On Learning Domain-Invariant Representations for Transfer Learning with Multiple Sources

Let $\mathcal{X}$ be a data space on which we endow a data distribution $\mathbb{P}$ with a corresponding density function $p(x)$. We consider the multi-class classification problem with the label set $\mathcal{Y}=\left[C\right]$, where $C$ is the number of classes and $\left[C\right]:=\{1,\ldots,C\}$. Denote $\mathcal{Y}_{\Delta}:=\left\{\alpha\in\mathbb{R}^{C}:\left\|\alpha\right\|_{1}=1\,\wedge\,\alpha\geq\mathbf{0}\right\}$ as the $C-$simplex label space, let $f:\mathcal{X}\mapsto\mathcal{Y}_{\Delta}$ be a probabilistic labeling function returning a $C$-tuple $f\left(x\right)=\left[f\left(x,i\right)\right]_{i=1}^{C}$, whose element $f\left(x,i\right)=p\left(y=i\mid x\right)$ is the probability to assign a data sample $x\sim\mathbb{P}$ to the class $i$ (i.e., $i\in\left\{1,...,C\right\}$). Moreover, a domain is denoted compactly as pair of data distribution and labeling function $\mathbb{D}:=\left(\mathbb{P},f\right)$. We note that given a data sample $x\sim\mathbb{P}$, its categorical label $y\in\mathcal{Y}$ is sampled as $y\sim Cat\left(f\left(x\right)\right)$ which a categorical distribution over $f\left(x\right)\in\mathcal{Y}_{\Delta}$.

Let $l:\mathcal{Y}_{\Delta}\times\mathcal{Y}\mapsto\mathbb{R}$ be a loss function, where $l\left(\hat{f}\left(x\right),y\right)$ with $\hat{f}\left(x\right)\in\mathcal{Y}_{\Delta}$ and $y\in\mathcal{Y}$ specifies the loss (e.g., cross-entropy, Hinge, L1, or L2 loss) to assign a data sample $x$ to the class $y$ by the hypothesis $\hat{f}$. Moreover, given a prediction probability $\hat{f}\left(x\right)$ w.r.t. the ground-truth prediction $f\left(x\right)$, we define the loss $\ell\left(\hat{f}\left(x\right),f\left(x\right)\right)=\mathbb{E}_{y\sim f\left(x\right)}\left[l\left(\hat{f}\left(x\right),y\right)\right]=\sum_{y=1}^{C}l\left(\hat{f}\left(x\right),y\right)f\left(x,y\right)$. This means $\ell\left(\cdot,\cdot\right)$ is convex w.r.t. the second argument. We further define the general loss caused by using a classifier $\hat{f}:\mathcal{X}\mapsto\mathcal{Y}_{\Delta}$ to predict $\mathbb{D}\equiv\left(\mathbb{P},f\right)$ as

We inspect the multiple source setting in which we are given multiple source domains $\{\mathbb{D}^{S,i}\}_{i=1}^{K}$ over the common data space $\mathcal{X}$, each of which consists of data distribution and its own labeling function $\mathbb{D}^{S,i}:=\left(\mathbb{P}^{S,i},f^{S,i}\right)$. Based on these source domains, we need to work out a learner or classifier that requires to evaluate on a target domain $\mathbb{D}^{T}:=\left(\mathbb{P}^{T},f^{T}\right)$. Depending on the knownness or unknownness of a target domain during training, we experience either multiple source DA (MSDA) [32, 30, 44, 19] or domain generalization (DG) [25, 29, 28, 35] setting.

One typical approach in MSDA and DG is to combine the source domains together [14, 29, 28, 35, 1, 11, 48] to learn DI representations with hope to generalize well to a target domain. When combining the source domains, we obtain a mixture of multiple source distributions denoted as $\mathbb{D}^{\pi}=\sum_{i=1}^{K}\pi_{i}\mathbb{D}^{S,i}$, where the mixing coefficients $\pi=\left[\pi_{i}\right]_{i=1}^{K}$ can be conveniently set to $\pi_{i}=\frac{N_{i}}{\sum_{j=1}^{K}N_{j}}$ with $N_{i}$ being the training size of the $i-$th source domain.

