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We seek algorithms for program learning that are both sample-efficient and computationally feasible. Classical results show that targets admitting short program descriptions (e.g., with short "python code") can be learned with a "small" number of examples (scaling with the size of the code) via length-first program enumeration, but the search is exponential in description length. Consequently, Gradient-based training avoids this cost yet can require exponentially many samples on certain short-program families.

To address this gap, we introduce LLM-ERM, a propose-and-verify framework that replaces exhaustive enumeration with an LLM-guided search over candidate programs while retaining ERM-style selection on held-out data. Specifically, we draw k candidates with a pretrained reasoning-augmented LLM, compile and check each on the data, and return the best verified hypothesis, with no feedback, adaptivity, or gradients. Theoretically, we show that coordinate-wise online mini-batch SGD requires many samples to learn certain short programs. Empirically, LLM-ERM solves tasks such as parity variants, pattern matching, and primality testing with as few as 200 samples, while SGD-trained transformers overfit even with 100,000 samples. These results indicate that language-guided program synthesis recovers much of the statistical efficiency of finite-class ERM while remaining computationally tractable, offering a practical route to learning succinct hypotheses beyond the reach of gradient-based training.