Despite significant advances in neural network architectures and optimization strategies for supervised learning, one of the long-standing challenges has been to effectively generalize classifier models to novel testing scenarios, typically characterized by unknown covariate shiftsHoffman et al. , changes in label distributions, or oblivious corruptions. In this paper, we consider the problem of unsupervised domain adaptation, wherein the goal is to utilize labeled data from a source domain to design a classifier that can generalize to an unlabeled target domain. We are especially interested in the case when no knowledge about the covariate shift is available.
Earlier approaches for unsupervised domain adaption, particularly in visual recognition, were based on countering the effects of distributional shifts by exploiting low-dimensional structures in data Fernando et al. , Gong et al. , Shrivastava et al. , Thopalli et al. . In other words, achieving domain invariance was posed as learning a mapping between simplified data representations, e.g. linear subspaces. Some of the key ideas here include not explicitly inferring the hypothesis that minimizes the generalization error, and reliance on simplifying assumptions on data geometry (e.g. single linear subspace for the entire dataset). Due to these reasons, these methods have fallen behind more recent approaches in terms of performance.
The foundational work of Ben-David et. al. Ben-David et al.  established an upper bound for target-error, on target data , that can be achieved using a hypothesis as the sum of three terms:
where, the first term denotes the error in the source domain , the second term is the discrepancy between the source-target pair (-divergence), and the third term measures the optimal error achievable in both the domains (often assumed to be negligible). Under this context, there are two broad categories of methods – ones that assume there exists a single hypothesis that can perform well in both domains (conservative), and those that do not make that assumption (non-conservative) Shu et al. .
More recent solutions for domain adaptation attempt to infer domain-invariant data representations by minimizing the discrepancy between feature distributions from the two domains. In particular, domain adversarial learning, which seeks to find a common representation where the two domains are indistinguishable, is at the core of several state-of-the-art methods Tzeng et al. , Hoffman et al. , Long et al. , and. V. Lempitsky. Y , GaninY et al. . However, it has recently been shown that domain adversarial training can be ineffective when working with a high-capacity feature extractor Shu et al. . High-capacity networks allow for learning arbitrary transformations that can reduce domain mismatch (in terms of feature distributions), yet might have no bearing on the final classifier performance Shu et al. .
The non-trivial interaction between the terms in (1) has motivated the inclusion of a variety of consistency-enforcing losses into the domain adversarial learning formulation. For example, GaninY et al. , Tzeng et al.  employ both feature and semantic losses for feature-level adaptation, while Liu and Tuzel , Bousmalis et al.  perform pixel-level adaptation via pixel and semantic consistency losses. More recently, Hoffman et al. Hoffman et al.  proposed to enforce cyclical consistency based on all the aforementioned losses, while Shu et al. Shu et al.  introduced a virtual adversarial loss to better regularize domain adversarial learning.
Key insights: The above discussion leads us to our core idea that one must try to blend the representational convenience of simplified data geometries, while not being constrained by analytic solutions for alignment. Analytic solutions for alignment while powerful, can cause error due to geometry mismatch to propagate downstream. We strike a balance between the following factors: a) assume tractable data geometries in source and target domains, which can be analytically leveraged for data alignment, b) synergistically adapt certain parameters derived from the analytic solution to alignment, in a manner that maximizes performance on the primary task of classification. This approach can be seen as inspired by meta-learning Finn et al. , specifically designed for handling interactions between domain alignment and hypothesis inference.
Contributions and findings: In this paper, we leverage the observation that explicit domain alignment behaves more as an auxiliary task, whose fidelity can be carefully adjusted to maximize the quality of the primary task, i.e., performance of the classifier on both source and target domains. This approach can be said to fall under the category of non-conservative adaptation, and hence we include explicit information-invariance losses for the unlabeled target domain, similar to Shu et al. . We make the following major findings:
With a disjoint primary-auxiliary formulation, we find that even a naïve global subspace based alignment Fernando et al.  with a fixed feature extractor, achieves higher or similar performance compared to state-of-the-art approaches on several benchmarks.
Moving from here, we define adaptable subspace alignment as the auxiliary task, which uses gradients from the primary task, to adjust the domain alignment. This is seen to improve performance much more significantly across benchmarks.
In summary, our findings show that by viewing domain alignment as an auxiliary task, we are able to entirely dispense the need for adversarial learning, consistency-enforcing regularizers, and other extensive hyper-parameter choices.
