Stanley Wei

Stanley Wei, Alex Damian, Jason D. Lee

Significant recent work has studied the ability of gradient descent to recover a hidden planted direction $θ^\star \in S^{d-1}$ in different high-dimensional settings, including tensor PCA and single-index models. The key quantity that governs the ability of gradient descent to traverse these landscapes is the information exponent $k^\star$ (Ben Arous et al., (2021)), which corresponds to the order of the saddle at initialization in the population landscape. Ben Arous et al., (2021) showed that $n \gtrsim d^{\max(1, k^\star-1)}$ samples were necessary and sufficient for online SGD to recover $θ^\star$, and Ben Arous et al., (2020) proved a similar lower bound for Langevin dynamics. More recently, Damian et al., (2023) showed it was possible to circumvent these lower bounds by running gradient descent on a smoothed landscape, and that this algorithm succeeds with $n \gtrsim d^{\max(1, k^\star/2)}$ samples, which is optimal in the worst case. This raises the question of whether it is possible to achieve the same rate without explicit smoothing. In this paper, we show that Langevin dynamics can succeed with $n \gtrsim d^{ k^\star/2 }$ samples if one considers the average iterate, rather than the last iterate. The key idea is that the combination of noise-injection and iterate averaging is able to emulate the effect of landscape smoothing. We apply this result to both the tensor PCA and single-index model settings. Finally, we conjecture that minibatch SGD can also achieve the same rate without adding any additional noise.

Noam Razin, Zixuan Wang, Hubert Strauss et al. · princeton

The success of Reinforcement Learning from Human Feedback (RLHF) critically depends on the quality of the reward model. However, while this quality is primarily evaluated through accuracy, it remains unclear whether accuracy fully captures what makes a reward model an effective teacher. We address this question from an optimization perspective. First, we prove that regardless of how accurate a reward model is, if it induces low reward variance, then the RLHF objective suffers from a flat landscape. Consequently, even a perfectly accurate reward model can lead to extremely slow optimization, underperforming less accurate models that induce higher reward variance. We additionally show that a reward model that works well for one language model can induce low reward variance, and thus a flat objective landscape, for another. These results establish a fundamental limitation of evaluating reward models solely based on accuracy or independently of the language model they guide. Experiments using models of up to 8B parameters corroborate our theory, demonstrating the interplay between reward variance, accuracy, and reward maximization rate. Overall, our findings highlight that beyond accuracy, a reward model needs to induce sufficient variance for efficient~optimization.

Zihan Zheng, Zerui Cheng, Zeyu Shen et al.

Recent reports claim that large language models (LLMs) now outperform elite humans in competitive programming. Drawing on knowledge from a group of medalists in international algorithmic contests, we revisit this claim, examining how LLMs differ from human experts and where limitations still remain. We introduce LiveCodeBench Pro, a benchmark composed of problems from Codeforces, ICPC, and IOI that are continuously updated to reduce the likelihood of data contamination. A team of Olympiad medalists annotates every problem for algorithmic categories and conducts a line-by-line analysis of failed model-generated submissions. Using this new data and benchmark, we find that frontier models still have significant limitations: without external tools, the best model achieves only 53% pass@1 on medium-difficulty problems and 0% on hard problems, domains where expert humans still excel. We also find that LLMs succeed at implementation-heavy problems but struggle with nuanced algorithmic reasoning and complex case analysis, often generating confidently incorrect justifications. High performance appears largely driven by implementation precision and tool augmentation, not superior reasoning. LiveCodeBench Pro thus highlights the significant gap to human grandmaster levels, while offering fine-grained diagnostics to steer future improvements in code-centric LLM reasoning.

Shuning Shang, Hubert Strauss, Stanley Wei et al.

Training language models via reinforcement learning often relies on imperfect proxy rewards, since ground truth rewards that precisely define the intended behavior are rarely available. Standard metrics for assessing the quality of proxy rewards, such as ranking accuracy, treat incorrect rewards as strictly harmful. In this work, however, we highlight that not all deviations from the ground truth are equal. By theoretically analyzing which outputs attract probability during policy gradient optimization, we categorize reward errors according to their effect on the increase in ground truth reward. The analysis establishes that reward errors, though conventionally viewed as harmful, can also be benign or even beneficial by preventing the policy from stalling around outputs with mediocre ground truth reward. We then present two practical implications of our theory. First, for reinforcement learning from human feedback (RLHF), we develop reward model evaluation metrics that account for the harmfulness of reward errors. Compared to standard ranking accuracy, these metrics typically correlate better with the performance of a language model after RLHF, yet gaps remain in robustly evaluating reward models. Second, we provide insights for reward design in settings with verifiable rewards. A key theme underlying our results is that the effectiveness of a proxy reward function depends heavily on its interaction with the initial policy and learning algorithm.

Stanley Wei, Sadhika Malladi, Sanjeev Arora et al.

Machine unlearning algorithms are increasingly important as legal concerns arise around the provenance of training data, but verifying the success of unlearning is often difficult. Provable guarantees for unlearning are often limited to supervised learning settings. In this paper, we provide the first theoretical guarantees for unlearning in the pre-training and fine-tuning paradigm by studying topic models, simple bag-of-words language models that can be adapted to solve downstream tasks like retrieval and classification. First, we design a provably effective unlearning algorithm for topic models that incurs a computational overhead independent of the size of the original dataset. Our analysis additionally quantifies the deletion capacity of the model -- i.e., the number of examples that can be unlearned without incurring a significant cost in model performance. Finally, we formally extend our analyses to account for adaptation to a given downstream task. In particular, we design an efficient algorithm to perform unlearning after fine-tuning the topic model via a linear head. Notably, we show that it is easier to unlearn pre-training data from models that have been fine-tuned to a particular task, and one can unlearn this data without modifying the base model.

Zixuan Wang, Stanley Wei, Daniel Hsu et al.

The transformer architecture has prevailed in various deep learning settings due to its exceptional capabilities to select and compose structural information. Motivated by these capabilities, Sanford et al. proposed the sparse token selection task, in which transformers excel while fully-connected networks (FCNs) fail in the worst case. Building upon that, we strengthen the FCN lower bound to an average-case setting and establish an algorithmic separation of transformers over FCNs. Specifically, a one-layer transformer trained with gradient descent provably learns the sparse token selection task and, surprisingly, exhibits strong out-of-distribution length generalization. We provide empirical simulations to justify our theoretical findings.