In term of representation learning, input is mapped into a latent space $\mathcal{Z}$ by a feature map $g:\mathcal{X}\mapsto\mathcal{Z}$ and then a classifier $\hat{h}:\mathcal{Z}\mapsto\mathcal{Y}_{\Delta}$ is trained based on the representations $g\left(\mathcal{X}\right)$. Let $f:\mathcal{X}\mapsto\mathcal{Y}_{\Delta}$ be the original labeling function. To facilitate the theory developed for latent space, we introduce representation distribution being the pushed-forward distribution $\mathbb{P}_{g}:=g_{\#}\mathbb{P}$, and the labeling function $h:\mathcal{Z}\mapsto\mathcal{Y}_{\Delta}$ induced by $g$ as $h(z)=\frac{\int_{g^{-1}\left(z\right)}f(x)p(x)dx}{\int_{g^{-1}\left(z\right)}p(x)dx}$ [22]. Going back to our multiple source setting, the source mixture becomes $\mathbb{D}_{g}^{\pi}=\sum_{i}\pi_{i}\mathbb{D}_{g}^{S,i}$, where each source domain is $\mathbb{D}_{g}^{S,i}=\left(\mathbb{P}_{g}^{S,i},h^{S,i}\right)$, and the target domain is $\mathbb{D}_{g}^{T}=\left(\mathbb{P}_{g}^{T},h^{T}\right)$.

Finally, in our theory development, we use Hellinger divergence between two distributions defined as $\begin{aligned} D_{1/2}\left(\mathbb{P}^{1},\mathbb{P}^{2}\right)=2\int\left(\sqrt{p^{1}(x)}-\sqrt{p^{2}(x)}\right)^{2}dx\end{aligned}$, whose squared $d_{1/2}=\sqrt{D_{1/2}}$ is a proper metric.

Hinted by the theoretical bounds developed in [3], DI representations learning, in which feature extractor $g$ maps source and target data distributions to a common distribution on the latent space, is well-grounded for the DA setting. However, this task becomes significantly challenging for the MSDA and DG settings due to multiple source domains and potential unknownness of target domain. Similar to the case of DA, it is desirable to develop theoretical bounds that directly motivate definitions of DI representations for the MSDA and DG settings, as being done in the next theorem.

The bound in Equation 1 implies that the target loss in the input or latent space depends on three terms: (i) representation discrepancy: $d_{1/2}\left(\mathbb{P}_{g}^{T},\mathbb{P}_{g}^{\pi}\right)$, (ii) the label shift: $\max_{i\in[K]}\mathbb{E}_{\mathbb{P}^{S,i}}\left[\|\Delta p^{i}(y|x)\|_{1}\right]$, and (iii) the general source loss: $\sum_{i=1}^{K}\pi_{i}\mathcal{L}\left(\hat{f},\mathbb{D}^{S,i}\right)$. To minimize the target loss in the left side, we need to minimize the three aforementioned terms. First, the label shift term is a natural characteristics of domains, hence almost impossible to tackle. Secondly, the representation discrepancy term can be explicitly tackled for the MSDA setting, while almost impossible for the DG setting. Finally, the general source loss term is convenient to tackle, where its minimization results in a feature extractor $g$ and a classifier $\hat{h}$.

Contrary to previous works in DA and MSDA [3, 32, 57, 7] that consider both losses and data discrepancy on data space, our bound connects losses on data space to discrepancy on representation space. Therefore, our theory provides a natural way to analyse representation learning, especially feature alignment in deep learning practice. Note that although DANN [14] explains their feature alignment method using theory developed by Ben-david et al. [3], it is not rigorous. In particular, while application of the theory to representation space yield a representation discrepancy term, the loss terms are also on that feature space, and hence minimizing these losses is not the learning goal. Finally, our setting is much more general, which extends to multilabel, stochastic labeling setting, and any bounded loss function.

From the upper bound, we turn to first type of DI representations for the MSDA and DG settings. Here, the objective of $g$ is to map samples onto representation space in a way that the common classifier $\hat{h}$ can effectively and correctly classifies them, regardless of which domain the data comes from.

The definition of general DI representations for the DG setting in Definition 2 is transparent in light of Theorem 1, wherein we aim to find $g$ and $\hat{h}$ to minimize the general source loss $\sum_{i=1}^{K}\pi_{i}\mathcal{L}\left(\hat{f},\mathbb{D}^{S,i}\right)$ due to the unknownness of $\mathbb{P}^{T}$. Meanwhile, regarding the general DI representations for the MSDA setting, we aim to find $g^{*}$ satisfying $\mathbb{P}_{g^{*}}^{T}=\mathbb{P}_{g^{*}}^{\pi}$ and $\hat{h}$ to minimize the general source loss $\sum_{i=1}^{K}\pi_{i}\mathcal{L}\left(\hat{f},\mathbb{D}^{S,i}\right)$. Practically, to find general representations for MSDA, we solve

where $\lambda>0$ is a trade-off parameter and $D$ can be any divergence (e.g., Jensen Shannon (JS) divergence [16], $f$-divergence [40], MMD distance [28], or WS distance [44]).

In addition, the general DI representations have been exploited in some works [60, 27] from the practical perspective and previously discussed in [2] from the theoretical perspective. Despite the similar definition to our work, general DI representation in [2] is not motivated from minimization of a target loss bound.

