| BackBone | Model category | Model | Cifar10(4k labels) |
| WRN-28-2 | Disagreement | Temporal Ensembling(TE) (Laine & Aila,2017) | 16.37 |
| Mean Teacher(Tarvainen & Valpola,2017) | 15.87 |
| 13.13 |
| VAT+EntMin(Miyato et al., 2019) MixMatch(Berthelot et al., 2019) | 6.37 |
| GS-BadGANt*(Li et al., 2019) | 17.11 |
| WRN-28-10 Generative | Generative | OSPOT-VAE | 8.51(±0.32) |
| AutoAugment(Cubuk et al., 2019) | 14.1 |
| Disagreement | Temporal Ensembing(Laine & Aila, 2017) | 12.16 |
| MixMatch*(Berthelot et al., 2019) | 4.95 |
| GS-BadGANt*(Li et al., 2019) GAN combine TE‡*(Wei et al., 2018) | 14.41 |
+
+Table 2: Error rate in Cifar10. $\dagger$ denotes the best semi-supervised generative approach result. $\ddagger$ denotes the model ensemble two categories. $^ *$ denotes the corresponding backbone is not exactly WideResNet (Zagoruyko & Komodakis, 2016), but belongs to one kind of its variations with a comparable amount of parameters.
+
+of $\mathbb { E } _ { q _ { \phi } ( \mathbf { z } | \mathbf { X } ) , q _ { \phi } ( \mathbf { c } | \mathbf { X } ) } \log p _ { \theta } ( \mathbf { X } | \mathbf { z } , \mathbf { c } )$ is difficult. To this end, we follow the work of (Rezende et al., 2014) and (Jang et al., 2017), using the reparameterization trick as
+
+$$
+\begin{array} { r l } & { \nabla _ { \theta , \phi } \mathbb { E } _ { q _ { \phi } ( \mathbf { z } | \mathbf { X } ) } \log p _ { \theta } ( \mathbf { X } | \mathbf { z } ) = \mathbb { E } _ { \mathcal { N } ( \epsilon ; \mathbf { 0 } , \mathbf { I } ) } \nabla _ { \theta , \phi } \log p _ { \theta } ( \mathbf { X } | \mu + \sigma \cdot \epsilon ) } \\ & { \mathbb { E } _ { \mathrm { G u m b e l } ( \epsilon ; \mathbf { 0 } , \mathbf { 1 } ) } ) \nabla _ { \theta , \phi } \log p _ { \theta } ( \mathbf { X } | \mathrm { S o f t m a x } ( \frac { \log \pi + \epsilon } { \tau } ) ) \to \nabla _ { \theta , \phi } \mathbb { E } _ { q _ { \phi } ( \mathbf { c } | \mathbf { X } ) } \log p _ { \theta } ( \mathbf { X } | \mathbf { c } ) ( \tau \to 0 ) } \end{array}
+$$
+
+Note that with (20), the algorithmic complexity of one-stage semi-supervised VAE is independent with the class number $K$ , making it easier to extend to large-scale classification tasks.
+
+# 4 EXPERIMENTS
+
+In this section, we demonstrate the 3 contributions of our OSPOT-VAE model with sufficient experiments on 4 standard SSL benchmark datasets, that is, MNIST, SVHN, Cifar10, and Cifar100. In Section 4.1, we show the validity of our one-stage semi-supervised VAE objective (13) by comparing with other one-stage and two-stage VAE models. Then, we evaluate the performance of OSPOT-VAE under “WideResNet”(Zagoruyko & Komodakis, 2016) backbone and compare with other state-of-the-art SSL models mentioned in Section 2. Besides, We provide an ablation study to verify the contribution of the optimal transport estimation. As an additional application, we show that good generative models and semi-supervised results can be obtained at the same time by OSPOT-VAE (Section 4.3). The source code is available at https: //github.com/PaperCodeSubmission/OSPOT-VAE; more details are available in Appendix A.6.
+
+# 4.1 ONE-STAGE SEMI-SUPERVISED VAE
+
+We evaluate the effectiveness of the one-stage semi-supervised VAE objective on 2 standard benchmarks, MNIST and SVHN. As for baseline models, we consider two VAE-based SSL models, which are one-stage disentangled VAE (Narayanaswamy et al., 2017) and two-stage VAE $[ \mathbf { M } 1 + \mathbf { M } 2 ]$ ) (Kingma et al., 2014). For fairness, except the target loss functions, all models use the same structure as is used in $\mathbf { M } 1 { + } \mathbf { M } 2$ (Kingma et al., 2014). The results are presented in Table 1, and our model achieves the best performance.
+
+Table 3: Error in Cifar100. $\dagger$ and $^ *$ have the same meaning as described in Table 2.
+
+| Algorithm1:Cramér GANLosses. |
| Parameter. Gradient penalty coefficient 入. Sample xr ~ P,𝑥g,xg~ Q,∈~ Uniform(0,1). Interpolate real and generated samples: 𝑥=∈xr+(1-∈)xg Sample generator loss (12): |
| Lg= |/h(xr)-h(xg)ll2+|/h(xr)-h(𝑥g)ll2 |
| -|h(xg)-h(xg)ll2 Sample surrogate generator loss (13) and critic loss: |
| Ls(u,v)= |h(xr)-h(u)ll2-|/h(xr)ll2 -/h(u)-h(ν)ll2+ h(u)ll2 Ls=1[Ls(xg,xg)+Ls(xg,xg)] |
+
+
+Figure 4: Approximate Wasserstein distances between CelebA test set and the generators. $N _ { u }$ is the number critic updates per generator update.
+
+# 5.1 CRAMÉR GAN
+
+We now consider the Generative Adversarial Networks (GAN) framework (Goodfellow et al., 2014), in particular issues arising in the Wasserstein GAN (Arjovsky et al., 2017), and propose a better GAN based on the Cramér distance. A GAN is composed of a generative model $Q$ (in our experiments, over images), called the generator, a target source $P$ , and a trainable loss function called a discriminator or critic. GANs are particularly interesting because we can establish a direct comparison between the two distances. Our choice of name reflects this fact, and we prefer Cramér GAN to the perhaps more technically correct, but less palatable Energy Distance GAN. In theory, the Wasserstein GAN algorithm requires training the critic until convergence, but this is rarely achievable: we would require a critic that is a very powerful network to approximate the Wasserstein distance well (Arora et al., 2017). Simultaneously, training this critic to convergence would overfit the empirical distribution of the training set, which is undesirable.
+
+Our proposed loss function allows for useful learning with imperfect critics by combining the energy distance with a transformation function $h : \mathbb { R } ^ { d } \mathbb { R } ^ { k }$ , where $d$ is the input dimensionality and $k ~ = ~ 2 5 6$ in our experiments. The generator then seeks to minimize the energy distance of the transformed variables $\mathcal { E } ( h ( X ) , h ( Y ) )$ , where $X$ is a real sample and $Y$ is a generated sample. The critic itself seeks to maximize this same distance by changing the parameters of $h$ , subject to a soft constraint (the gradient penalty used by Gulrajani et al., 2017). Specifically, the critic maximizes a surrogate loss whose gradient can be estimated from a single real sample. The Cramér GAN losses are summarized in Algorithm 1, with additional design choices detailed in Appendix C.
+
+We note that MMDs such as the energy distance have in the last year become an appealing tool for training GANs. Among others, the squared MMD is used within Generative Moment Matching Networks (Li et al., 2015; Dziugaite et al., 2015); Bouchacourt et al. (2016) trained a model to minimize the energy distance for hand pose estimation. Our use of the tranformation $h ( x )$ reflects our anecdotal finding that the direct minimization of the energy distance over raw images does not work well (see Figure 10 in appendix). Similar findings can be found in the work of Mroueh et al. (2017) and the independently developed MMD GAN (Li et al., 2017), which additionally uses an auto-encoder loss to make the transformation injective.
+
+The Cramér GAN we present here complements our comparison of the Wasserstein and Cramér distance from previous sections. At the same time, our experiments also provide novel GAN-related contributions, including the ability to perform conditional modelling using a surrogate generator loss, which lets us train the critic even when only one independent sample from $P$ is available. We note also that in our experiments, $\| x - y \| _ { 2 }$ distances were more stable than distances generated by Gaussian or Laplacian kernels.
+
+# 5.2 CRAMÉR GAN EXPERIMENTS
+
+We now show that, compared to the improved Wasserstein GAN (WGAN-GP) of Gulrajani et al. (2017), the Cramér GAN leads to more stable learning and increased diversity in the generated samples. In both cases we train generative models that predict the right half of an image given the left half; samples from unconditional models are provided in the appendix (Figure 10). The dataset we use here is the CelebA $6 4 \times 6 4$ dataset (Liu et al., 2015) of celebrity faces.
+
+Increased diversity. In our first experiment, we compare the qualitative diversity of completed faces by showing three sample completions generated by either model given the left half of a validation set image (Figure 3). We observe that the completions produced by WGAN-GP are almost deterministic. Our findings echo those of Isola et al. (2016), who observed that “the generator simply learned to ignore the noise.” By contrast, the completions produced by Cramér GAN are fairly diverse, including different hairstyles, accessories, and backgrounds. We view this lack of diversity in WGAN-GP as undesirable given that the main requirement of a generative model is that it should provide a variety of outputs.
+
+Theorem 1 provides a clue as to what may be happening here. We know that minimizing the sample Wasserstein loss will find the wrong minimum. In particular, when the target distribution has low entropy, the sample Wasserstein minimizer may actually be a deterministic distribution. But a good generative model of images must lie in this “almost deterministic” regime, since the space of natural images makes up but a fraction of all possible pixel combinations and hence there is little perpixel entropy. We hypothesize that the increased diversity in the Cramér GAN comes exactly from learning these almost deterministic predictions.
+
+More stable learning. In a second experiment, we varied the number of critic updates $( N _ { u } )$ per generator update. To compare performance between the two architectures, we measured the loss computed by an independent WGAN-GP critic trained on the validation set, following a similar evaluation previously done by Danihelka et al. (2017). Figure 4 shows the independent Wasserstein critic distance between each generator and the test set during the course of training. Echoing our results with the toy experiment and ordinal regression, the plot shows that when a single critic update is used, WGAN-GP performs particularly poorly. We note that additional critic updates also improve Cramér GAN. This indicates that it is helpful to keep adapting the $h ( x )$ transformation.
+
+# 6 CONCLUSION
+
+There are many situations in which the KL divergence, which is commonly used as a loss function in machine learning, is not suitable. The desirable alternatives, as we have explored, are the divergences that are ideal and allow for unbiased estimators: they allow geometric information to be incorporated into the optimization problem; because they are scale-sensitive and sum-invariant, they possess the convergence properties we require for efficient learning; and the correctness of their sample gradients means we can deploy them in large-scale optimization problems. Among open questions, we mention deriving an unbiased estimator that minimizes the Wasserstein distance, and variance analysis and reduction of the Cramér distance gradient estimate.
+
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+
+# A PROOFS
+
+# A.1 PROPERTIES OF A DIVERGENCE
+
+Proof (Proposition 1 and 2). The statement regarding (U) for the KL divergence is well-known, and forms the basis of most stochastic gradient algorithms for classification. Chung & Sobel (1987) have shown that the total variation does not have property (S); by Pinsker’s inequality, it follows that the same holds for the KL divergence. A proof of (I) and (S) for the Wasserstein metric is given by Bickel & Freedman (1981), while the lack of (U) is shown in the proof of Theorem 1. □
+
+# A.2 BIASED ESTIMATE
+
+Proof (Theorem $^ { l }$ ). Minimax bias: Consider $P = B ( { \theta } ^ { * } )$ , a Bernoulli distribution of parameter $\theta ^ { * }$ and $Q _ { \theta } = B ( \theta )$ a Bernoulli of parameter $\theta$ . The empirical distribution $\hat { P } _ { m }$ is a Bernoulli with parameter $\begin{array} { r } { \hat { \theta } : = \frac { 1 } { m } \sum _ { i = 1 } ^ { m } X _ { i } } \end{array}$ . Note that with $P$ and $Q _ { \theta }$ both Bernoulli distributions, the p $p ^ { t h }$ powers -Wasserstein metrics are equal, i.e. $w _ { 1 } ( P , Q _ { \theta } ) = w _ { p } ^ { p } ( P , Q _ { \theta } )$ . This gives us an easy way to prove the stronger result that all $p$ -Wasserstein metrics have biased sample gradients. The gradient of the loss $w _ { p } ^ { p } ( P , Q _ { \theta } )$ is, for $\theta \neq \theta ^ { * }$ ,
+
+$$
+\begin{array} { r } { g : = \nabla w _ { p } ^ { p } ( P , Q _ { \theta } ) = \nabla \Big [ \big | \theta ^ { * } - \theta \big | \Big ] = \mathrm { s g n } ( \theta - \theta ^ { * } ) , } \end{array}
+$$
+
+and similarly, the gradient of the sample loss is, for $\theta \neq { \hat { \theta } }$ ,
+
+$$
+\boldsymbol { \hat { g } } : = \nabla w _ { p } ^ { p } ( \boldsymbol { \hat { P } } _ { m } , Q _ { \theta } ) = \nabla \Big [ \big | \boldsymbol { \hat { \theta } } - \boldsymbol { \theta } \big | \Big ] = \mathrm { s g n } ( \theta - \boldsymbol { \hat { \theta } } ) .
+$$
+
+Notice that this estimate is biased for any $m \geq 1$ since
+
+$$
+\begin{array} { r } { \mathbb { E } \hat { g } = 2 \operatorname* { P r } \{ \hat { \theta } < \theta \} - 1 , } \end{array}
+$$
+
+which is different from $g$ for any $\theta ^ { * } \in ( 0 , 1 )$ . In particular for $m = 1$ , $\mathbb { E } _ { P } \hat { g } = 1 - 2 \theta ^ { * }$ does not depend on $\theta$ , thus a gradient descent using a one-sample gradient estimate has no chance of minimizing the Wasserstein loss as it will converge to either 1 or 0 instead of $\theta ^ { * }$ .
+
+Now observe that for $m \geq 2$ , and any $\textstyle \theta > { \frac { m - 1 } { m } }$ ,
+
+$$
+\operatorname* { P r } \{ \hat { \theta } < \theta \} = \operatorname* { P r } \{ \exists i \ { \mathrm { s . t . } } \ X _ { i } = 0 \} = 1 - ( \theta ^ { * } ) ^ { m } ,
+$$
+
+and therefore
+
+$$
+\mathbb { E } \hat { g } = 1 - 2 ( \theta ^ { * } ) ^ { m } .
+$$
+
+Taking $\textstyle \theta ^ { * } = { \frac { m - 1 } { m } }$ , we find that
+
+$$
+g - \mathbb { E } \hat { g } = 1 - [ 1 - 2 ( \theta ^ { * } ) ^ { m } ] = 2 \left( 1 - \frac { 1 } { m } \right) ^ { m } \ge 2 e ^ { - 2 } .
