Learning to Optimize

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Overview

Learning to Optimize studies how data can improve the algorithms and decisions used to solve optimization problems, especially when similar problems recur or standard solvers are costly. Approaches include meta-learning and reinforcement learning for update rules and optimizer selection, unrolled or differentiable methods such as learned ADMM, and learned heuristics or warm starts for mixed-integer and other solvers. These methods are used in settings ranging from neural-network training and black-box search to scheduling, power systems, and model predictive control, with work increasingly combining fast learned predictions with exact optimization or explicit feasibility checks. A central tension is whether methods trained on particular instances can generalize reliably to new problems while preserving solution quality, constraint satisfaction, and the computational guarantees of classical solvers.

Latest papers

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139 papers27 papers in the last four weeks, up 125% on the four weeks before. 0.2% of all new papers.

May 7, 2026

When Descent Is Too Stable: Event-Triggered Hamiltonian Learning to Optimize

Fixed-budget nonconvex optimization can fail not because local descent is unstable, but because it is too stable: after reaching a nearby stationary point, an optimizer may spend the remaining evaluations refining an uninformative local minimum. We formulate this failure mode as a control problem over optimizer dynamics, where the learner must decide when to descend, when to exploit a promising basin, and when stagnation should trigger movement elsewhere. We introduce SHAPE, a structured adaptive port-Hamiltonian task-family optimizer for event-triggered minima hunting under local information. Starting from gradient-descent dynamics, SHAPE lifts optimization to an augmented phase space (q,p)(q, p), where the primal state qq represents the candidate solution, the cotangent variable pp carries directional sensitivity, and a controller uu provides processed information from current gradient oracle. Within each stage, a learned Hamiltonian vector field induces structured local descent; across stages, a fixed event clock in the implementation updates ports and memory when local equilibria are detected, with stage-dependent horizons treated in the analysis as a direct generalization. This design preserves a passivity-compatible structure while allowing the same trained policy to use clean, stochastic, or estimated gradient inputs. Experiments on fixed-budget nonconvex optimization tasks show that SHAPE improves best-so-far performance compared with fixed-policy optimizers. These results suggest that adaptive Hamiltonian energy shaping provides a principled mechanism for balancing descent, exploration, and budget allocation in difficult optimization landscapes.

Yi Wang, Chandrajit Bajaj

1Oden Institute · Department of Computer Science

Dynamical SystemsLearning to OptimizeNonconvex OptimizationEvent-Triggered ControlPort-Hamiltonian SystemsHamiltonian Dynamics
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May 7, 2026

Distributionally-Robust Learning to Optimize

We propose a distributionally robust approach to learning hyperparameters for first-order methods in convex optimization. Given a dataset of problem instances, we minimize a Wasserstein distributionally robust version of the performance estimation problem (PEP) over algorithm parameters such as step sizes. Our framework unifies two extremes: as the robustness radius vanishes, we recover classical learning to optimize (L2O); as it grows, we recover worst-case optimal algorithm design via PEP. We solve the resulting problem with stochastic gradient descent, differentiating through the solution of an inner semidefinite program at each step. We prove high-probability bounds showing that the true risk of the learned algorithm is at most the in-sample L2O optimum plus a slack that shrinks with the sample size, and is no worse than the worst-case PEP bound. On unconstrained quadratic minimization, LASSO, and linear programming benchmarks, our learned algorithms achieve strong out-of-sample performance with certifiable robustness, outperforming both worst-case optimal and vanilla L2O baselines.

Vinit Ranjan, Jisun Park, Bartolomeo Stellato

Department of Operations Research and Financial Engineering, Princeton University. · Department of Operations Research and Financial Engineering, Princeton University, and Research Institute of Mathematics, Seoul National University. · Research Institute of Mathematics, Seoul National University.

