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ADD-OPT: Accelerated Distributed Directed Optimization
Chenguang Xi, Ran Xin, Usman A. Khan
TL;DR
The paper addresses distributed minimization of summed local objectives over strongly-connected directed graphs, where asymmetric communication prevents the direct use of doubly-stochastic methods. It proposes ADD-OPT, analyzes its consensus and optimization behavior through linear relations, and compares its step-size range with DEXTRA. Under strongly-convex objectives with globally Lipschitz-continuous gradients, ADD-OPT converges linearly and supports arbitrarily small positive step-sizes.
Problem
The problem is minimizing a sum of local objective functions over a strongly-connected directed network when each agent has only local-function information.
Method
ADD-OPT combines directed-network information exchange with gradient-based state updates and analyzes convergence through a linear relation involving G and H_k.
Results
O(µ^k), 0 < µ < 1, is the reported convergence rate for ADD-OPT under strongly-convex objectives with globally Lipschitz-continuous gradients.
Takeaways & Limitations
ADD-OPT converges with arbitrarily small positive step-sizes, widening the applicable range relative to existing directed-graph work.
Abstract
from arXiv · showhide
In this paper, we consider distributed optimization problems where the goal is to minimize a sum of objective functions over a multi-agent network. We focus on the case when the inter-agent communication is described by a strongly-connected, \emph{directed} graph. The proposed algorithm, ADD-OPT (Accelerated Distributed Directed Optimization), achieves the best known convergence rate for this class of problems,~$O(μ^{k}),0<μ<1$, given strongly-convex, objective functions with globally Lipschitz-continuous gradients, where~$k$ is the number of iterations. Moreover, ADD-OPT supports a wider and more realistic range of step-sizes in contrast to existing work. In particular, we show that ADD-OPT converges for arbitrarily small (positive) step-sizes. Simulations further illustrate our results.
I. INTRODUCTION
The paper studies distributed minimization of summed local objectives over strongly-connected directed networks, where agents communicate using asymmetric information flow. ADD-OPT addresses limitations of existing directed-graph methods by combining linear convergence under strong convexity and smoothness with arbitrarily small positive step-sizes.
- Problem setting: The objective is for networked agents to minimize the sum of local functions using information exchanged with neighboring agents.Each agent knows only its own objective function.
- Prior work: For strongly-convex objectives with globally Lipschitz-continuous gradients, prior undirected methods can achieve linear convergence.The paper contrasts these results with the more difficult directed-graph setting.
- Directed-network challenge: Directed graphs may lack doubly-stochastic weights, so methods designed for undirected networks do not directly apply.Row-stochasticity supports consensus, while column-stochasticity ensures equal contribution of local gradients.
- Prior work: Existing gradient-based directed methods using diminishing step-sizes converge relatively slowly at O(√(ln k/k)).The cited methods include Gradient-Push and directed distributed gradient descent.
- Gap and contribution: DEXTRA achieves linear convergence on directed graphs but requires a step-size interval whose positive lower bound may require global knowledge.ADD-OPT is proposed to remove this lower-bound restriction.
- Assumptions: ADD-OPT targets strongly-connected digraphs under strongly-convex, smooth local objectives, with agents knowing their out-degrees.The optimal solution is assumed to exist, be bounded, and be unique.
A. ADD-OPT Algorithm
ADD-OPT uses per-agent state variables and weighted exchanges over a directed graph to combine network information with local gradient updates. Its construction uses column-stochastic weights that agents can form from out-degree information.
- State variables: Each agent maintains three vector variables and a scalar variable indexed by the iteration counter.The excerpt identifies the state variables as x, additional vector states, and y.
- Communication: At each iteration, agents assign weights to their states and send the weighted information to their out-neighbors.The weights are associated with directed communication links.
- Updates: Each agent receives information from its in-neighbors and updates its local state using the received values.The step-size α is a positive number whose admissible interval is established later.
- Initialization: The algorithm initializes the scalar states with y_i^0 = 1 and permits arbitrary initial vectors.Under strong connectivity, the scalar states remain nonzero.
- Matrix representation: For analysis, the agent updates are rewritten in matrix form using a weighted adjacency matrix and Kronecker products.The resulting matrix A is column-stochastic, and Y_k is invertible under the stated graph and initialization conditions.
B. Interpretation of ADD-OPT
The paper interprets ADD-OPT’s limiting behavior as simultaneous consensus and optimality, then relates the algorithm to DEXTRA through modified weighting matrices. These modifications yield a wider admissible step-size range.
- Consensus interpretation: Assuming the generated sequences converge, the limiting state satisfies a relation that places the agents’ estimates in a consensus subspace.The argument uses the limiting relation for y and the resulting structure of x.
- Optimality interpretation: Summing the update relations yields the optimality condition for the distributed problem once the limiting state is in consensus.Thus the limiting estimate both reaches consensus and solves the optimization problem.
- Relation to DEXTRA: DEXTRA and ADD-OPT differ in the weighting matrices used for the x-update.ADD-OPT replaces (I + A) with 2A and replaces the modified matrix with A^2.
- Step-size comparison: ADD-OPT has a zero greatest lower bound for its step-size, whereas DEXTRA’s greatest lower bound is strictly positive.The paper links DEXTRA’s positive lower bound to its inability to use the identity matrix without eliminating communication.
III. MAIN RESULT
ADD-OPT’s convergence analysis reduces to a linear relation governed by G and H_k, then establishes linear convergence under a sufficiently small positive step-size. The resulting iterates converge to the unique optimizer at rate O(μ^k), while the admissible upper bound may require global-knowledge estimation.
