching loss J SM up
to multiplicative constants and a fixed constant offset.
The main idea is to estimate the
time-derivative of the Wasserstein metric by using the form of the continuity equations.
• (Theorem 2)
We prove J SM ≤ J DSM under suitable assumptions, which gives us a more
practical upper bound since J DSM is much easier to estimate in practice and is widely used as a loss function to train score-based models.• (Theorem 3)
We analyze how the model distribution changes if the data distribution isperturbed initially.
This guarantees that the score based-model is robust under noise.• (Section 4)
Our numerical experiments support the upper bound of our analysis.
We also
explore the effect on the weight decay coefficient in training and different choices of thenumber of total timesteps T .
Notations: x = ( x 1 , x 2 , · · · , x d ) ⊤ ∈ R d and t ∈ [0 , T ] denote the spatial variable and the timevariable, respectively.
Here, R d is the d -dimensional Euclidean space and T > 0 is the time horizon.
The gradient of a real-valued function ρ with respect to the spatial variable and the time-derivativeof ρ are denoted by ∇ ρ = (cid:16) ∂ρ∂x 1 , ∂ρ∂x 2 , · · · , ∂ρ∂x d (cid:17) ⊤ , and ∂ t ρ , respectively.
The Laplacian of ρ isgiven as ∆ ρ = ∇ · ( ∇ ρ ) .
Here, ∇ · F = ∂F 1 ∂x 1 + ∂F 2 ∂x 2 + · · · + ∂F d ∂x d denotes the divergence of
F = ( F 1 , F 2 , · · · , F d ) with respect to the spatial variable x . N ( µ, Σ) indicates Gaussian distribution with mean µ and covariance Σ .
∥ · ∥ indicates L 2 norm.
We denote the data distribution and thedistribution of the generated samples by p 0 and q 0 .
Preliminaries and related works
In this section, we analyze the forward process and the reverse process in the score-based modelthrough Partial differential equations (PDEs).
In particular, we investigate the time evolution ofmarginal distributions through the two processes.
2.1 The forward process
As in [37], the forward process can be modeled by the standard diffusion process { x ( t ) }
Tt =0 fort ∈ [0 , T ] such that x (0) ∼ p 0 and
[Equation]
[Equation] Here, w is the standard Wiener process, f : R d × [0 , T ] → R d is the drift coefficient of x ( t ) , andg : [0 , T ] → R is the diffusion coefficient of x ( t ) .
Let p t = p ( · , t ) be a probability density function of the random variable x ( t ) on R d at time t ∈ [0 , T ] .
Then, the function p : R d × [0 , T ] → [0 , + ∞ ) satisfies the following partial differential equations(PDEs), namely the Fokker–Planck equation or the forward Kolmogorov equation as in [4, Theorem2.2]:
[Equation] [Equation]
) where the pdf p 0 represents the data distribution.
Under suitable assumptions on f and g , a solutionto the above initial value problem uniquely exists.
We refer the readers to [5] for more details.
2.2 Understanding the reverse process through PDEs
To generate samples from t
= T to 0 , the score-based model uses the reverse-time SDE [2]:
[Equation]
Here, ¯ w is a standard Wiener process when time flows backward from t =
T to t = 0 .
As in the previous section, for given x ( T ) ∼ p T , a probability density function of the random variablex ( t ) from in the above process (3) satisfies the final value problem:
[Equation]
Using this simple identity, ∆ p = ∇ · ( ∇ p ) = ∇ · (cid:16) p ∇ pp (cid:17) =
∇ · ( p ∇ (log p )) , it can be seen that thedifferential equations in (2) and (4) are indeed identical.
2.3 The objective function
The score of the probability distribution, ∇ log p t , is approximated by a function s θ : R d × [0 , T ] → 98 R d .
For instance, in [31], the authors consider the weighted mean squared error (MSE):
[Equation] [Equation]
where λ : [0 , T ] → (0 , ∞ ) is a positive weighting function.
Replacing ∇ log p in (4) with s θ , we deduce a new reverse process.
