Hedging on the Frontier: Learning New Tasks with Few Samples

Modern machine learning pipelines increasingly rely on reusing pretrained and foundation models across downstream tasks. These pretrained models can differ not only in performance but also in how they can be used: some only provide black-box predictions, while others may permit white-box access to internal representations that can be probed or fine-tuned. When deployed to the target domain in the presence of distribution shift, no single strategy, including zero-shot application, fine-tuning, or directly training a target-specific model, is uniformly the best. In this work, we propose Frontier Learning, a framework that treats a library of candidate models spanning different training histories and access regimes as complementary sources of information rather than mutually exclusive alternatives. Frontier Learning constructs a unified target-domain feature by concatenating internal representations from white-box candidates as well as prediction outputs from black-box candidates, then fits a lightweight, regularized supervised learner on this concatenated representation using labeled target data. Because the resulting hypothesis class contains predictors obtained by zero-shot reuse, fine-tuning, and direct training as special cases, empirical risk minimization over the frontier learner is guaranteed to be no worse, on the training sample, than any individual baseline. We evaluate the framework in simulations spanning varying degrees of source-target compatibility and in two real-world distribution-shift settings: visual domain adaptation on DomainNet/VisDA and clinical mortality prediction across intensive care unit domains using MIMIC-IV-Notes. Across all settings, Frontier Learning matches or outperforms the strongest individual reuse strategy, with the largest gains arising precisely when no single baseline is reliable across the range of shift considered.

Transfer learning is an essential technique for many machine learning/AI models of complex structures such as large language models and generative AI. The essence of transfer learning is to leverage knowledge from resolved source tasks for a new target task, especially when the sample size $m$ of the training data for the latter is low. In this work, we rigorously analyze the potential benefit of transfer learning in terms of sample efficiency. Specifically, taking an optimal transport viewpoint of transfer learning, we find that when the data dimension $d$ is higher than $3$, the sample complexity for transfer learning is $O(m^{-(α+1)/d})$, with $α$ indicating the smoothness of the data distribution, as opposed to the $O(m^{-p/d})$ sample complexity for direct learning with $p$ indicating the smoothness of the optimal target model. Our finding theoretically supports a better sample efficiency for transfer learning, when the target task is optimizing over a family of not-so-smooth models (i.e., highly complex networks with the possible use of non-smooth activation functions). Using image classification as an example, we numerically demonstrate the sample efficiency for transfer learning, that is, in the data hungry regime, the model performance can be significantly improved by transfer learning.

Reinforcement learning (RL) has become a central post-training paradigm for eliciting reasoning capabilities in large language models, yet uniform task sampling allocates compute without regard to differences in how tasks respond to optimization. Existing task-valuation methods mostly rely on snapshot-based signals such as current pass rate or reward, which estimate how solvable a task is under the current policy. However, tasks with similar current solvability can still differ substantially in how positively they respond to further training. We study this residual axis as task learnability: a regime-conditional measure of expected positive response to continued training under a fixed RL post-training regime. By analyzing per-task reward trajectories, we find that learnability is reproducible across independently sampled training contexts and predictive of downstream utility. To make this signal practical before training begins, we propose TrajVal, a lightweight probe-based estimator that approximates per-task learnability from a short probe run and two endpoint evaluations. TrajVal can be used either as a standalone static prior for task sampling or as a multiplicative prior for existing online schedulers. Experiments on mathematical and logical reasoning benchmarks across multiple model scales show that TrajVal improves data efficiency over uniform sampling and provides complementary gains when combined with online scheduling methods.