Learning theory mostly addresses the standard learning paradigm, assuming the availability of complete and correct supervision signals for large amounts of data. However, in practice, machine learning researchers and practitioners acquire and make use of a range of incidental supervision signals that only have statistical associations with the gold supervision. This paper addresses the question: Can one quantify models' performance when learning with such supervision signals, without going through an exhaustive experimentation process with various supervision signals and learning protocols? To quantify the benefits of various incidental supervision signals, we propose a unified PAC-Bayesian Informativeness measure (PABI), characterizing the reduction in uncertainty that incidental supervision signals provide. We then demonstrate PABI's use in quantifying various types of incidental signals such as partial labels, noisy labels, constraints, cross-domain signals, and some combinations of these. Experiments on named entity recognition and question answering show that PABI correlates well with learning performance, providing a promising way to determine, ahead of learning, which supervision signals would be beneficial.READ FULL TEXT VIEW PDF
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The standard learning paradigm, where direct supervision signals are assumed to be available in high-quality and large amounts, has been struggling to fulfill the needs in many real-world AI applications. As a result, researchers and practitioners often resort to datasets that are not collected directly for the target task but, hopefully, capture some phenomena useful for it (Roth, 2017; Kolesnikov et al., 2019). However, it remains unclear how to predict the benefits of these incidental signals on our target task beforehand, so the common practice is often trial-and-error: do experiments with different combinations of datasets and learning protocols, often exhaustively, to achieve improvement on a target task (Liu et al., 2019; He et al., 2020; Khashabi et al., 2020). Not only this is very costly, this trial-and-error approach can also be hard to interpret: if we don’t see improvements, is it because the incidental signals themselves are not useful for our target task, or is it because the learning protocols we have tried are inappropriate?
The difficulties of foreshadowing the benefits of incidental supervision signals are two-fold. First, it is hard to provide a unified measure because of the intrinsic differences among various incidental signals (e.g., the difference between noisy labels and constraints). Second, it is hard to provide a practical measure along with theoretical guidance. Previous attempts are either not practical or without theoretical guidance (Baxter, 1998; Ben-David et al., 2010; Thrun and O’Sullivan, 1998; Gururangan et al., 2020). In this paper, we propose a unified PAC-Bayesian based informativeness measure (PABI
) to quantify the value of incidental signals. We suggest that the informativeness of various incidental signals can be uniformly characterized by the reduction in the original concept class uncertainty they provide. Specifically, in the PAC-Bayesian framework, the informativeness is based on the KL divergence between the prior and the posterior, where incidental signals are used to estimate a better prior (closer to the gold posterior) to achieve better generalization performance. Furthermore, we provide a more practical entropy-based approximation ofPABI.
We have been in need of a unified informativeness measure like PABI. For instance, it might be obvious that we can expect better learning performance if the training data are less noisy and more complete, but what if we want to compare the benefits of a noisy dataset and that of a partial dataset? PABI
enables this kind of comparisons beforehand, on a range of incidental signals such as partial labels, noisy labels, constraints, auxiliary signals, cross-domain signals, and some combinations of them, for sequence tagging tasks in natural language processing (NLP). A specific example of named entity recognition (NER) is shown in Fig.1.
Finally, our experiments on two NLP tasks, NER and question answering (QA), show that there is a strong positive correlation between PABI and the relative improvement for various incidental signals. This strong positive correlation indicates that the proposed unified, theory-motivated measure PABI can serve as a good indicator of the final learning performance, providing a promising way to know which signals are helpful for a target task beforehand.
There has been a line of theoretical work that attempts to exploit incidental supervision signals. Balcan and Blum (2010) propose to use unlabeled data to reduce the concept class with the help of incompatibility. Similarly, Abu-Mostafa (1993) analyzes the concept class reduction provided by invariance hints. Natarajan et al. (2013)
, instead, proposes to directly learn from noisy labels by adjusting the loss function.Van Rooyen and Williamson (2017) further consider more types of corrupted labels, such as noisy labels and partial labels. Others propose to use incidental supervision signals to learn the bias of the environment (Abu-Mostafa, 1993; Baxter, 1998) or quantify the dependencies in the structure (London et al., 2016; Ciliberto et al., 2019). As for exploiting cross-domain signals, Ben-David et al. (2010) provide a uniform convergence learning bound for learning from different domains. Our proposal PABI is different from these lines of work in several ways: First, PABI is a unified measure that applies to multiple signals, while earlier analysies mainly focused on one or two types of signals; second, as we show, PABI can be easily used in practice, while earlier theoretical work was not easy to use.
