This paper is concerned with multi-agent optimization problem. A distributed randomized gradient-free mirror descent (DRGFMD) method is developed by introducing a randomized gradient-free oracle in the mirror descent scheme where the non-Euclidean Bregman divergence is used. The classical gradient descent method is generalized without using subgradient information of objective functions. The proposed algorithm is the first distributed non-Euclidean zeroth-order method which achieves an O( 1 √ T ) convergence rate, recovering the best known optimal rate of distributed compact constrained convex optimization. Also, the DRGFMD algorithm achieves an O( ln T T ) convergence rate for the strongly convex constrained optimization case. The rate matches the best known non-compact constraint result. Moreover, a decentralized reciprocal weighted average approximating sequence is investigated and first used in distributed algorithm. A class of convergence rates are also achieved for the algorithm with weighted averaging (DRGFMD-WA). The technique on constructing the decentralized weighted average sequence provides new insight in searching for minimizers in distributed algorithms.

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Regularization schemes for regression have been widely studied in learning theory and inverse problems. In this paper, we study regularized distribution regression (DR) which involves two stages of sampling, and aims at regressing from probability measures to real-valued responses by regularization over a reproducing kernel Hilbert space. Many important tasks in statistical learning and inverse problems can be treated in this framework. Examples include multi-instance learning and point estimation for problems without analytical solutions. Recently, theoretical analysis on DR has been carried out via kernel ridge regression and several interesting learning behaviors have been observed. However, the topic has not been explored and understood beyond the least squares based DR. By introducing a robust loss function l σ for two-stage sampling problems, we present a novel robust distribution regression (RDR) scheme. With a windowing function V and a scaling parameter σ which can be appropriately chosen, l σ can include a wide range of commonly used loss functions that enrich the theme of DR. Moreover, the loss l σ is not necessarily convex, which enlarges the regression class (least squares) in the literature of DR. Learning rates in different regularity ranges of the regression function are comprehensively studied and derived via integral operator techniques. The scaling parameter σ is shown to be crucial in providing robustness and satisfactory learning rates of RDR.

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