+$$
+
+Thus for any $m$ , there exists $P = B ( \theta ^ { * } )$ and $Q _ { \theta } = B ( \theta )$ with $\textstyle \theta ^ { * } = { \frac { m - 1 } { m } } < \theta < 1$ such that the bias $\boldsymbol { g } - \mathbb { E } \hat { \boldsymbol { g } }$ is lower-bounded by a numerical constant. Thus the minimax bias does not vanish with the number of samples $m$ .
+
+Notice that a similar argument holds for $\theta ^ { * }$ and $\theta$ being close to 0. In both situations where $\theta ^ { * }$ is close to 0 or 1, the bias is non vanishing when $| \theta ^ { * } - \theta |$ is of order $\textstyle { \frac { 1 } { m } }$ . However this is even worse when $\theta ^ { * }$ is away from the boundaries. For example chosing $\theta ^ { * } = \textstyle { \frac { 1 } { 2 } }$ , we can prove that the bias is non vanishing even when $| \theta ^ { * } - \theta |$ is (only) of order $\frac { 1 } { \sqrt { m } }$ .
+
+Indeed, using the anti-concentration result of Veraar (2010) (Proposition 2), we have that for a sequence $Y _ { 1 } , \dots , Y _ { m }$ of Rademacher random variables (i.e. $+ / - 1$ with equal probability),
+
+$$
+\operatorname* { P r } \left( { \frac { 1 } { n } } \sum _ { i = 1 } ^ { m } Y _ { i } \geq \epsilon \right) \geq ( 1 - m \epsilon ^ { 2 } ) ^ { 2 } / 3 .
+$$
+
+This means that for samples $X _ { 1 } , \ldots , X _ { m }$ drawn from a Bernoulli $\begin{array} { r } { B ( \theta ^ { * } = \frac { 1 } { 2 } ) } \end{array}$ (i.e., $Y _ { i } = 2 X _ { i } - 1$ are Rademacher), we have
+
+$$
+\operatorname* { P r } \left( \hat { \theta } \geq \theta ^ { * } + \epsilon / 2 \right) \geq ( 1 - m \epsilon ^ { 2 } ) ^ { 2 } / 3 ,
+$$
+
+
+Figure 5: Wasserstein loss (black curve) $\theta \mapsto | \theta ^ { * } - \theta |$ versus expected sample Wasserstein loss (red curve) $\theta \mapsto \mathbb { E } [ | \hat { \theta } - \theta | ]$ , for different values of $m$ and $\theta ^ { * }$ and $p = 1$ . Left: $m = 1$ , $\theta ^ { * } = 0 . 6$ . A stochastic gradient using a one-sample Wasserstein gradient estimate will converge to 1 instead of $\theta ^ { * }$ . Middle: $m = 6$ , $\theta ^ { * } = 0 . 6$ . The minimum of the expected sample Wasserstein loss is the median of $\hat { \theta }$ which is here $\tilde { \theta } = { \textstyle \frac { 2 } { 3 } } \ne \theta ^ { * } = 0 . 6$ . Right: $m = 5$ , $p = 0 . 9$ . The minimum of the expected sample Wasserstein is $\tilde { \theta } = 1$ and not $\theta ^ { * } = 0 . 9$ .
+
+thus for $1 / 2 = \theta ^ { \ast } < \theta < \theta ^ { \ast } + 1 / \sqrt { 8 m }$ we have the following lower bound on the bias:
+
+$$
+g - \mathbb { E } \hat { g } = 2 \operatorname* { P r } \left( \hat { \theta } \geq \theta \right) \geq 1 / 6 .
+$$
+
+Thus the bias is lower-bounded by a constant (independent of $m$ ) when $\theta ^ { * } = \textstyle { \frac { 1 } { 2 } }$ and $\left| \theta ^ { * } - \theta \right| =$ $O ( 1 / \sqrt { m } )$ .
+
+Wrong minimum: From (5), we deduce that a stochastic gradient descent algorithm based on the sample Wasserstein gradient will converge to a $\tilde { \theta }$ such that $\begin{array} { r } { \mathrm { \tilde { P r } } \{ \hat { \theta } < \tilde { \theta } \} = \frac { 1 } { 2 } } \end{array}$ , i.e., $\tilde { \theta }$ is the median of the distribution over $\hat { \theta }$ , whereas $\theta ^ { * }$ is the mean of that distribution. Since $\hat { \theta }$ follows a (normalized) binomial distribution with parameters $m$ and $\theta ^ { * }$ , we know that the median $\tilde { \theta }$ and the mean $\theta ^ { * }$ do not necessarily coincide, and can actually be as far as $\frac { 1 } { 2 m }$ -away from each other. For example for any odd $m$ and any $\theta ^ { * } \in \left( { \frac { 1 } { 2 } } , { \frac { 1 } { 2 } } - { \frac { 1 } { 2 m } } \right)$ the median is $\theta ^ { * } - \frac { 1 } { 2 m }$ .
+
+It follows that the minimum of the expected sample Wasserstein loss (the fixed point of the stochastic gradient descent using the sample Wasserstein gradient) is different from the minimum of the true Wasserstein loss:
+
+$$
+\underset { \theta } { \arg \operatorname* { m i n } } \mathbb { E } [ w _ { p } ^ { p } ( \hat { P } _ { m } , Q _ { \theta } ) ] \neq \underset { \theta } { \arg \operatorname* { m i n } } [ w _ { p } ^ { p } ( P , Q _ { \theta } ) ] .
+$$
+
+This is illustrated in Figure 5.
+
+Notice that the fact that the minima of these losses differ is worrisome as it means that minimizing the sample Wasserstein loss using (finite) samples will not converge to the correct solution.
+
+Deterministic solutions: Consider the specific case where $( 1 / 2 ) ^ { 1 / n } < \theta ^ { * } < 1$ (illustrated in the right plot of Figure 5). Then the expected sample gradient $\nabla \mathbb { E } [ w _ { p } ^ { p } ( \hat { P } _ { m } , Q _ { \theta ^ { * } } ) ] = \mathbb { E } \hat { g } = 1 - 2 ( \theta ^ { * } ) ^ { n } <$ 0 for any $\theta$ , so a gradient descent algorithm will converge to 1 instead of $\theta ^ { * }$ . Notice that a symmetric argument applies for $\theta ^ { * }$ close to 0.
+
+In this simple example, minimizing the sample Wasserstein loss may lead to degenerate solutions (i.e., deterministic) when our target distributions have low (but not zero) entropy. □
+
+# A.3 CONSISTENCY OF THE SAMPLE 1-WASSERSTEIN GRADIENT
+
+We provide an additional result here showing that the sample 1-Wasserstein gradient converges to the true gradient as $m \infty$ .
+
+Theorem 3. Let $P$ and $Q _ { \theta }$ be probability distributions, with $Q _ { \theta }$ parametrized by $\theta$ . Assume that the set $\{ x \in X$ , such that $F _ { P } ( x ) = F _ { Q _ { \theta } } ( x ) \big \}$ has measure zero, and that for any $x \in X$ , the map $\tilde { \theta } \mapsto F _ { Q _ { \tilde { \theta } } } ( x )$ is differentiable in a neighborhood $\mathcal { V } ( \boldsymbol { \theta } )$ of $\theta$ with a uniformly bounded derivative
+
+(for $\tilde { \theta } \in \mathcal { V } ( \theta )$ and $x \in X ,$ ). Let $\begin{array} { r } { \hat { P } _ { m } = \frac { 1 } { m } \sum _ { i = 1 } ^ { m } \delta _ { X _ { i } } } \end{array}$ be the empirical distribution derived from m independent samples $X _ { 1 } , \ldots , X _ { m }$ drawn from $P$ . Then
+
+We note that the measure requirement is strictly to keep the proof simple, and does not subtract from the generality of the result.
+
+Proof. Let $\nabla : = \nabla _ { \theta }$ . Since $p = 1$ the Wasserstein distance $w _ { 1 } ( P , Q )$ measures the area between the curves defined by the distribution function of $P$ and $Q$ , thus $\begin{array} { r } { w _ { 1 } ( P , Q ) = l _ { 1 } ( P , Q ) = \int \left| F _ { P } ( x ) - \frac { } { } \right. } \end{array}$ $F _ { Q } ( x ) { \left| { d x } \right. }$ and
+
+$$
+\begin{array} { l l l } { \nabla w _ { 1 } ( P , Q _ { \theta } ) } & { = } & { \displaystyle \operatorname* { l i m } _ { \Delta \to 0 } \frac { w _ { 1 } ( P , Q _ { \theta + \Delta } ) - w _ { 1 } ( P , Q _ { \theta } ) } { \Delta } } \\ & { = } & { \displaystyle \operatorname* { l i m } _ { \Delta \to 0 } \int \frac { 1 } { \Delta } \Big ( \big | F _ { P } ( x ) - F _ { Q _ { \theta + \Delta } } ( x ) \big | - \big | F _ { P } ( x ) - F _ { Q _ { \theta } } ( x ) \big | \Big ) d x . } \end{array}
+$$
+
+Now since we have assumed that for any $x \in X$ , the map $\theta \mapsto F _ { Q _ { \theta } } ( x )$ is differentiable in a neighborhood $\mathcal { V } ( \boldsymbol { \theta } )$ of $\theta$ and its derivative is uniformly (over $\mathcal { V } ( \boldsymbol { \theta } )$ and $x$ ) bounded by $M$ , we have
+
+$$
+\frac { 1 } { \Delta } \Big | \big | F _ { P } ( x ) - F _ { Q _ { \theta + \Delta } } ( x ) \big | - \big | F _ { P } ( x ) - F _ { Q _ { \theta } } ( x ) \big | \Big | \quad \le \quad \frac { 1 } { \Delta } \big | F _ { Q _ { \theta + \Delta } } ( x ) - F _ { Q _ { \theta } } ( x ) \big | \le M .
+$$
+
+Thus the dominated convergence theorem applies and
+
+$$
+\begin{array} { r c l } { \nabla w _ { 1 } ( P , Q _ { \theta } ) } & { = } & { \displaystyle \int \operatorname* { l i m } _ { \Delta \to 0 } \frac { 1 } { \Delta } \Big ( \big | F _ { P } ( x ) - F _ { Q _ { \theta + \Delta } } ( x ) \big | - \big | F _ { P } ( x ) - F _ { Q _ { \theta } } ( x ) \big | \Big ) d x } \\ & { = } & { \displaystyle \int \nabla \big | F _ { P } ( x ) - F _ { Q _ { \theta } } ( x ) \big | d x } \\ & { = } & { \displaystyle \int \mathrm { s g n } \big ( F _ { P } ( x ) - F _ { Q _ { \theta } } ( x ) \big ) \nabla F _ { Q _ { \theta } } ( x ) d x , } \end{array}
+$$
+
+since we have assumed that the set of $x \in X$ such that $F _ { P } ( x ) = F _ { Q _ { \theta } } ( x )$ has measure zero.
+
+Now, using the same argument for $w _ { 1 } \big ( \hat { P } _ { m } , Q _ { \theta } \big )$ we deduce that
+
+$$
+\begin{array} { r l r } { \nabla w _ { 1 } ( \hat { P } _ { m } , Q _ { \theta } ) } & { = } & { \displaystyle \int \underbrace { \operatorname* { l i m } _ { \Delta \to 0 } \frac { 1 } { \Delta } \Big ( \big | F _ { \hat { P } _ { m } } ( x ) - F _ { Q _ { \theta + \Delta } } ( x ) \big | - \big | F _ { \hat { P } _ { m } } ( x ) - F _ { Q _ { \theta } } ( x ) \big | \Big ) } _ { A ( x ) } d x . } \end{array}
+$$
+
+Let us decompose this integral over $X$ as the sum of two integrals, one over $X \setminus \Omega _ { m }$ and the other one over $\Omega _ { m }$ , where $\Omega _ { m } = \big \{ x \in X , F _ { \hat { P } _ { m } } ( x ) = F _ { Q _ { \theta } } ( x ) \big \}$ . We have
+
+$$
+\int _ { X \setminus \Omega _ { m } } A ( x ) d x = \int _ { X \setminus \Omega _ { m } } \operatorname { s g n } \bigl ( F _ { \hat { P } _ { m } } ( x ) - F _ { Q _ { \theta } } ( x ) \bigr ) \nabla F _ { Q _ { \theta } } ( x ) d x ,
+$$
+
+and
+
+$$
+\begin{array} { r c l } { \Big | \displaystyle \int _ { \Omega _ { m } } A ( x ) d x \Big | } & { \le } & { \displaystyle \int _ { \Omega _ { m } } \operatorname* { l i m } _ { \Delta \to 0 } \frac { 1 } { \Delta } \Big ( \big | F _ { Q _ { \theta + \Delta } } ( x ) - F _ { Q _ { \theta } } ( x ) \big | \Big ) d x } \\ & { \le } & { M | \Omega _ { m } | . } \end{array}
+$$
+
+Now from the strong law of large numbers, we have that for any $x$ , the empirical cumulative distribution function $F _ { \hat { P } _ { m } } ( x )$ converges to the cumulative distribution $F _ { P } ( x )$ almost surely. We deduce that $\Omega _ { m }$ converges to the set $\left\{ x , F _ { P } ( x ) = F _ { Q _ { \theta } } ( x ) \right\}$ which has measure zero, thus $| \Omega _ { m } | \to 0$ and
+
+$$
+\begin{array} { r l r } { \displaystyle \operatorname* { l i m } _ { m \to \infty } \nabla w _ { 1 } ( \hat { P } _ { m } , Q _ { \theta } ) } & { = } & { \displaystyle \operatorname* { l i m } _ { m \to \infty } \int _ { X } \mathrm { s g n } \big ( { \cal F } _ { \hat { P } _ { m } } ( x ) - { \cal F } _ { Q _ { \theta } } ( x ) \big ) \nabla { \cal F } _ { Q _ { \theta } } ( x ) d x . } \end{array}
+$$
+
+Now, since $| \nabla F _ { Q _ { \theta } } ( x ) | \leq M$ , we can use once more the dominated convergence theorem to deduce that
+
+$$
+\begin{array} { l } { \displaystyle \operatorname* { l i m } _ { m \infty } \nabla w _ { 1 } ( \hat { P } _ { m } , Q _ { \theta } ) = \int _ { X } \displaystyle \operatorname* { l i m } _ { m \infty } \mathrm { s g n } \big ( F _ { \hat { P } _ { m } } ( x ) - F _ { Q _ { \theta } } ( x ) \big ) \nabla F _ { Q _ { \theta } } ( x ) d x } \\ { \displaystyle = \int _ { X } \mathrm { s g n } \big ( F _ { P } ( x ) - F _ { Q _ { \theta } } ( x ) \big ) \nabla F _ { Q _ { \theta } } ( x ) d x } \\ { \displaystyle = \nabla w _ { 1 } ( P , Q _ { \theta } ) . } \end{array}
+$$
+
+The following lemma will be useful in proving that the Cramér distance has property (U).