As pointed out in the next section, learning general DI representations increases the span of latent representations $g\left(x\right)$ on the latent space which might help to reduce $d_{1/2}\left(\mathbb{P}_{g}^{T},\mathbb{P}_{g}^{\pi}\right)$ for a general target domain. On the other hand, many other works [28, 29, 15, 52, 11] have also explored the possibility of enhancing generalization by finding common representation among source distributions. The following theorem motivates the latter, which is the second kind of DI representations.

Evidently, Theorem 3 suggests another kind of representation learning, where source representation distributions are aligned in order to lower the target loss’s bound as concretely defined as follows.

In the previous section, two kinds of DI representations are introduced, where each of them originates from minimization of different terms in the upper bound of target loss. In what follows, we examine and discuss their benefits and drawbacks. We start with the novel development of hypothesis-aware divergence for multiple distributions , which is necessary for our theory developed later.

Similar to the theoretical finding in Zhao et al. [56] developed for DA, we theoretically find that compression does come with a cost for MSDA and DG. We investigate the representation trade-off, typically how compressed DI representation affects classification loss. Specifically, we consider a data processing chain $\mathcal{X}\stackrel{{\scriptstyle g}}{{\longmapsto}}\mathcal{Z}\stackrel{{\scriptstyle\hat{h}}}{{\longmapsto}}\mathcal{Y}_{\Delta}$, where $\mathcal{X}$ is the common data space, $\mathcal{Z}$ is the latent space induced by a feature extractor $g$, and $\hat{h}$ is a hypothesis on top of the latent space. We define $\mathbb{P}_{\mathcal{Y}}^{\pi}$ and $\mathbb{P}_{\mathcal{Y}}^{T}$ as two distribution over $\mathcal{Y}$ in which to draw $y\sim\mathbb{P}_{\mathcal{Y}}^{\pi}$, we sample $k\sim Cat\left(\pi\right)$, $x\sim\mathbb{P}^{S,k}$, and $y\sim f^{S,k}\left(x\right)$, while similar to draw $y\sim\mathbb{P}_{\mathcal{Y}}^{T}$. Our theoretical bounds developed regarding the trade-off of learning DI representations are relevant to $d_{1/2}\left(\mathbb{P}_{\mathcal{Y}}^{\pi},\mathbb{P}_{\mathcal{Y}}^{T}\right)$.

Remark. Compared to the trade-off bound in the work of Zhao et al. [56], our context is more general, concerning MSDA and DG problems with multiple source domains and multi-class probabilistic labeling functions, rather than single source DA with binary-class and deterministic setting. Moreover, the Hellinger distance is more universal, in the sense that it does not depend on the choice of classifier family $\mathcal{H}$ and loss function $\ell$ as in the case of $\mathcal{H}$-divergence in [56].

We base on the first inequality of Theorem 8 to analyze the trade-off of learning general DI representations. The first term on the left hand side is the source mixture’s loss, which is controllable and tends to be small when enforcing learning general DI representations. With that in mind, if two label marginal distributions $\mathbb{P}_{\mathcal{Y}}^{\pi}$ and $\mathbb{P}_{\mathcal{Y}}^{T}$ are distant (i.e., $d_{1/2}\left(\mathbb{P}_{\mathcal{Y}}^{\pi},\mathbb{P}_{\mathcal{Y}}^{T}\right)$ is high), the sum $d_{1/2}\left(\mathbb{P}_{g}^{T},\mathbb{P}_{g}^{\pi}\right)+\mathcal{L}\left(\hat{h}\circ g,f^{T},\mathbb{P}^{T}\right)^{1/2}$ tends to be high. This leads to 2 possibilities. The first scenario is when the representation discrepancy $d_{1/2}\left(\mathbb{P}_{g}^{T},\mathbb{P}_{g}^{\pi}\right)$ has small value, e.g., it is minimized in MSDA setting, or it happens to be small by pure chance in DG setting. In this case, the lower bound of target loss $\mathcal{L}\left(\hat{h}\circ g,f^{T},\mathbb{P}^{T}\right)$ is high, possibly hurting model’s generalization ability. On the other hand, if the discrepancy $d_{1/2}\left(\mathbb{P}_{g}^{T},\mathbb{P}_{g}^{\pi}\right)$ is large for some reasons, the lower bound of target loss will be small, but its upper-bound is higher, as indicated Theorem 1.

