The paper introduces an operator calculus for population-based optimization, treating each update rule—such as mutation, selection, or recombination—as an operator. By establishing explicit regularity and small‑step conditions, the authors show that the leading effects of these operators add, allowing convergence guarantees to be reused as modular building blocks. The framework differentiates between finding and retaining good solutions, reducing the population’s mean objective, and concentrating candidates near an optimizer, and it identifies additional approximation conditions required for finite‑budget guarantees. Applications to distribution adaptation, recombinative evolution, and consensus dynamics are verified, including nonconvex cases, and controlled experiments illustrate how component effects vary with population geometry and performance measures.
By Pekka Malo, Lauri Viitasaari, Patrik Nummi, Antti Suominen, Ankur Sinha, Olli Tahvonen
The paper introduces Transport REINFORCE, a model‑free policy‑gradient method for discrete‑time mean‑field control that captures both the dynamics and population‑distribution effects of a policy. It perturbs the population distribution—directly on the simplex for finite state spaces and via a Gaussian‑mixture transport map for continuous spaces—to estimate the missing mean‑field contribution. The authors prove consistency of the perturbed objective and gradient, provide bias and mean‑square error bounds, and demonstrate improved performance over standard REINFORCE on several benchmarks.
By Adonis Jamal, Samy Mekkaoui, Yadh Hafsi, Huy\^en Pham
arXiv:2605. 08488v2 Announce Type: replace-cross Abstract: We develop a unified Lyapunov-integral quadratic constraint (IQC) framework for establishing uniform stability of first-order accelerated optimization algorithms in the $\beta$-smooth and $\gamma$-strongly convex regime.
By Don Li, Dacian Daescu
arXiv:2604. 08580v2 Announce Type: replace-cross Abstract: Reward fine-tuning of diffusion and flow models and sampling from tilted or Boltzmann distributions can both be formulated as stochastic optimal control (SOC) problems, where learning an optimal generative dynamics corresponds to optimizing a control under SDE constraints.
By Carles Domingo-Enrich, Jiequn Han
arXiv:2607. 11005v1 Announce Type: cross Abstract: This paper develops a model-free reinforcement learning framework for continuous--time extended mean field control problems, where both the dynamics and reward may depend on the joint distribution of states and controls.
By Ziheng Cheng, Xin Guo, Huy\^en Pham, Yufei Zhang
arXiv:2606. 17523v1 Announce Type: cross Abstract: Information-Geometric Optimization (IGO) provides a unified framework for black-box optimization by interpreting the adaptation of a search distribution as a natural gradient update.
By Ryosuke Kimura, Youhei Akimoto
arXiv:2307. 10053v5 Announce Type: replace-cross Abstract: In this paper, we focus on providing convergence guarantees for stochastic subgradient methods in minimizing nonsmooth nonconvex functions.
By Nachuan Xiao, Xiaoyin Hu, Kim-Chuan Toh
arXiv:2608. 02844v1 Announce Type: cross Abstract: We develop a class of diffusion-based stochastic particle optimisation methods for loss functions with intractable gradients.
By Jiechen Jackie Zhang, O. Deniz Akyildiz
arXiv:2610.18739v1 Announce Type: cross
Abstract: We develop a rate-transfer method that converts finite-time Lyapunov estimates for an associated stochastic differential equation (SDE) into finite-t...
By Neil Walton
The paper presents a theoretical study of Adam in non‑stationary stochastic optimization, distinguishing two regimes: Euclidean tracking under adaptive strong monotonicity and high‑probability projected stationarity for general smooth objectives. It derives finite‑time bounds that decompose into initialization, objective drift, first‑moment tracking error (β₁), and preconditioner perturbation (β₂), and characterizes burn‑in times for constant and step‑decay schedules. The analysis reveals a noise–drift tradeoff, showing that in noise‑dominated settings Adam’s adaptive mechanisms can improve guarantees, while in drift‑dominated settings they may worsen tracking, potentially making vanilla SGD preferable.
By Sharan Sahu, Abir Sarkar, Cameron J. Hogan, Martin T. Wells
The paper develops a diffusion approximation for stochastic gradient descent (SGD) when the optimization target is a functional on the Wasserstein space ℝ2. By lifting the problem to a Hilbert space via Lions differentiability, the authors construct a Gaussian random-field approximation whose velocity field matches the mean and covariance of the original stochastic gradient. They prove that this Gaussian approximation achieves second‑order weak accuracy, providing a rigorous basis for replacing sample‑driven randomness with analytically tractable Gaussian fluctuations in stochastic optimization over probability measures.
By Maria Oprea, Qin Li, Yunan Yang
The paper introduces a new framework for genetic algorithms where mutation and recombination operators are guided by machine‑learning optimization rather than random changes. It shows that such operators can improve objective values but at higher computational cost, and demonstrates three key phenomena: the necessity of solution‑pool diversity for parity learning, the simultaneous need for generation, mutation, and recombination to achieve near‑optimal solutions, and a phase transition in Gaussian settings where positive drift yields exponential speedup.
By Anna Brandenberger, Ilan Doron-Arad, Elchanan Mossel