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We study the complexity of sampling, rounding, and integrating arbitrary logconcave functions given an evaluation oracle. Our new approach provides the first complexity improvements in nearly two decades for general logconcave functions for all three problems, and matches the best-known complexities for the special case of uniform distributions on convex bodies. For the sampling problem, our output guarantees are significantly stronger than previously known, and lead to a streamlined analysis of statistical estimation based on dependent random samples. 

We present a new random walk for uniformly sampling high-dimensional convex bodies. It achieves state-of-the-art runtime complexity with stronger guarantees on the output than previously known, namely in Rényi divergence (which implies TV, KL etc.). The proof departs from known approaches for polytime algorithms for the problem - we utilize a stochastic diffusion perspective to show contraction to the target distribution with the rate of convergence determined by functional isoperimetric constants of the stationary density. 

The connections between (convex) optimization and (logconcave) sampling have been considerably enriched in the past decade with many conceptual and mathematical analogies. For instance, the Langevin algorithm can be viewed as a sampling analogue of gradient descent and has condition-number-dependent guarantees on its performance. In the early 1990s, Nesterov and Nemirovski developed the Interior-Point Method (IPM) for convex optimization based on self-concordant barriers, providing efficient algorithms for structured convex optimization, often faster than the general method. This raises the following question: can we develop an analogous IPM for structured sampling problems? In 2012, Kannan and Narayanan proposed the Dikin walk for uniformly sampling polytopes, and an improved analysis was given in 2020 by Laddha-Lee-Vempala. The Dikin walk uses a local metric defined by a self-concordant barrier for linear constraints. Here we generalize this approach by developing and adapting IPM machinery together with the Dikin walk for poly-time sampling algorithms. Our IPM-based sampling framework provides an efficient warm start and goes beyond uniform distributions and linear constraints. We illustrate the approach on important special cases, in particular giving the fastest algorithms to sample uniform, exponential, or Gaussian distributions on a truncated PSD cone. The framework is general and can be applied to other sampling algorithms. 

We study the convergence rate of discretized Riemannian Hamiltonian Monte Carlo on sampling from distributions in the form of e^{−f(x)} on a convex body M ⊂ R^n. We show that for distributions in the form of e−^{a x} on a polytope with m constraints, the convergence rate of a family of commonly-used integrators is independent of ∥a∥_2 and the geometry of the polytope. In particular, the implicit midpoint method (IMM) and the generalized Leapfrog method (LM) have a mixing time of mn^3 to achieve ϵ total variation distance to the target distribution. These guarantees are based on a general bound on the convergence rate for densities of the form e^{−f(x)} in terms of parameters of the manifold and the integrator. Our theoretical guarantee complements the empirical results of our old result, which shows that RHMC with IMM can sample ill-conditioned, non-smooth and constrained distributions in very high dimension efficiently in practice. 

Graph compression or sparsification is a basic information-theoretic and computational question. A major open problem in this research area is whether $$(1+\epsilon)$$-approximate cut-preserving vertex sparsifiers with size close to the number of terminals exist. As a step towards this goal, we initiate the study of a thresholded version of the problem: for a given parameter $$c$$, find a smaller graph, which we call \emph{connectivity-$$c$$ mimicking network}, which preserves connectivity among $$k$$ terminals exactly up to the value of $$c$$. We show that connectivity-$$c$$ mimicking networks of size $O(kc^4)$ exist and can be found in time $$m(c\log n)^{O(c)}$$. We also give a separate algorithm that constructs such graphs of size $$k \cdot O(c)^{2c}$$ in time $$mc^{O(c)}\log^{O(1)}n$$. These results lead to the first offline data structures for answering fully dynamic $$c$$-edge-connectivity queries for $$c \ge 4$$ in polylogarithmic time per query as well as more efficient algorithms for survivable network design on bounded treewidth graphs.