For given noise distribution q T , aprobability density function q t = q ( · , t ) at time t ∈ [0 , T ] solves
[Equation]
Here, we introduced a new function q : R d × [0 , T ] → [0 , + ∞ ) to illustrate the reverse process.
2.4 Related works
Optimal transport The Wasserstein distance has a solid connection to the Fokker–Planck equation.
In particular, the Fokker–Planck equation can be understood as the gradient flow in the space ofprobability measures equipped with the 2 -Wasserstein distance [1, 18, 26].
For two probabilitymeasures µ and ν on R d , we define the 2 -Wasserstein or Monge-Kantorovich distance as
[Equation]
Here, Π( µ, ν ) is the set of all couplings π of µ and ν where a Borel probability measure π on R d × R dsatisfies π ( A × R d ) = µ ( A ) and π ( R d × B ) =
ν ( B ) for all measurable subsets A, B ⊂ R d .
Under suitable assumptions on the drift coefficient and the diffusion coefficient, a solution to the
Fokker–Planck equation satisfies the so-called Wasserstein contraction property [6, 7, 28], whichplays an important role in our analysis.
More discussions can be found in Section 3.3 and Section 5.
Score matching and diffusion probabilistic models Score matching models [34, 36] use the scorefunction to reverse the noise corruption in a continuous time range by solving stochastic differentialequations (SDEs).
Diffusion probabilistic models [29, 17] adopt a similar idea of modeling tractabledenoising process to reverse each discrete step of noise corruption with probabilistic models.
In fact,[36] show that diffusion models are score matching models with certain forms of SDE, which unifiesthe whole framework.
Diffusion models are gaining increasing attention: recent advances [30, 25, 12]show that diffusion models can generate even better images than GANs while enjoying a more stabletraining process, which encourages it to be widely applied in various tasks, such as text-to-image andimage editing [24, 23], audio generation [9, 21], etc.
Approximation in the Wasserstein space
In this section, we estimate the Wasserstein distance between a given data distribution p 0 and themarginal of the generated samples q 0 obtained by the score-based model.
3.1 The main results
We present our main results under the following assumptions.(A1)
The drift coefficient f : R d × [0 , T ] → R d is Lipschitz continuous in the space variable x : thereexists a positive constant L f ( t ) ∈ (0 , ∞ ) , depending on t ∈ [0 , T ] , such that for all x, y ∈ R d
[Equation]
(A2) s θ : R d × [0 , T ] → R d satisfies the one-sided Lipschitz condition [13, Definition 2.1]: thereexists a constant L s ( t ) ∈ R , depending on t ∈ [0 , T ] , satisfying
[Equation]
The usual Lipschitz constant of s θ satisfies the above inequality (9).
Unlike the usual Lipschitzconstant, the one-sided Lipschitz constant L s ( t ) is not necessarily nonnegative (see Appendix B).
More technical assumptions can be found in Appendix A.
We now present a general result for any choice of loss functions. Let [Equation] and an integrating factor I : [0 , T ] → (0 , + ∞ ) ,
[Equation]
[Equation]
Theorem 1.
Let p 0 and q 0 be the data distribution and a pdf obtained by the score-based model asin Section 2.
Then the Wasserstein distance between p 0 and q 0 is bounded as follows:
[Equation]
The main idea of the proofs is to estimate the derivatives of the Wasserstein distance along with thetime evolution, which is illustrated in Section 5.
Applying the Cauchy-Schwarz inequality, we obtainthe upper bound in terms of the weighted MSE J SM .
Corollary 1.
Let p 0 and q 0 be given in Theorem 1.
If λ ( t ) =
g ( t ) 2 for all t ∈ [0 , T ] , then thefollowing inequality holds:
[Equation]
We defer the proofs of Theorem 1 and Corollary 1 to Appendix A.
Remark 1.
From the above result, the Wasserstein distance converges to zero as the loss function
J SM goes to zero if p T = q T .
However, due to the competition between I ( T ) and W 2 ( p T , q T ) , ageneral convergence result remains open (see Section 6 and Appendix G).
Remark 2.
If g 4 I 2 λ − 1 is integrable in [0 , T ] , the similar inequality holds for other loss functions.
Remark 3.
Note that the above upper bound depends on the time horizon T .
