In recent years, large language models (LLMs) have achieved unprecedented success across various disciplines, including natural language processing, computer vision, and reinforcement learning. This success has spurred a flourishing body of research aimed at understanding these models, from both theoretical perspectives such as representation and optimization, and scientific approaches such as interpretability.

To understand LLMs, an important research theme in the machine learning community is to model the input as mathematically structured data (e.g. Markov chains), where we have complete knowledge and control of the data properties. The goal is to use this controlled input to gain valuable insights into what solutions LLMs learn and how they learn them (e.g. induction head). This understanding is crucial, given the increasing ubiquity of the models, especially in safety-critical applications, and our limited understanding of them.

While the aforementioned works using this structured approach provide valuable insights into the inner workings of LLMs, the breadth and diversity of the field make it increasingly challenging for both experts and non-experts to stay abreast. To address this, our tutorial aims to provide a unifying perspective on recent advances in the analysis of LLMs, from a representational-cum-learning viewpoint. To this end, we focus on the two predominant classes of language models that have driven the AI revolution: transformers and recurrent models such as state-space models (SSMs). For these models, we discuss several concrete results, including their representational capacities, optimization landscape, and mechanistic interpretability. Building upon these perspectives, we outline several important future directions in this field, aiming to foster a clearer understanding of language models and to aid in the creation of more efficient architectures.