Broader interpretation: Our results find additional corroboration from analogous findings in Liu et al. 
, where meta-learning style optimization is used to automatically construct an auxiliary classification task so as to provide additional pseudo-supervisory guidance to the primary task of building a classifier. Viewed under the lens of meta-learning, our results indicate that, atleast for visual recognition, a single global domain alignment is sufficient, when coupled with an appropriately chosen primary task. While the proposed approach is highly effective in generalizing classifiers under covariate shifts, its effectiveness in other adaptation tasks such as image-to-image translation remains to be studied.
2 Related work
In this section, we briefly review the prior art in unsupervised domain adaptation. Furthermore, we will also discuss about meta auxiliary learning, which is closely related to the proposed approach.
Unsupervised Domain Adaptation: Unsupervised domain adaptation has been an important problem of research in multiple application areas and a wide variety of solutions have been developed. Earlier works such as Saenko et al. , GongB et al. , Pan et al. , Sun et al. , Fernando et al. , Sun and Saenko  focused on adapting the features of source and target domains by minimizing a notion of statistical divergence between them. These works can be analyzed through the work of Ben-David et al. , which provides an upper bound on target error in (1
). Building upon this intuition, successful state-of-the art methods use powerful feature extractors such as convolutional neural networks (CNNs), and aim to jointly minimize source error along with domain divergence error. Adversarial learningGoodfellowI et al. has been the workhorse of these solutions, implemented with different additional regularizers GaninY et al. , Long et al. , Hoffman et al. , Liu et al. , IsolaP et al. .
Subspace-based Alignment: The key idea behind this class of methods is to compute lower dimensional subspaces of source and target, align them and subsequently project the ambient data onto the aligned subspace. A classifier is finally trained on the newly computed lower dimensional source data and evaluated on target data. The most relevant works for our approach are Gong et al. , Gopalan et al. , Fernando et al. , Sun and Saenko . Geodesic-based methods Gopalan et al. , Gong et al.  compute a path along the manifold of subspaces (Grassmannian), and either project the source and target onto points along that path Gopalan et al.  or compute a linear map that projects source samples directly onto the target subspace Gong et al. . Furthermore, works such as Fernando et al. , Sun and Saenko  align the source and target subspaces by finding an affine transformation that decreases the Frobenius norm between them Fernando et al. , or by considering distributional statistics along with subspace basis Sun and Saenko .
Meta Auxiliary Learning: Meta-learning has been a recently successful approach in generalizing knowledge across related tasks Finn et al. . Broadly, meta-learning techniques can be grouped into three categories Finn et al.  – metric-based Koch et al. , Vinyals et al. , model-based Santoro et al. , Munkhdalai and Yu  and optimization-based Finn et al. , Ravi and Larochelle . Auxiliary learning on the other hand essentially focuses on increasing the performance of a primary task through the help of another related auxiliary task(s). This methodology has been applied to areas such as speech recognition Toshniwal et al. 
, depth estimation, semantic segmentationLiebel and Körner 
, and reinforcement learningJaderberg et al. . The work closely related to ours is meta-auxiliary learning Liu et al. , which aims to improve -class image classification performance (primary task) by solving a -class classification problem (auxiliary task). This is done by establishing a functional relationship between the classes. In contrast, we formulate subspace-based domain alignment as the auxiliary to the primary task of achieving a generalizable classifier that works well in both source and target domains.
3 Proposed Approach
In this section, we describe the proposed method for unsupervised domain adaptation. An overview of the approach can be found in Figure 1. Given data from the labeled source and unlabeled target domains, and denoted as and respectively, our algorithm progresses by iteratively updating the primary and auxiliary networks. In the rest of this paper, we use to indicate the latent features for source and target domains from a pre-trained feature extractor, , such as ResNet50 HeK et al. . The primary network updates the classifier, given the source and source-aligned target features, such that the inferred model is effective for both source and target domains. The auxiliary network solves for subspace-based domain alignment, by leveraging the loss from the primary network. The resulting alignment is sub-optimal in terms of the pure alignment cost, but results in an optimal alignment conditioned on the primary classification task.