+
+Lemma 1. Let $\mathbf { X } _ { m } : = X _ { 1 } , \ldots , X _ { m }$ be independent samples from $P$ , and let $\begin{array} { r } { \hat { P } _ { m } : = \frac { 1 } { m } \sum _ { i } \delta _ { X _ { i } } } \end{array}$ . Then
+
+$$
+\begin{array} { r } { \underset { \mathbf { X } _ { m } \sim P } { \mathbb { E } } F _ { \hat { P } _ { m } } ( x ) = F _ { P } ( x ) . } \end{array}
+$$
+
+Proof. Because the $X _ { i }$ ’s are independent,
+
+$$
+F _ { \hat { P } _ { m } } ( x ) = \int _ { - \infty } ^ { x } \hat { P } _ { m } ( \mathrm { d } x ) = \frac { 1 } { m } \sum _ { i = 1 } ^ { m } \mathbb { I } \left[ X _ { i } \le x \right] .
+$$
+
+Now, taking the expectation w.r.t. ${ \bf { X } } _ { m }$ ,
+
+$$
+\begin{array} { l } { \displaystyle \underset { { \bf X } _ { m } \sim P } { \mathbb { E } } F _ { \hat { P } _ { m } } ( x ) = \underset { { \bf X } _ { m } \sim P } { \mathbb { E } } \frac { 1 } { m } \sum _ { i = 1 } ^ { m } { \mathbb { I } } \left[ X _ { i } \leq x \right] } \\ { \displaystyle = \frac { 1 } { m } \sum _ { i = 1 } ^ { m } X _ { i \sim P } ^ { { \mathbb { E } } } \mathbb { I } \left[ X _ { i } \leq x \right] } \\ { \displaystyle = \frac { 1 } { m } \sum _ { i = 1 } ^ { m } \mathrm { P r } \{ X _ { i } \leq x \} } \\ { \displaystyle = F _ { P } ( x ) , } \end{array}
+$$
+
+since the $X _ { i }$ are identically distributed according to $P$ .
+
+Proof (Theorem 2). Like the Wasserstein metrics, the $l _ { p }$ metrics have dual forms as integral probability metrics (see Dedecker & Merlevède, 2007, for a proof):
+
+$$
+l _ { p } ( P , Q ) = \operatorname* { s u p } _ { f \in \mathbb { F } _ { q } } \big | \operatorname* { \mathbb { E } } _ { x \sim P } f ( x ) - \operatorname* { \mathbb { E } } _ { x \sim Q } f ( x ) \big | ,
+$$
+
+where $\mathbb { F } _ { q } : = \{ f : f$ is absolutely continuous, $\begin{array} { r } { \left\| \frac { \mathrm { d } f } { \mathrm { d } x } \right\| _ { q } \leq 1 \} } \end{array}$ and $q$ is the conjugate exponent of $p$ , i.e.
+$p ^ { - 1 } + q ^ { - 1 } = 1$ .3 We will use this dual form below.
+
+We will prove that $l _ { p }$ has properties (I) and (S) for $p \in [ 1 , \infty )$ ; the case $p = \infty$ follows by a similar argument. Begin by observing that
+
+$$
+\begin{array} { c } { { F _ { c X } ( x ) = P r \{ c X \leq x \} } } \\ { { = P r \left\{ X \leq \displaystyle \frac { x } { c } \right\} } } \\ { { = F _ { X } \left( \displaystyle \frac { x } { c } \right) . } } \end{array}
+$$
+
+Then we may rewrite $l _ { p } ^ { p } ( c X , c Y )$ as
+
+$$
+l _ { p } ^ { p } ( c X , c Y ) = \int _ { - \infty } ^ { \infty } { \left| F _ { X } \left( \frac { x } { c } \right) - F _ { Y } \left( \frac { x } { c } \right) \right| ^ { p } } \mathrm { d } x
+$$
+
+3This relationship is the reason for the notation $\mathbb { F } _ { \infty }$ in the definition the dual of the 1-Wasserstein (2).
+
+where $( a )$ uses a change of variables $z = x / c$ . Taking both sides to the power $1 / p$ proves that the $l _ { p }$ metric possesses property (S) of order $1 / p$ . For (I), we use the IPM formulation (6):
+
+$$
+\begin{array} { r l } & { l _ { p } \big ( A + X , A + Y \big ) = \underset { f \in \mathcal { F } _ { q } } { \operatorname* { s u p } } \bigg | _ { A + X } f ( x ) - \underset { A + Y } { \mathbb { E } } f ( y ) \bigg | } \\ & { \stackrel { ( a ) } { = } \underset { f \in \mathcal { F } _ { q } } { \operatorname* { s u p } } \bigg | \mathbb { E } _ { A } \mathbb { E } _ { X } f ( x + a ) - \mathbb { E } _ { A } \mathbb { E } _ { Y } f ( y + a ) \bigg | } \\ & { \stackrel { ( b ) } { = } \underset { f \in \mathcal { F } _ { q } } { \operatorname* { s u p } } \bigg | \mathbb { E } _ { A } \big [ \mathbb { E } _ { X } f ( x + a ) - \mathbb { E } _ { Y } f ( y + a ) \big ] \bigg | } \\ & { \stackrel { ( b ) } { \leq } \mathbb { E } _ { A } \underset { f \in \mathcal { F } _ { q } } { \operatorname* { s u p } } \bigg | \mathbb { E } _ { X } f ( x + a ) - \mathbb { E } _ { Y } f ( y + a ) \bigg | , } \end{array}
+$$
+
+where $( a )$ is by independence of $A$ and $X , Y$ , and $( b )$ is by Jensen’s inequality. Next, recall that $\mathcal { F } _ { q }$ is the set of absolutely continuous functions whose derivative has bounded $L _ { q }$ norm. Hence if $f \in \mathcal { F } _ { q }$ , then also for all $a$ the translate $g _ { a } ( x ) : = f ( x + a )$ is also in $\mathcal { F } _ { q }$ . Therefore,
+
+$$
+\begin{array} { r l } & { l _ { p } ( A + X , A + Y ) \leq \mathbb { E } _ { A } \underset { f \in \mathcal { F } _ { q } } { \mathrm { \mathbb { E } } } \bigg | \mathbb { E } _ { X } f ( x + a ) - \mathbb { E } _ { Y } f ( y + a ) \bigg | } \\ & { \qquad = \mathbb { E } _ { A } \underset { g \in \mathcal { F } _ { q } } { \mathrm { \mathbb { E } } } \bigg | \mathbb { E } _ { X } g ( x ) - \mathbb { E } _ { Y } g ( y ) \bigg | } \\ & { \qquad = \underset { g \in \mathcal { F } _ { q } } { \mathrm { \operatorname* { s u p } } } \bigg | \mathbb { E } _ { X } g ( x ) - \mathbb { E } _ { Y } g ( y ) \bigg | } \\ & { \qquad = l _ { p } ( X , Y ) . } \end{array}
+$$
+
+Now, to prove (U). Here we make use of the introductory requirement that “all expectations under consideration are finite.” Specifically, we require that the mean under $P$ $\mathbb { \lambda } , \mathbb { E } _ { x \sim P } [ x ]$ , is well-defined and finite, and similarly for $Q _ { \theta }$ . In this case,
+
+$$
+\underset { x \sim P } { \mathbb { E } } [ x ] = \int _ { 0 } ^ { \infty } ( 1 - F _ { P } ( x ) ) \mathrm { d } x - \int _ { - \infty } ^ { 0 } F _ { P } ( x ) \mathrm { d } x .
+$$
+
+This mild requirement guarantees that the tails of the distribution function $F _ { P }$ are light enough to avoid infinite Cramér distances and expected gradients (a similar condition was set by Dedecker & Merlevède (2007)). Now, by definition,
+
+$$
+\begin{array} { r l } { \nabla \theta _ { i } ^ { 2 } ( P , Q _ { \theta } ) = \nabla \theta \displaystyle \int _ { - \infty } ^ { \infty } \left( F _ { Q _ { \theta } } ( x ) - F _ { P } ( x ) \right) ^ { 2 } \mathrm { d } z } \\ { \boldsymbol { \stackrel { \cdot } { = } } } & { \underset { 0 \leq i } { \iint } \int _ { - \infty } ^ { \infty } \nabla \theta \left( F _ { Q _ { \theta } } ( x ) - F _ { P } ( x ) \right) ^ { 2 } \mathrm { d } z } \\ { \boldsymbol { \stackrel { \cdot } { = } } } & { \underset { 0 \leq i } { \iint } ( F _ { Q _ { \theta } } ( x ) - F _ { P } ( x ) ) \nabla _ { \theta } F _ { Q _ { \theta } } ( x ) \mathrm { d } x } \\ { \boldsymbol { \stackrel { \cdot } { = } } } & { \underset { 0 \leq i } { \iint } \int _ { - \infty } ^ { \infty } 2 \left( F _ { Q _ { \theta } } ( x ) - \mathbb { E } _ { \mathbf { x } _ { m } } F _ { \hat { \mu } _ { \infty } } ( x ) \right) \nabla _ { \theta } F _ { P } ( y _ { \infty } ( x ) \mathrm { d } x ) } \\ { \boldsymbol { \stackrel { \cdot } { = } } } & { \underset { 0 \leq i } { \iint } \int _ { - \infty } ^ { \infty } \left( F _ { Q _ { \theta } } ( x ) - F _ { \hat { \mu } _ { \infty } } ( x ) \right) \nabla _ { \theta } F _ { Q _ { \theta } } ( x ) \mathrm { d } x } \\ { \boldsymbol { \stackrel { \cdot } { = } } } & { \underset { 0 \leq i } { \iint } \sum _ { \mathbf { R } \setminus \mathbf { x } _ { m } } \left( F _ { Q _ { \theta } } ( x ) - F _ { \hat { \mu } _ { \infty } } ( x ) \right) \nabla _ { \theta } F _ { Q _ { \theta } } ( x ) \mathrm { d } x } \\ { \boldsymbol { \stackrel { \cdot } { = } } } & { \underset { 0 \leq i } { \iint } \sum _ { \mathbf { R } \setminus \mathbf { x } _ { m } } \int _ { - \infty } ^ { \infty } \left( F _ { Q _ { \theta } } ( x ) - F _ { \hat { \mu } _ { \infty } } ( x ) \right) \nabla _ { \theta } F _ { Q _ { \theta } } ( x ) \mathrm { d } x } \\ { \boldsymbol { \stackrel { \cdot } { = } } } & \underset { 0 \leq i } { \iint } \int _ { \mathbf { R } \setminus \mathbf { x } _ { m } } f \end{array}
+$$
+
+where (a) follows from the hypothesis (7) (the convergence of the squares follows from the convergence of the ordinary values), (b) follows from Lemma 1 and (c) follows from Fubini’s theorem, again invoking (7).
+
+Finally, we prove that of all the $l _ { p } ^ { p }$ distances $1 \le p \le \infty$ ) only the Cramér distance, $l _ { 2 } ^ { 2 }$ , has the (U) property.
+
+Without loss of generality, let us suppose $P$ is not a Dirac, and further suppose that for any $\mathbf { X } _ { m } \sim { \cal P }$ , $F _ { Q _ { \theta } } ( x ) \geq F _ { \hat { P } _ { m } } \bar { ( x ) }$ everywhere. For example, when $Q _ { \theta }$ has bounded support we can take $P$ to be a sufficiently translated version of $Q _ { \theta }$ , such that the two distributions’ supports do not overlap.
+
+We have already established that the 1-Wasserstein does not have the (U) property, and is equivalent to $l _ { p } ^ { p }$ for $p = 1$ . We will thus assume that $p > 1$ , and also that $p < \infty$ , the latter being recovered through standard limit arguments. Begin with the gradient for $l _ { p } ^ { p } ( P , Q _ { \theta } )$ ,
+
+$$
+\begin{array} { r l } { { \nabla _ { \theta } l _ { p } ^ { p } ( P , Q _ { \theta } ) = \nabla _ { \theta } \int _ { - \infty } ^ { \infty } | F _ { Q _ { \theta } } ( x ) - F _ { P } ( x ) | ^ { p } \mathrm { d } x } } \\ & { \stackrel { ( a ) } { = } p \int _ { - \infty } ^ { \infty } \big ( F _ { Q _ { \theta } } ( x ) - F _ { P } ( x ) \big ) ^ { p - 1 } \nabla _ { \theta } F _ { Q _ { \theta } } ( x ) \mathrm { d } x } \\ & { = p \int _ { - \infty } ^ { \infty } \phi _ { p } ( F _ { Q _ { \theta } } ( x ) - F _ { P } ( x ) ) \nabla _ { \theta } F _ { Q _ { \theta } } ( x ) \mathrm { d } x } \\ & { = p \int _ { - \infty } ^ { \infty } \phi _ { p } \Big ( \mathbb { E } _ { \mathbf { X } _ { m } } ( F _ { Q _ { \theta } } ( x ) - F _ { \hat { P } _ { m } } ( x ) ) \Big ) \nabla _ { \theta } F _ { Q _ { \theta } } ( x ) \mathrm { d } x , } \end{array}
+$$
+
+for $\phi _ { p } ( z ) = z ^ { p - 1 }$ ; in (a) we used the same argument as in Theorem 3.
+
+Now, $\phi _ { p }$ is convex on $[ 0 , \infty )$ when $p \geq 2$ and concave on the same interval when $1 < p < 2$ . From Jensen’s inequality we know that for a convex (concave) function $\phi$ and a random variable $Z$ , $\mathbb { E } \phi ( Z )$ is greater than (less than) or equal to $\phi ( \mathbb { E } Z )$ , with equality if and only if $\phi$ is linear or $Z$ is deterministic. By our first assumption we have ruled out the latter. By our second assumption $F _ { Q _ { \theta } } ( x ) \geq F _ { \hat { P } _ { m } } ( x )$ , we can apply Jensen’s inequality at every $x$ to deduce that
+
+$$
+\begin{array} { r l } & { \mathbb { E } _ { \mathbf { X } _ { m } } \left[ \nabla _ { \theta } l _ { p } ^ { p } ( \hat { P } _ { m } , Q _ { \theta } ) \right] < \nabla _ { \theta } l _ { p } ^ { p } ( P , Q _ { \theta } ) , \quad \mathrm { i f ~ } 1 < p < 2 , } \\ & { \mathbb { E } _ { \mathbf { X } _ { m } } \left[ \nabla _ { \theta } l _ { p } ^ { p } ( \hat { P } _ { m } , Q _ { \theta } ) \right] > \nabla _ { \theta } l _ { p } ^ { p } ( P , Q _ { \theta } ) , \quad \mathrm { i f ~ } p > 2 , } \\ & { \mathbb { E } _ { \mathbf { X } _ { m } } \left[ \nabla _ { \theta } l _ { p } ^ { p } ( \hat { P } _ { m } , Q _ { \theta } ) \right] = \nabla _ { \theta } l _ { p } ^ { p } ( P , Q _ { \theta } ) , \quad \mathrm { i f ~ } p = 2 . } \end{array}
+$$
+
+We conclude that of the $l _ { p } ^ { p }$ distances, only the Cramér distance has unbiased sample gradients.
+
+Proposition 3. The energy distance $\mathcal { E } ( P , Q )$ has properties (I), (S), and $( U )$ .