There has also been work on providing practical informativeness measures for incidental signals. Ning et al. (2019) propose to use the concaveness of the mutual information with different percentage of annotations to quantify the strength of structures. Gururangan et al. (2020) propose to use vocabulary overlap to estimate the domain similarity of texts. Thrun and O’Sullivan (1998)
propose to cluster tasks based on the similarity between the task-optimal distance metric of k-nearest neighbors (KNN). The differences between this line of empirical work andPABI are two-fold: First, PABI is a unified measure for various signals, while earlier analysis mainly focused on one type of signals; second, PABI is based on PAC-Bayesian theory, while earlier proposals lacked theoretical guidance.
Let be the input space, be the label space, and be the prediction space. We assume the underlying distribution on is . Let be the loss function that we use to evaluate learning algorithms. Given a set of training samples generated i.i.d. from , we want to learn a predictor such that it generalizes well to unseen data with respect to , measured by the generalization error . The empirical error over is . The concept class we consider is denoted by . When is finite, its size is . Let
denote the space of probability distributions over.
General Bayesian learning algorithms (Zhang and others, 2006) in the PAC-Bayesian framework (McAllester, 1999b, a; Seeger, 2002; McAllester, 2003a, b; Maurer, 2004; Guedj, 2019) aim to choose a posterior over the concept class based on a prior and training data , where is a hyper parameter that controls the tradeoff between the prior and the data likelihood. Since we aim to choose a distribution over concepts instead of a particular concept, the training error and the generalization error needs to be modified as and respectively. When the gold posterior is one-hot (exactly one entry of is and the remaining entries are ), we have the original definitions of training error and generalization error in the PAC framework (Valiant, 1984).
where is the posterior distribution with the optimal , is the ideal posterior, denotes the KL divergence from to , and are two constants. This is based on the Theorem 2 in Guedj (2019).
As shown in the generalization bound, the generalization error is bounded by the KL divergence from the prior distribution to the gold posterior distribution. Therefore, we propose to utilize incidental signals to improve the prior distribution from to so that it is closer to the ideal posterior distribution . The corresponding informativeness measure, PABI, is defined as follows
Note that if , while if , then . This result is consistent with our intuition that the closer is to , the more benefits we can gain from incidental signals. In our following analysis, the original prior is uniform (without any information), and is one-hot with the true concept . The square root function is used in PABI for two reasons: First, the generalization bounds in both PAC-Bayesian and PAC (see Sec. 2.2) frameworks have the square root function; second, in our later experiments, we find that square root function can significantly improve the Pearson correlation between the relative performance improvement and PABI.
However, is unknown in practice, which makes the informativeness hard to be computed in some complex cases. A reasonable approximation is
where is the entropy function and is the one-hot distribution concentrated on . The intuition behind is that we can use the prior to estimate the distribution of the gold concept. We will show this is a reasonable approximation in Sec. 2.2. Therefore, in practice, we can simply use as a proxy.
For the uniform prior and over finite concept class , we have and , where denotes the reduced concept class using incidental signals and is the gold one-hot posterior. In this case, improving the prior from to means deterministically reducing the concept class from to . Therefore, PABI can be written as:
Remark: can be also derived by the generalization bound in the PAC framework (Mohri et al., 2018) which says with probability over ,
However, the PAC framework without an affiliated probability measure (equivalent to uniform priors in the PAC-Bayesian framework) cannot handle the probabilistic cases. For example, incidental signals can reduce the probability of some concepts, though the concept class is not reduced. In this example, the informativeness measure is zero, but we actually benefit from incidental signals. It is worthwhile to notice that and are equivalent in the finite concept class with uniform priors, indicating that our entropy-based approximation is reasonable. Some analysis on the extensions and general limitations of PABI can be found in Appx. A.1 and A.2. We need to notice that the size of concept class also plays an important role in the lower bound on the generalization error (more details in Appx. A.3.), indicating that PABI based on the reduction of the concept class is a reasonable measure.
In this section, we show some examples of sequence tagging tasks222Given an input generated from a distribution , the task aims to get the corresponding label , where is the vocabulary of input words, and is the label set for the task. in NLP for PABI. More examples and details can be found in Appx. A.4
. Similar to the categorization of transfer learning(Pan and Yang, 2009), we use inductive signals to denote the signals with the same marginal distribution of as gold signals but a different task from gold signals, such as noisy and auxiliary signals, and transductive signals to denote the signals with the same task as gold signals but a different marginal distribution of from gold signals, such as cross-domain and cross-lingual signals. In our following analysis, we focus on the tasks with finite concept class333Note that tasks with infinite concept class can also be approximated by the finite concept class in practice because of memory and time limit. which is quite common in NLP. Another assumption is that the number of incidental signals is large enough, and more discussions can be found in Sec. 5.