Learning to OptimizeDistributionally Robust OptimizationRobust OptimizationWasserstein Distributionally Robust Optimization
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Apr 30, 2026

NLPOpt-Net: A Learning Method for Nonlinear Optimization with Feasibility Guarantees

Nonlinear Parametric Optimization Network (NLPOpt-Net) is an unsupervised learning architecture to solve constrained nonlinear programs (NLP). Given the structure of an NLP, it learns the parametric solution maps with guaranteed constraint satisfaction. The architecture consists of a backbone neural network (NN) followed by a multilayer (kk-layered) projection. While the NN drives toward optimality through a loss function consisting of a modified Lagrangian augmented with a consistency loss, the projection ensures feasibility by projecting the NN predictions in the original constraint manifold. Instead of typical distance minimization, our projection exploits local quadratic approximations of the original NLP. Under certain conditions (such as convexity), the projection has a descent property, which improves the NN predictions further. NLPOpt-Net deploys an inversion-free, modified Chambolle-Pock algorithm to solve the constrained quadratic projections during the forward pass and uses the implicit function theorem for efficient backpropagation. The fixed structure of the projection further allows decoupling of the NN and the projection once the training is complete. NLPOpt-Net solves large-scale convex QP, QCQP, NLP, and nonconvex problems with near zero optimality gap and constraint violations reduced to machine precision. Additionally, it provides near accurate prediction of the active sets and corresponding dual variables, thereby enabling a scalable approach for multiparametric programming. Compiling the projection in C provides order of magnitude improvement in inference time compared to JAX. We provide the codes and NLPOpt-Net as a ready to use package that includes GPU support.

Bimol Nath Roy, Rahul Golder, MM Faruque Hasan

Texas A&M University

Neural Network OptimizationLearning to OptimizeConstrained OptimizationDifferentiable Optimization
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Apr 29, 2026

Learning Over-Relaxation Policies for ADMM with Convergence Guarantees

The Alternating Direction Method of Multipliers (ADMM) is a widely used method for structured convex optimization, and its practical performance depends strongly on the choice of penalty and relaxation parameters. Motivated by settings such as Model Predictive Control (MPC), where one repeatedly solves related optimization problems with fixed structure and changing parameter values, we propose learning online updates of the relaxation parameter to improve average performance on problem classes of interest, while guaranteeing that asymptotic convergence is not compromised for the worst-case realization of such problems. This choice is computationally attractive in the Operator Splitting Quadratic Program (OSQP)-like architectures, since adapting relaxation does not trigger the matrix refactorizations associated with penalty updates. We establish convergence guarantees for ADMM with time-varying penalty and relaxation parameters under mild assumptions, and show on benchmark quadratic programs that the resulting learned policies improve both iteration count and wall-clock time on average over baseline OSQP.

Junan Lin, Paul J. Goulart, Luca Furieri

Department of Engineering Science, University of Oxford, OX1 3PJ Oxford, UK

Learning to OptimizeAlternating Direction Method of MultipliersConstrained Optimization
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Apr 28, 2026

Accelerating Regularized Attention Kernel Regression for Spectrum Cartography

Spectrum cartography reconstructs spatial radio fields from sparse and heterogeneous wireless measurements, underpinning many sensing and optimization tasks in wireless networks. Attention mechanisms have recently enabled adaptive measurement aggregation via attention kernel-based formulations. However, the resulting exponential kernels exhibit severe spectral imbalance, inducing large condition numbers that render standard iterative solvers ineffective for regularized attention kernel regression. This paper proposes a Learning-based Attention Kernel Regression (LAKER) algorithm for accelerating regularized attention kernel regression in spectrum cartography. The key idea is to learn a data-dependent preconditioner that captures the inverse spectral structure of the attention kernel system, directly reducing the condition number bottleneck. The preconditioner is obtained by solving a regularized maximum-likelihood estimation problem via a shrinkage-regularized convex--concave procedure, and is integrated with a preconditioned conjugate gradient solver for efficient optimization, whose solution is used for radio map reconstruction. Extensive experiments demonstrate that LAKER significantly reduces condition numbers by up to three orders of magnitude, accelerates convergence by over twenty-fold compared to baselines, and maintains high reconstruction accuracy, establishing learning-based preconditioning as an effective approach for attention kernel regression in spectrum cartography.

