Steven Cao

Steven Cao, Percy Liang, Gregory Valiant · berkeley

Given only a few observed entries from a low-rank matrix $X$, matrix completion is the problem of imputing the missing entries, and it formalizes a wide range of real-world settings that involve estimating missing data. However, when there are too few observed entries to complete the matrix, what other aspects of the underlying matrix can be reliably recovered? We study one such problem setting, that of "one-sided" matrix completion, where our goal is to recover the right singular vectors of $X$, even in the regime where recovering the left singular vectors is impossible, which arises when there are more rows than columns and very few observations. We propose a natural algorithm that involves imputing the missing values of the matrix $X^TX$ and show that even with only two observations per row in $X$, we can provably recover $X^TX$ as long as we have at least $Ω(r^2 d \log d)$ rows, where $r$ is the rank and $d$ is the number of columns. We evaluate our algorithm on one-sided recovery of synthetic data and low-coverage genome sequencing. In these settings, our algorithm substantially outperforms standard matrix completion and a variety of direct factorization methods.

Steven Cao, Gregory Valiant, Percy Liang

We study the problem of entropy calibration, which asks whether a language model's entropy over generations matches its log loss on human text. Past work found that models are miscalibrated, with entropy per step increasing (and text quality decreasing) as generations grow longer. This error accumulation is a fundamental problem in autoregressive models, and the standard solution is to truncate the distribution, which improves text quality at the cost of diversity. In this paper, we ask: is miscalibration likely to improve with scale, and is it theoretically possible to calibrate without tradeoffs? To build intuition, we first study a simplified theoretical setting to characterize the scaling behavior of miscalibration with respect to dataset size. We find that the scaling behavior depends on the power law exponent of the data distribution -- in particular, for a power law exponent close to 1, the scaling exponent is close to 0, meaning that miscalibration improves very slowly with scale. Next, we measure miscalibration empirically in language models ranging from 0.5B to 70B parameters. We find that the observed scaling behavior is similar to what is predicted by the simplified setting: our fitted scaling exponents for text are close to 0, meaning that larger models accumulate error at a similar rate as smaller ones. This scaling (or, lack thereof) provides one explanation for why we sample from larger models with similar amounts of truncation as smaller models, even though the larger models are of higher quality. However, truncation is not a satisfying solution because it comes at the cost of increased log loss. In theory, is it even possible to reduce entropy while preserving log loss? We prove that it is possible, if we assume access to a black box which can fit models to predict the future entropy of text.