Aman Bhargava, Cameron Witkowski, Shi-Zhuo Looi et al.
Prompt engineering is crucial for deploying LLMs but is poorly understood mathematically. We formalize LLM systems as a class of discrete stochastic dynamical systems to explore prompt engineering through the lens of control theory. We offer a mathematical analysis of the limitations on the controllability of self-attention as a function of the singular values of the parameter matrices. We present complementary empirical results on the controllability of a panel of LLMs, including Falcon-7b, Llama-7b, and Falcon-40b. Given initial state $\mathbf x_0$ from Wikitext and prompts of length $k \leq 10$ tokens, we find that the "correct" next token is reachable at least 97% of the time, and that the top 75 most likely next tokens are reachable at least 85% of the time. Intriguingly, short prompt sequences can dramatically alter the likelihood of specific outputs, even making the least likely tokens become the most likely ones. This control-theoretic analysis of LLMs demonstrates the significant and poorly understood role of input sequences in steering output probabilities, offering a foundational perspective for enhancing language model system capabilities.
The introduction of the transformer architecture and the self-attention mechanism has led to an explosive production of language models trained on specific downstream tasks and data domains. With over 200, 000 models in the Hugging Face ecosystem, users grapple with selecting and optimizing models to suit multifaceted workflows and data domains while addressing computational, security, and recency concerns. There is an urgent need for machine learning frameworks that can eliminate the burden of model selection and customization and unleash the incredible power of the vast emerging model library for end users. Here, we propose a context-aware routing system, Tryage, that leverages a language model router for optimal selection of expert models from a model library based on analysis of individual input prompts. Inspired by the thalamic router in the brain, Tryage employs a perceptive router to predict down-stream model performance on prompts and, then, makes a routing decision using an objective function that integrates performance predictions with user goals and constraints that are incorporated through flags (e.g., model size, model recency). Tryage allows users to explore a Pareto front and automatically trade-off between task accuracy and secondary goals including minimization of model size, recency, security, verbosity, and readability. Across heterogeneous data sets that include code, text, clinical data, and patents, the Tryage framework surpasses Gorilla and GPT3.5 turbo in dynamic model selection identifying the optimal model with an accuracy of 50.9% , compared to 23.6% by GPT 3.5 Turbo and 10.8% by Gorilla. Conceptually, Tryage demonstrates how routing models can be applied to program and control the behavior of multi-model LLM systems to maximize efficient use of the expanding and evolving language model ecosystem.
Aman Bhargava, Cameron Witkowski, Alexander Detkov et al.
Two primary ways to change LLM behavior are prompting and weight updates (e.g., fine-tuning). Prompting LLMs is simple and effective, specifying the desired changes explicitly in natural language, whereas weight updates provide more expressive and permanent behavior changes, specified implicitly via training on large datasets. We present a technique for "baking" prompts into the weights of an LLM. Prompt Baking converts a prompt $u$ and initial weights $θ$ to a new set of weights $θ_u$ such that new "baked" LLM behaves like the original prompted LLM. Mathematically, we minimize the KL divergence between $P_θ(\cdot | u)$ and $P_{θ_u}(\cdot)$, where $P$ is the LLM's probability distribution over token sequences. Across all our experiments, we find prompts can be readily baked into weight updates. Baking chain-of-thought prompts improves zero-shot performance on GSM8K, ASDiv, MBPP, ARC-Easy, ARC-Challenge, and CommonsenseQA benchmarks. Baking news headlines directly updates an LLM's knowledge. And baking instructions & personas alleviates "prompt forgetting" over long sequences. Furthermore, stopping baking early creates "half-baked" models, continuously scaling prompt strength. Baked models retain their sensitivity to further prompting and baking, including re-prompting with the baked-in prompt. Surprisingly, the re-prompted models yield further performance gains in instruction following, as well as math reasoning and coding benchmarks. Taking re-prompting and re-baking to the limit yields a form of iterative self-improvement we call Prompt Pursuit, and preliminary results on instruction following exhibit dramatic performance gains. Finally, we discuss implications for AI safety, continuous model updating, enhancing real-time learning capabilities in LLM-based agents, and generating more stable AI personas.
Machine learning problems have an intrinsic geometric structure as central objects including a neural network's weight space and the loss function associated with a particular task can be viewed as encoding the intrinsic geometry of a given machine learning problem. Therefore, geometric concepts can be applied to analyze and understand theoretical properties of machine learning strategies as well as to develop new algorithms. In this paper, we address three seemingly unrelated open questions in machine learning by viewing them through a unified framework grounded in differential geometry. Specifically, we view the weight space of a neural network as a manifold endowed with a Riemannian metric that encodes performance on specific tasks. By defining a metric, we can construct geodesic, minimum length, paths in weight space that represent sets of networks of equivalent or near equivalent functional performance on a specific task. We, then, traverse geodesic paths while identifying networks that satisfy a second objective. Inspired by the geometric insight, we apply our geodesic framework to 3 major applications: (i) Network sparsification (ii) Mitigating catastrophic forgetting by constructing networks with high performance on a series of objectives and (iii) Finding high-accuracy paths connecting distinct local optima of deep networks in the non-convex loss landscape. Our results are obtained on a wide range of network architectures (MLP, VGG11/16) trained on MNIST, CIFAR-10/100. Broadly, we introduce a geometric framework that unifies a range of machine learning objectives and that can be applied to multiple classes of neural network architectures.
