  Let $p$ denote a generative language model. Let $r$ denote a reward model
that returns a scalar that captures the degree at which a draw from $p$ is
preferred. The goal of language model alignment is to alter $p$ to a new
distribution $\phi$ that results in a higher expected reward while keeping
$\phi$ close to $p.$ A popular alignment method is the KL-constrained
reinforcement learning (RL), which chooses a distribution $\phi_\Delta$ that
maximizes $E_{\phi_{\Delta}} r(y)$ subject to a relative entropy constraint
$KL(\phi_\Delta || p) \leq \Delta.$ Another simple alignment method is
best-of-$N$, where $N$ samples are drawn from $p$ and one with highest reward
is selected. In this paper, we offer a closed-form characterization of the
optimal KL-constrained RL solution. We demonstrate that any alignment method
that achieves a comparable trade-off between KL divergence and reward must
approximate the optimal KL-constrained RL solution in terms of relative
entropy. To further analyze the properties of alignment methods, we introduce
two simplifying assumptions: we let the language model be memoryless, and the
reward model be linear. Although these assumptions may not reflect complex
real-world scenarios, they enable a precise characterization of the asymptotic
behavior of both the best-of-$N$ alignment, and the KL-constrained RL method,
in terms of information-theoretic quantities. We prove that the reward of the
optimal KL-constrained RL solution satisfies a large deviation principle, and
we fully characterize its rate function. We also show that the rate of growth
of the scaled cumulants of the reward is characterized by a proper Renyi cross
entropy. Finally, we show that best-of-$N$ is asymptotically equivalent to
KL-constrained RL solution by proving that their expected rewards are
asymptotically equal, and concluding that the two distributions must be close
in KL divergence.
