Human feedback is commonly utilized to finetune AI assistants. But human feedback may also encourage model responses that match user beliefs over truthful ones, a behaviour known as sycophancy. We investigate the prevalence of sycophancy in models whose finetuning procedure made use of human feedback, and the potential role of human preference judgments in such behavior. We first demonstrate that five state-of-the-art AI assistants consistently exhibit sycophancy across four varied free-form text-generation tasks. To understand if human preferences drive this broadly observed behavior, we analyze existing human preference data. We find that when a response matches a user's views, it is more likely to be preferred. Moreover, both humans and preference models (PMs) prefer convincingly-written sycophantic responses over correct ones a non-negligible fraction of the time. Optimizing model outputs against PMs also sometimes sacrifices truthfulness in favor of sycophancy. Overall, our results indicate that sycophancy is a general behavior of state-of-the-art AI assistants, likely driven in part by human preference judgments favoring sycophantic responses.

In this paper, we explain the universal approximation capabilities of deep residual neural networks through geometric nonlinear control. Inspired by recent work establishing links between residual networks and control systems, we provide a general sufficient condition for a residual network to have the power of universal approximation by asking the activation function, or one of its derivatives, to satisfy a quadratic differential equation. Many activation functions used in practice satisfy this assumption, exactly or approximately, and we show this property to be sufficient for an adequately deep neural network with $2n$ states to approximate arbitrarily well, on a compact set and with respect to the supremum norm, any continuous function from $\mathbb{R}^n$ to $\mathbb{R}^n$. We further show this result to hold for very simple architectures for which the weights only need to assume two values. The first key technical contribution consists of relating the universal approximation problem to controllability of an ensemble of control systems corresponding to a residual network and to leverage classical Lie algebraic techniques to characterize controllability. The second technical contribution is to identify monotonicity as the bridge between controllability of finite ensembles and uniform approximability on compact sets.