Partial labels. The labels for each example in sequence tagging tasks is a sequence and some of them are unknown in this case. Assuming that the data are with unknown labels and gold labels, the size of the reduced concept class will be . Therefore, the PABI is .
Noisy labels. For each token, is determined by (i.e. and the probability of other labels are all ). We can get the corresponding probability distribution of labels over the tokens in all inputs ( over the concept class). In this way, the corresponding informativeness is .
For transductive signals, such as cross-domain signals, we can first extend the concept class to the extended concept class with the corresponding extended input space . After that, we can use incidental signals to estimate a better prior distribution over the extended concept class , and then get the corresponding over the original concept class by restricting the concept from to . In this way, the informativeness of transductive signals can still be measured by or . The restriction step is similar to Roth and Zelenko (2000).
However, how to compute is still unclear. For simplicity, we use to denote the gold system on the gold signals, to denote the perfect system on the incidental signals, and to denote the silver system trained on the incidental signals. Source domain (target domain) denotes the domain of incidental signals (gold signals). To compute PABI, we add two assumptions here: I. is a noisy version of with a noise ratio in both source and target domain; II. is a noisy version of with a noise ratio in both source and target domains. In practice, is unknown but it can be estimated by in the source domain and (the noise rate of the silver system on the target domain where is the marginal distribution of ) as follows:
Let be a concept class of VC dimension for binary classification. Let be a labeled sample of size generated by drawing points () from according to and points () from (the distribution of incidental signals) according to . If is the empirical joint error minimizer, and is the target error minimizer, is the joint error minimizer, under assumption I, and assume that is expressive enough so that both the target error minimizer and the joint error minimizer can achieve zero errors, then for any , with probability at least ,
Remark: The take-away is that the target error of the empirical joint error minimizer is related to .
The proof of Theorem 3.1 can be found in Appx. A.6. The corresponding informativeness of transductive signals can be then computed as . Theorem 3.1 indicates that PABI is a reasonable measure to quantify the benefits from transductive signals. Although the computation cost of PABI for transductive signals is higher than that for inductive signals, it is still much cheaper than building combined models with joint training.
The mix of partial and noisy labels. The corresponding informativeness for the mix of partial and noisy labels is , where denotes the ratio of unlabeled tokens, and denotes the noise ratio.
The mix of partial labels and constraints. For the BIO constraint with partial labels, we can use dynamic programming to estimate the average size of concept class by sampling as Ning et al. (2019).
In this subsection, we analyze the informativeness of inductive signals for NER. We use Ontonotes NER ( types of named entities) (Hovy et al., 2006) as the main task. We randomly sample sentences ( words) of the development set as the small gold signals, sentences ( words) of the development set as the large incidental signals. We use a two-layer NNs with 5-gram features as our basic model. The lower bound for our experiments is the result of the model with small gold Ontonotes NER annotations and bootstrapped on the unlabeled texts of the large gold Ontonotes NER, which is F1, and the upper bound is the result of the model with both small gold Ontonotes NER annotations and the large gold Ontonotes NER annotations, which is F1.
To utilize inductive signals, we propose a new bootstrapping based algorithm CWBPP (Algorithm 1), where inductive signals are used to improve the inference stage by approximating a better prior. The algorithm is an extension of CoDL (Chang et al., 2007) with various inductive signals.
NER with individual inductive signals. We first experiment on individual inductive signals, including partial labels, noisy labels, auxiliary labels. For partial labels, we experiment on NER with four different partial rates: , , , and . For noisy labels, we experiment on NER with seven different noisy rates: . For auxiliary labels, we experiment on two auxiliary tasks: named entity detection and coarse NER (CoNLL annotations with types of named entities (Sang and De Meulder, 2003)). As shown in Fig. 2(a)-2(c), we can see that there is a strong correlation between the relative improvement and PABI for three types of inductive signals separately.