+
+Proof. As before, write $\mathcal { E } ( X , Y ) : = \mathcal { E } ( P , Q )$ . Recall that
+
+$$
+\mathcal { E } ( X , Y ) = 2 \mathbb { E } \left\| X - Y \right\| _ { 2 } - \mathbb { E } \left\| X - X ^ { \prime } \right\| _ { 2 } - \mathbb { E } \left\| Y - Y ^ { \prime } \right\| _ { 2 } .
+$$
+
+Consider a random variable $A$ independent of $X$ and $Y$ . First, we want to prove property (I):
+
+$$
+{ \mathcal { E } } ( A + X , A + Y ) \leq { \mathcal { E } } ( X , Y ) .
+$$
+
+We will use Proposition 2 from Székely & Rizzo (2013) to express the energy distance in terms of characteristic functions $\phi _ { X } , \phi _ { Y }$ of $d$ -dimensional random variables $X$ and $Y$ :
+
+$$
+{ \mathcal { E } } ( X , Y ) = { \frac { 1 } { c _ { d } } } \int _ { R ^ { d } } { \frac { | \phi _ { X } ( t ) - \phi _ { Y } ( t ) | ^ { 2 } } { | t | ^ { d + 1 } } } d t
+$$
+
+where
+
+$$
+c _ { d } = \frac { \pi ^ { ( d + 1 ) / 2 } } { \Gamma ( \frac { d + 1 } { 2 } ) } .
+$$
+
+The proof then uses properties of characteristic functions $( | \phi _ { A } ( t ) | \leq 1$ and $\phi _ { A + X } ( t ) = \phi _ { A } ( t ) \phi _ { X } ( t )$ for independent variables $A$ and $X$ ) to show:
+
+$$
+\begin{array} { l } { \displaystyle \mathcal { E } ( A + X , A + Y ) = \frac { 1 } { c _ { d } } \int _ { R ^ { d } } \frac { | \phi _ { A + X } ( t ) - \phi _ { A + Y } ( t ) | ^ { 2 } } { | t | ^ { d + 1 } } d t } \\ { \displaystyle \qquad = \frac { 1 } { c _ { d } } \int _ { R ^ { d } } \frac { | \phi _ { A } ( t ) \phi _ { X } ( t ) - \phi _ { A } ( t ) \phi _ { Y } ( t ) | ^ { 2 } } { | t | ^ { d + 1 } } d t } \\ { \displaystyle \qquad = \frac { 1 } { c _ { d } } \int _ { R ^ { d } } \frac { | \phi _ { X } ( t ) - \phi _ { Y } ( t ) | ^ { 2 } } { | t | ^ { d + 1 } } | \phi _ { A } ( t ) | ^ { 2 } d t } \\ { \displaystyle \qquad \leq \frac { 1 } { c _ { d } } \int _ { R ^ { d } } \frac { | \phi _ { X } ( t ) - \phi _ { Y } ( t ) | ^ { 2 } } { | t | ^ { d + 1 } } d t } \\ { \displaystyle \qquad = \mathcal { E } ( X , Y ) . } \end{array}
+$$
+
+This proves (I). Next, consider a real value $c > 0$ . We have
+
+$$
+\begin{array} { r l } & { \mathcal { E } ( c X , c Y ) = 2 \mathbb { E } \left\| c X - c Y \right\| _ { 2 } - \mathbb { E } \left\| c X - c X ^ { \prime } \right\| _ { 2 } - \mathbb { E } \left\| c Y - c Y ^ { \prime } \right\| _ { 2 } } \\ & { \qquad = 2 c \mathbb { E } \left\| X - Y \right\| _ { 2 } - c \mathbb { E } \left\| X - X ^ { \prime } \right\| _ { 2 } - c \mathbb { E } \left\| Y - Y ^ { \prime } \right\| _ { 2 } } \\ & { \qquad = c \mathcal { E } ( X , Y ) . } \end{array}
+$$
+
+This proves (S). Finally, suppose that $Y$ is distributed according to $Q _ { \theta }$ parametrized by $\theta$ . Let $\mathbf { X } _ { m } = X _ { 1 } , \ldots , X _ { m }$ be drawn from $P$ , and let $\begin{array} { r } { \hat { P } _ { m } : = \frac { 1 } { m } \sum _ { i = 1 } ^ { m } \delta _ { X _ { i } } } \end{array}$ . Let $\hat { X }$ be the random variable distributed according to $\hat { P } _ { m }$ , and ${ \hat { X } } ^ { \prime }$ an independent copy of $\hat { X }$ . Then
+
+$$
+\mathcal { E } ( \hat { P } _ { m } , Q _ { \theta } ) = \mathcal { E } ( \hat { X } , Y ) = 2 \mathbb { E } \left\| \hat { X } - Y \right\| _ { 2 } - \mathbb { E } \left\| \hat { X } - \hat { X } ^ { \prime } \right\| _ { 2 } - \mathbb { E } \left\| Y - Y ^ { \prime } \right\| _ { 2 } .
+$$
+
+The gradient of the true loss w.r.t. $\theta$ is
+
+$$
+\nabla _ { \theta } \mathcal { E } ( X , Y ) = 2 \nabla _ { \theta } \mathbb { E } \left\| X - Y \right\| _ { 2 } - \nabla _ { \theta } \mathbb { E } \left\| Y - Y ^ { \prime } \right\| _ { 2 } .
+$$
+
+Now, taking the gradient of the sample loss w.r.t. $\theta$ ,
+
+$$
+\nabla _ { \theta } \mathcal { E } ( \hat { X } , Y ) = 2 \nabla _ { \theta } \mathbb { E } \left. \hat { X } - Y \right. _ { 2 } - \nabla _ { \theta } \mathbb { E } \left. Y - Y ^ { \prime } \right. _ { 2 } .
+$$
+
+Since the second terms of the gradients match, all we need to show is that the first terms are equal, in expectation. Assuming that $\nabla _ { \theta }$ and the expectation over $\mathbf { X } _ { m }$ commute, we write
+
+$$
+\begin{array} { r l } & { \underset { { \mathbf { X } } _ { m } } { \mathbb { E } } \nabla _ { \theta } \mathbb { E } \| \hat { X } - Y \| _ { 2 } = \nabla _ { \theta } \underset { { \mathbf { X } } _ { m } } { \mathbb { E } } \mathbb { E } \| \hat { X } - Y \| _ { 2 } } \\ & { \qquad = \nabla _ { \theta } \underset { { \mathbf { X } } _ { m } } { \mathbb { E } } \underset { { \mathbf { X } } \sim \hat { P } _ { m } } { \mathbb { E } } \| x - Y \| _ { 2 } , } \end{array}
+$$
+
+by independence of $X$ and $Y$ . But now we know that the expected empirical distribution is $P$ , that is
+
+$$
+\begin{array} { r } { \underset { \mathbf { X } _ { m } } { \mathbb { E } } \underset { x \sim \hat { P } _ { m } } { \mathbb { E } } \left\| x - Y \right\| _ { 2 } = \underset { x \sim P } { \mathbb { E } } \left\| x - Y \right\| _ { 2 } = \mathbb { E } \left\| X - Y \right\| _ { 2 } . } \end{array}
+$$
+
+It follows that the first terms of (8) and (9) are also equal, in expectation w.r.t. $\mathbf { X } _ { m }$ . Hence we conclude that the energy distance has property (U), that is
+
+$$
+\underset { { \substack { \mathbf { X } _ { m } \sim P } } } { \mathbb { E } } \nabla _ { \theta } \mathcal { E } ( \hat { P } _ { m } , Q _ { \theta } ) = \nabla _ { \theta } \mathcal { E } ( P , Q _ { \theta } ) .
+$$
+
+# B COMPARISON WITH THE WASSERSTEIN DISTANCE
+
+Figure 2 (left) provides learning curves for the toy experiment described in Section 4.2.
+
+
+Figure 6: Wasserstein while training to minimize different loss functions (Wasserstein, KL, Cramér). Averaged over 10 random initializations. Error-bands indicate one standard deviation. Note the different y-axes.
+
+
+Figure 7: Ordinal regression on the year prediction MSD dataset. Each loss function trained with various minibatch sizes. Training progress shown in terms of: Left. RMSE, Middle. Wasserstein distance, Right. Negative log-likelihood.
+
+# B.1 ORDINAL REGRESSION
+
+We compare the different losses on an ordinal regression task using the Year Prediction MSD dataset from (Lichman, 2013). The task is to predict the year of a song (taking on values from 1922 to 2011), from 90-dimensional feature representation of the song.4 Previous work has used this dataset for benchmarking regression performance (Hernández-Lobato & Adams, 2015), treating the target as a continuous value. Following Hernández-Lobato & Adams (2015), we train a network with a single hidden layer with 100 units and ReLU non-linearity, using SGD with 40 passes through the training data, using the standard train-test split for this dataset (Lichman, 2013). Unlike (Hernández-Lobato & Adams, 2015), the network outputs a probability distribution over the years (90 possible years from 1922-2011).
+
+We train models using either the 1-Wasserstein loss, the Cramér loss, or the KL loss, the latter of which reduces the ordinal regression problem to a classification problem. In all cases, we compare performance for three different minibatch sizes, i.e. the number of input-target pairs per gradient step. Note that the minibatch size only affects the gradient estimation, but has otherwise no direct relation to the number of samples $m$ previously discussed, since each sample corresponds to a different input vector. We report results as a function of number of passes over the training data so that our results are comparable with previous work, but note that smaller batch sizes get more updates.
+
+The results are shown in Figure 2. Training using the Cramér loss results in the lowest root mean squared error (RMSE) and the final RMSE value of 8.89 is comparable to regression (HernándezLobato & Adams, 2015) which directly optimizes for MSE. We further observe that minimizing the Wasserstein loss trains relatively slowly and leads to significantly higher KL loss. Interestingly, larger minibatch sizes do seem to improve the performance of the Wasserstein-based method somewhat, suggesting that there might be some beneficial bias reduction from combining similar inputs. By contrast, using with the Cramér loss trains significantly faster and is more robust to choice of minibatch size.
+
+| Algorithm1:Cramér GANLosses. |
| Parameter. Gradient penalty coefficient 入. Sample xr ~ P,𝑥g,xg~ Q,∈~ Uniform(0,1). Interpolate real and generated samples: 𝑥=∈xr+(1-∈)xg Sample generator loss (12): |
| Lg= |/h(xr)-h(xg)ll2+|/h(xr)-h(𝑥g)ll2 |
| -|h(xg)-h(xg)ll2 Sample surrogate generator loss (13) and critic loss: |
| Ls(u,v)= |h(xr)-h(u)ll2-|/h(xr)ll2 -/h(u)-h(ν)ll2+ h(u)ll2 Ls=1[Ls(xg,xg)+Ls(xg,xg)] |
",
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+ "img_path": "images/10c3321ab14ddd89420fde4333c1de5a5e166d15df5415254b4af109a6ace48b.jpg",
+ "image_caption": [
+ "Figure 4: Approximate Wasserstein distances between CelebA test set and the generators. $N _ { u }$ is the number critic updates per generator update. "
+ ],
+ "image_footnote": [],
+ "bbox": [
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+ {
+ "type": "text",
+ "text": "5.1 CRAMÉR GAN ",
+ "text_level": 1,
+ "bbox": [
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+ "type": "text",
+ "text": "We now consider the Generative Adversarial Networks (GAN) framework (Goodfellow et al., 2014), in particular issues arising in the Wasserstein GAN (Arjovsky et al., 2017), and propose a better GAN based on the Cramér distance. A GAN is composed of a generative model $Q$ (in our experiments, over images), called the generator, a target source $P$ , and a trainable loss function called a discriminator or critic. GANs are particularly interesting because we can establish a direct comparison between the two distances. Our choice of name reflects this fact, and we prefer Cramér GAN to the perhaps more technically correct, but less palatable Energy Distance GAN. In theory, the Wasserstein GAN algorithm requires training the critic until convergence, but this is rarely achievable: we would require a critic that is a very powerful network to approximate the Wasserstein distance well (Arora et al., 2017). Simultaneously, training this critic to convergence would overfit the empirical distribution of the training set, which is undesirable. ",
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+ "type": "text",
+ "text": "Our proposed loss function allows for useful learning with imperfect critics by combining the energy distance with a transformation function $h : \\mathbb { R } ^ { d } \\mathbb { R } ^ { k }$ , where $d$ is the input dimensionality and $k ~ = ~ 2 5 6$ in our experiments. The generator then seeks to minimize the energy distance of the transformed variables $\\mathcal { E } ( h ( X ) , h ( Y ) )$ , where $X$ is a real sample and $Y$ is a generated sample. The critic itself seeks to maximize this same distance by changing the parameters of $h$ , subject to a soft constraint (the gradient penalty used by Gulrajani et al., 2017). Specifically, the critic maximizes a surrogate loss whose gradient can be estimated from a single real sample. The Cramér GAN losses are summarized in Algorithm 1, with additional design choices detailed in Appendix C. ",
+ "bbox": [
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+ },
+ {
+ "type": "text",
+ "text": "We note that MMDs such as the energy distance have in the last year become an appealing tool for training GANs. Among others, the squared MMD is used within Generative Moment Matching Networks (Li et al., 2015; Dziugaite et al., 2015); Bouchacourt et al. (2016) trained a model to minimize the energy distance for hand pose estimation. Our use of the tranformation $h ( x )$ reflects our anecdotal finding that the direct minimization of the energy distance over raw images does not work well (see Figure 10 in appendix). Similar findings can be found in the work of Mroueh et al. (2017) and the independently developed MMD GAN (Li et al., 2017), which additionally uses an auto-encoder loss to make the transformation injective. ",
+ "bbox": [
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+ },
+ {
+ "type": "text",
+ "text": "The Cramér GAN we present here complements our comparison of the Wasserstein and Cramér distance from previous sections. At the same time, our experiments also provide novel GAN-related contributions, including the ability to perform conditional modelling using a surrogate generator loss, which lets us train the critic even when only one independent sample from $P$ is available. We note also that in our experiments, $\\| x - y \\| _ { 2 }$ distances were more stable than distances generated by Gaussian or Laplacian kernels. ",
+ "bbox": [
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+ },
+ {
+ "type": "text",
+ "text": "5.2 CRAMÉR GAN EXPERIMENTS ",
+ "text_level": 1,
+ "bbox": [
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+ "type": "text",
+ "text": "We now show that, compared to the improved Wasserstein GAN (WGAN-GP) of Gulrajani et al. (2017), the Cramér GAN leads to more stable learning and increased diversity in the generated samples. In both cases we train generative models that predict the right half of an image given the left half; samples from unconditional models are provided in the appendix (Figure 10). The dataset we use here is the CelebA $6 4 \\times 6 4$ dataset (Liu et al., 2015) of celebrity faces. ",
+ "bbox": [
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+ },
+ {
+ "type": "text",
