High-dimensional Analysis of Knowledge Distillation: Weak-to-Strong Generalization and Scaling Laws

A growing number of machine learning scenarios rely on knowledge distillation where one uses the output of a surrogate model as labels to supervise the training of a target model. In this work, we provide a sharp characterization of this process for ridgeless, high-dimensional regression, under two settings: (i) model shift, where the surrogate model is arbitrary, and (ii) distribution shift, where the surrogate model is the solution of empirical risk minimization with out-of-distribution data. In both cases, we characterize the precise risk of the target model through non-asymptotic bounds in terms of sample size and data distribution under mild conditions. As a consequence, we identify the form of the optimal surrogate model, which reveals the benefits and limitations of discarding weak features in a data-dependent fashion. In the context of weak-to-strong (W2S) generalization, this has the interpretation that (i) W2S training, with the surrogate as the weak model, can provably outperform training with strong labels under the same data budget, but (ii) it is unable to improve the data scaling law. We validate our results on numerical experiments both on ridgeless regression and on neural network architectures.

Mechanics of Next Token Prediction with Self-Attention

Transformer-based language models are trained on large datasets to predict the next token given an input sequence. Despite this simple training objective, they have led to revolutionary advances in natural language processing. Underlying this success is the self-attention mechanism. In this work, we ask: $\textit{What}$ $\textit{does}$ $\textit{a}$ $\textit{single}$ $\textit{self-attention}$ $\textit{layer}$ $\textit{learn}$ $\textit{from}$ $\textit{next-token}$ $\textit{prediction?}$ We show that training self-attention with gradient descent learns an automaton which generates the next token in two distinct steps: $\textbf{(1)}$ $\textbf{Hard}$ $\textbf{retrieval:}$ Given input sequence, self-attention precisely selects the $\textit{high-priority}$ $\textit{input}$ $\textit{tokens}$ associated with the last input token. $\textbf{(2)}$ $\textbf{Soft}$ $\textbf{composition:}$ It then creates a convex combination of the high-priority tokens from which the next token can be sampled. Under suitable conditions, we rigorously characterize these mechanics through a directed graph over tokens extracted from the training data. We prove that gradient descent implicitly discovers the strongly-connected components (SCC) of this graph and self-attention learns to retrieve the tokens that belong to the highest-priority SCC available in the context window. Our theory relies on decomposing the model weights into a directional component and a finite component that correspond to hard retrieval and soft composition steps respectively. This also formalizes a related implicit bias formula conjectured in [Tarzanagh et al. 2023]. We hope that these findings shed light on how self-attention processes sequential data and pave the path toward demystifying more complex architectures.

From Self-Attention to Markov Models: Unveiling the Dynamics of Generative Transformers

Modern language models rely on the transformer architecture and attention mechanism to perform language understanding and text generation. In this work, we study learning a 1-layer self-attention model from a set of prompts and associated output data sampled from the model. We first establish a precise mapping between the self-attention mechanism and Markov models: Inputting a prompt to the model samples the output token according to a context-conditioned Markov chain (CCMC) which weights the transition matrix of a base Markov chain. Additionally, incorporating positional encoding results in position-dependent scaling of the transition probabilities. Building on this formalism, we develop identifiability/coverage conditions for the prompt distribution that guarantee consistent estimation and establish sample complexity guarantees under IID samples. Finally, we study the problem of learning from a single output trajectory generated from an initial prompt. We characterize an intriguing winner-takes-all phenomenon where the generative process implemented by self-attention collapses into sampling a limited subset of tokens due to its non-mixing nature. This provides a mathematical explanation to the tendency of modern LLMs to generate repetitive text. In summary, the equivalence to CCMC provides a simple but powerful framework to study self-attention and its properties.

Transformers as Algorithms: Generalization and Implicit Model Selection in In-context Learning

In-context learning (ICL) is a type of prompting where a transformer model operates on a sequence of (input, output) examples and performs inference on-the-fly. This implicit training is in contrast to explicitly tuning the model weights based on examples. In this work, we formalize in-context learning as an algorithm learning problem, treating the transformer model as a learning algorithm that can be specialized via training to implement-at inference-time-another target algorithm. We first explore the statistical aspects of this abstraction through the lens of multitask learning: We obtain generalization bounds for ICL when the input prompt is (1) a sequence of i.i.d. (input, label) pairs or (2) a trajectory arising from a dynamical system. The crux of our analysis is relating the excess risk to the stability of the algorithm implemented by the transformer, which holds under mild assumptions. Secondly, we use our abstraction to show that transformers can act as an adaptive learning algorithm and perform model selection across different hypothesis classes. We provide numerical evaluations that (1) demonstrate transformers can indeed implement near-optimal algorithms on classical regression problems with i.i.d. and dynamic data, (2) identify an inductive bias phenomenon where the transfer risk on unseen tasks is independent of the transformer complexity, and (3) empirically verify our theoretical predictions.