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...optimize some objective function known by all agents (possibly up to some additive noise). More recently, numerous works extended this kind of algorithm to more involved multiagent scenarios, see [7]–=-=[19]-=- as a non-exhaustive list. In this context, one seeks to minimize a sum of local private cost functions fi of the agents: min θ∈Rd N∑ i=1 fi(θ) , (3) where for all i, the function fi is supposed to be...

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...}‖V−1‖1/2αn βn = ‖V‖1/2‖(1 + 2‖U‖1/2‖L‖)βn. (5.26) Thus, ∑ n∈N √ λnαn < +∞ and ∑ n∈N λnβn < +∞. Finally, (5.16) and (g) guarantee that supn∈N(1 + τn)γn < 2ϑ. All the assumptions of Proposition 4.4 are therefore satisfied for algorithm (5.17), which concludes the proof. Remark 5.4 (i) Algorithm 5.10 can be viewed as a stochastic version of the primal-dual algorithm investigated in [20, Example 6.4] when the metric is fixed in the latter. Particular cases of such fixed metric primal-algorithm can be found in [12, 16, 30, 34, 35]. (ii) The same type of primal-dual algorithm is investigated in [5, 43] in a different context since in those papers the stochastic nature of the algorithms stems from the random activation of blocks of variables. 22 5.2 Example We illustrate an implementation of Algorithm 5.2 in a simple scenario with H = RN by constructing an example in which the gradient approximation conditions are fulfilled. For every k ∈ {1, . . . , q} and every n ∈ N, set sk,n = ∇j∗k(vk,n) and suppose that (yn)n∈N is almost surely bounded. This assumption is satisfied, in particular, if dom f and (bn)n∈N are bounded. In addition, let (∀n ∈ N) Xn = σ ( x0,v0, (Kn′ , zn′)06n′<mn , (bn′ , cn′...