NER with mixed inductive signals. A more complex case is the comparison between the mixed inductive signals. We consider two types of mixed inductive signals: incidental signals with both partial and noisy labels, and incidental signals with both partial labels and constraints. For the first type of mixed signals, we experiment on the combination between three unknown partial rates (, , and ) and four noisy rates (, , , and ). As for the second type of mixed signals, we experiment on the combination between the BIO constraint and five unknown partial rates (, , , , and ). As shown in Fig. 2(d)-2(e), there is a strong correlation between the relative improvement and PABI for mixed inductive signals whose benefits cannot be quantified by existing frameworks.
NER with various inductive signals. After we put the three types of individual inductive signals and the two types of mixed inductive signals together, we still see a correlation between PABI and the relative performance improvement in experiments in Fig. 2(f). This strong positive correlation indicates that it is feasible to foreshadow the potential contribution of various incidental supervision signals with the help of PABI, which cannot be addressed by existing frameworks.
NER with cross-domain signals We consider four NER datasets, Ontonotes, CoNLL, twitter(Strauss et al., 2016), and GMB (Bos et al., 2017). We aim to detect the person names here because the only shared type of the four datasets is the person. In our experiments, the twitter NER serves as the main dataset and other three datasets are cross-domain datasets. Because we only focus on the person names, a lot of sentences in the original dataset will not include any entities. We random sample sentences to keep that sentences without entities and sentences with at least one entity. There are sentences in the small gold training set, sentences ( times of the gold signals) in the large incidental training set, and sentences in the test set.
We use BERT (Devlin et al., 2019) as our basic model and use the joint training strategy to make use of incidental signals. The lower bound for our experiments is the result with only small gold twitter annotations, which is F1, and the upper bound is the result with both small gold twitter annotations and large gold twitter annotations, which is . and is computed by using sentence-level accuracy.
The relation between the relative improvement and naive/our informativeness measure is shown in Fig. 3(a). We can see that there is a strong positive correlation between the relative improvement and PABI for cross-domain NER. The adjustment from (the noise of the imperfect system) is crucial (Eq. (1)), especially for the GMB dataset, indicating that directly using is not a good choice.
QA with cross-domain signals. We consider SQuAD (Rajpurkar et al., 2016), QAMR (Michael et al., 2017), Large QA-SRL (FitzGerald et al., 2018), QA-RE (Levy et al., 2017), NewsQA (Trischler et al., 2017), TriviaQA (Joshi et al., 2017). In our experiments, the SQuAD dataset servers as the main dataset and other datasets are cross-domain datasets. We randomly sample QA pairs as the small gold signals, about QA pairs as the large incidental signals ( times of the small gold signals), and QA pairs as the test data. For consistency, we only keep one answer for each question in all datasets.
We use BERT as our basic model and consider two strategies to make use of incidental signals: joint training and pre-training. The lower bound for our experiments is the result with only small gold SQuAD annotations, which is exact match. The upper bound for the joint training is the result with both small gold SQuAD annotations and large SQuAD annotations, which is exact match. Similarily, the upper bound for the pre-training is exact match.
The relation between the relative improvement (pre-training or joint training) and the informativeness (the PABI based on in Eq. (1) or naive baseline ) are shown in Fig. 3(b)-(c). We can see that there is a strong correlation between the relative improvement and PABI, and PABI works better than the naive baseline. Another thing worthwhile to notice is that the most informative QA dataset is not always the same for different main QA datasets. For example, for NewsQA, the most informative QA dataset is SQuAD, while the most informative QA dataset for SQuAD is QAMR.
Supported by PAC-Bayesian theory, this paper proposes a unified framework, PABI, to characterize incidental supervision signals by how much uncertainty they can reduce in one’s hypothesis space. We demonstrate the effectiveness of PABI in foreshadowing the benefits of various signals (e.g., partial labels, noisy labels, auxiliary labels, constraints, cross-domain signals and combinations of them) for solving NER and QA. As the recent success of natural language modeling has given rise to many explorations in knowledge transfer across tasks and corpora (Bjerva, 2017; Phang et al., 2018; Zhu et al., 2019; Liu et al., 2019; He et al., 2020; Khashabi et al., 2020) , PABI is a concrete step towards explaining some of these observations.
PABI can also provide guidance in designing learning protocols. For instance, in a B/I/O sequence chunking task,444B/I/O indicates if a token is the begin/inside/outside of a text span. missing labels make it a partial annotation problem, while treating missing labels as O introduces noise. Since the informativeness of partial signals is larger than that of noisy signals with the same partial/noisy rate (see details in Sec. 3.1), PABI suggests us not to treat missing labels as O, and this is exactly what Mayhew et al. (2019) prove to us via their experiments.
There are another two key factors one should bear in mind when using PABI.