+ "text": "Increased diversity. In our first experiment, we compare the qualitative diversity of completed faces by showing three sample completions generated by either model given the left half of a validation set image (Figure 3). We observe that the completions produced by WGAN-GP are almost deterministic. Our findings echo those of Isola et al. (2016), who observed that “the generator simply learned to ignore the noise.” By contrast, the completions produced by Cramér GAN are fairly diverse, including different hairstyles, accessories, and backgrounds. We view this lack of diversity in WGAN-GP as undesirable given that the main requirement of a generative model is that it should provide a variety of outputs. ",
+ "bbox": [
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+ "page_idx": 7
+ },
+ {
+ "type": "text",
+ "text": "Theorem 1 provides a clue as to what may be happening here. We know that minimizing the sample Wasserstein loss will find the wrong minimum. In particular, when the target distribution has low entropy, the sample Wasserstein minimizer may actually be a deterministic distribution. But a good generative model of images must lie in this “almost deterministic” regime, since the space of natural images makes up but a fraction of all possible pixel combinations and hence there is little perpixel entropy. We hypothesize that the increased diversity in the Cramér GAN comes exactly from learning these almost deterministic predictions. ",
+ "bbox": [
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+ "page_idx": 7
+ },
+ {
+ "type": "text",
+ "text": "More stable learning. In a second experiment, we varied the number of critic updates $( N _ { u } )$ per generator update. To compare performance between the two architectures, we measured the loss computed by an independent WGAN-GP critic trained on the validation set, following a similar evaluation previously done by Danihelka et al. (2017). Figure 4 shows the independent Wasserstein critic distance between each generator and the test set during the course of training. Echoing our results with the toy experiment and ordinal regression, the plot shows that when a single critic update is used, WGAN-GP performs particularly poorly. We note that additional critic updates also improve Cramér GAN. This indicates that it is helpful to keep adapting the $h ( x )$ transformation. ",
+ "bbox": [
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+ },
+ {
+ "type": "text",
+ "text": "6 CONCLUSION ",
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+ ],
+ "page_idx": 7
+ },
+ {
+ "type": "text",
+ "text": "There are many situations in which the KL divergence, which is commonly used as a loss function in machine learning, is not suitable. The desirable alternatives, as we have explored, are the divergences that are ideal and allow for unbiased estimators: they allow geometric information to be incorporated into the optimization problem; because they are scale-sensitive and sum-invariant, they possess the convergence properties we require for efficient learning; and the correctness of their sample gradients means we can deploy them in large-scale optimization problems. Among open questions, we mention deriving an unbiased estimator that minimizes the Wasserstein distance, and variance analysis and reduction of the Cramér distance gradient estimate. ",
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+ "text": "Yossi Rubner, Carlo Tomasi, and Leonidas J Guibas. The earth mover’s distance as a metric for image retrieval. International journal of computer vision, 40(2):99–121, 2000. ",
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+ "text": "Tim Salimans, Ian Goodfellow, Wojciech Zaremba, Vicki Cheung, Alec Radford, and Xi Chen. Improved techniques for training GANs. In Advances in Neural Information Processing Systems, pp. 2234–2242, 2016. ",
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+ "text": "Dino Sejdinovic, Bharath Sriperumbudur, Arthur Gretton, Kenji Fukumizu, et al. Equivalence of distance-based and RKHS-based statistics in hypothesis testing. The Annals of Statistics, 41(5): 2263–2291, 2013. ",
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+ "text": "Christian Szegedy, Vincent Vanhoucke, Sergey Ioffe, Jon Shlens, and Zbigniew Wojna. Rethinking the inception architecture for computer vision. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 2818–2826, 2016. ",
+ "bbox": [
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+ },
+ {
+ "type": "text",
+ "text": "Gabor J. Székely. E-statistics: The energy of statistical samples. Technical Report 02-16, Bowling Green State University, Department of Mathematics and Statistics, 2002. ",
+ "bbox": [
+ 176,
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+ "page_idx": 9
+ },
+ {
+ "type": "text",
+ "text": "Gábor J Székely and Maria L Rizzo. Energy statistics: A class of statistics based on distances. Journal of statistical planning and inference, 143(8):1249–1272, 2013. \nAaron Van den Oord, Nal Kalchbrenner, and Koray Kavukcuoglu. Pixel recurrent neural networks. In Proceedings of the International Conference on Machine Learning, 2016. \nMark Veraar. On Khintchine inequalities with a weight. Proceedings of the American Mathematical Society, 138(11):4119–4121, 2010. \nVladimir M. Zolotarev. Metric distances in spaces of random variables and their distributions. Sbornik: Mathematics, 30(3):373–401, 1976. ",
+ "bbox": [
+ 171,
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+ 826,
+ 246
+ ],
+ "page_idx": 10
+ },
+ {
+ "type": "text",
+ "text": "A PROOFS ",
+ "text_level": 1,
+ "bbox": [
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+ ],
+ "page_idx": 11
+ },
+ {
+ "type": "text",
+ "text": "A.1 PROPERTIES OF A DIVERGENCE",
+ "text_level": 1,
+ "bbox": [
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+ "page_idx": 11
+ },
+ {
+ "type": "text",
+ "text": "Proof (Proposition 1 and 2). The statement regarding (U) for the KL divergence is well-known, and forms the basis of most stochastic gradient algorithms for classification. Chung & Sobel (1987) have shown that the total variation does not have property (S); by Pinsker’s inequality, it follows that the same holds for the KL divergence. A proof of (I) and (S) for the Wasserstein metric is given by Bickel & Freedman (1981), while the lack of (U) is shown in the proof of Theorem 1. □ ",
+ "bbox": [
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+ "page_idx": 11
+ },
+ {
+ "type": "text",
+ "text": "A.2 BIASED ESTIMATE ",
+ "text_level": 1,
+ "bbox": [
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+ "page_idx": 11
+ },
+ {
+ "type": "text",
+ "text": "Proof (Theorem $^ { l }$ ). Minimax bias: Consider $P = B ( { \\theta } ^ { * } )$ , a Bernoulli distribution of parameter $\\theta ^ { * }$ and $Q _ { \\theta } = B ( \\theta )$ a Bernoulli of parameter $\\theta$ . The empirical distribution $\\hat { P } _ { m }$ is a Bernoulli with parameter $\\begin{array} { r } { \\hat { \\theta } : = \\frac { 1 } { m } \\sum _ { i = 1 } ^ { m } X _ { i } } \\end{array}$ . Note that with $P$ and $Q _ { \\theta }$ both Bernoulli distributions, the p $p ^ { t h }$ powers -Wasserstein metrics are equal, i.e. $w _ { 1 } ( P , Q _ { \\theta } ) = w _ { p } ^ { p } ( P , Q _ { \\theta } )$ . This gives us an easy way to prove the stronger result that all $p$ -Wasserstein metrics have biased sample gradients. The gradient of the loss $w _ { p } ^ { p } ( P , Q _ { \\theta } )$ is, for $\\theta \\neq \\theta ^ { * }$ , ",
+ "bbox": [
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+ },
+ {
+ "type": "equation",
+ "img_path": "images/2bfd99fda74f94da8c223dcc6cdcfdd7bd068c4e3cdba6e2fc3983408df21ae2.jpg",
+ "text": "$$\n\\begin{array} { r } { g : = \\nabla w _ { p } ^ { p } ( P , Q _ { \\theta } ) = \\nabla \\Big [ \\big | \\theta ^ { * } - \\theta \\big | \\Big ] = \\mathrm { s g n } ( \\theta - \\theta ^ { * } ) , } \\end{array}\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 11
+ },
+ {
+ "type": "text",
+ "text": "and similarly, the gradient of the sample loss is, for $\\theta \\neq { \\hat { \\theta } }$ , ",
+ "bbox": [
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+ "page_idx": 11
+ },
+ {
+ "type": "equation",
+ "img_path": "images/267a8544b1dcb4a3a637c5a15261136e0289c42cc4324f0ae4d3b985335b516e.jpg",
+ "text": "$$\n\\boldsymbol { \\hat { g } } : = \\nabla w _ { p } ^ { p } ( \\boldsymbol { \\hat { P } } _ { m } , Q _ { \\theta } ) = \\nabla \\Big [ \\big | \\boldsymbol { \\hat { \\theta } } - \\boldsymbol { \\theta } \\big | \\Big ] = \\mathrm { s g n } ( \\theta - \\boldsymbol { \\hat { \\theta } } ) .\n$$",
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+ "bbox": [
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+ },
+ {
+ "type": "text",
+ "text": "Notice that this estimate is biased for any $m \\geq 1$ since ",
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+ },
+ {
+ "type": "equation",
+ "img_path": "images/9677888b0ba1e7875c40bffb9671f25df67c676068d8723af840509e750474c4.jpg",
+ "text": "$$\n\\begin{array} { r } { \\mathbb { E } \\hat { g } = 2 \\operatorname* { P r } \\{ \\hat { \\theta } < \\theta \\} - 1 , } \\end{array}\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 11
+ },
+ {
+ "type": "text",
+ "text": "which is different from $g$ for any $\\theta ^ { * } \\in ( 0 , 1 )$ . In particular for $m = 1$ , $\\mathbb { E } _ { P } \\hat { g } = 1 - 2 \\theta ^ { * }$ does not depend on $\\theta$ , thus a gradient descent using a one-sample gradient estimate has no chance of minimizing the Wasserstein loss as it will converge to either 1 or 0 instead of $\\theta ^ { * }$ . ",
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+ },
+ {
+ "type": "text",
+ "text": "Now observe that for $m \\geq 2$ , and any $\\textstyle \\theta > { \\frac { m - 1 } { m } }$ , ",
+ "bbox": [
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+ {
+ "type": "equation",
+ "img_path": "images/2c9b5e6acc36d0e01f35da41b0d1e1cd56e753b6dc30a41c4ca8c90febdaef03.jpg",
+ "text": "$$\n\\operatorname* { P r } \\{ \\hat { \\theta } < \\theta \\} = \\operatorname* { P r } \\{ \\exists i \\ { \\mathrm { s . t . } } \\ X _ { i } = 0 \\} = 1 - ( \\theta ^ { * } ) ^ { m } ,\n$$",
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+ {
+ "type": "text",
+ "text": "and therefore ",
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+ "type": "equation",
+ "img_path": "images/d7b70f248d919c8559926ab24717d85cc5d9c5f30a858aa09d5c60f6e00e299c.jpg",
+ "text": "$$\n\\mathbb { E } \\hat { g } = 1 - 2 ( \\theta ^ { * } ) ^ { m } .\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 11
+ },
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+ "type": "text",
+ "text": "Taking $\\textstyle \\theta ^ { * } = { \\frac { m - 1 } { m } }$ , we find that ",
+ "bbox": [
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+ "type": "equation",
+ "img_path": "images/02cb24e85ba41e10316436b7f80fc3691de03d3e689b3d1ae4b48633bfe5f4da.jpg",
+ "text": "$$\ng - \\mathbb { E } \\hat { g } = 1 - [ 1 - 2 ( \\theta ^ { * } ) ^ { m } ] = 2 \\left( 1 - \\frac { 1 } { m } \\right) ^ { m } \\ge 2 e ^ { - 2 } .\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 11
+ },
+ {
+ "type": "text",
+ "text": "Thus for any $m$ , there exists $P = B ( \\theta ^ { * } )$ and $Q _ { \\theta } = B ( \\theta )$ with $\\textstyle \\theta ^ { * } = { \\frac { m - 1 } { m } } < \\theta < 1$ such that the bias $\\boldsymbol { g } - \\mathbb { E } \\hat { \\boldsymbol { g } }$ is lower-bounded by a numerical constant. Thus the minimax bias does not vanish with the number of samples $m$ . ",
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+ },
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+ "type": "text",
+ "text": "Notice that a similar argument holds for $\\theta ^ { * }$ and $\\theta$ being close to 0. In both situations where $\\theta ^ { * }$ is close to 0 or 1, the bias is non vanishing when $| \\theta ^ { * } - \\theta |$ is of order $\\textstyle { \\frac { 1 } { m } }$ . However this is even worse when $\\theta ^ { * }$ is away from the boundaries. For example chosing $\\theta ^ { * } = \\textstyle { \\frac { 1 } { 2 } }$ , we can prove that the bias is non vanishing even when $| \\theta ^ { * } - \\theta |$ is (only) of order $\\frac { 1 } { \\sqrt { m } }$ . ",
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+ "type": "text",
+ "text": "Indeed, using the anti-concentration result of Veraar (2010) (Proposition 2), we have that for a sequence $Y _ { 1 } , \\dots , Y _ { m }$ of Rademacher random variables (i.e. $+ / - 1$ with equal probability), ",
+ "bbox": [
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+ "img_path": "images/6ff9fe04d426b3cdfcaf6e18e0bec70c72343dfd1f19b2572be132280f9207fc.jpg",
+ "text": "$$\n\\operatorname* { P r } \\left( { \\frac { 1 } { n } } \\sum _ { i = 1 } ^ { m } Y _ { i } \\geq \\epsilon \\right) \\geq ( 1 - m \\epsilon ^ { 2 } ) ^ { 2 } / 3 .\n$$",
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+ "type": "text",
+ "text": "This means that for samples $X _ { 1 } , \\ldots , X _ { m }$ drawn from a Bernoulli $\\begin{array} { r } { B ( \\theta ^ { * } = \\frac { 1 } { 2 } ) } \\end{array}$ (i.e., $Y _ { i } = 2 X _ { i } - 1$ are Rademacher), we have ",
+ "bbox": [
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+ "type": "equation",
+ "img_path": "images/64359c419b91dcc888649f22a191f6176071d5dda613f3d906d529500dd42ce3.jpg",
