On the Learn-to-Optimize Capabilities of Transformers in In-Context Sparse Recovery

An intriguing property of the Transformer is its ability to perform in-context learning (ICL), where the Transformer can solve different inference tasks without parameter updating based on the contextual information provided by the corresponding input-output demonstration pairs. It has been theoretically proved that ICL is enabled by the capability of Transformers to perform gradient-descent algorithms (Von Oswald et al., 2023a; Bai et al., 2024). This work takes a step further and shows that Transformers can perform learning-to-optimize (L2O) algorithms. Specifically, for the ICL sparse recovery (formulated as LASSO) tasks, we show that a K-layer Transformer can perform an L2O algorithm with a provable convergence rate linear in K. This provides a new perspective explaining the superior ICL capability of Transformers, even with only a few layers, which cannot be achieved by the standard gradient-descent algorithms. Moreover, unlike the conventional L2O algorithms that require the measurement matrix involved in training to match that in testing, the trained Transformer is able to solve sparse recovery problems generated with different measurement matrices. Besides, Transformers as an L2O algorithm can leverage structural information embedded in the training tasks to accelerate its convergence during ICL, and generalize across different lengths of demonstration pairs, where conventional L2O algorithms typically struggle or fail. Such theoretical findings are supported by our experimental results.

Federated Representation Learning in the Under-Parameterized Regime

Federated representation learning (FRL) is a popular personalized federated learning (FL) framework where clients work together to train a common representation while retaining their personalized heads. Existing studies, however, largely focus on the over-parameterized regime. In this paper, we make the initial efforts to investigate FRL in the under-parameterized regime, where the FL model is insufficient to express the variations in all ground-truth models. We propose a novel FRL algorithm FLUTE, and theoretically characterize its sample complexity and convergence rate for linear models in the under-parameterized regime. To the best of our knowledge, this is the first FRL algorithm with provable performance guarantees in this regime. FLUTE features a data-independent random initialization and a carefully designed objective function that aids the distillation of subspace spanned by the global optimal representation from the misaligned local representations. On the technical side, we bridge low-rank matrix approximation techniques with the FL analysis, which may be of broad interest. We also extend FLUTE beyond linear representations. Experimental results demonstrate that FLUTE outperforms state-of-the-art FRL solutions in both synthetic and real-world tasks.