Base model performance. In the generalization bound in both PAC and PAC-Bayesian, we can see that the relative improvement in the generalization bound from reducing is small if is large. In practice, the relative improvement is the real improvement with some noise. Therefore, we can see that the real improvement is dominant if is small and the noise is dominant if is large. Therefore, PABI may not work well when is large and when the performance on the target task is already good enough.
The impact of the size of incidental signals on PABI. Our previous analysis is based on a strong assumption that incidental signals are large enough (ideally ) A more realistic PABI is based on with examples as , where denotes the restricted concept class of on the examples, and so does . (1) When is large enough, . (2) When the sizes of different incidental signals are all , the relative improvement is independent of (), and . Our experiments are based on this case and does not really rely on the assumption that incidental signals are large enough. (3) The incidental signals we are comparing are not large enough and have different sizes, we need to use to incorporate that difference.
In this paper, we propose a unified informativeness measure, PABI, to foreshadow the benefits of incidental supervision. On the one hand, PABI can be used to alleviate annotation cost (task-specific supervised signals) and computation cost (trail-and-error fashion) for the machine learning community. On the other hand, PABI is hard to compute for some complex incidental supervision signals, limiting its usage in some complicated real-world applications.
Proceedings of the twelfth annual conference on Computational learning theory, pp. 164–170. Cited by: §2.1.
Glove: global vectors for word representation. In Proceedings of the 2014 conference on empirical methods in natural language processing (EMNLP), pp. 1532–1543. Cited by: §A.7.
Proc. of the Conference on Artificial Intelligence (AAAI), pp. 639–644. Cited by: §3.2.
Proceedings of the 2nd Workshop on Noisy User-generated Text (WNUT), pp. 138–144. Cited by: §4.2.
In practice, algorithms are often based on parametric concept class. The two informativeness measures in the PAC-Bayesian framework, and , can be easily adapted to handle the cases in parametric concept class. Given parametric space , we can easily change the probability distribution over the parametric concept class to the probability distribution over the finite concept class by clustering concepts in the parametric space according to their outputs on all inputs. The concepts in each cluster have the same outputs on all inputs as outputs of one concept in the finite concept class . We then merge the probabilities of concepts in the same cluster to get the probability distribution over the finite concept class . This merging approach can be applied to any concept class which is not equal to the finite concept class
, including non-parametric and semi-parametric concept class. In practice, we can use sampling algorithms, such as Markov chain Monte Carlo (MCMC) methods, to simulate this clustering strategy.
Different informativeness measures are based on different assumptions, so we analyze their limitations in detail to understand their limitations in applications.
For the informativeness measure , it cannot handle probabilistic signals or infinite concept classes. There are various probabilistic incidental signals, such as soft constraints and probabilistic co-occurrences between an auxiliary task and the main task. An example of probabilistic co-occurrences between part-of-speech (PoS) tagging and NER is that the adjectives have a probability to have the label in NER. As for the infinite concept class, most classifiers are based on infinite parametric spaces. Thus, cannot be applied to these classifiers.
The informativeness measure is hard to be computed for some complex cases. In practice, we can use the estimated posterior distribution over the gold data, which is asymptotically unbiased, to estimate it. Another approximation is to use the informativeness measure . However, it is not directly linked to the generalization bound, so more work is needed to guarantee its reliability for some complex probabilistic cases. We postpone to provide the theoretical guarantees for on more complex cases as our future work.
In the following theorem, we show that the VC dimension (size of concept class) also plays an important role in the lower bund for the generalization error, indicating that PABI based on the reduction of the concept class is a reasonable measure.
Let be a concept class with VC dimension . Then, for any and any learning algorithm , there exists a distribution over and a target concept such that
where is a consistent concept with S returned by . This is the Theorem 3.20 in Chapter 3.4 of Mohri et al. (2018).
In this subsection, we show more examples with incidental signals, including within-sentence constraints, cross-sentence constraints, auxiliary labels, cross-lingual signals, cross-modal signals, and the mix of cross-domian signals and constraints.
Within-Sentence Constraints. As for within-sentence constraints, we show three types of common constraints in NLP, which are BIO constraints, assignment constraints, and ranking constraints.
BIO constraints are widely used in sequence tagging tasks, such as NER. For BIO constraints, I-X must follow B-X or I-X, where “X” is finer types such as PER (person) and LOC (location). We consider a simple case here: there are only B, I, O three labels. We have for the BIO constraint. Therefore, . This value can be approximated by the dynamic programming as Ning et al. (2019).
