Nicholas Pischke

Morenikeji Neri, Nicholas Pischke, Thomas Powell

We prove a novel and general result on the asymptotic behavior of stochastic processes which conform to a certain relaxed supermartingale condition. Our result provides quantitative information in the form of an explicit and effective construction of a rate of convergence for this process, both in mean and almost surely, that is moreover highly uniform in that it only depends on very few data of the surrounding objects involved in the iteration. We then apply this result to derive new quantitative versions of well-known concepts and theorems from stochastic approximation, in particular providing effective rates for a variant of the Robbins-Siegmund theorem, Dvoretzky's convergence theorem, as well as the convergence of stochastic quasi-Fejér monotone sequences, the latter of which formulated in a novel and highly general metric context. We utilize the classic and widely studied Robbins-Monro procedure as a template to evaluate our quantitative results and their applicability in greater detail. We conclude by illustrating the breadth of potential further applications with a brief discussion on a variety of other well-known iterative procedures from stochastic approximation. Throughout, we isolate and discuss special cases of our results which allow for the construction of fast, and in particular linear, rates.

Nicholas Pischke, Thomas Powell

We provide abstract, general and highly uniform rates of asymptotic regularity for a generalized stochastic Halpern-style iteration, which incorporates a second mapping in the style of a Krasnoselskii-Mann iteration. This iteration is general in two ways: First, it incorporates stochasticity in a completely abstract way rather than fixing a sampling method; secondly, it includes as special cases stochastic versions of various schemes from the optimization literature, including Halpern's iteration as well as a Krasnoselskii-Mann iteration with Tikhonov regularization terms in the sense of Boţ, Csetnek and Meier. For these specific cases, we in particular obtain linear rates of asymptotic regularity, matching (or improving) the currently best known rates for these iterations in stochastic optimization, and quadratic rates of asymptotic regularity are obtained in the context of inner product spaces for the general iteration. At the end, we briefly sketch how the schemes presented here can be instantiated in the context of reinforcement learning to yield novel methods for Q-learning.

Nicholas Pischke

We define a stochastic variant of the proximal point algorithm in the general setting of nonlinear (separable) Hadamard spaces for approximating zeros of the mean of a stochastically perturbed monotone vector field and prove its convergence under a suitable strong monotonicity assumption, together with a probabilistic independence assumption and a separability assumption on the tangent spaces. As a particular case, our results transfer previous work by P. Bianchi on that method in Hilbert spaces for the first time to Hadamard manifolds. Moreover, our convergence proof is fully effective and allows for the construction of explicit rates of convergence for the iteration towards the (unique) solution both in mean and almost surely. These rates are moreover highly uniform, being independent of most data surrounding the iteration, space or distribution. In that generality, these rates are novel already in the context of Hilbert spaces. Linear nonasymptotic guarantees under additional second-moment conditions on the Yosida approximates and special cases of stochastic convex minimization are discussed.