3.1 Primary Task: Classifier Design
We construct the primary task with the goal of achieving effective class discrimination in both source and target domains. With inputs as source/target images directly, or latent features extracted from a pre-trained feature extractor, we learn the parameters for a classifier network parameterized by . The losses used for the optimization include: (i) standard categorical cross-entropy loss for the labeled source data, (ii) conditional entropy Shu et al.  loss on the softmax predictions for target data, (iii) class-balance loss French et al.  for the unlabeled target domain. Note, the second and third loss terms are used as regularizers to counter the assumption that a single hypothesis might not be effective for both domains, i.e. non-conservative. In its simplest form, this formulation should work if there is no covariate shift between the domains. However, in our setup, in order to account for unknown shifts (if they exist), we formulate an auxiliary task for domain alignment. Formally, let , represent the cross entropy loss on the source, and conditional entropy loss on the target respectively, i.e.
Let French et al. 
denote the class balance loss, implemented as binary cross-entropy loss between the mean of predictions of the network over a mini-batch to that of a uniform probability vector – this loss regularizes network behavior when the data exhibits large class imbalance. The overall loss function is thus defined as.
3.2 Auxiliary Task: Domain Alignment
We posit that a meta-learning style training between a generalizable classification task, and an auxiliary domain alignment task, relaxes the requirements of the alignment step such that even simple alignment strategies can provide sufficient information to improve the classifier. In order to test this idea, we assume a simplified data geometry, in the form of low-dimensional linear subspaces Fernando et al. . Note that, as a generative model for a dataset, a single linear subspace or even a union of linear subspaces is a poor choice on its own. However, when coupled with an appropriate primary task using a sufficiently high capacity classifier, we will show it can be highly effective in domain adaptation.
Formulation: Let us denote the basis vectors for the -dimensional subspaces inferred from source and target domains as and
respectively. The subspaces are inferred using singular value decomposition of the featuresfrom the feature extractor . The alignment between two subspaces can be parameterized as an affine transformation , i.e.,
where, denotes the Frobenius norm. The solution to (3) can be obtained in closed-form Fernando et al.  as . This implies that the adjusted coordinate system, also referred as the source-aligned target subspace can be constructed as
Since the primary task invokes the classifier optimization using features in the ambient space, we need to re-project the target features using , i.e.,
where denotes the modified target features. The solution to this optimization problem can be obtained in closed-form as , where, and is computed from (3). When the alignment loss is linearly combined with the primary task objective, there exists no closed-form solution and the objective function becomes non-convex. We will construct an approach that takes in gradients from the primary task to adjust .
Given the primary and auxiliary task formulations, we can adopt different training strategies to combine their estimates: (i) Independent: This is the classical subspace alignment strategy, where the aligned features are directly used to optimize the classifier parameters, (ii) Joint: Similar to domain adversarial training methods, we can jointly optimize for both steps together, (iii) Alternating: This meta-learning style optimization solves for the primary task with the current estimate of the alignment, and subsequently updates the auxiliary network with both primary and auxiliary losses. As we will show later, the meta optimization strategy works the best in comparison to the other two. Now, we describe the algorithm for the proposed approach in detail.
Initialization phase: Before applying the proposed meta-optimization strategy, we need to initialize the parameters for both the primary and auxiliary tasks. First, we pre-train the feature extractor and the classifier using the losses described in section 3.1, without any explicit domain alignment. In the experiments section, we refer to this initialization as no adaptation. We then fit -dimensional subspaces, and , to the features obtained using for both the source and target domains. Note that the feature extractor is not updated for the rest of the training process, and hence the subspaces are fixed. The alignment matrix between the two subspaces is obtained using equation (3).
Training phase: In order to enable information flow between the two tasks, we propose to allow the auxiliary task to utilize gradients from the primary task. Similarly, the estimated alignment is applied to the target data while updating the classifier parameters in the primary task. To enable this flow, we define a subspace alignment network, that parameterizes as a linear layer with neurons. This parameterization allows to directly solve equation (3), when the losses from the primary task are taken into consideration. The primary and auxiliary tasks are solved alternatively until convergence – during the auxiliary task optimization, we freeze the classifier parameters and use the source/target losses from along with the alignment cost, in order to update . Since the feature extractor is fixed, there is no need to recompute the subspaces. It is important to note that, similar to existing meta-learning strategies Finn et al. , the auxiliary task is optimized using a held-out validation set, distinct from that used for the primary task. We find this critical to the effective convergence of our algorithm. Upon estimation of an updated , the classifier network is refined using source features and source-aligned target features obtained with the new alignment. Upon convergence (typically within iterations on all datasets considered), optimal values for both and are returned. Following the model-agnostic meta learning (MAML), we could perform the meta optimization using gradients-through-gradients. However, even without that, our approach produces highly effective generalization on all benchmark datasets.
