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.

TREACLE: Thrifty Reasoning via Context-Aware LLM and Prompt Selection

Recent successes in natural language processing have led to the proliferation of large language models (LLMs) by multiple providers. Each LLM offering has different inference accuracy, monetary cost, and latency, and their accuracy further depends on the exact wording of the question (i.e., the specific prompt). At the same time, users often have a limit on monetary budget and latency to answer all their questions, and they do not know which LLMs to choose for each question to meet their accuracy and long-term budget requirements. To navigate this rich design space, we propose TREACLE (Thrifty Reasoning via Context-Aware LLM and Prompt Selection), a reinforcement learning policy that jointly selects the model and prompting scheme while respecting the user's monetary cost and latency constraints. TREACLE uses the problem context, including question text embeddings (reflecting the type or difficulty of a query) and the response history (reflecting the consistency of previous responses) to make smart decisions. Our evaluations on standard reasoning datasets (GSM8K, CSQA, and LLC ) with various LLMs and prompts show that TREACLE enables cost savings of up to 85% compared to baselines while maintaining high accuracy. Importantly, it provides the user with the ability to gracefully trade off accuracy for cost.