Offset Rademacher complexities have been shown to provide tight upper bounds for the square loss in a broad class of problems including improper statistical learning and online learning. We show that the offset complexity can be generalized to any loss that satisfies a certain general convexity condition. Further, we show that this condition is closely related to both exponential concavity and self-concordance, unifying apparently disparate results. By a novel geometric argument, many of our bounds translate to improper learning in a non-convex class with Audibert's star algorithm. Thus, the offset complexity provides a versatile analytic tool that covers both convex empirical risk minimization and improper learning under entropy conditions. Applying the method, we recover the optimal rates for proper and improper learning with the p-loss for 1

A.1 Statistics of correlations between different regions and the center pixel We calculate the correlations between image pixels in different log-polar regions and the center pixels on the training set of CIFAR-100. Specifically, for each pixel in each image, we divide its 11 11 neighboring area into different regions by LPSC with 3 distance levels, 8 direction levels, and a growth rate of 2. The center pixels of all areas form the center set. The pixels at the same position of all areas also form a pixel set. For each position, we calculate the correlation score between the corresponding pixel set and the center set. The correlation scores of positions in the same region of all training images are averaged to obtain the correlation score between the region and the center pixel.

Specifically, robustness with only ACM loss is 48.38%, the addition of soft-labels improves it to 49.53%, the addition of mixup improves it to 52.29%, and the addition of both of these components make final robustness to 56.65%. Also, note that only soft labels are not enough to transfer robustness in this case, as shown by KDOnly column. This is in line with the observations of Goldblum et al. [4]. A.4.2 Role of Intermediate Features To understand the role of low, mid, and high-level features, we performed experiments on CIFAR-10 by progressively changing blocks used for distillation. For this ablation study, we kept all the standard settings reported in the Section A.1. Our correspondence of blocks and features is as follows: block 2: low-level features; block 3: mid-level features; block 4: high-level features. Please note that block 1 corresponds to the output of the first layer only. Therefore, we do not call it low-level features.