A theoretically grounded characterization of feature representations 

A large body of work has explored how learned feature representations can be useful for a variety of downstream tasks . This is true even when the downstream tasks differ greatly from the actual objective used to ( pre ) train the feature representation . This observation underlies the success of , e.g. , few-shot learning , transfer learning and self-supervised learning , among others . However , very little is understood about why such transfer is successful , and more importantly , how one should choose the pre-training task . As a first step towards this understanding , we ask : what makes a feature representation good for a target task ? We present simple , intuitive measurements of the feature space that are good predictors of downstream task performance . We present theoretical results showing how these measurements can be used to bound the error of the downstream classifiers , and show empirically that these bounds correlate well with actual downstream performance . Finally , we show that our bounds are practically useful for choosing the right pre-trained representation for a target task . 1 INTRODUCTION . Since the ( re ) -discovery of neural networks for visual recognition , a plethora of work has observed that the feature representations trained on ImageNet generalize to many downstream tasks , even to new domains ( Donahue et al. , 2014 ; Kornblith et al. , 2019 ; Kolesnikov et al. , 2020 ; Wallace & Hariharan , 2020 ) . This observation , and the resulting gain in accuracy even with very limited labels , has heralded new research directions into other ways of learning representations , such as self-supervised learning ( Chen et al. , 2020 ) or meta-learning ( Snell et al. , 2017 ) . This growing field of representation learning has yielded ostensibly better and better feature representations . However , a closer look reveals many mysterious results . For example , meta-learning methods do not transfer across domains ( Guo et al. , 2020 ) and self-supervised representations struggle with fine-grained recognition ( Wallace & Hariharan , 2020 ) . These empirical results are extremely valuable , but do not provide a deeper understanding of the corresponding phenomena . On the other side of the spectrum , theoretical work on neural representations is illuminating , but often makes assumptions about models or tasks ( Du et al. , 2020 ; Arora et al. , 2019b ) . We lack a general understanding of which neural representations work well for a given task and why . In this paper , we take a first step towards such an understanding by developing lower and upper bounds on classifier accuracy based on data-driven properties of the feature space . Classical theoretical bounds focus entirely on the complexity of the classifier and ignore the feature space . In contrast , our bounds are based on two intuitive properties of the feature representation ( Figure 1 ) : ( a ) Local alignment , which is the degree to which nearby data points share labels , and ( b ) local congregation which is the degree to which data points embed close to each other . Intuitively , if the feature representation is locally aligned to the task , then any smooth classifier will be able to model the task well . If it is additionally congregated , then most test points will have nearby training instances , so the classifier will generalize well from limited data . We show that our bounds are not only intuitive and theoretically justified , but also predictive of actual performance in practical settings : on a large dataset of realistic few-shot tasks , we can use our bounds to pick the best pre-trained representation without any training . Taken together , our work is a first step towards a general , intuitive characterization of the feature space that is predictive of downstream classifier performance . 2 RELATED WORK . Analyzing transferability : The motivations of our work lie in understanding the empirical effectiveness of transferring feature representations from supervised ( Donahue et al. , 2014 ; Zhai et al. , 2019 ; Kolesnikov et al. , 2020 ) or self-supervised ( Goyal et al . ; Wallace & Hariharan , 2020 ; Chen et al. , 2020 ) tasks . With these successes on transfer , and with the availability of a large number of pretrained features trained on a wide variety of domains , there has been an increasing interest in predicting transferability , or in selecting the right features for a particular target task . The simplest approach is to train a classifier with every available pre-trained representation and pick the best performer ( Zamir et al. , 2018 ; You et al. , 2021 ) . This is both computationally expensive and requires lots of labeled data for the target task . If the pre-training task is a classification task with a known label space then the conditional distribution of targets given pre-training task labels is informative of transfer performance ( Tran et al. , 2019 ; Nguyen et al. , 2020 ) . However , this approach is difficult to apply if the pre-training task is not classification , or is inaccessible ( as with models trained on proprietary data ) . This inaccessibility of pre-training data is also a problem for approaches that match pre-training and target distributions ( Gao & Chaudhari , 2021 ) , or explicitly adapt the pre-training images or label space ( Cui et al. , 2018 ) . In contrast to these approaches , we focus on directly analyzing the pre-trained feature representation , which allows us to be agnostic to the actual task it was trained on . Our work is most similar in setup and motivation to work on measuring the alignment between feature representations and the target labels ( Huang et al. , 2021 ; Bao et al. , 2019 ) , but in addition to allowing model selection , yields intuitive and general bounds on accuracy . Our work is orthogonal to work on characterizing tasks and measuring task similarity ; this frequently requires pre-trained features in the first place ( Achille et al. , 2019 ; Wallace et al. , 2021 ; Song et al. , 2020 ; Dwivedi et al. , 2020 ; Dwivedi & Roig , 2019 ) . Analyzing feature representations : Our work is also related to research on understanding feature representations in general . Some of this work has focused on understanding invariance properties of convnet features ( Aubry & Russell , 2015 ) . Others have looked at what individual feature dimensions mean ( Agrawal et al. , 2014 ; Zeiler & Fergus , 2014 ; Szegedy et al. , 2013 ) . Still others have explored if and when