Aligning language models with human preferences is essential for ensuring their safety and reliability. Although most existing approaches assume specific human preference models such as the Bradley-Terry model, this assumption may fail to accurately capture true human preferences, and consequently, these methods lack statistical consistency, i.e., the guarantee that language models converge to the true human preference as the number of samples increases. In contrast, direct density ratio optimization (DDRO) achieves statistical consistency without assuming any human preference models. DDRO models the density ratio between preferred and non-preferred data distributions using the language model, and then optimizes it via density ratio estimation. However, this density ratio is unstable and often diverges, leading to training instability of DDRO. In this paper, we propose a novel alignment method that is both stable and statistically consistent. Our approach is based on the relative density ratio between the preferred data distribution and a mixture of the preferred and non-preferred data distributions. Our approach is stable since this relative density ratio is bounded above and does not diverge. Moreover, it is statistically consistent and yields significantly tighter convergence guarantees than DDRO. We experimentally show its effectiveness with Qwen 2.5 and Llama 3.

In detail, we choose UMAP [15] as the projection algorithm and train the projecting function in Hopper using 64000 transitions sampled by the expert agent. To evaluate a policy, we sample the same number of transitions, and then project them onto a 2-dimensional space by the trained projectingfunction. For empirical estimation, we subsequently discretize the projected 2-dimensional state space into small grid regions, and estimated the distribution via Kernel Density Estimation (KDE) [19]with Gaussian kernel. These twohyperparameters affect the experimental results more significantly. Moreover, as mentioned in Section 6.3, they can be tuned based onthedistribution ofthedataset.

Leveraging distributed computing resources and decentralized data iscrucial, ifnot necessary,for large-scale machine learning applications.

Assume thatthere exists r > 0 such that κ(x,x) r2 for all x X.