Based on the second inequality of Theorem 8, we observe that if two label marginal distributions $\mathbb{P}_{\mathcal{Y}}^{\pi}$ and $\mathbb{P}_{\mathcal{Y}}^{T}$ are distant while enforcing learning compressed DI representations (i.e., both source loss and source-source feature discrepancy $\left[\sum_{i=1}^{K}\pi_{i}\mathcal{L}\left(\hat{h}\circ g,f^{S,i},\mathbb{P}^{S,i}\right)\right]^{1/2}+\sum_{i=1}^{K}\sum_{j=1}^{K}\frac{\sqrt{\pi_{j}}}{K}d_{1/2}\left(\mathbb{P}_{g}^{S,i},\mathbb{P}_{g}^{S,j}\right)$ are low), the sum $\sum_{i=1}^{K}\sum_{j=1}^{K}\frac{\sqrt{\pi_{j}}}{K}d_{1/2}\left(\mathbb{P}_{g}^{T},\mathbb{P}_{g}^{S,i}\right)+\mathcal{L}\left(h\circ g,f^{T},\mathbb{P}^{T}\right)^{1/2}$ is high. For the MSDA setting, the discrepancy $\sum_{i=1}^{K}\sum_{j=1}^{K}\frac{\sqrt{\pi}_{j}}{K}d_{1/2}\left(\mathbb{P}_{g}^{T},\mathbb{P}_{g}^{S,i}\right)$ is trained to get smaller, meaning that the lower bound of target loss $\mathcal{L}\left(\hat{h}\circ g,f^{T},\mathbb{P}^{T}\right)$ is high, hurting the target performance. Similarly, for the DG setting, if the trained feature extractor $g$ occasionally reduces $\sum_{i=1}^{K}\sum_{j=1}^{K}\frac{\sqrt{\pi}_{j}}{K}d_{1/2}\left(\mathbb{P}_{g}^{T},\mathbb{P}_{g}^{S,i}\right)$ for some unseen target domain, it certainly increases the target loss $\mathcal{L}\left(h\circ g,f^{T},\mathbb{P}^{T}\right)$. In contrast, if for some target domains, the discrepancy $\sum_{i=1}^{K}\sum_{j=1}^{K}\frac{\sqrt{\pi}_{j}}{K}d_{1/2}\left(\mathbb{P}_{g}^{T},\mathbb{P}_{g}^{S,i}\right)$ is high by some reasons, by linking to the upper bound in Theorem 3, the target general loss has a high upper-bound, hence is possibly high.

This trade-off between representation discrepancy and target loss suggests a sweet spot for just-right feature alignment. In that case, the target loss is most likely to be small.

We consider target domain which is similar to source domain in this experiment. To govern the trade-off between two kinds of DI representations, we gradually increase the source compression strength $\lambda$ with noting that $\lambda=0$ means only focusing on learning general DI representations. According to our theory and as shown in Figure 1, learning only general DI representations ($\lambda=0$) maximizes the cross-domain class divergence by encouraging the classes of source domains more separate arbitrarily, while by increasing $\lambda$, we enforce compressing the latent representations of source domains together. Therefore, for an appropriate $\lambda$, the class separation from general DI representations and the source compression from compressed DI representations (i.e., just-right compressed DI representations), are balanced as in the case (b) of Figure 1, while for overly high $\lambda$, source compression from compressed DI representations dominates and compromises the class separation from general DI representations (i.e., overly-compressed DI representations).

Figure 2a shows the source validation and target accuracies when increasing $\lambda$ (i.e., encouraging the source compression). It can be observed that both source validation accuracy and target accuracy have the same pattern: increasing when setting appropriate $\lambda$ for just-right compressed DI representations and compromising when setting overly high values $\lambda$ for overly-compressed DI representations. Figure 2b shows in detail the variation of the source validation accuracy for each specific $\lambda$. In practice, we should encourage learning two kinds of DI representations simultaneously by finding an appropriate trade-off to balance them for working out just-right compressed DI representations.

In this experiment, we inspect the generalization capacity of the models trained on the source domains with different compression strengths $\lambda$ to different target domains (i.e., close and far ones). As shown in Figure 3, the target accuracy is lower for the far target domain than for the close one regardless of the source compression strengths $\lambda$. This observation aligns with our upper-bounds developed in Theorems 1 and 3 in which larger representation discrepancy increases the upper-bounds of the general target loss, hence more likely hurting the target performance.

Another interesting observation is that for the far target domain, the target accuracy for $\lambda=0$ peaks. This can be partly explained as the general DI representations help to increase the span of source latent representations for more chance to match a far target domain (cf. Section 2.3.2). Specifically, the source-target discrepancy $d_{1/2}\left(\mathbb{P}_{g}^{T},\mathbb{P}_{g}^{\pi}\right)$ could be small and the upper bound in Theorem 1 is lower. In contrast, for the close target domain, the compressed DI representations help to really improve the target performance, while over-compression degrades the target performance, similar to result on source domains. This is because the target domain is naturally close to the source domains, the source and target latent representations are already mixed up, but by encouraging the compressed DI representations, we aim to learning more elegant representations for improving target performance.