Gradient is All You Need? How Consensus-Based Optimization can be Interpreted as a Stochastic Relaxation of Gradient Descent

arXiv:2306.09778 · cs.LG, cs.NA, math.NA, math.OC, stat.ML · Submitted 2023-06-16 · Read on arXiv

cs.LG, cs.NA, math.NA, math.OC, stat.ML

Submitted: 2023-06-16

Updated: 2026-02-27

Comments: 49 pages, 5 figures

Journal ref: SIAM J. Math. Data Sci. 8.3 (2026)

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Terminology

• Objective Function — 23× in this paper · explained in 2 episodes
• Langevin Dynamics — 18× in this paper · explained in 2 episodes
• Gradient Flow — 8× in this paper · explained in 2 episodes
• Federated Learning — 6× in this paper · explained in 14 episodes
• Generalization — 5× in this paper · explained in 13 episodes
• Loss Function — 5× in this paper · explained in 2 episodes
• Wasserstein Distance — 5× in this paper · explained in 4 episodes
• Adam — 3× in this paper · explained in 2 episodes
• Adaptation — 3× in this paper · explained in 1 episode
• Benchmark — 3× in this paper · explained in 2 episodes
• Reinforcement Learning — 3× in this paper · explained in 4 episodes
• Stochastic Gradient Descent (SGD) — 3× in this paper · explained in 5 episodes
• Convergence Guarantees — 2× in this paper · explained in 3 episodes
• Optimal Transport — 2× in this paper · explained in 5 episodes
• Accuracy — 1× in this paper · explained in 1 episode
• Curse of Dimensionality — 1× in this paper · explained in 6 episodes
• Diffusion Processes — 1× in this paper · explained in 2 episodes
• Initial Data — 1× in this paper · explained in 2 episodes
• Pruning — 1× in this paper · explained in 4 episodes
• Regularization — 1× in this paper · explained in 3 episodes
• Sparsity — 1× in this paper · explained in 4 episodes
• Theoretical Guarantees — 1× in this paper · explained in 2 episodes

Sources

• From Consensus-Based Optimization to Evolution Strategies: Proof of Global Convergence
• CB$^2$O: Consensus-Based Bi-Level Optimization
• No bad local minima: Data independent training error guarantees for multilayer neural networks
• Consensus-based optimization (CBO): Towards Global Optimality in Robotics

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