Adaptive Markov chains are an important class of Monte Carlo methods for sampling from probability distributions. The time evolution of adaptive algorithms depends on past samples, and thus these algorithms are non-Markovian. Although there has been previous work establishing conditions for their ergodicity, not much is known theoretically about their finite sample properties. In this paper, using a notion of discrete Ricci curvature for Markov kernels introduced by Ollivier, we establish concentration inequalities and finite sample bounds for a class of adaptive Markov chains. After establishing some general results, we give quantitative bounds for `multi-level' adaptive algorithms such as the equi-energy sampler. We also provide the first rigorous proofs that the finite sample properties of an equi-energy sampler are superior to those of related parallel tempering and Metropolis-Hastings samplers after a learning period comparable to their mixing times.