The invariance principle from causality is at the heart of notable approaches such as invariant risk minimization (IRM) that seek to address out-of-distribution (OOD) generalization failures. Despite the promising theory, invariance principle-based approaches fail in common classification tasks, where invariant (causal) features capture all the information about the label. Are these failures due to the methods failing to capture the invariance? Or is the invariance principle itself insufficient? To answer these questions, we revisit the fundamental assumptions in linear regression tasks, where invariance-based approaches were shown to provably generalize OOD. In contrast to the linear regression tasks, we show that for linear classification tasks we need much stronger restrictions on the distribution shifts, or otherwise OOD generalization is impossible. Furthermore, even with appropriate restrictions on distribution shifts in place, we show that the invariance principle alone is insufficient. We prove that a form of the information bottleneck constraint along with invariance helps address key failures when invariant features capture all the information about the label and also retains the existing success when they do not. We propose an approach that incorporates both of these principles and demonstrate its effectiveness in several experiments.
We present a smoothly broken power law functional form that accurately models and extrapolates the scaling behaviors of deep neural networks (i.e. how the evaluation metric of interest varies as the amount of compute used for training, number of model parameters, training dataset size, or upstream performance varies) for each task within a large and diverse set of upstream and downstream tasks, in zero-shot, prompted, and fine-tuned settings. This set includes largescale vision and unsupervised language tasks, diffusion generative modeling of images, arithmetic, and reinforcement learning. When compared to other functional forms for neural scaling behavior, this functional form yields extrapolations of scaling behavior that are considerably more accurate (root mean squared log error of its extrapolations are 0.86 times that of previous state-of-the-art on average) on this set. Moreover, this functional form accurately models and extrapolates scaling behavior that other functional forms are incapable of expressing such as the non-monotonic transitions present in the scaling behavior of phenomena such as double descent and the delayed, sharp inflection points present in the scaling behavior of tasks such as arithmetic. Code is available at https: //github.com/ethancaballero/broken_neural_scaling_laws
Empirical science of neural scaling laws is a rapidly growing area of significant importance to the future of machine learning, particularly in the light of recent breakthroughs achieved by large-scale pre-trained models such as GPT-3, CLIP and DALL-e. Accurately predicting the neural network performance with increasing resources such as data, compute and model size provides a more comprehensive evaluation of different approaches across multiple scales, as opposed to traditional point-wise comparisons of fixed-size models on fixed-size benchmarks, and, most importantly, allows for focus on the best-scaling, and thus most promising in the future, approaches. In this work, we consider a challenging problem of few-shot learning in image classification, especially when the target data distribution in the few-shot phase is different from the source, training, data distribution, in a sense that it includes new image classes not encountered during training. Our current main goal is to investigate how the amount of pre-training data affects the few-shot generalization performance of standard image classifiers. Our key observations are that (1) such performance improvements are well-approximated by power laws (linear log-log plots) as the training set size increases, (2) this applies to both cases of target data coming from either the same or from a different domain (i.e., new classes) as the training data, and (3) few-shot performance on new classes converges at a faster rate than the standard classification performance on previously seen classes. Our findings shed new light on the relationship between scale and generalization.
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