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Communication-efficient algorithms for decentralized and stochastic optimization

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Abstract

We present a new class of decentralized first-order methods for nonsmooth and stochastic optimization problems defined over multiagent networks. Considering that communication is a major bottleneck in decentralized optimization, our main goal in this paper is to develop algorithmic frameworks which can significantly reduce the number of inter-node communications. Our major contribution is to present a new class of decentralized primal–dual type algorithms, namely the decentralized communication sliding (DCS) methods, which can skip the inter-node communications while agents solve the primal subproblems iteratively through linearizations of their local objective functions. By employing DCS, agents can find an \(\epsilon \)-solution both in terms of functional optimality gap and feasibility residual in \({{\mathcal {O}}}(1/\epsilon )\) (resp., \({{\mathcal {O}}}(1/\sqrt{\epsilon })\)) communication rounds for general convex functions (resp., strongly convex functions), while maintaining the \({{\mathcal {O}}}(1/\epsilon ^2)\) (resp., \(\mathcal{O}(1/\epsilon )\)) bound on the total number of intra-node subgradient evaluations. We also present a stochastic counterpart for these algorithms, denoted by SDCS, for solving stochastic optimization problems whose objective function cannot be evaluated exactly. In comparison with existing results for decentralized nonsmooth and stochastic optimization, we can reduce the total number of inter-node communication rounds by orders of magnitude while still maintaining the optimal complexity bounds on intra-node stochastic subgradient evaluations. The bounds on the (stochastic) subgradient evaluations are actually comparable to those required for centralized nonsmooth and stochastic optimization under certain conditions on the target accuracy.

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Notes

  1. We can define the norm associated with \({{\mathbf {X}}}\) in a more general way, e.g., \( \Vert {\mathbf {x}}\Vert ^2:={\sum }_{i=1}^m p_i\Vert x_i\Vert _{X_i}^2, \ \forall {\mathbf {x}}=(x_1,\ldots ,x_m) \in {{\mathbf {X}}}, \) for some \(p_i > 0\), \(i = 1, \ldots ,m\). Accordingly, the prox-function \({\mathbf {V}}(\cdot ,\cdot )\) can be defined as \( {\mathbf {V}}({\mathbf {x}},{\mathbf {u}}):={\sum }_{i=1}^m p_iV_i(x_i,u_i), \ \forall {\mathbf {x}},{\mathbf {u}}\in {{\mathbf {X}}}. \) This setting gives us flexibility to choose \(p_i\)’s based on the information of individual \(X_i\)’s, and the possibility to further refine the convergence results.

  2. Observed that we only need condition (3.32) when \(\mu >0\), in other words, the objective functions \(f_i\)’s are strongly convex.

  3. We added the subscript i to emphasize that this inequality holds for any agent \(i \in {\mathcal {N}}\) with \(\phi = f_i\). More specifically, \(\varPhi _i(u_i) := \langle w_i, u_i\rangle + f_i(u_i) + \eta V_i(x_i,u_i)\).

  4. We added the superscript k in \(\delta _i^{t-1,k}\) to emphasize that this error is generated at the k-th outer loop.

  5. We implemented the Erhos-Renyi algorithm based on a MATLAB function written by Pablo Blider, which can be found in https://www.mathworks.com/matlabcentral/fileexchange/4206.

  6. This real dataset can be downloaded from https://www.csie.ntu.edu.tw/~cjlin/libsvmtools/datasets/.

  7. We choose the average of iterates obtained by the first agent as the output solution for distributed dual averaging method in all three problems.

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  1. Department of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA, 30332, USA

    Guanghui Lan, Soomin Lee & Yi Zhou

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  1. Guanghui Lan
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  2. Soomin Lee
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Correspondence to Guanghui Lan.

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This work was funded by National Science Foundation Grants 1637473 and 1637474, and Office of Naval Research grant N00014-16-1-2802.

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Lan, G., Lee, S. & Zhou, Y. Communication-efficient algorithms for decentralized and stochastic optimization. Math. Program. 180, 237–284 (2020). https://doi.org/10.1007/s10107-018-1355-4

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