Constrained and Composite Sampling via Proximal Sampler

We study two log-concave sampling problems: constrained sampling and composite sampling. First, we consider sampling from a target distribution with density proportional to $\exp(-f(x))$ supported on a convex set $K \subset \mathbb R^d$, where $f$ is convex. The main challenge is enforcing feasibility without degrading mixing. Using an epigraph transformation, we reduce this task to sampling from a uniform distribution up to an one-dimensional exponential tilt over a lifted convex set in $\mathbb R^{d+1}$. We then solve the lifted problem using a proximal sampler. Assuming only a separation oracle for $K$ and a subgradient oracle for $f$, we develop an implementation of the proximal sampler based on the cutting-plane method and rejection sampling. Second, we study composite sampling, where the target is proportional to $\exp(-f(x)-h(x))$ with closed and convex $f$ and $h$. This composite structure is standard in Bayesian inference with $f$ modeling data fidelity and $h$ encoding prior information. We reduce composite sampling via an epigraph lifting of $h$ to constrained sampling in $\mathbb R^{d+1}$, which allows direct application of the constrained sampling algorithm developed in the first part. This reduction results in a double epigraph lifting formulation in $\mathbb R^{d+2}$, on which we apply a proximal sampler. By keeping $f$ and $h$ separate, we further demonstrate how different combinations of oracle access (such as subgradient and proximal) can be leveraged to construct separation oracles for the lifted problem. For both sampling problems, we establish mixing time bounds measured in Rényi and $\chi^2$ divergences. Finally, assuming exact RGO access, we also revisit the proximal sampler and provide a novel variational analysis of its contraction. We formulate the interpolation curves associated with its half-steps as variational problems and identify their solutions through optimal control and strong duality, which leads to contraction of the proximal sampler in Rényi and $\chi^2$ divergences.