Towards Few-Shot Adaptation of Foundation Models via Multitask Finetuning

Foundation models have emerged as a powerful tool for many AI problems. Despite the tremendous success of foundation models, effective adaptation to new tasks, particularly those with limited labels, remains an open question and lacks theoretical understanding. An emerging solution with recent success in vision and NLP involves finetuning a foundation model on a selection of relevant tasks, before its adaptation to a target task with limited labeled samples. In this paper, we study the theoretical justification of this multitask finetuning approach. Our theoretical analysis reveals that with a diverse set of related tasks, this multitask finetuning leads to reduced error in the target task, in comparison to directly adapting the same pretrained model. We quantify the relationship between finetuning tasks and target tasks by diversity and consistency metrics, and further propose a practical task selection algorithm. We substantiate our theoretical claims with extensive empirical evidence. Further, we present results affirming our task selection algorithm adeptly chooses related finetuning tasks, providing advantages to the model performance on target tasks. We believe our study shed new light on the effective adaptation of foundation models to new tasks that lack abundant labels. Our code is available at https://github.com/OliverXUZY/Foudation-Model_Multitask.

Fourier Circuits in Neural Networks: Unlocking the Potential of Large Language Models in Mathematical Reasoning and Modular Arithmetic

In the evolving landscape of machine learning, a pivotal challenge lies in deciphering the internal representations harnessed by neural networks and Transformers. Building on recent progress toward comprehending how networks execute distinct target functions, our study embarks on an exploration of the underlying reasons behind networks adopting specific computational strategies. We direct our focus to the complex algebraic learning task of modular addition involving $k$ inputs. Our research presents a thorough analytical characterization of the features learned by stylized one-hidden layer neural networks and one-layer Transformers in addressing this task. A cornerstone of our theoretical framework is the elucidation of how the principle of margin maximization shapes the features adopted by one-hidden layer neural networks. Let $p$ denote the modulus, $D_p$ denote the dataset of modular arithmetic with $k$ inputs and $m$ denote the network width. We demonstrate that a neuron count of $ m \geq 2^{2k-2} \cdot (p-1) $, these networks attain a maximum $ L_{2,k+1} $-margin on the dataset $ D_p $. Furthermore, we establish that each hidden-layer neuron aligns with a specific Fourier spectrum, integral to solving modular addition problems. By correlating our findings with the empirical observations of similar studies, we contribute to a deeper comprehension of the intrinsic computational mechanisms of neural networks. Furthermore, we observe similar computational mechanisms in the attention matrix of the Transformer. This research stands as a significant stride in unraveling their operation complexities, particularly in the realm of complex algebraic tasks.

Provable Guarantees for Neural Networks via Gradient Feature Learning

Neural networks have achieved remarkable empirical performance, while the current theoretical analysis is not adequate for understanding their success, e.g., the Neural Tangent Kernel approach fails to capture their key feature learning ability, while recent analyses on feature learning are typically problem-specific. This work proposes a unified analysis framework for two-layer networks trained by gradient descent. The framework is centered around the principle of feature learning from gradients, and its effectiveness is demonstrated by applications in several prototypical problems, such as mixtures of Gaussians and parity functions. The framework also sheds light on interesting network learning phenomena such as feature learning beyond kernels and the lottery ticket hypothesis.

When and How Does Known Class Help Discover Unknown Ones? Provable Understanding Through Spectral Analysis

Novel Class Discovery (NCD) aims at inferring novel classes in an unlabeled set by leveraging prior knowledge from a labeled set with known classes. Despite its importance, there is a lack of theoretical foundations for NCD. This paper bridges the gap by providing an analytical framework to formalize and investigate when and how known classes can help discover novel classes. Tailored to the NCD problem, we introduce a graph-theoretic representation that can be learned by a novel NCD Spectral Contrastive Loss (NSCL). Minimizing this objective is equivalent to factorizing the graph's adjacency matrix, which allows us to derive a provable error bound and provide the sufficient and necessary condition for NCD. Empirically, NSCL can match or outperform several strong baselines on common benchmark datasets, which is appealing for practical usage while enjoying theoretical guarantees.