Liping Tao, Chee Wei Tan

College of Computing and Data Science, Nanyang Technological University, Singapore 639798

Learning to Optimize
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Apr 23, 2026

Transferable SCF-Acceleration through Solver-Aligned Initialization Learning

The cost of Kohn-Sham density functional theory (KS-DFT) calculations scales with the number of solver iterations, which depends on the quality of the initial guess. Machine learning methods that predict initial guesses from molecular geometry can reduce this cost, but matrix-prediction models fail when extrapolating to larger molecules, degrading rather than accelerating convergence [Liu et al., 2025]. We show that this failure is a supervision problem, not an extrapolation problem: models trained on ground-state targets fit those targets well out of distribution, yet produce initial guesses that slow convergence. Solver-Aligned Initialization Learning (SAIL) resolves this for both Hamiltonian and density matrix models by differentiating through the self-consistent field (SCF) solver end-to-end. We introduce the Effective Relative Iteration Count (ERIC), a correction to the commonly used RIC that accounts for hidden Fock-build overhead. On QM40, which contains molecules up to 4×\times larger than the training distribution, SAIL reduces ERIC by 37% (PBE), 33% (SCAN), and 28% (B3LYP), more than doubling the previous state-of-the-art reduction on B3LYP. On QMugs molecules 10×\times larger than the training set, SAIL delivers a 1.35×\times wall-time speedup at the hybrid level of theory, extending ML SCF acceleration to large drug-like molecules.

Eike S. Eberhard, Viktor Kotsev, Timm Güthle +1

1Technical University of Munich (TUM) · 2Munich Data Science Institute (MDSI) · 3Munich Center for Machine Learning (MCML)

Quantum ChemistryDensity Functional TheoryLearning to OptimizeScientific MLComputational Materials ScienceComputational Quantum Chemistry
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Apr 22, 2026

Learning to Evolve: A Self-Improving Framework for Multi-Agent Systems via Textual Parameter Graph Optimization

Designing and optimizing multi-agent systems (MAS) is a complex, labor-intensive process of "Agent Engineering." Existing automatic optimization methods, primarily focused on flat prompt tuning, lack the structural awareness to debug the intricate web of interactions in MAS. More critically, these optimizers are static; they do not learn from experience to improve their own optimization strategies. To address these gaps, we introduce Textual Parameter Graph Optimization (TPGO), a framework that enables a multi-agent system to learn to evolve. TPGO first models the MAS as a Textual Parameter Graph (TPG), where agents, tools, and workflows are modular, optimizable nodes. To guide evolution, we derive "textual gradients," structured natural language feedback from execution traces, to pinpoint failures and suggest granular modifications. The core of our framework is Group Relative Agent Optimization (GRAO), a novel meta-learning strategy that learns from historical optimization experiences. By analyzing past successes and failures, GRAO becomes progressively better at proposing effective updates, allowing the system to learn how to optimize itself. Extensive experiments on complex benchmarks like GAIA and MCP-Universe show that TPGO significantly enhances the performance of state-of-the-art agent frameworks, achieving higher success rates through automated, self-improving optimization.

Shan He, Runze Wang, Zhuoyun Du +4

Future Living Lab of Alibaba

Multi-Agent System OptimizationLearning to OptimizeMulti-Agent SystemsSelf-Improving Agents
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Apr 21, 2026

Decision-Focused Federated Learning Under Heterogeneous Objectives and Constraints

We consider Decision-Focused Federated Learning (DFFL), a predict-then-optimize setting in which multiple clients collaboratively train predictive models for downstream linear optimization problems without exchanging raw data. Besides the data heterogeneity typical of standard federated learning, clients may also have different objective functions and feasible regions. Building on the SPO+ surrogate loss, we derive heterogeneity bounds that separate objective shift, measured through cost-vector distances, from feasible-set shift, measured through support-function and shape-distance terms. We show that, for general compact feasible sets, small objective perturbations can still induce nonvanishing decision-focused loss discrepancies, while strongly convex feasible regions yield sharper stability-based bounds. We then lift these pointwise bounds to a local-versus-federated excess-risk comparison, showing that federation is beneficial when the statistical advantage of pooling exceeds a client-specific heterogeneity penalty. Computational experiments on polyhedral and strongly convex problems confirm that federation is substantially more robust under strongly convex feasible regions. Finally, we evaluate a simple validation-based interpolation between local and federated DFFL models. This interpolation mitigates the theoretical tradeoff and reduces aggregate regret and worst-client harm in both synthetic experiments and a PJM energy-pricing case study.

