MANIFOLD CHARACTERISTICS THAT PREDICT DOWNSTREAM TASK PERFORMANCE

Abstract

Pretraining methods are typically compared by evaluating the accuracy of linear classifiers, transfer learning performance, or visually inspecting the representation manifold's (RM) lower-dimensional projections. We show that the differences between methods can be understood more clearly by investigating the RM directly, which allows for a more detailed comparison. To this end, we propose a framework and new metric to measure and compare different RMs. We also investigate and report on the RM characteristics for various pretraining methods. These characteristics are measured by applying sequentially larger local alterations to the input data, using white noise injections and Projected Gradient Descent (PGD) adversarial attacks, and then tracking each datapoint. We calculate the total distance moved for each datapoint and the relative change in distance between successive alterations. We show that self-supervised methods learn an RM where alterations lead to large but constant size changes, indicating a smoother RM than fully supervised methods. We then combine these measurements into one metric, the Representation Manifold Quality Metric (RMQM), where larger values indicate larger and less variable step sizes, and show that RMQM correlates positively with performance on downstream tasks.

1. INTRODUCTION

Understanding why deep neural networks generalise so well remains a topic of intense research, despite the practical successes that have been achieved with such networks. Less ambitiously than aiming for a complete understanding, we can search for characteristics that indicate good generalisation. Knowledge of such characteristics can then be incorporated into training methods and open more research avenues. These characteristics can also be used to evaluate and compare networks. Arguably the most successful current theories of generalisation focus on the flatness of the loss surface at the minima (Hochreiter & Schmidhuber, 1997; Dziugaite & Roy, 2017; Dherin et al., 2021) (even though the most straightforward measures of flatness are known to be deficient Dinh et al. ( 2017)). Petzka et al. (2021) expands on this argument and shows that these methods correlate strongly with model performance, and reflect the assumption that the labels are locally constant in feature space. A thorough survey by Jiang et al. (2020) shows that some recent methods are, in fact, negatively correlated with generalisation. To our knowledge, no theory looks at the structural characteristics of the learned Representation Manifold (RM) as a predictor for generalisation. We investigate whether structural characteristics in the RMs correlate with generalisation to task performance. To illustrate the intuition behind our investigation, consider Figure 1 , which represents two RMs, A and B. Assume that each RM is produced by the same architecture, trained on the same dataset; both have a flat minima but are trained with different methods. In the case of A, where the manifold is smooth, the sample representations of the Green class are, on average, closer to other Green class's points. Likewise, presentations of the Red class will, on average, be closer to other Red class's samples. On the other hand, if we consider RM B, there are chasms in the manifold that lead to some sample representations being closer to samples of the other class rather than samples of their own class, as illustrated in the blue patch.

