Understanding In-Context Learning for Nonlinear Regression with Transformers: Attention as Featurizer

In-context learning is a remarkable property of transformers and has recently received a lot of interest. In many studies of in-context learning, it has been shown that transformers are capable of implementing solver for linear and non-linear regression problems, in which the most of them implement gradient descent algorithm. However, it is still unclear whether those implementations have actually been acquired through training. In this paper, we construct a transformer with linear self-attention, which in-context learns the least squares estimate in a simple regression task. The point here is that the closed form (analytical) solution is approximately obtained by using layer normalization rather than an approximate solution based on gradient descent algorithm. Then, we show an experimental example, in which our implementation is mainly used in the transformer trained with l1 regularization when the target output is the least squares estimate.

Test-time training (TTT) enhances model performance by explicitly updating designated parameters prior to each prediction to adapt to the test data. While TTT has demonstrated considerable empirical success, its theoretical underpinnings remain limited, particularly for nonlinear models. In this paper, we investigate the combination of TTT with in-context learning (ICL), where the model is given a few examples from the target distribution at inference time. We analyze this framework in the setting of single-index models, where the feature vector is drawn from a hidden low-dimensional subspace. For single-layer transformers trained with gradient-based algorithms and adopting TTT, we establish an upper bound on the prediction risk. Our theory reveals that TTT enables the single-layer transformers to adapt to both the feature vector and the link function, which vary across tasks. This creates a sharp contrast with ICL alone, which is theoretically difficult to adapt to shifts in the link function. Moreover, we provide the convergence rate with respect to the data length, showing the predictive error can be driven arbitrarily close to the noise level as the context size and the network width grow.

Transformers have demonstrated remarkable in-context learning (ICL) capabilities. The strong ICL performance of transformers is commonly believed to arise from their ability to implicitly execute certain algorithms on the context, thereby enhancing prediction and generation. In this work, we investigate how transformers with softmax attention perform in-context learning on linear classification data. We first construct a class of multi-layer transformers that can perform in-context logistic regression, with each layer exactly performing one step of normalized gradient descent on an in-context loss. Then, we show that our constructed transformer can be obtained through (i) training a single self-attention layer supervised by one-step gradient descent, and (ii) recurrently applying the trained layer to obtain a looped model. Training convergence guarantees of the self-attention layer and out-of-distribution generalization guarantees of the looped model are provided. Our results advance the theoretical understanding of ICL mechanism by showcasing how softmax transformers can effectively act as in-context learners.