- III. MAIN RESULT: ADD-OPT’s key linear relation reduces convergence analysis to the spectral radius of G and the decay of H_k.If ρ(G)<1 and H_k decays linearly, the error sequence converges linearly.
- III. MAIN RESULT: ρ(G_α)<1 when the step-size satisfies α ∈ (0, ᾱ).The result follows after characterizing the step-sizes for which G_α can have an eigenvalue equal to one.
- III. MAIN RESULT: Although convergence holds for sufficiently small step-sizes, the upper bound ᾱ depends on global knowledge and may require distributed estimation.Agents must also agree on the same step-size, which may be pre-programmed.
- III. MAIN RESULT: The auxiliary bounds use constants γ_1, γ_2, and γ with values below one to control the relevant matrix powers and error terms.Lemma 5 combines the decay properties associated with Y^k and G.
- III. MAIN RESULT: Under the stated assumptions and α ∈ (0, ᾱ), ADD-OPT’s sequence z^k converges exactly to the unique optimizer z* at a linear rate.The theorem provides a positive constant M bounding the convergence error for every iteration k.
- III. MAIN RESULT: The analysis yields ||t^k||_2 = O(μ^k) for any μ ∈ (γ,1), which supplies the paper’s linear convergence characterization.The theorem transfers this bound to the optimizer sequence through the stated relation between ||z^k−z*||_2 and ||t^k||_2.
IV. AUXILIARY RELATIONS
The auxiliary results establish relations for the average sequences and bounds used in the convergence proof. They also invoke the contraction of fixed-step gradient descent for smooth, strongly-convex objectives.
- Lemma 9 states that fixed-step gradient descent on a smooth, strongly-convex function contracts distance to the optimizer by a fixed ratio.For 0 < α < 2/(nl), the ratio is η = max(|1 − αnl|, |1 − αns|).
- Lemma 7 derives iterative equations governing the average sequences xk and wk.
- The proof uses the initial condition w0 = ∇f0, equivalently w0 = g0, to establish the stated relation.
- Lemma 8 provides inequalities derived from Eq. (24), including a bound involving the convergence of Yk toward Y∞.
V. CONVERGENCE ANALYSIS
The convergence analysis bounds consensus error, optimization error, and gradient-tracking error by recursions involving their previous values and the algorithm’s coefficient matrices.
- The proof bounds consensus, optimization, and gradient-tracking errors linearly in terms of their past values and ∥xk−1∥2.The recursion coefficients are entries of G and Hk−1.
- Step 1 bounds the disagreement ∥xk − Y∞xk∥ using the update relation and Lemma 7(b).
- Step 2 bounds ∥xk − z∗∥2 by interpreting x+ = xk−1 − αhk−1 as a centralized gradient-descent step.The contraction follows from the smooth, strongly-convex gradient-descent result in Lemma 9.
- The analysis uses Lipschitz continuity to control gradient differences and related error terms.
- Step 4 combines the bounds from the preceding steps to complete the convergence proof.
VI. NUMERICAL EXPERIMENTS
The numerical study evaluates ADD-OPT on distributed logistic regression over a directed graph, with each agent holding local labeled training examples.
- The experiments analyze ADD-OPT on a distributed logistic regression problem over a directed graph.
- Each agent has local feature-label pairs, and its objective function is defined from those training examples.
- The experimental setting uses n = 10 agents, mi = 10 examples per agent, and p = 3 features.
A. Convergence rate
The convergence-rate experiment compares ADD-OPT with several distributed optimization algorithms on a directed network. ADD-OPT and DEXTRA converge linearly, whereas the other compared methods converge sub-linearly.
- A. Convergence rate: The experiment compares ADD-OPT, DEXTRA, Gradient-Push, Directed-Distributed Gradient Descent, and Weight Balanced-Distributed Gradient Descent.
- A. Convergence rate: The network topology is the strongly-connected directed network shown in Fig. 1.
- A. Convergence rate: ADD-OPT and DEXTRA have a fast linear convergence rate, while the other methods are sub-linear.
B. Step-size range
ADD-OPT supports arbitrarily small positive step-sizes, while its practical upper bound can substantially exceed the theoretical estimate in the given setting.
- 0.2 is approximately DEXTRA’s greatest lower-bound step-size, whereas ADD-OPT works with sufficiently small step-sizes.
- 8.7/9.57 is the estimated theoretical upper bound for ADD-OPT’s step-size under the stated parameter setting.
- 1.12 is the practical upper bound observed for ADD-OPT’s step-size in the given setting.
- 1/(10l) is the estimated step-size bound when τ = ϵ = y = y− = 1, σ = 0.9, and s ≤ l.
C. Convergence rate vs. step-sizes
ADD-OPT’s convergence behavior depends on its step-size through the spectral radius of Gα, with the fastest observed convergence at α = 0.3.
- α = 0.3 gives the best observed convergence rate because it minimizes the spectral radius ρ(Gα).The residual at the 200th iteration is also evaluated against α.
- Fig. 3 compares ADD-OPT and DEXTRA according to their step-size ranges.
- Fig. 4 presents the range of ADD-OPT’s step-size.
- ρ(Gα) < 1 approximately for α in (0, 0.3), matching the theoretical step-size bound.
- The spectral radius satisfies ρ(G) = 1 at α = 0 and α = α, and ρ(G) < 1 for α ∈ (0, α).
D. Convergence rate vs graph sparsity
For three strongly-connected directed graphs, residuals decrease faster as the number of edges increases from Ga to Gb to Gc.
- Residuals decrease faster from Ga to Gb to Gc as the number of edges increases.
- More communication channels are associated with faster convergence in the compared directed graphs.
- Fig. 6 shows three examples of strongly-connected directed graphs used in the sparsity experiment.