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Asynchronous Distributed Optimization using a Randomized Alternating Direction Method of Multipliers
Franck Iutzeler, Pascal Bianchi, Philippe Ciblat, Walid Hachem
TL;DR
The paper addresses distributed optimization with private agent costs when synchronous coordination is undesirable. It develops randomized Gauss-Seidel iterations of a Douglas-Rachford operator to obtain asynchronous ADMM-like updates, and proves almost-sure convergence to minimizers under the stated assumptions. Numerical results report that the proposed algorithm outperforms Distributed Gradient Descent in squared error versus primal updates.
Problem
The paper studies how networked agents with private convex costs can reach consensus on the aggregate-cost minimizer despite the limitations of synchronous coordination.
Method
The method applies randomized Gauss-Seidel iterations to a Douglas-Rachford operator associated with two maximal monotone operators, yielding asynchronous ADMM-like updates.
Results
The iterates converge almost surely to a minimizer of the dual problem, and the numerical algorithm outperforms Distributed Gradient Descent in squared error versus primal updates.
Takeaways & Limitations
Random component activation can extend ADMM-style distributed optimization to an asynchronous setting while retaining convergence under the paper's assumptions.
Abstract
from arXiv · showhide
Consider a set of networked agents endowed with private cost functions and seeking to find a consensus on the minimizer of the aggregate cost. A new class of random asynchronous distributed optimization methods is introduced. The methods generalize the standard Alternating Direction Method of Multipliers (ADMM) to an asynchronous setting where isolated components of the network are activated in an uncoordinated fashion. The algorithms rely on the introduction of randomized Gauss-Seidel iterations of a Douglas-Rachford operator for finding zeros of a sum of two monotone operators. Convergence to the sought minimizers is provided under mild connectivity conditions. Numerical results sustain our claims.
I. INTRODUCTION
The paper studies distributed optimization in which agents with private convex costs seek consensus, and motivates asynchronous updates because agents may have heterogeneous data sizes and computational abilities. It introduces randomized asynchronous methods related to proximal splitting and ADMM.
- Problem setting: Agents seek a consensus minimizer of an aggregate objective while each agent knows only its own convex cost function.The local cost can represent the price paid by an agent for the global network state.
- Motivation: Distributed optimization requires agents to update local estimates using private objectives and exchange information with neighbors.The motivating application is cloud learning with data distributed across virtual machines.
- Motivation: Synchronous algorithms require all agents to finish local computations and merge results at each external-clock tick.Variable local-data sizes and heterogeneous computational abilities make this coordination difficult.
- Related methods: Existing methods include first-order optimization with gossip, distributed Newton-Raphson methods, and proximal-splitting methods such as ADMM.The paper focuses on the proximal-splitting class.
- Contribution: The paper introduces asynchronous algorithms in which randomly awakened network components perform local updates without the coordination required by synchronous methods.The algorithms extend distributed optimization toward uncoordinated component activation.
II. DISTRIBUTED OPTIMIZATION ON A GRAPH
The paper models agents on a graph with private convex costs and reformulates consensus optimization over connected components. Under the connectivity assumptions, the reformulated problem has exactly the constant consensus minimizers of the original problem.
- Problem formulation: Each graph agent has a private proper closed convex cost function on a Euclidean space.The network is represented by a finite nonoriented graph.
- Assumptions: The assumptions require each cost to be proper, closed, and convex, with the original infimum finite and attained.These conditions are stated in Assumption 1.
- Assumptions: Every vertex must belong to at least one component, and the union of the induced component subgraphs must be connected.Distinct components may overlap, while the connectivity condition implies that the full graph is connected.
- Reformulation: The reformulation uses a function that is zero when each component variable is constant and infinite otherwise, together with restrictions of a global variable to the components.This encodes consensus within each component.
- Equivalence: Under the connectivity assumption, minimizers of the reformulated problem are exactly constant network assignments whose common value minimizes the original objective.This establishes equivalence between the graph formulation and the original consensus problem.
III. SYNCHRONOUS ADMM
The paper applies standard ADMM to a constrained reformulation of the distributed optimization problem. Under the stated assumptions, the resulting primal sequence converges to a minimizer of the reformulated problem.
- ADMM formulation: The unconstrained problem is rewritten as minimizing f(x) + g(z) subject to z = Mx.The augmented Lagrangian introduces a constant parameter ρ > 0.
- ADMM iterations: ADMM alternates updates of x, z, and the dual variable using the augmented Lagrangian.The iterations are the standard ADMM updates for the constrained formulation.
- Convergence: Under Assumption 1, the sequence generated by the primal update converges to a minimizer of the reformulated problem.This convergence result is cited as an immediate consequence of the standard ADMM theory.
B. Decentralized Implementation
The synchronous ADMM updates can be decomposed into local computations over agents and components, with component-level averaging providing the required coordination. Edge-based components make this averaging especially simple.
- Local decomposition: The primal update is obtained agent by agent by removing terms independent of each agent's variable.The derivation introduces constants that simplify the local optimization.
- Local decomposition: The component update has a decomposable form, enabling implementation across individual components.The paper derives this form after simplifying the second ADMM update.
- Local decomposition: Dual updates are computed for every component and every agent belonging to that component.The componentwise expression follows from the ADMM constraint update.
- Algorithm: The resulting synchronous algorithm has agents compute local primal quantities and components compute their updates at each iteration.The paper lists separate agent and component operations for the implementation.
- Communication: The distributed implementation requires each component to compute an average of the relevant local quantities.For edge components, the two incident vertices can share estimates and agree on their average.
IV. A RANDOMIZED PROXIMAL ALGORITHM
The section introduces monotone operators, resolvents, and proximal point iterations as the framework for analyzing optimization algorithms. For maximal monotone operators, proximal iterations converge whenever the fixed-point set is nonempty.
