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DOFFO_TR: a Decentralized Objective Function-Free Optimization method with Trust-Region
Stefania Bellavia, Greta Malaspina, Benedetta Morini
TL;DR
The paper addresses decentralized optimization when stepsize tuning, objective-function access, communication, and privacy are limiting factors. It proposes an objective function-free trust-region method within a penalty-based consensus framework, using adaptive steps and local-gradient monitoring. The method is theoretically analyzed and numerically shown to provide effective accuracy–communication trade-offs, with second-order variants offering better performance in ill-conditioned settings.
Problem
Decentralized optimization requires adaptive stepsizes despite unavailable aggregated objective information, while gradient sharing can threaten privacy.
Method
The paper reformulates nonconvex optimization through quadratic penalties and solves each subproblem using an objective function-free trust-region procedure that monitors gradient norms.
Results
The method provides worst-case complexity of order ε^-2|log(ε)| for some variants and ε^-4|log(ε)| for others, while second-order variants outperform the first-order variant under ill-conditioning.
Takeaways & Limitations
The approach achieves effectiveness with a trade-off between iteration count, accuracy, and communication overhead, without requiring agents to share local gradients.
Takeaways & Limitations
The penalty theory assumes connected networks and communication matrices from a finite collection satisfying the stated assumptions.
Abstract
from arXiv · showhide
In this paper, we propose a novel objective function-free trust-region method designed to solve optimization problems over decentralized networks. Unlike traditional approaches that often rely on stepsize tuning, our framework employs a function-free trust-region procedure that enables adaptive selection of the step length. Our approach accommodates first- and second-order models and eliminates the need to share local function values and gradients among agents, thereby enhancing privacy and computational efficiency. On the theoretical side, we establish provable iteration complexity guarantees that, for some variants, match those established for classical centralized trust-region methods. Numerical evaluations demonstrate that our approach achieves a favorable trade-off between performance and efficiency, requiring only moderate communication overhead compared to state-of-the-art methods in the literature.
1 Introduction
The paper targets nonconvex decentralized optimization under limited local information, communication, and privacy constraints. It introduces an objective function-free trust-region framework with adaptive stepsizes, first- or second-order models, and gradient-norm monitoring.
- Problem setting: Decentralized agents cooperatively optimize privately held local functions over a communication network while limiting computational cost and communication traffic.The setting covers applications including robotics, machine learning, federated learning, wireless sensor networks, and IoT.
- Challenges: Fixed stepsizes require unavailable problem- and network-dependent constants, making practical tuning difficult and potentially risking divergence.Classical line-search and trust-region methods also require aggregated objective values that local information cannot provide.
- Privacy: Gradient sharing can expose private data, motivating methods that avoid sharing local gradients in decentralized federated learning.The proposed approach requires only local-gradient norms rather than local gradients themselves.
- Contribution: The proposed method reformulates nonconvex optimization as consensus problems solved through a quadratic penalty sequence and an objective function-free trust-region procedure.The trust-region solver is presented as the main contribution and is described as an initial decentralized adaptation of this scheme.
- Contribution: Trust regions adapt step sizes and permit first- and second-order models, with second-order models requiring more computation but potentially accelerating convergence.The method balances iteration counts against communication overhead through decentralized monitoring and trust-region updates.
- Evaluation: The paper provides iteration-complexity analysis, theoretical support for the penalty approach, and numerical comparison with Gradient Tracking methods.The reported numerical results characterize effectiveness and communication trade-offs.
2 The penalty approach
The penalty approach converts the consensus-constrained decentralized problem into a sequence of unconstrained problems. Under connected-network and communication-matrix assumptions, progressively accurate stationary points yield KKT accumulation points.
- Consensus formulation: The decentralized problem introduces local variables and consensus constraints requiring all agents’ copies to agree.The consensus formulation is equivalent to the original problem through the communication matrix.
- Penalty approach: The penalty method transforms the constrained consensus problem into unconstrained quadratic-penalty subproblems that exploit communication-matrix structure and sparsity.Each subproblem uses a positive penalty parameter βs and communication matrix Ws.
- Assumptions: The network may change with the penalty parameter, provided each network remains connected and its communication matrix belongs to a finite collection satisfying the stated assumptions.The matrices are instantiated, for example, using Metropolis weights.
- Algorithm 2.1: Algorithm 2.1 successively computes an εs-approximate stationary point of Φβs, then decreases both βs and εs.Thus, penalty subproblems become increasingly accurate as the penalty parameter diminishes.
- Guarantee: Every accumulation point of the generated sequence is a KKT optimality point for the consensus problem under the theorem’s network, matrix, and limiting assumptions.The result assumes βs and εs converge to zero and the communication matrices satisfy the prescribed finite-collection condition.
3 A distributed function-free method for the penalty subproblem
DOFFO TR solves each penalty subproblem through decentralized, objective-function-free trust-region updates. Its variants use local models and exchange scalar gradient-norm information rather than full gradients, with periodic or local parameter updates.
- Communication pattern: The aggregated-gradient norm requires network-wide information, whereas local-gradient norms are available without communication after local gradients are formed.The DSF procedure distributes scalar information needed for the aggregated norm.
- Communication pattern: Communication cost for the norm exchange is proportional to the number of nodes and independent of the problem dimension n because nodes share norms instead of gradient vectors.This reduces the payload relative to exchanging full n-dimensional gradients.
- Distributed trust-region iteration: DOFFO TR forms separable local trust-region models from block-diagonal matrices, allowing each node to compute its step independently after model formation.The method supports first-order models with B_i^k=0 and Newton-like models when B_i^k is nonzero.