Assignment constraints can be used in various types of semantic parsing tasks, such as semantic role labeling (SRL). Assume we need to assign agents with tasks such that the agent nodes and the task nodes form a bipartite graph (without loss of generality, assume ). Each agent is represented by a feature vector in . We have . This informativeness doesn’t rely on the choice of where that denotes discrete feature space for arguments.
Ranking constraints can be used in ranking problems, such as temporal relation extraction. For a ranking problem with items, there are pairwise comparisons in total. Its structure is a chain following the transitivity constraints, i.e., if and , then . In this way, we have . This informativeness doesn’t rely on the choice of where denotes discrete feature space for events.
Cross-sentence Constraints. For cross-sentence constraints, we consider a common example, global statistics based on -tuple of tokens, i.e. pairs of tokens in different sentences must have the same labels. We can group words into groups with probability . In this way, we have . The approximation holds as long as , , and n are not all too small. For example, as shown in Table 1, the percentage of 5-gram words with unique NER labels is , so ideally the corresponding PABI will be . It is worthwhile to note that the k-gram words with unique labels can also be caused by the low frequency of the appearance of the k-grams. In our experiments, we only consider the k-grams with unique labels that appear at least twice in the data. We experiment on NER with three types of cross-sentence constraints: uni-gram words with unique NER labels, bi-gram words with unique NER labels, and 5-gram Part-of-Speech (PoS) tags with unique NER labels555Here we use PoS tags as a special type of cross-sentence constraints by specifying the labels of tokens whose PoS tags have unique NER labels.. The results are shown in Fig. 4.
Auxiliary labels. For auxiliary labels, we show two examples as follows:
For a multi-class sequence tagging task, we use the corresponding detection task as auxiliary signals. Given a multi-class sequence tagging task with labels in the BIO format (Ramshaw and Marcus, 1999), we will have labels for the detection and labels for the classification. Thus, , where is the percentage of the label O among all labels.
Coarse-grained NER for Fine-grained NER. We have four types, PER, ORG, LOC and MISC for CoNLL NER and types for Ontonotes NER. The mapping between CoNLL NER and Ontonotes NER is as follows: PER (PERSON), ORG (ORG), LOC(LOC, FAC, GPE), MISC(NORP, PRODUCT, EVENT, LANGUAGE), O(WORF_OF_ART, LAW, DATE, TIME, PERCENT, MONEY, QUANTITY, ORDINAL, CARDINAL, O) (Augenstein et al., 2017). In the BIO setting, we have , where , , are the percentage of LOC(including B-LOC and I-LOC), MISC (including B-MISC and I-MISC), and O among all possible labels.
Cross-lingual signals. For cross-lingual signals, we can use multilingual BERT to get in the extended input space . After that, and can be computed accordingly.
Cross-modal signals. For cross-modal signals, we only consider the case where labels of gold and incidental signals are same and inputs of gold and incidental are aligned. A common situation is that a video has visual, acoustic, and textual information. In this case, the images and speech related to the texts can be used as cross-modal information. We can use cross-modal mapping between speech/images and texts (e.g. Chung et al. (2018)) to estimate the and for cross-modal signals.
The mix of cross-domain signals and constraints. Let denote the perfect system on cross-domain signals and satisfying constrains on inputs of gold signals, and denote the model trained on cross-domain signals and satisfying constraints on inputs of gold signals. In this way, we can estimate and by forcing constraints in their inference stage.
For simplicity, we use to denote , to denote , and to denote . We then re-write the definitions of , and as , and . Note that is the label set for the task. Considering all three systems in the target domain, we have
Therefore, we have .
A concept is a function : . The probability according to the distribution that a concept disagrees with a labeling function (which can also be a concept) is defined as
Note that here is the loss function and where is the gold label for . We denote () the corresponding weighted combination of true source and target errors, measured with respect to and as follows:
Let be a concept in concept class . Then
where , , and .
For a fixed concept from with VC dimension , if a random labeled sample () of size is generated by drawing points () from and points () from , and labeling them according to and respectively, then for any with probability at least (over the choice of the samples),
where and is the natural number.
Proof. Given Lemma 5 in Ben-David et al. (2010), which says for any , with probability (over the choice of the samples),
According to the Vapnik-Chervonenkis theory (Vapnik and Chervonenkis, 2015), we have with probability ,
This is the standard generalization bound with an adjust term (see more in Chapter 3.3 of Mohri et al. (2018)). ∎
Proof of Theorem 3.1. Let , then