Konstantinos Ziliaskopoulos, Alexander Vinel

Auburn University

Predict-Then-OptimizeLearning to OptimizeDecision-Focused LearningConstrained OptimizationFederated LearningHeterogeneous Federated Learning
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Apr 16, 2026

RELOAD: A Robust and Efficient Learned Query Optimizer for Database Systems

Recent advances in query optimization have shifted from traditional rule-based and cost-based techniques towards machine learning-driven approaches. Among these, reinforcement learning (RL) has attracted significant attention due to its ability to optimize long-term performance by learning policies over query planning. However, existing RL-based query optimizers often exhibit unstable performance at the level of individual queries, including severe performance regressions, and require prolonged training to reach the plan quality of expert, cost-based optimizers. These shortcomings make learned query optimizers difficult to deploy in practice and remain a major barrier to their adoption in production database systems. To address these challenges, we present RELOAD, a robust and efficient learned query optimizer for database systems. RELOAD focuses on (i) robustness, by minimizing query-level performance regressions and ensuring consistent optimization behavior across executions, and (ii) efficiency, by accelerating convergence to expert-level plan quality. Through extensive experiments on standard benchmarks, including Join Order Benchmark, TPC-DS, and Star Schema Benchmark, RELOAD demonstrates up to 2.4x higher robustness and 3.1x greater efficiency compared to state-of-the-art RL-based query optimization techniques.

Seokwon Lee, Jaeyoung Sim, Sihyun Kim +3

Yonsei University BDAI Lab, Seoul 03722, South Korea · Microsoft Gray Systems Lab, Redmond, USA

Learning to OptimizeDatabase SystemsRobust RLRelational Databases
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Apr 14, 2026

Optimizing Earth Observation Satellite Schedules under Unknown Operational Constraints: An Active Constraint Acquisition Approach

Earth Observation (EO) satellite scheduling (deciding which imaging tasks to perform and when) is a well-studied combinatorial optimization problem. Existing methods typically assume that the operational constraint model is fully specified in advance. In practice, however, constraints governing separation between observations, power budgets, and thermal limits are often embedded in engineering artefacts or high-fidelity simulators rather than in explicit mathematical models. We study EO scheduling under unknown constraints: the objective is known, but feasibility must be learned interactively from a binary oracle. Working with a simplified model restricted to pairwise separation and global capacity constraints, we introduce Conservative Constraint Acquisition~(CCA), a domain-specific procedure designed to identify justified constraints efficiently in practice while limiting unnecessary tightening of the learned model. Embedded in the Learn&Optimize framework, CCA supports an interactive search process that alternates optimization under a learned constraint model with targeted oracle queries. On synthetic instances with up to 50tasks and dense constraint networks, L&O improves over a no-knowledge greedy baseline and uses far fewer main oracle queries than a two-phase acquire-then-solve baseline (FAO). For n≤30n\leq 30, the average gap drops from 65--68% (Priority Greedy) to 17.7--35.8% using L&O. At n=50n{=}50, where the CP-SAT reference is the best feasible solution found in 120s, L&O improves on FAO on average (17.9% vs.\ 20.3%) while using 21.3 main queries instead of 100 and about 5×5\times less execution time.

Mohamed-Bachir Belaid

NILU, Climate and environmental research institute, Kjeller, Norway

Constrained OptimizationLearning to OptimizeRemote Sensing
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Apr 14, 2026

HUANet: Hard-Constrained Unrolled ADMM for Constrained Convex Optimization

This paper presents HUANet, a constrained deep neural network architecture that unrolls the Alternating Direction Method of Multipliers (ADMM) into a trainable neural network for accelerating parametric constrained convex optimization. Existing end-to-end learning methods operate as black-box mappings from parameters to solutions, often without explicitly incorporating optimality principles or guaranteeing constraint satisfaction. To address these limitations, HUANet embeds a hard-constrained neural network within each unrolled ADMM iteration, where a differentiable correction stage enforces the affine equalities of the primal subproblem. Furthermore, we incorporate first-order optimality conditions into a self-supervised training loss to promote the convergence of the proposed unrolled algorithm. Extensive numerical experiments for benchmark optimization problems and a control application demonstrate and validate the effectiveness of HUANet in accelerating constrained convex optimization solving.

