Managing interprocessor delays in distributed recursive algorithms

Borkar, VS; Phansalkar, V.V.

doi:10.1007/bf02743940

Cited by 6 publications

(6 citation statements)

References 6 publications

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“…Provided that {b(n)} also satisfy the conditions stipulated in [6], the delays do not affect the conclusions for the same reasons as in [6]. This underscores the role of decreasing step-sizes in controlling errors due to delays, first highlighted in [11]. Under (A8), one can mimic the arguments of the preceding section with minor differences to conclude that Theorem 3 continues to hold.…”

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confidence: 57%

Nonlinear Gossip

Mathkar¹,

Borkar²

2016

SIAM J. Control Optim.

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We consider a gossip-based distributed stochastic approximation scheme wherein processors situated at the nodes of a connected graph perform stochastic approximation algorithms, modified further by an additive interaction term equal to a weighted average of iterates at neighboring nodes along the lines of "gossip" algorithms. We allow these averaging weights to be modulated by the iterates themselves. The main result is a Benaim-type meta-theorem characterizing the possible asymptotic behavior in terms of a limiting o.d.e. In particular, this ensures "consensus," which we further strengthen to a form of "dynamic consensus" which implies that they asymptotically track a single common trajectory belonging to an internally chain transitive invariant set of a common o.d.e. that we characterize. We also consider a situation where this averaging is replaced by a fully nonlinear operation and extend the results to this case, which in particular allows us to handle certain projection schemes.1. Introduction. In [30], Tsitsiklis, Athans, and Bertsekas introduced a novel framework for distributed algorithms, ideally suited for distributed stochastic optimization. This included in particular a distributed stochastic approximation scheme with a decreasing step-size combined with an averaging across the processors to ensure "consensus." There has been a lot of development since on such schemes and related variants; see, e.g., [14], [19], [21], [29], [27], to mention a few representative works. The pure averaging component has seen even more explosive development under the rubric of "gossip" or "consensus" algorithms; see [28] for a survey. Related "second order" dynamics also appears in the literature on flocking of mobile agents [32] and synchonization [33] (see [24] for a survey). In this work, we go a step further in a different direction, viz., to replace the simple (linear) averaging mechanism by a nonlinear operation that subsumes the classical case. Specifically, we consider two distinct cases. The first is what may be considered a state-dependent averaging in which the averaging stochastic matrix is modulated by the current values of the iterates. This adds another layer of complication to the analysis and is well motivated by some applications. We establish a dynamic version of consensus characterizing the possible common trajectories which the component iterations jointly track in a potentially nonconvergent scenario. In the second case, we replace averaging by a fully nonlinear operation satisfying some key hypotheses. Just as consensus in averaging leads to a constant vector in the limit, i.e., convergence to the one-dimensional invariant subspace of stochastic matrices, the nonlinear case leads to convergence to

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confidence: 57%

Nonlinear Gossip

Mathkar¹,

Borkar²

2016

SIAM J. Control Optim.

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“…If the simulation-based algorithm is implemented in a distributed manner using several processors, it is natural to have interprocessor communication delays. Mercifully, under very mild conditions these delays do not affect the convergence analysis, as shown in Borkar and Phansalkar [4]. Intuitively, the reason is that the time scaling n -> /" is sublinear and concave and thus squeezes the time intervals of any fixed length into smaller and smaller time intervals as time goes on.…”

Section: Extensionsmentioning

confidence: 93%

“…The analysis of Borkar and Phansalkar [4] allows, in fact, more general delays, namely, random delays satisfying a mild conditional moment condition. The reader is referred to Borkar and Phansalkar [4] for details.…”

Section: Extensionsmentioning

confidence: 99%

Multiscale Stochastic Approximation for Parametric Optimization of Hidden Markov Models

Bhatnagar

Borkar

1997

Prob. Eng. Inf. Sci.

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“…Since (57) holds, the proof for Theorem 3.4 i) is still applicable. As a result, {X k } is a.s. bounded and there exists a positive integer k 0 such that the compact form (15) holds. Then by (55) we obtain that ∞ k=0 γ k ε i,k < ∞ a.s., and hence A5 holds for almost all ω.…”

Section: A Distributed Pcamentioning

confidence: 99%

“…Besides, the performance gap between the distributed and the centralized stochastic approximation algorithms is investigated in [14]. The asynchronous and distributed stochastic approximation algorithms are also addressed in [15], [16] with the components separately estimated at different processors. However, is noticed that almost all aforementioned SA-based distributed algorithms require rather restrictive conditions to guarantee convergence.…”

Section: Introductionmentioning

confidence: 99%

Distributed Stochastic Approximation Algorithm With Expanding Truncations

Lei¹,

Chen

2020

IEEE Trans. Automat. Contr.

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In this work, a novel distributed stochastic approximation algorithm (DSAA) is proposed to seek roots of the sum of local functions, each of which is associated with an agent from the multiple agents connected over a network. At each iteration, each agent updates its estimate for the root utilizing the noisy observations of its local function and the information derived from the neighboring agents. The key difference of the proposed algorithm from the existing ones consists in the expanding truncations (so it is called as DSAAWET), by which the boundedness of the estimates can be guaranteed without imposing the growth rate constraints on the local functions. The estimates generated by DSAAWET are shown to converge almost surely (a.s.) to a consensus set, which belongs to a connected subset of the root set of the sum function. In comparison with the existing results, we impose weaker conditions on the local functions and on the observation noise. We then apply the proposed algorithm to two applications, one from signal processing and the other one from distributed optimization, and establish the almost sure convergence. Numerical simulation results are also included.

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Managing interprocessor delays in distributed recursive algorithms

Abstract: For a class of distributed recursive algorithms, it is shown that a stochastic approximation-like tapering stepsize routine suppresses the effects of interprocessor delays.

Cited by 6 publications

References 6 publications

Nonlinear Gossip

Nonlinear Gossip

Multiscale Stochastic Approximation for Parametric Optimization of Hidden Markov Models

Distributed Stochastic Approximation Algorithm With Expanding Truncations

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