Lambda-Skip Connections: the architectural component that prevents Rank Collapse

Rank collapse, a phenomenon where embedding vectors in sequence models rapidly converge to a uniform token or equilibrium state, has recently gained attention in the deep learning literature. This phenomenon leads to reduced expressivity and potential training instabilities due to vanishing gradients. Empirical evidence suggests that architectural components like skip connections, LayerNorm, and MultiLayer Perceptrons (MLPs) play critical roles in mitigating rank collapse. While this issue is well-documented for transformers, alternative sequence models, such as State Space Models (SSMs), which have recently gained prominence, have not been thoroughly examined for similar vulnerabilities. This paper extends the theory of rank collapse from transformers to SSMs using a unifying framework that captures both architectures. We study how a parametrized version of the classic skip connection component, which we call \emph{lambda-skip connections}, provides guarantees for rank collapse prevention. Through analytical results, we present a sufficient condition to guarantee prevention of rank collapse across all the aforementioned architectures. We also study the necessity of this condition via ablation studies and analytical examples. To our knowledge, this is the first study that provides a general guarantee to prevent rank collapse, and that investigates rank collapse in the context of SSMs, offering valuable understanding for both theoreticians and practitioners. Finally, we validate our findings with experiments demonstrating the crucial role of architectural components such as skip connections and gating mechanisms in preventing rank collapse.

Understanding the differences in Foundation Models: Attention, State Space Models, and Recurrent Neural Networks

Softmax attention is the principle backbone of foundation models for various artificial intelligence applications, yet its quadratic complexity in sequence length can limit its inference throughput in long-context settings. To address this challenge, alternative architectures such as linear attention, State Space Models (SSMs), and Recurrent Neural Networks (RNNs) have been considered as more efficient alternatives. While connections between these approaches exist, such models are commonly developed in isolation and there is a lack of theoretical understanding of the shared principles underpinning these architectures and their subtle differences, greatly influencing performance and scalability. In this paper, we introduce the Dynamical Systems Framework (DSF), which allows a principled investigation of all these architectures in a common representation. Our framework facilitates rigorous comparisons, providing new insights on the distinctive characteristics of each model class. For instance, we compare linear attention and selective SSMs, detailing their differences and conditions under which both are equivalent. We also provide principled comparisons between softmax attention and other model classes, discussing the theoretical conditions under which softmax attention can be approximated. Additionally, we substantiate these new insights with empirical validations and mathematical arguments. This shows the DSF's potential to guide the systematic development of future more efficient and scalable foundation models.

Linearly Controlled Language Generation with Performative Guarantees

The increasing prevalence of Large Language Models (LMs) in critical applications highlights the need for controlled language generation strategies that are not only computationally efficient but that also enjoy performance guarantees. To achieve this, we use a common model of concept semantics as linearly represented in an LM's latent space. In particular, we take the view that natural language generation traces a trajectory in this continuous semantic space, realized by the language model's hidden activations. This view permits a control-theoretic treatment of text generation in latent space, in which we propose a lightweight, gradient-free intervention that dynamically steers trajectories away from regions corresponding to undesired meanings. Crucially, we show that this intervention, which we compute in closed form, is guaranteed (in probability) to steer the output into the allowed region. Finally, we demonstrate on a toxicity avoidance objective that the intervention steers language away from undesired content while maintaining text quality.

State Space Models as Foundation Models: A Control Theoretic Overview

In recent years, there has been a growing interest in integrating linear state-space models (SSM) in deep neural network architectures of foundation models. This is exemplified by the recent success of Mamba, showing better performance than the state-of-the-art Transformer architectures in language tasks. Foundation models, like e.g. GPT-4, aim to encode sequential data into a latent space in order to learn a compressed representation of the data. The same goal has been pursued by control theorists using SSMs to efficiently model dynamical systems. Therefore, SSMs can be naturally connected to deep sequence modeling, offering the opportunity to create synergies between the corresponding research areas. This paper is intended as a gentle introduction to SSM-based architectures for control theorists and summarizes the latest research developments. It provides a systematic review of the most successful SSM proposals and highlights their main features from a control theoretic perspective. Additionally, we present a comparative analysis of these models, evaluating their performance on a standardized benchmark designed for assessing a model's efficiency at learning long sequences.

NARRATE: Versatile Language Architecture for Optimal Control in Robotics

The impressive capabilities of Large Language Models (LLMs) have led to various efforts to enable robots to be controlled through natural language instructions, opening exciting possibilities for human-robot interaction The goal is for the motor-control task to be performed accurately, efficiently and safely while also enjoying the flexibility imparted by LLMs to specify and adjust the task through natural language. In this work, we demonstrate how a careful layering of an LLM in combination with a Model Predictive Control (MPC) formulation allows for accurate and flexible robotic control via natural language while taking into consideration safety constraints. In particular, we rely on the LLM to effectively frame constraints and objective functions as mathematical expressions, which are later used in the motor-control module via MPC. The transparency of the optimization formulation allows for interpretability of the task and enables adjustments through human feedback. We demonstrate the validity of our method through extensive experiments on long-horizon reasoning, contact-rich, and multi-object interaction tasks. Our evaluations show that NARRATE outperforms current existing methods on these benchmarks and effectively transfers to the real world on two different embodiments. Videos, Code and Prompts at narrate-mpc.github.io