Aman Bhargava

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.

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.

Pantelis Vafidis, Aman Bhargava, Antonio Rangel

Intelligent perception and interaction with the world hinges on internal representations that capture its underlying structure (''disentangled'' or ''abstract'' representations). Disentangled representations serve as world models, isolating latent factors of variation in the world along approximately orthogonal directions, thus facilitating feature-based generalization. We provide experimental and theoretical results guaranteeing the emergence of disentangled representations in agents that optimally solve multi-task evidence accumulation classification tasks, canonical in the neuroscience literature. The key conceptual finding is that, by producing accurate multi-task classification estimates, a system implicitly represents a set of coordinates specifying a disentangled representation of the underlying latent state of the data it receives. The theory provides conditions for the emergence of these representations in terms of noise, number of tasks, and evidence accumulation time. We experimentally validate these predictions in RNNs trained to multi-task, which learn disentangled representations in the form of continuous attractors, leading to zero-shot out-of-distribution (OOD) generalization in predicting latent factors. We demonstrate the robustness of our framework across autoregressive architectures, decision boundary geometries and in tasks requiring classification confidence estimation. We find that transformers are particularly suited for disentangling representations, which might explain their unique world understanding abilities. Overall, our framework establishes a formal link between competence at multiple tasks and the formation of disentangled, interpretable world models in both biological and artificial systems, and helps explain why ANNs often arrive at human-interpretable concepts, and how they both may acquire exceptional zero-shot generalization capabilities.