+ "text": "$$\n\\operatorname* { P r } \\left( \\hat { \\theta } \\geq \\theta ^ { * } + \\epsilon / 2 \\right) \\geq ( 1 - m \\epsilon ^ { 2 } ) ^ { 2 } / 3 ,\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ {
+ "type": "image",
+ "img_path": "images/a67267df7cc310db4629c2ccc9bd29f80be80a577a57c7c0486e836a94a6d02e.jpg",
+ "image_caption": [
+ "Figure 5: Wasserstein loss (black curve) $\\theta \\mapsto | \\theta ^ { * } - \\theta |$ versus expected sample Wasserstein loss (red curve) $\\theta \\mapsto \\mathbb { E } [ | \\hat { \\theta } - \\theta | ]$ , for different values of $m$ and $\\theta ^ { * }$ and $p = 1$ . Left: $m = 1$ , $\\theta ^ { * } = 0 . 6$ . A stochastic gradient using a one-sample Wasserstein gradient estimate will converge to 1 instead of $\\theta ^ { * }$ . Middle: $m = 6$ , $\\theta ^ { * } = 0 . 6$ . The minimum of the expected sample Wasserstein loss is the median of $\\hat { \\theta }$ which is here $\\tilde { \\theta } = { \\textstyle \\frac { 2 } { 3 } } \\ne \\theta ^ { * } = 0 . 6$ . Right: $m = 5$ , $p = 0 . 9$ . The minimum of the expected sample Wasserstein is $\\tilde { \\theta } = 1$ and not $\\theta ^ { * } = 0 . 9$ . "
+ ],
+ "image_footnote": [],
+ "bbox": [
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+ "page_idx": 12
+ },
+ {
+ "type": "text",
+ "text": "thus for $1 / 2 = \\theta ^ { \\ast } < \\theta < \\theta ^ { \\ast } + 1 / \\sqrt { 8 m }$ we have the following lower bound on the bias: ",
+ "bbox": [
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+ "type": "equation",
+ "img_path": "images/af2f5317d6b45b5f77278c9e7bdbed829c60d03aaebfc7eee5b1d4f199d810dc.jpg",
+ "text": "$$\ng - \\mathbb { E } \\hat { g } = 2 \\operatorname* { P r } \\left( \\hat { \\theta } \\geq \\theta \\right) \\geq 1 / 6 .\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ },
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+ "type": "text",
+ "text": "Thus the bias is lower-bounded by a constant (independent of $m$ ) when $\\theta ^ { * } = \\textstyle { \\frac { 1 } { 2 } }$ and $\\left| \\theta ^ { * } - \\theta \\right| =$ $O ( 1 / \\sqrt { m } )$ . ",
+ "bbox": [
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+ "page_idx": 12
+ },
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+ "type": "text",
+ "text": "Wrong minimum: From (5), we deduce that a stochastic gradient descent algorithm based on the sample Wasserstein gradient will converge to a $\\tilde { \\theta }$ such that $\\begin{array} { r } { \\mathrm { \\tilde { P r } } \\{ \\hat { \\theta } < \\tilde { \\theta } \\} = \\frac { 1 } { 2 } } \\end{array}$ , i.e., $\\tilde { \\theta }$ is the median of the distribution over $\\hat { \\theta }$ , whereas $\\theta ^ { * }$ is the mean of that distribution. Since $\\hat { \\theta }$ follows a (normalized) binomial distribution with parameters $m$ and $\\theta ^ { * }$ , we know that the median $\\tilde { \\theta }$ and the mean $\\theta ^ { * }$ do not necessarily coincide, and can actually be as far as $\\frac { 1 } { 2 m }$ -away from each other. For example for any odd $m$ and any $\\theta ^ { * } \\in \\left( { \\frac { 1 } { 2 } } , { \\frac { 1 } { 2 } } - { \\frac { 1 } { 2 m } } \\right)$ the median is $\\theta ^ { * } - \\frac { 1 } { 2 m }$ . ",
+ "bbox": [
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+ "page_idx": 12
+ },
+ {
+ "type": "text",
+ "text": "It follows that the minimum of the expected sample Wasserstein loss (the fixed point of the stochastic gradient descent using the sample Wasserstein gradient) is different from the minimum of the true Wasserstein loss: ",
+ "bbox": [
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+ },
+ {
+ "type": "equation",
+ "img_path": "images/6e10d0e680fdbb9cd4212ca9c56ef8c60dd5bc55f609fe325c715aebfeca79ce.jpg",
+ "text": "$$\n\\underset { \\theta } { \\arg \\operatorname* { m i n } } \\mathbb { E } [ w _ { p } ^ { p } ( \\hat { P } _ { m } , Q _ { \\theta } ) ] \\neq \\underset { \\theta } { \\arg \\operatorname* { m i n } } [ w _ { p } ^ { p } ( P , Q _ { \\theta } ) ] .\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ },
+ {
+ "type": "text",
+ "text": "This is illustrated in Figure 5. ",
+ "bbox": [
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+ "page_idx": 12
+ },
+ {
+ "type": "text",
+ "text": "Notice that the fact that the minima of these losses differ is worrisome as it means that minimizing the sample Wasserstein loss using (finite) samples will not converge to the correct solution. ",
+ "bbox": [
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+ "page_idx": 12
+ },
+ {
+ "type": "text",
+ "text": "Deterministic solutions: Consider the specific case where $( 1 / 2 ) ^ { 1 / n } < \\theta ^ { * } < 1$ (illustrated in the right plot of Figure 5). Then the expected sample gradient $\\nabla \\mathbb { E } [ w _ { p } ^ { p } ( \\hat { P } _ { m } , Q _ { \\theta ^ { * } } ) ] = \\mathbb { E } \\hat { g } = 1 - 2 ( \\theta ^ { * } ) ^ { n } <$ 0 for any $\\theta$ , so a gradient descent algorithm will converge to 1 instead of $\\theta ^ { * }$ . Notice that a symmetric argument applies for $\\theta ^ { * }$ close to 0. ",
+ "bbox": [
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+ "page_idx": 12
+ },
+ {
+ "type": "text",
+ "text": "In this simple example, minimizing the sample Wasserstein loss may lead to degenerate solutions (i.e., deterministic) when our target distributions have low (but not zero) entropy. □ ",
+ "bbox": [
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+ "page_idx": 12
+ },
+ {
+ "type": "text",
+ "text": "A.3 CONSISTENCY OF THE SAMPLE 1-WASSERSTEIN GRADIENT ",
+ "text_level": 1,
+ "bbox": [
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+ "page_idx": 12
+ },
+ {
+ "type": "text",
+ "text": "We provide an additional result here showing that the sample 1-Wasserstein gradient converges to the true gradient as $m \\infty$ . ",
+ "bbox": [
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+ "page_idx": 12
+ },
+ {
+ "type": "text",
+ "text": "Theorem 3. Let $P$ and $Q _ { \\theta }$ be probability distributions, with $Q _ { \\theta }$ parametrized by $\\theta$ . Assume that the set $\\{ x \\in X$ , such that $F _ { P } ( x ) = F _ { Q _ { \\theta } } ( x ) \\big \\}$ has measure zero, and that for any $x \\in X$ , the map $\\tilde { \\theta } \\mapsto F _ { Q _ { \\tilde { \\theta } } } ( x )$ is differentiable in a neighborhood $\\mathcal { V } ( \\boldsymbol { \\theta } )$ of $\\theta$ with a uniformly bounded derivative ",
+ "bbox": [
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+ "page_idx": 12
+ },
+ {
+ "type": "text",
+ "text": "(for $\\tilde { \\theta } \\in \\mathcal { V } ( \\theta )$ and $x \\in X ,$ ). Let $\\begin{array} { r } { \\hat { P } _ { m } = \\frac { 1 } { m } \\sum _ { i = 1 } ^ { m } \\delta _ { X _ { i } } } \\end{array}$ be the empirical distribution derived from m independent samples $X _ { 1 } , \\ldots , X _ { m }$ drawn from $P$ . Then ",
+ "bbox": [
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+ "page_idx": 13
+ },
+ {
+ "type": "text",
+ "text": "We note that the measure requirement is strictly to keep the proof simple, and does not subtract from the generality of the result. ",
+ "bbox": [
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+ "page_idx": 13
+ },
+ {
+ "type": "text",
+ "text": "Proof. Let $\\nabla : = \\nabla _ { \\theta }$ . Since $p = 1$ the Wasserstein distance $w _ { 1 } ( P , Q )$ measures the area between the curves defined by the distribution function of $P$ and $Q$ , thus $\\begin{array} { r } { w _ { 1 } ( P , Q ) = l _ { 1 } ( P , Q ) = \\int \\left| F _ { P } ( x ) - \\frac { } { } \\right. } \\end{array}$ $F _ { Q } ( x ) { \\left| { d x } \\right. }$ and ",
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+ "page_idx": 13
+ },
+ {
+ "type": "equation",
+ "img_path": "images/bd27d1595daf96948ae84a56e5b037a7966cffb3d62f0c62ebf5b75d1d335f56.jpg",
+ "text": "$$\n\\begin{array} { l l l } { \\nabla w _ { 1 } ( P , Q _ { \\theta } ) } & { = } & { \\displaystyle \\operatorname* { l i m } _ { \\Delta \\to 0 } \\frac { w _ { 1 } ( P , Q _ { \\theta + \\Delta } ) - w _ { 1 } ( P , Q _ { \\theta } ) } { \\Delta } } \\\\ & { = } & { \\displaystyle \\operatorname* { l i m } _ { \\Delta \\to 0 } \\int \\frac { 1 } { \\Delta } \\Big ( \\big | F _ { P } ( x ) - F _ { Q _ { \\theta + \\Delta } } ( x ) \\big | - \\big | F _ { P } ( x ) - F _ { Q _ { \\theta } } ( x ) \\big | \\Big ) d x . } \\end{array}\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 13
+ },
+ {
+ "type": "text",
+ "text": "Now since we have assumed that for any $x \\in X$ , the map $\\theta \\mapsto F _ { Q _ { \\theta } } ( x )$ is differentiable in a neighborhood $\\mathcal { V } ( \\boldsymbol { \\theta } )$ of $\\theta$ and its derivative is uniformly (over $\\mathcal { V } ( \\boldsymbol { \\theta } )$ and $x$ ) bounded by $M$ , we have ",
+ "bbox": [
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+ "page_idx": 13
+ },
+ {
+ "type": "equation",
+ "img_path": "images/a0f2f54b420d346643dff88c4e1f29148a027b9e07e203953f2732e97949f935.jpg",
+ "text": "$$\n\\frac { 1 } { \\Delta } \\Big | \\big | F _ { P } ( x ) - F _ { Q _ { \\theta + \\Delta } } ( x ) \\big | - \\big | F _ { P } ( x ) - F _ { Q _ { \\theta } } ( x ) \\big | \\Big | \\quad \\le \\quad \\frac { 1 } { \\Delta } \\big | F _ { Q _ { \\theta + \\Delta } } ( x ) - F _ { Q _ { \\theta } } ( x ) \\big | \\le M .\n$$",
+ "text_format": "latex",
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+ "text": "Thus the dominated convergence theorem applies and ",
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+ "text": "$$\n\\begin{array} { r c l } { \\nabla w _ { 1 } ( P , Q _ { \\theta } ) } & { = } & { \\displaystyle \\int \\operatorname* { l i m } _ { \\Delta \\to 0 } \\frac { 1 } { \\Delta } \\Big ( \\big | F _ { P } ( x ) - F _ { Q _ { \\theta + \\Delta } } ( x ) \\big | - \\big | F _ { P } ( x ) - F _ { Q _ { \\theta } } ( x ) \\big | \\Big ) d x } \\\\ & { = } & { \\displaystyle \\int \\nabla \\big | F _ { P } ( x ) - F _ { Q _ { \\theta } } ( x ) \\big | d x } \\\\ & { = } & { \\displaystyle \\int \\mathrm { s g n } \\big ( F _ { P } ( x ) - F _ { Q _ { \\theta } } ( x ) \\big ) \\nabla F _ { Q _ { \\theta } } ( x ) d x , } \\end{array}\n$$",
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+ "text": "since we have assumed that the set of $x \\in X$ such that $F _ { P } ( x ) = F _ { Q _ { \\theta } } ( x )$ has measure zero. ",
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+ "type": "text",
+ "text": "Now, using the same argument for $w _ { 1 } \\big ( \\hat { P } _ { m } , Q _ { \\theta } \\big )$ we deduce that ",
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+ "text": "$$\n\\begin{array} { r l r } { \\nabla w _ { 1 } ( \\hat { P } _ { m } , Q _ { \\theta } ) } & { = } & { \\displaystyle \\int \\underbrace { \\operatorname* { l i m } _ { \\Delta \\to 0 } \\frac { 1 } { \\Delta } \\Big ( \\big | F _ { \\hat { P } _ { m } } ( x ) - F _ { Q _ { \\theta + \\Delta } } ( x ) \\big | - \\big | F _ { \\hat { P } _ { m } } ( x ) - F _ { Q _ { \\theta } } ( x ) \\big | \\Big ) } _ { A ( x ) } d x . } \\end{array}\n$$",
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+ "text": "Let us decompose this integral over $X$ as the sum of two integrals, one over $X \\setminus \\Omega _ { m }$ and the other one over $\\Omega _ { m }$ , where $\\Omega _ { m } = \\big \\{ x \\in X , F _ { \\hat { P } _ { m } } ( x ) = F _ { Q _ { \\theta } } ( x ) \\big \\}$ . We have ",
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+ "text": "$$\n\\int _ { X \\setminus \\Omega _ { m } } A ( x ) d x = \\int _ { X \\setminus \\Omega _ { m } } \\operatorname { s g n } \\bigl ( F _ { \\hat { P } _ { m } } ( x ) - F _ { Q _ { \\theta } } ( x ) \\bigr ) \\nabla F _ { Q _ { \\theta } } ( x ) d x ,\n$$",
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+ "text": "and ",
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+ "text": "$$\n\\begin{array} { r c l } { \\Big | \\displaystyle \\int _ { \\Omega _ { m } } A ( x ) d x \\Big | } & { \\le } & { \\displaystyle \\int _ { \\Omega _ { m } } \\operatorname* { l i m } _ { \\Delta \\to 0 } \\frac { 1 } { \\Delta } \\Big ( \\big | F _ { Q _ { \\theta + \\Delta } } ( x ) - F _ { Q _ { \\theta } } ( x ) \\big | \\Big ) d x } \\\\ & { \\le } & { M | \\Omega _ { m } | . } \\end{array}\n$$",
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+ "text": "Now from the strong law of large numbers, we have that for any $x$ , the empirical cumulative distribution function $F _ { \\hat { P } _ { m } } ( x )$ converges to the cumulative distribution $F _ { P } ( x )$ almost surely. We deduce that $\\Omega _ { m }$ converges to the set $\\left\\{ x , F _ { P } ( x ) = F _ { Q _ { \\theta } } ( x ) \\right\\}$ which has measure zero, thus $| \\Omega _ { m } | \\to 0$ and ",
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+ "text": "$$\n\\begin{array} { r l r } { \\displaystyle \\operatorname* { l i m } _ { m \\to \\infty } \\nabla w _ { 1 } ( \\hat { P } _ { m } , Q _ { \\theta } ) } & { = } & { \\displaystyle \\operatorname* { l i m } _ { m \\to \\infty } \\int _ { X } \\mathrm { s g n } \\big ( { \\cal F } _ { \\hat { P } _ { m } } ( x ) - { \\cal F } _ { Q _ { \\theta } } ( x ) \\big ) \\nabla { \\cal F } _ { Q _ { \\theta } } ( x ) d x . } \\end{array}\n$$",
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+ "text": "Now, since $| \\nabla F _ { Q _ { \\theta } } ( x ) | \\leq M$ , we can use once more the dominated convergence theorem to deduce that ",