Numerical experimentsfor different choices of T are provided in Appendix F.
3.2 The relation between J SM and J DSM
The above results give the guarantee that J SM can upper-bound Wasserstein distance.
However, J SM is generally intractable due to the unknown quantity ∇ x log p t ( x ) [32], and is rarely used inpractice.
Alternatively, there is a conditional version of the score-matching loss J DSM , which isequivalent to J SM up to a constant as shown in [38].
J DSM is also widely used as a loss function totrain score-based models:
[Equation]
where p 0 t indicates the conditional probability of x at step t given x (0) .
Now we present our newresult, which shows that J DSM is also an upper bound of J SM under certain assumption.
Theorem 2.
If p 0 t satisfies
[Equation]
then we have
[Equation]
Corollary 2.
Let p ( x ( t ) | x ( t − 1))
=
N ( √ 1 − β t x
( t − 1) , β t I ) as in Denoising Diffusion Probabilistic Models (DDPM)
[17], which corresponds to f ( x, t ) = − 12 β t x , and g ( t ) = √ β t .
Here,β t > 0 is the noise schedule for all t , and let λ ( t ) =
g 2 ( t ) .
Then eq.
(16) holds.
The proofs of Theorem 2 and Corollary 2 are in Appendix D. We also provide numerical resultswhich show J SM ≤ J DSM for DDPM in Appendix E.
Remark 4.
Using Theorem 2 and Corollary 2, we can present an upper bound of the Wassersteindistance in terms of J DSM , where both W 2 ( p 0 , q 0 ) and J DSM are tractable.
This enables us toempirically verify this upper bound via DDPM in Section 4.
3.3 Perturbation and sensitivity analysis
Let ˜ p 0 be a small perturbation of the original data distribution p 0 .
For a given corrupted distribution˜ p 0 , denote the probability density functions in the forward process and the reverse process by ˜ p t and˜ q t , respectively.
In this section, we investigate how this perturbation changes the score-based model.
In particular, we are interested in the upper bound on W 2 ( q 0 , ˜ q 0 ) with respect to W 2 ( p 0 , ˜ p 0 ) .
It is well known that the Wasserstein distance between two probability distributions in the forwardprocess can be estimated as follows: for all t ∈ [0 , T ]
[Equation]
This is the so-called Wasserstein contraction property [8].
In the reverse process, the score functionin the drift coefficients may vary and thus the above inequality does not directly give us the upperbound on W 2 ( q 0 , ˜ q 0 ) .
Our analysis in the previous sections allows us to estimate W 2 ( q 0 , ˜ q 0 ) .
It is worth noting that the Wasserstein distance satisfies the triangle inequality, unlike other statisticalmeasures, including the Kullback–Leibler (KL) divergence. Combining our main results with W 2 ( q 0 , ˜ q 0 ) ≤ W 2 ( p 0 , ˜ p 0 ) + W 2 ( p 0 , q 0 )
+ W 2 (˜ p 0 , ˜ q 0 ) , we conclude the following theorem.
Theorem 3.
Let ˜ p 0 be the corrupted data distribution and ˜ q 0 be a probability density functionobtained by the score-based model starting from ˜ p 0 .
It holds that
[Equation]
[Equation]
Here, ˜ J SM and ˜ I denote the weighted MSE and the integrating factor from the perturbed model.
Remark 5.
The Wasserstein distance between the original data distribution p 0 and the modeldistribution ˜ q 0 starting from the corrupted data ˜ p 0 can also be estimated similarly.
Experiments
Here we present experiments on simulation datasets to verify the upper bound in eq.
4.1 Experiment settings
Datasets Here we adopt three 2D datasets for simulation: One cluster Gaussian N ( 0 , 0 .
1 I ) , twomoons in [27], and four clusters Gaussian mixture N (( ± 0 .
5 , ± 0 .
5) ⊤ , 0 .
01 I ) with equal weights foreach cluster.
Visualization for these datasets are in Fig 2.
For forward dynamics, we use discrete timesteps with T = 10 .