features from pre-trained networks transfer well between tasks as a function of the layer chosen ( Yosinski et al. , 2014 ) or the architecture ( Kornblith et al. , 2019 ) . Recently these explorations have been extended to other representation learning techniques , notably self supervised techniques ( Wang & Isola , 2020 ; Wallace & Hariharan , 2020 ) . The insights from these explorations inspire our results . However , these explorations have been primarily empirical , and are therefore limited by the diversity of real world benchmarks that they experiment with . On the other end of the spectrum , there is prior work on theoretical analysis of transfer learning . This prior work often follows the framework proposed by Baxter ( 2000 ) , and in so doing makes assumptions about the distributions of the different tasks ( Maurer , 2009 ; Du et al. , 2020 ; Galanti et al. , 2016 ; Pentina & Lampert , 2014 ) . In contrast , our approach eschews these assumptions in lieu of data-driven measurements . Data-driven complexity measures have been explored before for analyzing neural network training dynamics and generalization . These measures include eigenvectors of the gram matrix between data points ( Arora et al. , 2019a ) , the variance and separation of class-specific manifolds ( Cohen et al. , 2020 ) , the underlying intrinsic dimensionality of the task ( Lampinen & Ganguli , 2018 ) or the mutual information between representations and the inputs or labels ( Shwartz-Ziv & Tishby , 2017 ) . The corresponding results can be used for analyzing feature representations as well . Our proposed measurements are similar , but are simpler to measure and potentially more intuitive . 3 PROBLEM SETUP . Suppose we are interested in mapping a space of inputs X to a space of targets Y . There is an underlying distribution D over X × Y . We have a feature representation φ : X → Rf which might be pre-trained on another dataset or task , which is inaccessible to us . We will assume that ‖φ ( x ) ‖ ≤ B . For ease of exposition in the paper , we focus on the task of binary classification ; the case of multiclass classification is similar . For binary classification , we write the label space as Y = { −1 , 1 } . Our classifier will use a scoring function that operates on feature space , h : Rf → R. The predicted label will then be sign ( h ( φ ( x ) ) ) , where if h ( φ ( x ) ) is 0 , we will arbitrarily assign a label of -1 . The set of possible functions h defines the hypothesis class for the classifier ; denote this byH . For most of our analysis , we primarily care about the smoothness of the functions in H. We will assume that all functions inH are Lipschitz continuous with Lipschitz constant less than W . Thus : h ( φ ( x ) ) − h ( φ ( x′ ) ) ≤W‖φ ( x ) − φ ( x′ ) ‖ ∀h ∈ H ( 1 ) We note that one such hypothesis class which is commonly used in practice is the class of linear classifiers of bounded norm : W = { v 7→ wTv ; ‖w‖ ≤W } . Because the zero-one loss , l∗ ( h ( φ ( x ) ) , y ) = I [ sign ( h ( φ ( x ) ) ) 6= y ] is difficult to analyze , we will use the following continuous upper bound which is standard in theoretical treatments ( Mohri et al. , 2018 ) l ( h ( φ ( x ) ) , y ) = min ( 1 , max ( 0 , 1− yh ( φ ( x ) ) ) ) ≥ l∗ ( h ( φ ( x ) ) , y ) ( 2 ) Our focus in this paper is to understand how the properties of the feature extractor φ affect the loss of the classifiers l ( h ( φ ( x ) ) ) . In particular , we wish to understand how φ impacts ( a ) the lowest average error that one can achieve , and ( b ) the true error when one generalizes from a small training set . We begin by first identifying the key properties of feature representations . 4 TWO PROPERTIES OF FEATURE REPRESENTATIONS . What properties should we use to characterize feature representations ? First , we should use properties that are easy to measure , potentially with limited labeled data . Second , these properties should be easy for human developers and practitioners to reason about . Third , they should correlate well with the final accuracy of downstream classifiers . In sum , we want simple , intuitive measurements of the feature space that are predictive of downstream accuracy . One kind of intuitive measurement is to look at what the feature representation considers as similar . In particular , we could look at pairs of examples that embed close to each other in feature space and ask if they are indeed similar in terms of their ground truth labels . We call this property local alignment . In particular , we make the following definition Definition 1 . Suppose α > 0 . The local alignment of the feature space φ , denoted by pφa ( α ) is the probability that two data points ( x , y ) , ( x′ , y′ ) ∼ D share a label given that they embed within a distance of α : pφa ( α ) , P ( y = y ′ ∣∣ ‖φ ( x ) − φ ( x′ ) ‖ ≤ α , ( x , y ) , ( x′ , y′ ) ∼ D ) ( 3 ) α here is a hyperparameter which governs the resolution at which we do the analysis . Note that this notion of local alignment is different from the alignment proposed by Wang & Isola ( 2020 ) : the latter looks at how often two data points that are semantically similar embed close to each other , while the former looks at how often two data points that embed close to each other are semantically similar . A feature space that is locally aligned per our definition may not actually be aligned per the definition of Wang & Isola ( 2020 ) because two very similar images may be embedded far away from each other . Local alignment alone may be meaningless if data points do not generally embed close to each other . We also need the feature space to be such that data points generally congregate : Definition 2 . Suppose α > 0 . The degree of congregation of the feature space φ , denoted by pφc , is the probability that two points x , x′ sampled from D embed within a distance of α : pφc ( α ) , P ( ‖φ ( x ) − φ ( x′ ) ‖ ≤ α|x , x′ ∼ D ) ( 4 ) We have α as a hyperparameter here too . We now use these notions of local alignment and congregation to analyze the downstream error of classifiers .

Summary: The paper addresses the question of what makes a representation suitable or "good" for a particular task. The submission includes three data-dependent bounds: a lower bound on the average accuracy and two upper bounds on generalization. The analysis is based on the simple and intuitive concepts of local alignment and degree of congregation of the data points within a representational space and according to a binary labeling of the points. Two experiments support the theoretical claims.