Dominik Schildknecht, Anastasia N. Popova, Jack Stellwagen et al.
The control of far-from-equilibrium physical systems, including active materials, has emerged as an important area for the application of reinforcement learning (RL) strategies to derive control policies for physical systems. In active materials, non-linear dynamics and long-range interactions between particles prohibit closed-form descriptions of the system's dynamics and prevent explicit solutions to optimal control problems. Due to fundamental challenges in solving for explicit control strategies, RL has emerged as an approach to derive control strategies for far-from-equilibrium active matter systems. However, an important open question is how the mathematical structure and the physical properties of the active matter systems determine the tractability of RL for learning control policies. In this work, we show that RL can only find good strategies to the canonical active matter task of mixing for systems that combine attractive and repulsive particle interactions. Using mathematical results from dynamical systems theory, we relate the availability of both interaction types with the existence of hyperbolic dynamics and the ability of RL to find homogeneous mixing strategies. In particular, we show that for drag-dominated translational-invariant particle systems, hyperbolic dynamics and, therefore, mixing requires combining attractive and repulsive interactions. Broadly, our work demonstrates how fundamental physical and mathematical properties of dynamical systems can enable or constrain reinforcement learning-based control.
Living neural networks in our brains autonomously self-organize into large, complex architectures during early development to result in an organized and functional organic computational device. A key mechanism that enables the formation of complex architecture in the developing brain is the emergence of traveling spatio-temporal waves of neuronal activity across the growing brain. Inspired by this strategy, we attempt to efficiently self-organize large neural networks with an arbitrary number of layers into a wide variety of architectures. To achieve this, we propose a modular tool-kit in the form of a dynamical system that can be seamlessly stacked to assemble multi-layer neural networks. The dynamical system encapsulates the dynamics of spiking units, their inter/intra layer interactions as well as the plasticity rules that control the flow of information between layers. The key features of our tool-kit are (1) autonomous spatio-temporal waves across multiple layers triggered by activity in the preceding layer and (2) Spike-timing dependent plasticity (STDP) learning rules that update the inter-layer connectivity based on wave activity in the connecting layers. Our framework leads to the self-organization of a wide variety of architectures, ranging from multi-layer perceptrons to autoencoders. We also demonstrate that emergent waves can self-organize spiking network architecture to perform unsupervised learning, and networks can be coupled with a linear classifier to perform classification on classic image datasets like MNIST. Broadly, our work shows that a dynamical systems framework for learning can be used to self-organize large computational devices.
Living neural networks emerge through a process of growth and self-organization that begins with a single cell and results in a brain, an organized and functional computational device. Artificial neural networks, however, rely on human-designed, hand-programmed architectures for their remarkable performance. Can we develop artificial computational devices that can grow and self-organize without human intervention? In this paper, we propose a biologically inspired developmental algorithm that can 'grow' a functional, layered neural network from a single initial cell. The algorithm organizes inter-layer connections to construct a convolutional pooling layer, a key constituent of convolutional neural networks (CNN's). Our approach is inspired by the mechanisms employed by the early visual system to wire the retina to the lateral geniculate nucleus (LGN), days before animals open their eyes. The key ingredients for robust self-organization are an emergent spontaneous spatiotemporal activity wave in the first layer and a local learning rule in the second layer that 'learns' the underlying activity pattern in the first layer. The algorithm is adaptable to a wide-range of input-layer geometries, robust to malfunctioning units in the first layer, and so can be used to successfully grow and self-organize pooling architectures of different pool-sizes and shapes. The algorithm provides a primitive procedure for constructing layered neural networks through growth and self-organization. Broadly, our work shows that biologically inspired developmental algorithms can be applied to autonomously grow functional 'brains' in-silico.
The inverse statistical problem of finding direct interactions in complex networks is difficult. In the natural sciences, well-controlled perturbation experiments are widely used to probe the structure of complex networks. However, our understanding of how and why perturbations aid inference remains heuristic, and we lack automated procedures that determine network structure by combining inference and perturbation. Therefore, we propose a general mathematical framework to study inference with iteratively applied perturbations. Using the formulation of information geometry, our framework quantifies the difficulty of inference and the information gain from perturbations through the curvature of the underlying parameter manifold, measured by Fisher information. We apply the framework to the inference of spin network models and find that designed perturbations can reduce the sampling complexity by $10^6$-fold across a variety of network architectures. Physically, our framework reveals that perturbations boost inference by causing a network to explore previously inaccessible states. Optimal perturbations break spin-spin correlations within a network, increasing the information available for inference and thus reducing sampling complexity by orders of magnitude. Our active learning framework could be powerful in the analysis of complex networks as well as in the rational design of experiments.