For t ∈ [1 , 10] , p ( x ( t ) | x ( t − 1)) =
N ( √ 1 − β t x ( t − 1) , β t I ) as in [17], i.e., f ( x, t ) = − 12 β t x , and g ( t ) = √ β t .
For β t , we adopt asigmoid schedule from β 1 = 10 − 5 to β T = 10 − 2 .
Training and evaluation We use a 3-layer neural network as the score matching model, with ReLunonlinearity and skip-connection at the final output.
Each layer is composed of a linear layer with128 hidden neurons, and an embedding layer for 10 timesteps.
For optimizer, we use AdamW [22]with lr = 0 .
001 and weight decay coefficient 0 .
To evaluate Wasserstein distance, we use POT[15] to compute W 2 ( p 0 , q 0 ) at the end of each training epoch.
To avoid the term of W 2 ( p T , q T ) in eq.(16) for verification, we first perform forward pass on sampled x (0) to get x ( T ) , and perform datageneration on the same x ( T ) to approximate q T ≈ p T and W 2 ( p T , q T ) ≈ 0 .
For loss function, weuse J DSM with λ ( t ) =
g 2 ( t ) and batch size = 512.
To get both lower and higher losses to verify ourbound, we first maximize the loss for 15 epochs, and then minimize the loss till convergence.
4.2 Results
To verify the upper bound in eq.
(16), we take log on each side of the equation with W 2 ( p T , q T ) =
0and get:
[Equation]
We present a log-log plot to verify the slope and the intercept in the upper bound, respectively: see
Fig 2.
The log-log plots are obtained by evaluating Wasserstein distance and the loss function duringtraining.
The red dots are the sampled data from each dataset, the green dots are generated data afterloss maximization, and the purple dots are the generated data sampled at convergence.
From thelog-log plots, we can observe that the theoretical line (the black line) is a valid upper bound of theempirical one (the blue dots).
[Figure]
4.3 Exploring the upper bound
Here we explore different factors that can affect the tightness of the theoretical upper bound.
Asshown in Fig 2, the gap between the theoretical bound and the empirical one mainly results from theintercept in eq.
Recall our intercept is 12 log (cid:16)(cid:82) T
g 2 ( t ) I ( t ) 2 dt (cid:17) for I ( t ) given in (11).
4.3.1 Effect of weight regularization
Since g ( t ) and L f are fixed, only the one-sided Lipchitz constant L s ( r ) of the model s θ is subject tothe training process, which is estimated via grid search in our experiments.
Notice that the loss function J DSM aims to minimize the distance between s θ ( x ( t ) , t )
and∇ x log p t ( x | x (0)) .
Thus, L s at convergence is determined by dynamics ∇ x log p 0 t ( x | x (0)) .
Ifthe one-sided Lipschitz constant for ∇ x log p 0 t ( x | x (0)) is small, then the intercept would be expected to be small (see example in Fig 2a of Gaussian cluster with moderately high variance). Ifthe one-sided Lipschitz constant for ∇ x log p 0 t ( x | x (0)) is large, to minimize the loss, the interceptwould also be large (see example in Fig 2c of Gaussian clusters with lower variance).
However, it canbe possible to make the upper bound tighter while sacrificing the training loss by constraining L s .
To constrain L s , the simplest and most efficient way is to constrain the norm of the weights in thescore-based model. Here we present the results of training with L 2 regularization using differentcoefficients on the four clusters dataset, which is suffered from a large gap between empirical andtheoretical bounds on the intercept.
As we can observe in Fig 3a, the loss at convergence will increasewhen we impose stronger regularization on the weights, while the intercept decreases.
Importantly,the intercept approaches the empirical one without not exceeding it, even with a very large weightdecay coefficient.
We also present the log-log training plots for weight decay = 0 .
5 , 1 , 5 respectively:
In Fig 4, we can observe that as we impose stronger weight decay regularization, the theoretical upperbound will approach the empirical one, though the loss at convergence will increase .