Using multiple subspaces: The fidelity of the auxiliary task relies directly on the quality of the subspace approximation. For complex datasets, a single low-dimensional subspace is often a poor approximation. Hence, we propose to allow the complexity of the auxiliary model to be adjusted by using multiple target subspaces. To this end, we obtain independent bootstraps of the target data and fit a single low-dimensional subspace of dimension to each of them. While solving for the auxiliary task, we compute individual alignment matrices to the source with respect to the same classifier . During the update of the classifier , we pose this as a multi-task learning problem, wherein a single classifier is used with different source-aligned targets. This is valid since all (bootstrapped) subspaces are in the same ambient feature space. During test time, we treat the predictions obtained using features from different alignment matrices as an ensemble and perform majority voting.
We evaluate the proposed method on four widely used visual domain adaptation tasks – digits, ImageCLEF, VisDA-2017 challenge and Office-Home datasets, and present comparisons to several state-of-the-art domain adaptation techniques. Across all the experiments, an 80-20 random split of source and target training data is performed to update the primary and auxiliary tasks. All experiments were run using PyTorch frameworkPaszke et al.  on a Nvidia-TitanX GPU based computer.
Dataset: ImageCLEF111http://imageclef.org/2014/adaptation is organized by selecting common categories of images shared by three public image datasets (domains): ImageNet ILSVRC 2012 (I), Caltech-256 (C), and Pascal VOC 2012 (P). There are categories, with images each, resulting in a total of images in each domain. We conduct experiments by permuting the domains : I P, P I, I C, C I, C P, P C.
Model: Our feature extractor is based on the pre-trained ResNet-50 architecture HeK et al. , Russakovsky et al. . This model is fine-tuned using the strategy in Section 3.1 with and set at . We then use SALT on the latent features from the penultimate layer of the fine-tuned ResNet. Source and target subspaces of dimension are constructed from these -dimensional features using SVD. The classifier network is chosen to be the last fully connected layer, subsequently refined with a learning rate of 1e-4 using SGD optimizer with a momentum of 0.9. The subspace alignment network is trained with a learning rate of 1e-3 using the Adam optimizer Kingma and Ba . The proposed approach is compared against a number of baseline methods including Long et al. , LongM et al. , GaninY et al. , Long et al.  and the results are reported in table 1. The results clearly show that even a naïve alignment strategy can produce improved performance over sophisticated adversarial learning methods, with SALT’s alternating optimization strategy.
|I P||P I||I C||C I||C P||P C||Average|
|DAN LongM et al.||74.5||82.2||92.8||86.3||69.2||89.8||82.5|
|DANN GaninY et al.||75.0||86.0||96.2||87.0||74.3||91.5||85.0|
|JAN Long et al. ||76.8||88.0||94.7||89.5||74.2||91.7||85.8|
CDAN+E Long et al. 
Ablation Study: In order to understand the impact of the different components, we perform an ablation study on this dataset. We describe each setting in this experiment next:
No Adaptation: A baseline method where we use the classifier trained on the source directly on the target features without any adaptation.
Primary Only: We leave out the auxiliary task, but include all the losses used in the primary task described in equation (2).
Independent: Here, we use the closed form solution in subspace alignment from equation (4), and then solve for the primary task independently.
Joint Optimization: We employ a joint optimization strategy, wherein we jointly update the alignment , and the classifier together.
Alternating Optimization: This is our proposed strategy that updates and the classifier in an alternating fashion.
The results from the study are illustrated in figure 2(a). A key observation is that, since the alignment strategy is weak, when done independently it does not lead to any performance gains. However, the proposed optimization provides significant improvement over even a joint optimization strategy.