- A monotone operator is maximal when it is not strictly contained in another monotone operator.
- For a convex function, minimizing the function is equivalent to finding a zero of its subdifferential.
- The resolvent of T is JρT = (I + ρT)^−1, and maximal monotonicity is equivalent to firm non-expansiveness of this resolvent with domain Y.
- Proximal iterations ζk+1 = JρT(ζk) converge to a fixed point whenever T is maximal monotone and fix(JρT) is nonempty.
B. Random Gauss-Seidel iterations
The paper replaces simultaneous operator updates with randomized coordinate updates in a product space. Under independent activation with positive probability for every block, the resulting iterates converge almost surely to a fixed point of the original firmly non-expansive operator.
- Block updates: The product space Y is decomposed into L blocks, and each coordinate update replaces only the selected block with Sℓ(ζ).
- Random activation: Random Gauss-Seidel iterations apply one block operator ˆSξk+1 at each step instead of applying all block updates simultaneously.
- Random activation: The activation variables ξk are independent and identically distributed, with P[ξ1 = ℓ] = pℓ > 0 for every block.
- Convergence: Theorem 2 guarantees that, for any initial value, the randomized iterates converge almost surely to a random variable supported by fix(S).
V. RANDOM ADMM
The paper derives asynchronous ADMM by applying randomized Gauss-Seidel iterations to the Douglas-Rachford operator underlying standard ADMM. The resulting iterations can be expressed as a practical asynchronous ADMM-like algorithm.
- Standard ADMM is a special case of the Douglas-Rachford algorithm, which is itself a special case of a proximal point algorithm.
- Random Gauss-Seidel iterations of the Douglas-Rachford operator produce sequences that converge to the sought solutions and can be written as asynchronous ADMM-like updates.
A. Douglas-Rachford operator
The Douglas-Rachford construction reformulates the dual optimization problem as finding a zero of the sum of two maximal monotone operators. Zeros of this sum are obtained from zeros of a Douglas-Rachford operator through a resolvent step.
- The dual problem's optimality condition is expressed as finding zeros of T + U, with T = −M · ∂f∗ ◦ (−M∗) and U = ∂g∗.
- The Douglas-Rachford operator is constructed from the two monotone operators T and U using a parameter ρ > 0.
- Under Assumption 1, the operator R is maximal monotone and has a nonempty zero set.
- A zero ζ of R yields a zero JρU(ζ) of T + U, so solving the original monotone inclusion reduces to solving for a zero of R followed by a resolvent step.
B. Random Gauss-Seidel Iterations
Randomized Gauss-Seidel updates apply selected blocks of a Douglas-Rachford resolvent, yielding asynchronous iterates that converge almost surely to a minimizer under the stated assumptions. The construction represents each iterate through blockwise averages and resolvent components, then updates only the randomly selected component.
- Convergence: Theorem 3 guarantees that the iterates ζk+1 = Ŝξk+1(ζk) produce λk = JρU(ζk), which converges almost surely to a minimizer.This result holds under Assumptions 1, 2, and 3 for any initial value ζ0.
- Resolvent representation: Each ζ ∈ Z is uniquely represented by (λ, z) ∈ U with λ + ρz = ζ, and λ equals JρU(ζ).The representation also gives explicit component formulas for λ and z.
- Resolvent representation: For each component ℓ, λℓ is obtained by subtracting the component average from ζℓ, specifically λℓ = ζℓ − ¯ζℓ1Aℓ.The average ¯ζℓ is defined over the component Aℓ, and the corresponding z expression follows from z = (ζ − λ)/ρ.
- Operator update: The resolvent output is computed through the monotone operator relation involving T = −M · ∂f∗◦(−M∗), with the component update reduced to equation (13).The derivation uses the Fenchel-Young equality to express the required x through the subdifferential of f.
VI. IMPLEMENTATION EXAMPLE
The implementation activates network edges through randomized agent wake-ups and neighbor selection. Activated endpoint agents perform proximal operations, exchange updated values, and apply the corresponding asynchronous ADMM steps.
- Activation process: At time k + 1, agent v wakes with probability qv and selects a neighbor w uniformly from Nv, activating edge {v, w}.The activated edge corresponds to one component Aℓ in the network decomposition.
- Activation process: The probability that edge {v, w} is activated is qv/|Nv| + qw/|Nw|, which is positive under the stated connectivity condition.Either endpoint can wake and choose the other endpoint.
- Local computation: When an edge is activated, its two endpoint agents perform the specified proximal operations.These operations constitute the local computation associated with the asynchronous ADMM update.
- Communication: After the proximal steps, the two agents exchange xk+1(v) and xk+1(w) and perform the subsequent update operations.Communication is restricted to the activated pair.
- Communication: The communication scheme resembles Random Gossip, which was introduced for distributed averaging.
VII. NUMERICAL RESULTS
The numerical experiment compares synchronous ADMM, asynchronous ADMM, and distributed gradient descent with Random Gossip on a five-node network of agents with distinct quadratic objectives. The plotted squared error shows that the proposed asynchronous ADMM outperforms distributed gradient descent.
- Experimental setup: The experiment uses five agents connected by edges {1,2}, {2,3}, {3,4}, {4,5}, and {5,3}.Each agent maintains a different quadratic convex function and seeks consensus on the minimizer.
- Experimental setup: The comparison includes synchronous ADMM, asynchronous ADMM, and distributed gradient descent with a 1/k stepsize using Random Gossip.
- Numerical results: The figure plots squared error against the number of primal updates for the three considered algorithms.
- Numerical results: The asynchronous ADMM algorithm clearly outperforms distributed gradient descent in the experiment.