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Relating Misfit to Gain in Weak-to-Strong Generalization Beyond the Squared Loss

Abhijeet Mulgund, Chirag Pabbaraju

TL;DR

This work extends the misfit-based understanding of weak-to-strong generalization from regression with squared loss to tasks governed by Bregman divergences, including classification via cross-entropy. It shows that, under realizability and convexity, the gain in target performance is bounded below by the misfit between strong and weak predictors, with an additional error term that vanishes as the number of combined strong components $k$ grows. When the strong class is non-convex, the authors prove that convex combinations of $k$ strong components still yield near-misfit–gain, with an $O(\sqrt{c/k})$ correction for $c$ classes. The theory is validated through synthetic experiments and real NLP/vision tasks, demonstrating that the observed gain tracks the KL misfit and improves with larger $k$, supporting the proposed training recipe for weak-to-strong generalization in practical settings.

Abstract

The paradigm of weak-to-strong generalization constitutes the training of a strong AI model on data labeled by a weak AI model, with the goal that the strong model nevertheless outperforms its weak supervisor on the target task of interest. For the setting of real-valued regression with the squared loss, recent work quantitatively characterizes the gain in performance of the strong model over the weak model in terms of the misfit between the strong and weak model. We generalize such a characterization to learning tasks whose loss functions correspond to arbitrary Bregman divergences when the strong class is convex. This extends the misfit-based characterization of performance gain in weak-to-strong generalization to classification tasks, as the cross-entropy loss can be expressed in terms of a Bregman divergence. In most practical scenarios, however, the strong model class may not be convex. We therefore weaken this assumption and study weak-to-strong generalization for convex combinations of $k$ strong models in the strong class, in the concrete setting of classification. This allows us to obtain a similar misfit-based characterization of performance gain, upto an additional error term that vanishes as $k$ gets large. Our theoretical findings are supported by thorough experiments on synthetic as well as real-world datasets.

Relating Misfit to Gain in Weak-to-Strong Generalization Beyond the Squared Loss

TL;DR

This work extends the misfit-based understanding of weak-to-strong generalization from regression with squared loss to tasks governed by Bregman divergences, including classification via cross-entropy. It shows that, under realizability and convexity, the gain in target performance is bounded below by the misfit between strong and weak predictors, with an additional error term that vanishes as the number of combined strong components grows. When the strong class is non-convex, the authors prove that convex combinations of strong components still yield near-misfit–gain, with an correction for classes. The theory is validated through synthetic experiments and real NLP/vision tasks, demonstrating that the observed gain tracks the KL misfit and improves with larger , supporting the proposed training recipe for weak-to-strong generalization in practical settings.

Abstract

The paradigm of weak-to-strong generalization constitutes the training of a strong AI model on data labeled by a weak AI model, with the goal that the strong model nevertheless outperforms its weak supervisor on the target task of interest. For the setting of real-valued regression with the squared loss, recent work quantitatively characterizes the gain in performance of the strong model over the weak model in terms of the misfit between the strong and weak model. We generalize such a characterization to learning tasks whose loss functions correspond to arbitrary Bregman divergences when the strong class is convex. This extends the misfit-based characterization of performance gain in weak-to-strong generalization to classification tasks, as the cross-entropy loss can be expressed in terms of a Bregman divergence. In most practical scenarios, however, the strong model class may not be convex. We therefore weaken this assumption and study weak-to-strong generalization for convex combinations of strong models in the strong class, in the concrete setting of classification. This allows us to obtain a similar misfit-based characterization of performance gain, upto an additional error term that vanishes as gets large. Our theoretical findings are supported by thorough experiments on synthetic as well as real-world datasets.

Paper Structure

This paper contains 30 sections, 9 theorems, 67 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Related Work
  3. Preliminaries
  4. Weak-to-strong Generalization
  5. Notation
  6. Convexity
  7. Bregman Divergences
  8. Cross-Entropy and KL-Divergence
  9. Geometry of Bregman Divergences
  10. Expectations of Bregman Divergences
  11. Main Results
  12. Experiments
  13. Synthetic Data Experiments
  14. Pretraining.
  15. Weak Model Finetuning.
  16. ...and 15 more sections

Key Result

Theorem 4.1

Let $\psi: {\mathbb{R}}^n \to \overline{\mathbb{R}}^{+}$ be a proper convex function s.t. $U_\psi \neq \emptyset$. Let $h_s: \mathcal{X} \to {\mathbb{R}}^{d_s}$ and $h_w: \mathcal{X} \to {\mathbb{R}}^{d_w}$ be the strong and weak learner representations respectively. Let $f_w: {\mathbb{R}}^{d_w} \to then for any $\epsilon > 0$, there exists $\delta > 0$ such that for all $f_s \in \mathcal{F}$ that

Figures (4)

  • Figure 1: Synthetic data experiments. The Gain and Misfit closely track each other. For $c=100$, we see that the correlation between misfit and gain weakens, suggested also by the $O(\sqrt{c / k})$ error term from \ref{['thm:multi-main-result']}.
  • Figure 2: Weak model (gpt2) is trained once on true labels and thereafter fixed. Each strong model is trained on the weak labels (on a held-out set separate from that which weak model was trained on) for 10 random initializations of the $k$-convex combination of logistic regression heads; we plot the average test loss/misfit across these 10 runs, along with the standard deviations as the error bars.
  • Figure 3: (a), (b) Weak model (AlexNet) is trained once on true labels and thereafter fixed; we report numbers averaged over 10 runs of weak-to-strong training. (c) We observe that the difference between misfit and gain decreases as $k$ increases, and also that the decrease slows down with increasing $k$. (d) The test loss of the weak and strong model is measured not with respect to the ground truth test data, but instead with respect to the predictions of the best strong model on the test data. This is done to ensure realizability.
  • Figure 4: For all datasets except ImageNet, we observe that the difference between misfit and gain decreases as $k$ increases, and also that the decrease slows down with increasing $k$. Since ImageNet has $c=1000$ classes, and we only consider values of $k$ till $100$, we suspect that the $c$ term dominates in the $O(\sqrt{c/k})$ error.

Theorems & Definitions (21)

  • Definition 3.1: Bregman Divergence
  • Definition 3.3
  • Theorem 4.1: Bregman Misfit-Gain Inequality
  • proof : Proof Sketch
  • Corollary 4.2: Cross-Entropy Misfit-Gain Inequality
  • proof : Proof Sketch
  • Theorem 4.3
  • proof
  • Corollary A.1: Multi-Class Misfit-Gain Inequality
  • proof
  • ...and 11 more