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+ "text": "$$\n\\begin{array} { l } { \\displaystyle \\operatorname* { l i m } _ { m \\infty } \\nabla w _ { 1 } ( \\hat { P } _ { m } , Q _ { \\theta } ) = \\int _ { X } \\displaystyle \\operatorname* { l i m } _ { m \\infty } \\mathrm { s g n } \\big ( F _ { \\hat { P } _ { m } } ( x ) - F _ { Q _ { \\theta } } ( x ) \\big ) \\nabla F _ { Q _ { \\theta } } ( x ) d x } \\\\ { \\displaystyle = \\int _ { X } \\mathrm { s g n } \\big ( F _ { P } ( x ) - F _ { Q _ { \\theta } } ( x ) \\big ) \\nabla F _ { Q _ { \\theta } } ( x ) d x } \\\\ { \\displaystyle = \\nabla w _ { 1 } ( P , Q _ { \\theta } ) . } \\end{array}\n$$",
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+ "text": "The following lemma will be useful in proving that the Cramér distance has property (U). ",
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+ "text": "Lemma 1. Let $\\mathbf { X } _ { m } : = X _ { 1 } , \\ldots , X _ { m }$ be independent samples from $P$ , and let $\\begin{array} { r } { \\hat { P } _ { m } : = \\frac { 1 } { m } \\sum _ { i } \\delta _ { X _ { i } } } \\end{array}$ . Then ",
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+ "text": "$$\n\\begin{array} { r } { \\underset { \\mathbf { X } _ { m } \\sim P } { \\mathbb { E } } F _ { \\hat { P } _ { m } } ( x ) = F _ { P } ( x ) . } \\end{array}\n$$",
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+ "text": "Proof. Because the $X _ { i }$ ’s are independent, ",
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+ "text": "$$\nF _ { \\hat { P } _ { m } } ( x ) = \\int _ { - \\infty } ^ { x } \\hat { P } _ { m } ( \\mathrm { d } x ) = \\frac { 1 } { m } \\sum _ { i = 1 } ^ { m } \\mathbb { I } \\left[ X _ { i } \\le x \\right] .\n$$",
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+ "type": "text",
+ "text": "Now, taking the expectation w.r.t. ${ \\bf { X } } _ { m }$ , ",
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+ "text": "$$\n\\begin{array} { l } { \\displaystyle \\underset { { \\bf X } _ { m } \\sim P } { \\mathbb { E } } F _ { \\hat { P } _ { m } } ( x ) = \\underset { { \\bf X } _ { m } \\sim P } { \\mathbb { E } } \\frac { 1 } { m } \\sum _ { i = 1 } ^ { m } { \\mathbb { I } } \\left[ X _ { i } \\leq x \\right] } \\\\ { \\displaystyle = \\frac { 1 } { m } \\sum _ { i = 1 } ^ { m } X _ { i \\sim P } ^ { { \\mathbb { E } } } \\mathbb { I } \\left[ X _ { i } \\leq x \\right] } \\\\ { \\displaystyle = \\frac { 1 } { m } \\sum _ { i = 1 } ^ { m } \\mathrm { P r } \\{ X _ { i } \\leq x \\} } \\\\ { \\displaystyle = F _ { P } ( x ) , } \\end{array}\n$$",
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+ "type": "text",
+ "text": "since the $X _ { i }$ are identically distributed according to $P$ . ",
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+ "text": "Proof (Theorem 2). Like the Wasserstein metrics, the $l _ { p }$ metrics have dual forms as integral probability metrics (see Dedecker & Merlevède, 2007, for a proof): ",
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+ "text": "$$\nl _ { p } ( P , Q ) = \\operatorname* { s u p } _ { f \\in \\mathbb { F } _ { q } } \\big | \\operatorname* { \\mathbb { E } } _ { x \\sim P } f ( x ) - \\operatorname* { \\mathbb { E } } _ { x \\sim Q } f ( x ) \\big | ,\n$$",
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+ "type": "text",
+ "text": "where $\\mathbb { F } _ { q } : = \\{ f : f$ is absolutely continuous, $\\begin{array} { r } { \\left\\| \\frac { \\mathrm { d } f } { \\mathrm { d } x } \\right\\| _ { q } \\leq 1 \\} } \\end{array}$ and $q$ is the conjugate exponent of $p$ , i.e. \n$p ^ { - 1 } + q ^ { - 1 } = 1$ .3 We will use this dual form below. ",
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+ "type": "text",
+ "text": "We will prove that $l _ { p }$ has properties (I) and (S) for $p \\in [ 1 , \\infty )$ ; the case $p = \\infty$ follows by a similar argument. Begin by observing that ",
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+ "text": "$$\n\\begin{array} { c } { { F _ { c X } ( x ) = P r \\{ c X \\leq x \\} } } \\\\ { { = P r \\left\\{ X \\leq \\displaystyle \\frac { x } { c } \\right\\} } } \\\\ { { = F _ { X } \\left( \\displaystyle \\frac { x } { c } \\right) . } } \\end{array}\n$$",
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+ "text": "Then we may rewrite $l _ { p } ^ { p } ( c X , c Y )$ as ",
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+ "text": "$$\nl _ { p } ^ { p } ( c X , c Y ) = \\int _ { - \\infty } ^ { \\infty } { \\left| F _ { X } \\left( \\frac { x } { c } \\right) - F _ { Y } \\left( \\frac { x } { c } \\right) \\right| ^ { p } } \\mathrm { d } x\n$$",
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+ "type": "text",
+ "text": "3This relationship is the reason for the notation $\\mathbb { F } _ { \\infty }$ in the definition the dual of the 1-Wasserstein (2). ",
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+ "type": "text",
+ "text": "where $( a )$ uses a change of variables $z = x / c$ . Taking both sides to the power $1 / p$ proves that the $l _ { p }$ metric possesses property (S) of order $1 / p$ . For (I), we use the IPM formulation (6): ",
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+ "text": "$$\n\\begin{array} { r l } & { l _ { p } \\big ( A + X , A + Y \\big ) = \\underset { f \\in \\mathcal { F } _ { q } } { \\operatorname* { s u p } } \\bigg | _ { A + X } f ( x ) - \\underset { A + Y } { \\mathbb { E } } f ( y ) \\bigg | } \\\\ & { \\stackrel { ( a ) } { = } \\underset { f \\in \\mathcal { F } _ { q } } { \\operatorname* { s u p } } \\bigg | \\mathbb { E } _ { A } \\mathbb { E } _ { X } f ( x + a ) - \\mathbb { E } _ { A } \\mathbb { E } _ { Y } f ( y + a ) \\bigg | } \\\\ & { \\stackrel { ( b ) } { = } \\underset { f \\in \\mathcal { F } _ { q } } { \\operatorname* { s u p } } \\bigg | \\mathbb { E } _ { A } \\big [ \\mathbb { E } _ { X } f ( x + a ) - \\mathbb { E } _ { Y } f ( y + a ) \\big ] \\bigg | } \\\\ & { \\stackrel { ( b ) } { \\leq } \\mathbb { E } _ { A } \\underset { f \\in \\mathcal { F } _ { q } } { \\operatorname* { s u p } } \\bigg | \\mathbb { E } _ { X } f ( x + a ) - \\mathbb { E } _ { Y } f ( y + a ) \\bigg | , } \\end{array}\n$$",
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+ "type": "text",
+ "text": "where $( a )$ is by independence of $A$ and $X , Y$ , and $( b )$ is by Jensen’s inequality. Next, recall that $\\mathcal { F } _ { q }$ is the set of absolutely continuous functions whose derivative has bounded $L _ { q }$ norm. Hence if $f \\in \\mathcal { F } _ { q }$ , then also for all $a$ the translate $g _ { a } ( x ) : = f ( x + a )$ is also in $\\mathcal { F } _ { q }$ . Therefore, ",
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+ "text": "$$\n\\begin{array} { r l } & { l _ { p } ( A + X , A + Y ) \\leq \\mathbb { E } _ { A } \\underset { f \\in \\mathcal { F } _ { q } } { \\mathrm { \\mathbb { E } } } \\bigg | \\mathbb { E } _ { X } f ( x + a ) - \\mathbb { E } _ { Y } f ( y + a ) \\bigg | } \\\\ & { \\qquad = \\mathbb { E } _ { A } \\underset { g \\in \\mathcal { F } _ { q } } { \\mathrm { \\mathbb { E } } } \\bigg | \\mathbb { E } _ { X } g ( x ) - \\mathbb { E } _ { Y } g ( y ) \\bigg | } \\\\ & { \\qquad = \\underset { g \\in \\mathcal { F } _ { q } } { \\mathrm { \\operatorname* { s u p } } } \\bigg | \\mathbb { E } _ { X } g ( x ) - \\mathbb { E } _ { Y } g ( y ) \\bigg | } \\\\ & { \\qquad = l _ { p } ( X , Y ) . } \\end{array}\n$$",
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+ "type": "text",
+ "text": "Now, to prove (U). Here we make use of the introductory requirement that “all expectations under consideration are finite.” Specifically, we require that the mean under $P$ $\\mathbb { \\lambda } , \\mathbb { E } _ { x \\sim P } [ x ]$ , is well-defined and finite, and similarly for $Q _ { \\theta }$ . In this case, ",
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+ "text": "$$\n\\underset { x \\sim P } { \\mathbb { E } } [ x ] = \\int _ { 0 } ^ { \\infty } ( 1 - F _ { P } ( x ) ) \\mathrm { d } x - \\int _ { - \\infty } ^ { 0 } F _ { P } ( x ) \\mathrm { d } x .\n$$",
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+ "type": "text",
+ "text": "This mild requirement guarantees that the tails of the distribution function $F _ { P }$ are light enough to avoid infinite Cramér distances and expected gradients (a similar condition was set by Dedecker & Merlevède (2007)). Now, by definition, ",
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+ "img_path": "images/4c9dc6d58da61f8b24e502d7d4ba11e30bc4355b20153f5fedf84dcabdb63062.jpg",
+ "text": "$$\n\\begin{array} { r l } { \\nabla \\theta _ { i } ^ { 2 } ( P , Q _ { \\theta } ) = \\nabla \\theta \\displaystyle \\int _ { - \\infty } ^ { \\infty } \\left( F _ { Q _ { \\theta } } ( x ) - F _ { P } ( x ) \\right) ^ { 2 } \\mathrm { d } z } \\\\ { \\boldsymbol { \\stackrel { \\cdot } { = } } } & { \\underset { 0 \\leq i } { \\iint } \\int _ { - \\infty } ^ { \\infty } \\nabla \\theta \\left( F _ { Q _ { \\theta } } ( x ) - F _ { P } ( x ) \\right) ^ { 2 } \\mathrm { d } z } \\\\ { \\boldsymbol { \\stackrel { \\cdot } { = } } } & { \\underset { 0 \\leq i } { \\iint } ( F _ { Q _ { \\theta } } ( x ) - F _ { P } ( x ) ) \\nabla _ { \\theta } F _ { Q _ { \\theta } } ( x ) \\mathrm { d } x } \\\\ { \\boldsymbol { \\stackrel { \\cdot } { = } } } & { \\underset { 0 \\leq i } { \\iint } \\int _ { - \\infty } ^ { \\infty } 2 \\left( F _ { Q _ { \\theta } } ( x ) - \\mathbb { E } _ { \\mathbf { x } _ { m } } F _ { \\hat { \\mu } _ { \\infty } } ( x ) \\right) \\nabla _ { \\theta } F _ { P } ( y _ { \\infty } ( x ) \\mathrm { d } x ) } \\\\ { \\boldsymbol { \\stackrel { \\cdot } { = } } } & { \\underset { 0 \\leq i } { \\iint } \\int _ { - \\infty } ^ { \\infty } \\left( F _ { Q _ { \\theta } } ( x ) - F _ { \\hat { \\mu } _ { \\infty } } ( x ) \\right) \\nabla _ { \\theta } F _ { Q _ { \\theta } } ( x ) \\mathrm { d } x } \\\\ { \\boldsymbol { \\stackrel { \\cdot } { = } } } & { \\underset { 0 \\leq i } { \\iint } \\sum _ { \\mathbf { R } \\setminus \\mathbf { x } _ { m } } \\left( F _ { Q _ { \\theta } } ( x ) - F _ { \\hat { \\mu } _ { \\infty } } ( x ) \\right) \\nabla _ { \\theta } F _ { Q _ { \\theta } } ( x ) \\mathrm { d } x } \\\\ { \\boldsymbol { \\stackrel { \\cdot } { = } } } & { \\underset { 0 \\leq i } { \\iint } \\sum _ { \\mathbf { R } \\setminus \\mathbf { x } _ { m } } \\int _ { - \\infty } ^ { \\infty } \\left( F _ { Q _ { \\theta } } ( x ) - F _ { \\hat { \\mu } _ { \\infty } } ( x ) \\right) \\nabla _ { \\theta } F _ { Q _ { \\theta } } ( x ) \\mathrm { d } x } \\\\ { \\boldsymbol { \\stackrel { \\cdot } { = } } } & \\underset { 0 \\leq i } { \\iint } \\int _ { \\mathbf { R } \\setminus \\mathbf { x } _ { m } } f \\end{array}\n$$",
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+ "type": "text",
+ "text": "where (a) follows from the hypothesis (7) (the convergence of the squares follows from the convergence of the ordinary values), (b) follows from Lemma 1 and (c) follows from Fubini’s theorem, again invoking (7). ",
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+ "text": "Finally, we prove that of all the $l _ { p } ^ { p }$ distances $1 \\le p \\le \\infty$ ) only the Cramér distance, $l _ { 2 } ^ { 2 }$ , has the (U) property. ",
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+ "text": "Without loss of generality, let us suppose $P$ is not a Dirac, and further suppose that for any $\\mathbf { X } _ { m } \\sim { \\cal P }$ , $F _ { Q _ { \\theta } } ( x ) \\geq F _ { \\hat { P } _ { m } } \\bar { ( x ) }$ everywhere. For example, when $Q _ { \\theta }$ has bounded support we can take $P$ to be a sufficiently translated version of $Q _ { \\theta }$ , such that the two distributions’ supports do not overlap. ",
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+ "text": "We have already established that the 1-Wasserstein does not have the (U) property, and is equivalent to $l _ { p } ^ { p }$ for $p = 1$ . We will thus assume that $p > 1$ , and also that $p < \\infty$ , the latter being recovered through standard limit arguments. Begin with the gradient for $l _ { p } ^ { p } ( P , Q _ { \\theta } )$ , ",
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+ "text": "$$\n\\begin{array} { r l } { { \\nabla _ { \\theta } l _ { p } ^ { p } ( P , Q _ { \\theta } ) = \\nabla _ { \\theta } \\int _ { - \\infty } ^ { \\infty } | F _ { Q _ { \\theta } } ( x ) - F _ { P } ( x ) | ^ { p } \\mathrm { d } x } } \\\\ & { \\stackrel { ( a ) } { = } p \\int _ { - \\infty } ^ { \\infty } \\big ( F _ { Q _ { \\theta } } ( x ) - F _ { P } ( x ) \\big ) ^ { p - 1 } \\nabla _ { \\theta } F _ { Q _ { \\theta } } ( x ) \\mathrm { d } x } \\\\ & { = p \\int _ { - \\infty } ^ { \\infty } \\phi _ { p } ( F _ { Q _ { \\theta } } ( x ) - F _ { P } ( x ) ) \\nabla _ { \\theta } F _ { Q _ { \\theta } } ( x ) \\mathrm { d } x } \\\\ & { = p \\int _ { - \\infty } ^ { \\infty } \\phi _ { p } \\Big ( \\mathbb { E } _ { \\mathbf { X } _ { m } } ( F _ { Q _ { \\theta } } ( x ) - F _ { \\hat { P } _ { m } } ( x ) ) \\Big ) \\nabla _ { \\theta } F _ { Q _ { \\theta } } ( x ) \\mathrm { d } x , } \\end{array}\n$$",
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+ "type": "text",
+ "text": "for $\\phi _ { p } ( z ) = z ^ { p - 1 }$ ; in (a) we used the same argument as in Theorem 3. ",