4.3.2 Effect of the number of total timesteps
Notice that in the definition of J DSM and the intercept in eq. (19), they are both dependent on T ,which might also affect the tightness of the upper bound: as T increases, the minimal W 2 ( p 0 , q 0 ) is Weight decay coefficient - -    0. Minimal log(J_DSM)
Intercept
Minimal log(W(p_0,q_0)) (a) Minimal losses log( J during Number of total timesteps (T) - - Minimal log(J_DSM))
Minimal log(W(p_0,q_0)) Achievable upperbound ( [Figure]
Minimal losses log( [Figure] DSM ) during training, min- imal log Wasserstein distances during training and achievable upper bounds (Section 4.3.2) of different choices of T .
[Figure]
expected to decrease, but minimal J DSM would not necessarily decrease due to its dependence on T . To verify this, we conduct experiments in the four clusters dataset with T = 2 , 5 , 10 , 20 and 40 . Here we first define the term achievable upper bound as RHS of eq. (19) when plugging in theminimal J DSM in training.
As shown in Fig 3b, we can observe that when T increases, minimal W 2 ( p 0 , q 0 ) during training would decrease, and minimal J DSM during training would increase. Theachievable upper bound would also increase.
We include log-log plots of different choices of T in Appendix F, which shows that the intercept would also increase, and thus the upper bound would belooser when T increases.
The main idea of the proofs
We investigate the time evolution of probability measures arising in the score-based models.
Usingthe optimal transport theory, we differentiate the Wasserstein distances, W 2 ( p t , q t ) with respect totime.
The main idea is to express these equations (2), and (6) in the form of the continuity equation:
[Equation]
where v is a so-called velocity field.
Let us denote the velocity fields of p , and q by
[Equation]
Then, the equations (2), and (6) can be written in the form of (20) with v [ p ] and v [ q ] , respectively.
Using these vector fields, we compute the time derivative of the Wasserstein distance between twoevolving probability measures.
For velocity fields v [ ρ ( i ) ] depending on ρ ( i ) , let ρ ( i ) for i = 1 , 2 betwo solutions of the continuity equations ∂ t ρ ( i ) + ∇ · ( ρ ( i ) v [ ρ ( i ) ])
= 0 .
Then, the relation betweenthe Wasserstein distance and velocity fields arises from the theory of optimal transport:
[Equation]
(21) where π t is an optimal transport plan between ρ (1) t and ρ (2) t .
See [1, Theorem 8.4.7] and [28, Corollary5.25].
This equality plays an important role when we estimate the Wasserstein distance.
The followingproposition is a consequence of (21) and Lemma 1 in Appendix C.
Proposition 1.
Let p t and q t be solutions to (2) and (6) , respectively, in [0 , T ] .
Then,
[Equation]
[Equation]
b ( t ) :=
[Equation]
Discussions
Convergence of the Wasserstein distance If p T =
q T , then our result yields that the Wassersteindistance W 2 ( p 0 , q 0 ) converges to zero as J SM goes to zero.
However, if p T is not equal to q T , thenwe find that it is not easy to obtain such a convergence result.
This is due to the nature of the Langevindynamics in the reverse process.
Typically, additional assumptions such as the log-concavity arerequired to guarantee the Langevin dynamics convergence, for instance, [11].
In addition, the dependence of the neural network s θ makes us hard to evaluate the precise upperbound.
We expect that appropriate assumptions on s θ and the initial distribution p 0 may resolve thisissue, but we do not pursue them in this paper.
See our example in Appendix G.
Estimating L s
Estimating our theoretical bound requires estimating the one-sided Lipschitz constant L s , which is NP-hard if done via grid search [39], making it difficult to estimate in highdimensional data.
Note that the two-sided Lipschitz constant is an upper bound of the one-sidedone, which can be more efficiently estimated as shown in [14].
However, it would result in a loosertheoretical upper bound if we adopt the two-sided Lipschitz constant, and estimating the one-sided Lipschitz constant does not enjoy the semi-definiteness of estimating the two-sided one in [14].
Conclusion
In this paper, we mainly discuss the relationship between the loss function of score-based modelsand the Wasserstein distance.
With the help of the theory in optimal transport, we present a novelupper bound on Wasserstein distance in terms of the loss function used in practice.
We also providenumerical experiments to verify our theoretical bound with various simulation datasets.
Moreover,we explore factors that affect the estimation of the upper bound and show the corresponding influenceon the training process.
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