Two Heads are Better than One: Towards Better Adversarial Robustness by Combining Transduction and Rejection

Both transduction and rejection have emerged as important techniques for defending against adversarial perturbations. A recent work by Tram\`er showed that, in the rejection-only case (no transduction), a strong rejection-solution can be turned into a strong (but computationally inefficient) non-rejection solution. This detector-to-classifier reduction has been mostly applied to give evidence that certain claims of strong selective-model solutions are susceptible, leaving the benefits of rejection unclear. On the other hand, a recent work by Goldwasser et al. showed that rejection combined with transduction can give provable guarantees (for certain problems) that cannot be achieved otherwise. Nevertheless, under recent strong adversarial attacks (GMSA, which has been shown to be much more effective than AutoAttack against transduction), Goldwasser et al.'s work was shown to have low performance in a practical deep-learning setting. In this paper, we take a step towards realizing the promise of transduction+rejection in more realistic scenarios. Theoretically, we show that a novel application of Tram\`er's classifier-to-detector technique in the transductive setting can give significantly improved sample-complexity for robust generalization. While our theoretical construction is computationally inefficient, it guides us to identify an efficient transductive algorithm to learn a selective model. Extensive experiments using state of the art attacks (AutoAttack, GMSA) show that our solutions provide significantly better robust accuracy.

Stratified Adversarial Robustness with Rejection

Recently, there is an emerging interest in adversarially training a classifier with a rejection option (also known as a selective classifier) for boosting adversarial robustness. While rejection can incur a cost in many applications, existing studies typically associate zero cost with rejecting perturbed inputs, which can result in the rejection of numerous slightly-perturbed inputs that could be correctly classified. In this work, we study adversarially-robust classification with rejection in the stratified rejection setting, where the rejection cost is modeled by rejection loss functions monotonically non-increasing in the perturbation magnitude. We theoretically analyze the stratified rejection setting and propose a novel defense method -- Adversarial Training with Consistent Prediction-based Rejection (CPR) -- for building a robust selective classifier. Experiments on image datasets demonstrate that the proposed method significantly outperforms existing methods under strong adaptive attacks. For instance, on CIFAR-10, CPR reduces the total robust loss (for different rejection losses) by at least 7.3% under both seen and unseen attacks.

Domain Generalization via Nuclear Norm Regularization

The ability to generalize to unseen domains is crucial for machine learning systems deployed in the real world, especially when we only have data from limited training domains. In this paper, we propose a simple and effective regularization method based on the nuclear norm of the learned features for domain generalization. Intuitively, the proposed regularizer mitigates the impacts of environmental features and encourages learning domain-invariant features. Theoretically, we provide insights into why nuclear norm regularization is more effective compared to ERM and alternative regularization methods. Empirically, we conduct extensive experiments on both synthetic and real datasets. We show that nuclear norm regularization achieves strong performance compared to baselines in a wide range of domain generalization tasks. Moreover, our regularizer is broadly applicable with various methods such as ERM and SWAD with consistently improved performance, e.g., 1.7% and 0.9% test accuracy improvements respectively on the DomainBed benchmark.

The Trade-off between Universality and Label Efficiency of Representations from Contrastive Learning

Pre-training representations (a.k.a. foundation models) has recently become a prevalent learning paradigm, where one first pre-trains a representation using large-scale unlabeled data, and then learns simple predictors on top of the representation using small labeled data from the downstream tasks. There are two key desiderata for the representation: label efficiency (the ability to learn an accurate classifier on top of the representation with a small amount of labeled data) and universality (usefulness across a wide range of downstream tasks). In this paper, we focus on one of the most popular instantiations of this paradigm: contrastive learning with linear probing, i.e., learning a linear predictor on the representation pre-trained by contrastive learning. We show that there exists a trade-off between the two desiderata so that one may not be able to achieve both simultaneously. Specifically, we provide analysis using a theoretical data model and show that, while more diverse pre-training data result in more diverse features for different tasks (improving universality), it puts less emphasis on task-specific features, giving rise to larger sample complexity for down-stream supervised tasks, and thus worse prediction performance. Guided by this analysis, we propose a contrastive regularization method to improve the trade-off. We validate our analysis and method empirically with systematic experiments using real-world datasets and foundation models.

A Theoretical Analysis on Feature Learning in Neural Networks: Emergence from Inputs and Advantage over Fixed Features

An important characteristic of neural networks is their ability to learn representations of the input data with effective features for prediction, which is believed to be a key factor to their superior empirical performance. To better understand the source and benefit of feature learning in neural networks, we consider learning problems motivated by practical data, where the labels are determined by a set of class relevant patterns and the inputs are generated from these along with some background patterns. We prove that neural networks trained by gradient descent can succeed on these problems. The success relies on the emergence and improvement of effective features, which are learned among exponentially many candidates efficiently by exploiting the data (in particular, the structure of the input distribution). In contrast, no linear models on data-independent features of polynomial sizes can learn to as good errors. Furthermore, if the specific input structure is removed, then no polynomial algorithm in the Statistical Query model can learn even weakly. These results provide theoretical evidence showing that feature learning in neural networks depends strongly on the input structure and leads to the superior performance. Our preliminary experimental results on synthetic and real data also provide positive support.