Unraveling the Gradient Descent Dynamics of Transformers

While the Transformer architecture has achieved remarkable success across various domains, a thorough theoretical foundation explaining its optimization dynamics is yet to be fully developed. In this study, we aim to bridge this understanding gap by answering the following two core questions: (1) Which types of Transformer architectures allow Gradient Descent (GD) to achieve guaranteed convergence? and (2) Under what initial conditions and architectural specifics does the Transformer achieve rapid convergence during training? By analyzing the loss landscape of a single Transformer layer using Softmax and Gaussian attention kernels, our work provides concrete answers to these questions. Our findings demonstrate that, with appropriate weight initialization, GD can train a Transformer model (with either kernel type) to achieve a global optimal solution, especially when the input embedding dimension is large. Nonetheless, certain scenarios highlight potential pitfalls: training a Transformer using the Softmax attention kernel may sometimes lead to suboptimal local solutions. In contrast, the Gaussian attention kernel exhibits a much favorable behavior. Our empirical study further validate the theoretical findings.

To Supervise or Not: How to Effectively Learn Wireless Interference Management Models?

Machine learning has become successful in solving wireless interference management problems. Different kinds of deep neural networks (DNNs) have been trained to accomplish key tasks such as power control, beamforming and admission control. There are two popular training paradigms for such DNNs-based interference management models: supervised learning (i.e., fitting labels generated by an optimization algorithm) and unsupervised learning (i.e., directly optimizing some system performance measure). Although both of these paradigms have been extensively applied in practice, due to the lack of any theoretical understanding about these methods, it is not clear how to systematically understand and compare their performance. In this work, we conduct theoretical studies to provide some in-depth understanding about these two training paradigms. First, we show a somewhat surprising result, that for some special power control problem, the unsupervised learning can perform much worse than its supervised counterpart, because it is more likely to stuck at some low-quality local solutions. We then provide a series of theoretical results to further understand the properties of the two approaches. Generally speaking, we show that when high-quality labels are available, then the supervised learning is less likely to be stuck at a solution than its unsupervised counterpart. Additionally, we develop a semi-supervised learning approach which properly integrates these two training paradigms, and can effectively utilize limited number of labels to find high-quality solutions. To our knowledge, these are the first set of theoretical results trying to understand different training approaches in learning-based wireless communication system design.