- Distributed trust-region iteration: Each node computes local gradients, shares updated local iterates with neighbors, and updates adaptive parameters through global or local communication schemes.Global updates occur every C iterations; otherwise, nodes use local information and neighbor exchanges.
- Adaptive parameter updates: The decentralized update rules combine temporary local parameters with neighbors’ values to improve consensus among the adaptive parameters.When progress tests fail, temporary parameters increase according to local gradient information before neighbor averaging.
- Adaptive parameter updates: The condition ∥g(x^k)∥≤η1ω^{k−1} monitors progress toward optimality and identifies successful iterations used in the parameter-update analysis.The successful-iteration set contains indices where this condition holds.
4 Theoretical properties
The analysis establishes convergence and iteration-complexity bounds for DOFFO TR under smoothness, communication, and lower-boundedness assumptions. Complexity depends on the global-update frequency and local-update parameter, with η2=0 recovering the centralized-order bound.
- Assumptions and descent: Assuming continuously differentiable local objectives with Lipschitz gradients, the gradients of the aggregate objective and penalty function are also Lipschitz continuous.The paper uses LF=Σ_i L_i for the aggregate gradient and L=LF plus the consensus-penalty contribution.
- Convergence analysis: Successful iterations yield a decrease in the penalty-function gradient, providing the basis for relating progress to the number of successful iterations.The iteration-complexity analysis counts successful and unsuccessful iterations separately.
- Iteration complexity: The number of unsuccessful iterations between consecutive successful iterations is bounded by combining bounds for iterations with insufficient and sufficiently large adaptive parameters.The resulting bound uses the sets I1 and I2 and communication-mixing parameters.
- Theorem scope: The main complexity theorem applies under assumptions including bounded level sets, objective lower bounds, and conditions on the communication and adaptive-update parameters.A special η1=0 case does not require the additional boundedness and Assumption 4.3 conditions noted for the general result.
- Iteration complexity: O(ε^-4|log(ε)|) is the worst-case iteration order when C>1 and η2≠0, while η2=0 gives O(ε^-2|log(ε)|), analogous to the centralized method.The smaller C is, the smaller the multiplicative complexity constant, but forming the aggregated gradient norm requires more communication.
5 Numerical results
The numerical study evaluates DOFFO TR variants on quadratic and classification problems, including sensitivity to penalty parameters, model order, communication settings, and comparisons with GTA. Second-order variants generally provide the strongest accuracy–communication trade-offs, while network topology and hyperparameter choices affect performance.
- Experimental setup: The experiments test Algorithm 2.1 with first-order DOFFO TR1 and second-order DOFFO TR2 models under multiple communication settings.The tested configurations include C = 1 and C = N, with varying η2 for second-order variants.
- Experimental setup: The study compares DOFFO TR against Gradient Tracking Algorithms using gradient-evaluation budgets and communication traffic as efficiency measures.GTA configurations vary communication steps and local gradient computations, while its stepsize is treated as a hyperparameter.
- Quadratic problems: β0 = 10−2 gives the best quadratic-problem performance across all tested DOFFO TR variants.For β0 ∈ {10−2, 10−1, 1}, second-order procedures solve at least 98% of problems within a factor 10 of the best β choice; at β0 = 10, this holds for approximately 75% of runs.
- Quadratic problems: DOFFO TR2(1), DOFFO TR2(N, 0), and DOFFO TR2(N, 0.1) achieve similar Er performance, while C = N settings reduce communication traffic.Performance degrades for η2 = 0.5 and η2 = 0.9.
- Quadratic problems: DOFFO TR has worst-case iteration complexity O(ϵ−2|log(ϵ)|) for C = 1 or C = N with η2 = 0, versus O(ϵ−4|log(ϵ)|) when C = N and η2 > 0.The experiments also find DOFFO TR1(1) significantly worse than the tested second-order variants in final error and communication traffic.
- Classification problems: On classification problems, DOFFO TR1(1) is effective in classification error but less communication-efficient than DOFFO TR2(1) and DOFFO TR2(N, 0.1).The latter second-order variants offer a good compromise between accuracy and communication traffic, while DOFFO TR1(1) compares well with the tested GTA version among first-order methods.
- Comparison against GTA: DOFFO TR2(1) and DOFFO TR2(N, 0.1) perform well on both error and communication traffic, whereas GTA3(N, 1) has comparable accuracy but significantly higher communication traffic.GTA3(1, N) is the cheapest in total communications but is less accurate and has failures for all tested α values.
6 Conclusions
The paper proposes DOFFO TR for decentralized optimization, combining adaptive trust-region steps with a theoretical complexity analysis and numerical evaluation against Gradient Tracking methods.
- DOFFO TR addresses decentralized optimization using an objective function-free method with adaptive stepsize selection.The approach targets a balance between iteration count and communication overhead.
- The paper analyzes DOFFO TR's worst-case iteration complexity and compares it computationally with recently proposed Gradient Tracking methods.
- Numerical results show the effectiveness of the proposed approach.
A Appendix. Proofs from Sections 4
The appendix supplies proofs and auxiliary lemmas supporting the theoretical results in Section 4, including bounds under different parameter regimes.
- Auxiliary proofs: The appendix proves Lemma 4.6 as part of the theoretical development underlying DOFFO TR.
- Auxiliary lemmas: Lemma A.1 establishes bounds for the scalar bbk defined in (25) under the stated assumptions.
- Proof strategy: The appendix derives intermediate bounds by applying update rules, assumptions, and earlier lemmas across successive iterations.
- Case analysis: Lemma A.3 distinguishes the cases C = 1 or C > 1 with η2 = 0 from C > 1 with η2 > 0.
- Special case: When η1 = 0, Lemma A.4 follows from Lemma A.3 because the relevant set ST reduces to {k0} = {0}.