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+ "text": "Now, $\\phi _ { p }$ is convex on $[ 0 , \\infty )$ when $p \\geq 2$ and concave on the same interval when $1 < p < 2$ . From Jensen’s inequality we know that for a convex (concave) function $\\phi$ and a random variable $Z$ , $\\mathbb { E } \\phi ( Z )$ is greater than (less than) or equal to $\\phi ( \\mathbb { E } Z )$ , with equality if and only if $\\phi$ is linear or $Z$ is deterministic. By our first assumption we have ruled out the latter. By our second assumption $F _ { Q _ { \\theta } } ( x ) \\geq F _ { \\hat { P } _ { m } } ( x )$ , we can apply Jensen’s inequality at every $x$ to deduce that ",
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+ "text": "$$\n\\begin{array} { r l } & { \\mathbb { E } _ { \\mathbf { X } _ { m } } \\left[ \\nabla _ { \\theta } l _ { p } ^ { p } ( \\hat { P } _ { m } , Q _ { \\theta } ) \\right] < \\nabla _ { \\theta } l _ { p } ^ { p } ( P , Q _ { \\theta } ) , \\quad \\mathrm { i f ~ } 1 < p < 2 , } \\\\ & { \\mathbb { E } _ { \\mathbf { X } _ { m } } \\left[ \\nabla _ { \\theta } l _ { p } ^ { p } ( \\hat { P } _ { m } , Q _ { \\theta } ) \\right] > \\nabla _ { \\theta } l _ { p } ^ { p } ( P , Q _ { \\theta } ) , \\quad \\mathrm { i f ~ } p > 2 , } \\\\ & { \\mathbb { E } _ { \\mathbf { X } _ { m } } \\left[ \\nabla _ { \\theta } l _ { p } ^ { p } ( \\hat { P } _ { m } , Q _ { \\theta } ) \\right] = \\nabla _ { \\theta } l _ { p } ^ { p } ( P , Q _ { \\theta } ) , \\quad \\mathrm { i f ~ } p = 2 . } \\end{array}\n$$",
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+ "text": "We conclude that of the $l _ { p } ^ { p }$ distances, only the Cramér distance has unbiased sample gradients. ",
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+ "text": "Proposition 3. The energy distance $\\mathcal { E } ( P , Q )$ has properties (I), (S), and $( U )$ . ",
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+ "text": "Proof. As before, write $\\mathcal { E } ( X , Y ) : = \\mathcal { E } ( P , Q )$ . Recall that ",
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+ "text": "$$\n\\mathcal { E } ( X , Y ) = 2 \\mathbb { E } \\left\\| X - Y \\right\\| _ { 2 } - \\mathbb { E } \\left\\| X - X ^ { \\prime } \\right\\| _ { 2 } - \\mathbb { E } \\left\\| Y - Y ^ { \\prime } \\right\\| _ { 2 } .\n$$",
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+ "text": "Consider a random variable $A$ independent of $X$ and $Y$ . First, we want to prove property (I): ",
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+ "text": "$$\n{ \\mathcal { E } } ( A + X , A + Y ) \\leq { \\mathcal { E } } ( X , Y ) .\n$$",
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+ "text": "We will use Proposition 2 from Székely & Rizzo (2013) to express the energy distance in terms of characteristic functions $\\phi _ { X } , \\phi _ { Y }$ of $d$ -dimensional random variables $X$ and $Y$ : ",
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+ "text": "$$\n{ \\mathcal { E } } ( X , Y ) = { \\frac { 1 } { c _ { d } } } \\int _ { R ^ { d } } { \\frac { | \\phi _ { X } ( t ) - \\phi _ { Y } ( t ) | ^ { 2 } } { | t | ^ { d + 1 } } } d t\n$$",
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+ "text": "where ",
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+ "text": "$$\nc _ { d } = \\frac { \\pi ^ { ( d + 1 ) / 2 } } { \\Gamma ( \\frac { d + 1 } { 2 } ) } .\n$$",
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+ "text": "The proof then uses properties of characteristic functions $( | \\phi _ { A } ( t ) | \\leq 1$ and $\\phi _ { A + X } ( t ) = \\phi _ { A } ( t ) \\phi _ { X } ( t )$ for independent variables $A$ and $X$ ) to show: ",
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+ "text": "$$\n\\begin{array} { l } { \\displaystyle \\mathcal { E } ( A + X , A + Y ) = \\frac { 1 } { c _ { d } } \\int _ { R ^ { d } } \\frac { | \\phi _ { A + X } ( t ) - \\phi _ { A + Y } ( t ) | ^ { 2 } } { | t | ^ { d + 1 } } d t } \\\\ { \\displaystyle \\qquad = \\frac { 1 } { c _ { d } } \\int _ { R ^ { d } } \\frac { | \\phi _ { A } ( t ) \\phi _ { X } ( t ) - \\phi _ { A } ( t ) \\phi _ { Y } ( t ) | ^ { 2 } } { | t | ^ { d + 1 } } d t } \\\\ { \\displaystyle \\qquad = \\frac { 1 } { c _ { d } } \\int _ { R ^ { d } } \\frac { | \\phi _ { X } ( t ) - \\phi _ { Y } ( t ) | ^ { 2 } } { | t | ^ { d + 1 } } | \\phi _ { A } ( t ) | ^ { 2 } d t } \\\\ { \\displaystyle \\qquad \\leq \\frac { 1 } { c _ { d } } \\int _ { R ^ { d } } \\frac { | \\phi _ { X } ( t ) - \\phi _ { Y } ( t ) | ^ { 2 } } { | t | ^ { d + 1 } } d t } \\\\ { \\displaystyle \\qquad = \\mathcal { E } ( X , Y ) . } \\end{array}\n$$",
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+ "text": "This proves (I). Next, consider a real value $c > 0$ . We have ",
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+ "text": "$$\n\\begin{array} { r l } & { \\mathcal { E } ( c X , c Y ) = 2 \\mathbb { E } \\left\\| c X - c Y \\right\\| _ { 2 } - \\mathbb { E } \\left\\| c X - c X ^ { \\prime } \\right\\| _ { 2 } - \\mathbb { E } \\left\\| c Y - c Y ^ { \\prime } \\right\\| _ { 2 } } \\\\ & { \\qquad = 2 c \\mathbb { E } \\left\\| X - Y \\right\\| _ { 2 } - c \\mathbb { E } \\left\\| X - X ^ { \\prime } \\right\\| _ { 2 } - c \\mathbb { E } \\left\\| Y - Y ^ { \\prime } \\right\\| _ { 2 } } \\\\ & { \\qquad = c \\mathcal { E } ( X , Y ) . } \\end{array}\n$$",
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+ "text": "This proves (S). Finally, suppose that $Y$ is distributed according to $Q _ { \\theta }$ parametrized by $\\theta$ . Let $\\mathbf { X } _ { m } = X _ { 1 } , \\ldots , X _ { m }$ be drawn from $P$ , and let $\\begin{array} { r } { \\hat { P } _ { m } : = \\frac { 1 } { m } \\sum _ { i = 1 } ^ { m } \\delta _ { X _ { i } } } \\end{array}$ . Let $\\hat { X }$ be the random variable distributed according to $\\hat { P } _ { m }$ , and ${ \\hat { X } } ^ { \\prime }$ an independent copy of $\\hat { X }$ . Then ",
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+ "text": "$$\n\\mathcal { E } ( \\hat { P } _ { m } , Q _ { \\theta } ) = \\mathcal { E } ( \\hat { X } , Y ) = 2 \\mathbb { E } \\left\\| \\hat { X } - Y \\right\\| _ { 2 } - \\mathbb { E } \\left\\| \\hat { X } - \\hat { X } ^ { \\prime } \\right\\| _ { 2 } - \\mathbb { E } \\left\\| Y - Y ^ { \\prime } \\right\\| _ { 2 } .\n$$",
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+ "text": "The gradient of the true loss w.r.t. $\\theta$ is ",
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+ "text": "$$\n\\nabla _ { \\theta } \\mathcal { E } ( X , Y ) = 2 \\nabla _ { \\theta } \\mathbb { E } \\left\\| X - Y \\right\\| _ { 2 } - \\nabla _ { \\theta } \\mathbb { E } \\left\\| Y - Y ^ { \\prime } \\right\\| _ { 2 } .\n$$",
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+ "text": "Now, taking the gradient of the sample loss w.r.t. $\\theta$ , ",
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+ "text": "$$\n\\nabla _ { \\theta } \\mathcal { E } ( \\hat { X } , Y ) = 2 \\nabla _ { \\theta } \\mathbb { E } \\left. \\hat { X } - Y \\right. _ { 2 } - \\nabla _ { \\theta } \\mathbb { E } \\left. Y - Y ^ { \\prime } \\right. _ { 2 } .\n$$",
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+ "text": "Since the second terms of the gradients match, all we need to show is that the first terms are equal, in expectation. Assuming that $\\nabla _ { \\theta }$ and the expectation over $\\mathbf { X } _ { m }$ commute, we write ",
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+ "text": "$$\n\\begin{array} { r l } & { \\underset { { \\mathbf { X } } _ { m } } { \\mathbb { E } } \\nabla _ { \\theta } \\mathbb { E } \\| \\hat { X } - Y \\| _ { 2 } = \\nabla _ { \\theta } \\underset { { \\mathbf { X } } _ { m } } { \\mathbb { E } } \\mathbb { E } \\| \\hat { X } - Y \\| _ { 2 } } \\\\ & { \\qquad = \\nabla _ { \\theta } \\underset { { \\mathbf { X } } _ { m } } { \\mathbb { E } } \\underset { { \\mathbf { X } } \\sim \\hat { P } _ { m } } { \\mathbb { E } } \\| x - Y \\| _ { 2 } , } \\end{array}\n$$",
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+ "type": "text",
+ "text": "by independence of $X$ and $Y$ . But now we know that the expected empirical distribution is $P$ , that is ",
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+ "text": "$$\n\\begin{array} { r } { \\underset { \\mathbf { X } _ { m } } { \\mathbb { E } } \\underset { x \\sim \\hat { P } _ { m } } { \\mathbb { E } } \\left\\| x - Y \\right\\| _ { 2 } = \\underset { x \\sim P } { \\mathbb { E } } \\left\\| x - Y \\right\\| _ { 2 } = \\mathbb { E } \\left\\| X - Y \\right\\| _ { 2 } . } \\end{array}\n$$",
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+ "text": "It follows that the first terms of (8) and (9) are also equal, in expectation w.r.t. $\\mathbf { X } _ { m }$ . Hence we conclude that the energy distance has property (U), that is ",
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+ "text": "$$\n\\underset { { \\substack { \\mathbf { X } _ { m } \\sim P } } } { \\mathbb { E } } \\nabla _ { \\theta } \\mathcal { E } ( \\hat { P } _ { m } , Q _ { \\theta } ) = \\nabla _ { \\theta } \\mathcal { E } ( P , Q _ { \\theta } ) .\n$$",
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+ "type": "text",
+ "text": "B COMPARISON WITH THE WASSERSTEIN DISTANCE ",
+ "text_level": 1,
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+ "text": "Figure 2 (left) provides learning curves for the toy experiment described in Section 4.2. ",
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+ "img_path": "images/bb9eb8fc8453f3de2d2d611d050cc847c8987ed509f0f247ded575cc13ebd29f.jpg",
+ "image_caption": [
+ "Figure 6: Wasserstein while training to minimize different loss functions (Wasserstein, KL, Cramér). Averaged over 10 random initializations. Error-bands indicate one standard deviation. Note the different y-axes. "
+ ],
+ "image_footnote": [],
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+ "img_path": "images/91a73c66b75972f227e6f2e628ecc59ee0f795684aee37530bca503bdaa84b93.jpg",
+ "image_caption": [
+ "Figure 7: Ordinal regression on the year prediction MSD dataset. Each loss function trained with various minibatch sizes. Training progress shown in terms of: Left. RMSE, Middle. Wasserstein distance, Right. Negative log-likelihood. "
+ ],
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+ "type": "text",
+ "text": "B.1 ORDINAL REGRESSION ",
+ "text_level": 1,
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+ "text": "We compare the different losses on an ordinal regression task using the Year Prediction MSD dataset from (Lichman, 2013). The task is to predict the year of a song (taking on values from 1922 to 2011), from 90-dimensional feature representation of the song.4 Previous work has used this dataset for benchmarking regression performance (Hernández-Lobato & Adams, 2015), treating the target as a continuous value. Following Hernández-Lobato & Adams (2015), we train a network with a single hidden layer with 100 units and ReLU non-linearity, using SGD with 40 passes through the training data, using the standard train-test split for this dataset (Lichman, 2013). Unlike (Hernández-Lobato & Adams, 2015), the network outputs a probability distribution over the years (90 possible years from 1922-2011). ",
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+ "text": "We train models using either the 1-Wasserstein loss, the Cramér loss, or the KL loss, the latter of which reduces the ordinal regression problem to a classification problem. In all cases, we compare performance for three different minibatch sizes, i.e. the number of input-target pairs per gradient step. Note that the minibatch size only affects the gradient estimation, but has otherwise no direct relation to the number of samples $m$ previously discussed, since each sample corresponds to a different input vector. We report results as a function of number of passes over the training data so that our results are comparable with previous work, but note that smaller batch sizes get more updates. ",
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+ "type": "text",
+ "text": "The results are shown in Figure 2. Training using the Cramér loss results in the lowest root mean squared error (RMSE) and the final RMSE value of 8.89 is comparable to regression (HernándezLobato & Adams, 2015) which directly optimizes for MSE. We further observe that minimizing the Wasserstein loss trains relatively slowly and leads to significantly higher KL loss. Interestingly, larger minibatch sizes do seem to improve the performance of the Wasserstein-based method somewhat, suggesting that there might be some beneficial bias reduction from combining similar inputs. By contrast, using with the Cramér loss trains significantly faster and is more robust to choice of minibatch size. ",
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+ "type": "table",
+ "img_path": "images/9cc18472784234cd5d8af3e8ecd816130433e1649d716b6646c9faf5e2981349.jpg",
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+ "table_footnote": [],
+ "table_body": "| Algorithm1:Cramér GANLosses. |
| Parameter. Gradient penalty coefficient 入. Sample xr ~ P,𝑥g,xg~ Q,∈~ Uniform(0,1). Interpolate real and generated samples: 𝑥=∈xr+(1-∈)xg Sample generator loss (12): |
| Lg= |/h(xr)-h(xg)ll2+|/h(xr)-h(𝑥g)ll2 |
| -|h(xg)-h(xg)ll2 Sample surrogate generator loss (13) and critic loss: |
| Ls(u,v)= |h(xr)-h(u)ll2-|/h(xr)ll2 -/h(u)-h(ν)ll2+ h(u)ll2 Ls=1[Ls(xg,xg)+Ls(xg,xg)] |
",
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| Parameter. Gradient penalty coefficient 入. Sample xr ~ P,𝑥g,xg~ Q,∈~ Uniform(0,1). Interpolate real and generated samples: 𝑥=∈xr+(1-∈)xg Sample generator loss (12): |
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