On the interchange of subdifferentiation and conditional expectation for convex functionals

Citations in the corpus

An asynchronous mini-batch algorithm for regularized stochastic optimization (2015, background): “unction that may be nonsmooth and extended real-valued. Let us deﬁne f(x) := E ˘ F(x;˘) = Z F(x;˘)dP(˘): (2) Note that the expectation function f is convex, differentiable, and rf(x) = E ˘[r xF(x;˘)] [40]. We use X? to denote the set of optimal solutions of Problem (1) and ˚? to denote the correspon”
An Asynchronous Mini-Batch Algorithm for Regularized Stochastic Optimization (2016, unknown): “Note that the expectation function f is convex, differentiable, and ∇f(x) = Eξ[∇xF (x, ξ)] [34].”
Dual Averaging Methods for Regularized Stochastic Learning and Online Optimization (2009, unknown): “The above analysis of convergence rates matches that for the p-norm based SMIDAS (Stochastic MIrror Descent Algorithm made Sparse) algorithm developed in Shalev-Shwartz and Tewari (2009). However, like other algorithms of the mirror-descent type, including TG (Langford et al.”
Multi-agent mirror descent for decentralized stochastic optimization (2015, background): “We consider the setting where each agent has access to a stochastic first-order black-box oracle: the agent can query the oracle and receive an independent sample ⇠ from the distribution P(⇠), from which it can evaluate F (x, ⇠); this is equivalent to querying the oracle with a value x and…”