Juan Pablo Contreras

Juan Pablo Contreras, Cristóbal Guzmán, David Martínez-Rubio

We develop algorithms for the optimization of convex objectives that have Hölder continuous $q$-th derivatives by using a $q$-th order oracle, for any $q \geq 1$. Our algorithms work for general norms under mild conditions, including the $\ell_p$-settings for $1\leq p\leq \infty$. We can also optimize structured functions that allow for inexactly implementing a non-Euclidean ball optimization oracle. We do this by developing a non-Euclidean inexact accelerated proximal point method that makes use of an \emph{inexact uniformly convex regularizer}. We show a lower bound for general norms that demonstrates our algorithms are nearly optimal in high-dimensions in the black-box oracle model for $\ell_p$-settings and all $q \geq 1$, even in randomized and parallel settings. This new lower bound, when applied to the first-order smooth case, resolves an open question in parallel convex optimization.

Mario Bravo, Juan Pablo Contreras

We analyze the oracle complexity of the stochastic Halpern iteration with minibatch, where we aim to approximate fixed-points of nonexpansive and contractive operators in a normed finite-dimensional space. We show that if the underlying stochastic oracle has uniformly bounded variance, our method exhibits an overall oracle complexity of $\tilde{O}(\varepsilon^{-5})$, to obtain $\varepsilon$ expected fixed-point residual for nonexpansive operators, improving recent rates established for the stochastic Krasnoselskii-Mann iteration. Also, we establish a lower bound of $Ω(\varepsilon^{-3})$ which applies to a wide range of algorithms, including all averaged iterations even with minibatching. Using a suitable modification of our approach, we derive a $O(\varepsilon^{-2}(1-γ)^{-3})$ complexity bound in the case in which the operator is a $γ$-contraction to obtain an approximation of the fixed-point. As an application, we propose new model-free algorithms for average and discounted reward MDPs. For the average reward case, our method applies to weakly communicating MDPs without requiring prior parameter knowledge.