4.2 Digits classification
Datasets: We consider three data sources (domains) for the digits classification task: USPS Hull , MNIST LeCun et al. , and the Street View House Numbers (SVHN) Netzer et al.  dataset. Each of these datasets have 10 categories (digits from 0-9). The USPS dataset contains training and test grayscale images of handwritten images, each one of size pixels. The MNIST dataset contains training and testing grayscale images of size . The SVHN dataset contains house numbers extracted from Google Street View images. This dataset contains training images, and testing images of size . We perform the following three experiments in this task. a) MNIST USPS, b) USPS MNIST, and c) SVHN MNIST and report the accuracies on the standard target test sets
Model: The model used for all the tasks is based on the architecture from Bhushan Damodaran et al. . The model consists of six convolutional layers containing
filters; with ReLU activations and two fully-connected layers ofand (number of classes) hidden units. The Adam optimizer () was used to update the model using a mini-batch size of for the two domains. We compare our results with a number of state-of-the-art domain adaptation methods and the results are shown in table (a)a.
SALT achieves higher accuracy than the others in two out of three experiments. In the third experiment, we are close to the best performing DeepJdot Bhushan Damodaran et al. . With one of the tasks in this dataset, we demonstrate the effect of using multiple subspaces on the classification performance. As discussed earlier, allowing multiple target subspaces increases the complexity of the auxiliary task. As showed in figure 2(b), with the SVHN-MNIST DA task using 3 or more subspaces leads to significant performance gains. However, we found that increasing it further did not lead to additional improvements.
Dataset: VisDA-2017 is a difficult simulation-to-realworld dataset, with two highly distinct domains: Synthetic, renderings of 3D models from different angles and with different lightning conditions; Real which are natural images. This dataset contains over 280K images across 12 classes.
Model: Owing to this dataset’s complexity we choose ResNet-152 HeK et al. as our feature extractor and as in previous case, we fine tune it to obtain the -dimensional features and the subspace dimension is chosen to be . The classifier and subspace alignment network are trained with the same hyper-parameters as in section 4.1. From table (b)b, it can be clearly seen that our model comprehensively outperforms the results reported so far in the literature.
|Method||Ar Cl||Ar Pr||Ar RW||Cl Ar||Cl Pr||Cl Rw||Pr Ar||Pr Cl||Pr Rw||Rw Ar||Rw Cl||Rw Pr||Avg|
|DeepJdot Bhushan Damodaran et al. ||39.7||50.4||62.5||39.5||54.4||53.2||36.7||39.2||63.5||52.3||45.4||70.5||50.6|
|DAN LongM et al.||43.6||57.0||67.9||45.8||56.5||60.4||44.0||43.6||67.7||63.1||51.5||74.3||56.3|
|DANN GaninY et al.||45.6||59.3||70.1||47.0||58.5||60.9||46.1||43.7||68.5||63.2||51.8||76.8||57.6|
|JAN Long et al. ||45.9||61.2||68.9||50.4||59.7||61.0||45.8||43.4||70.3||63.9||52.4||76.8||58.3|
|CDAN Long et al. ||50.7||70.6||76||57.6||70||70||57.4||50.9||77.3||70.9||56.7||81.6||65.8|
Datasets: This challenging dataset VenkateswaraH et al. is comprised of 15,500 images in 65 classes from office and home settings, forming four extremely dissimilar domains: Artistic images (Ar), Clip Art (Cl), Product images (Pr), and Real-World images (Rw).
Model: Similar to section 4.1, we fine tune a pre-trained ResNet-50 and obtain the -d features and the subspace dimension is chosen to be . The classifier and subspace alignment network are trained with the same hyper-parameters as earlier. Comparisons to the state-of-the-art methods are reported in table 3. We observe that while SALT consistently outperforms baseline methods including the recent DeepJdot Bhushan Damodaran et al. , it is slightly inferior to Long et al.  in most of the tasks.
In this work, we present a principled and effective approach to tackle the problem of unsupervised domain adaptation, in the context of visual recognition. The proposed method – SALT– poses alignment as an auxiliary task to the primary task of maximizing performance on the source dataset. Building on insights from meta-learning literature, SALT proposes to solve domain alignment by utilizing gradients from the primary task. The alternating optimization between primary and auxiliary tasks, without refining the feature extractor, provides a venue for systematic control of domain alignment intended to achieve improved generalization to the target set. Through an extensive quantitative and qualitative evaluation, it is shown that SALT achieves performance that is comparable or higher than the state-of-the-art on multiple standard benchmarks. SALT is generic, and can be used in conjunction with any other feature extractor. Future work includes extending the SALT methodology to newer tasks such as as semantic segmentation, open-set classification Saito et al. , and image-to-image translation.
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