Trinh Tran, Binh Nguyen, Truong X. Nghiem

Department of Electrical and Computer Engineering, University of Central Florida, Orlando, FL 32816, USA

Learning to OptimizeAlternating Direction Method of MultipliersAlgorithm UnrollingConstrained Optimization
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Apr 13, 2026

Generalization Guarantees on Data-Driven Tuning of Gradient Descent with Langevin Updates

We study learning to learn through the lens of hyperparameter tuning. We propose the Langevin Gradient Descent Algorithm (LGD), which approximates the mean of the posterior distribution defined by the loss function and regularizer of a regression task with convex objective. For classification tasks, the LGD algorithm estimates the posterior probabilities of each class on the test set. We prove the existence of an optimal hyperparameter configuration for which the LGD algorithm achieves the Bayes' optimal solution for squared loss on regression tasks, and for which LGD closely approximates the posterior probabilities for well-specified classification tasks. Subsequently, we study generalization guarantees on meta learning optimal hyperparameters for the LGD algorithm from a given set of tasks in the data-driven setting. For a number of parameters dd and hyperparameter dimension hh, we show a pseudo-dimension bound of O(dh)O(dh), up to logarithmic terms under mild assumptions on LGD. This matches the dependence of the bounds on number of parameters obtained in prior work for linear regression using the elastic net, which only allows for h=2h=2 hyperparameters, and extends their bounds to regression on convex loss. Compared to bounds on regularized logistic regression that allow for only h=1h=1 hyperparameter, our bounds improve greatly on the dependence on samples per task at the cost of worse dependence on the number of parameters by accounting for hardware-aware procedures. Finally, we show empirical evidence of the success of LGD and the meta learning procedure for few-shot learning on linear and logistic regression using synthetically created datasets.

Saumya Goyal, Rohith Rongali, Ritabrata Ray +1

Machine Learning Department, Carnegie Mellon University · Electrical and Computer Engineering Department, Carnegie Mellon University.

Meta-LearningGradient DescentLearning to OptimizeStatistical Learning TheoryHyperparameter Optimization
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Apr 13, 2026

Deep Learning for Sequential Decision Making under Uncertainty: Foundations, Frameworks, and Frontiers

Artificial intelligence (AI) is moving increasingly beyond prediction to support decisions in complex, uncertain, and dynamic environments. This shift creates a natural intersection with operations research and management science (OR/MS), which has long provided methodological foundations for sequential decision making under uncertainty. At the same time, deep learning advances, including feedforward neural networks, recurrent architectures, transformers, large language models (LLMs), and deep reinforcement learning, have expanded data-driven modeling for large-scale decisions. This tutorial presents an OR/MS-centered perspective on deep learning for sequential decision making under uncertainty, bridging neural architectures and OR/MS approaches to decision making. Its premise: deep learning complements optimization rather than replacing it. Deep learning brings adaptability and scalable approximation, whereas OR/MS provides the mathematical rigor to represent constraints, recourse, uncertainty, and decision quality. The tutorial reviews key decision making foundations, connects them to the major neural architectures in modern AI, and organizes the field around three central themes: predict-then-optimize and decision-aware learning, learning-based decision generation under constraints for continuous and discrete problems with temporal coupling, and deep reinforcement learning for sequential and combinatorial decision making. Impact spans supply chains, service systems, healthcare and epidemic response, agriculture, energy, environmental sustainability, and autonomous operations. This tutorial frames these developments as part of a shift from predictive AI toward decision-capable AI, highlighting OR/MS's role in shaping the next generation of integrated learning--optimization systems.

I. Esra Buyuktahtakin

Grado Department of Industrial and Systems Engineering, Virginia Tech Blacksburg, VA 24061, USA

Reinforcement LearningPredict-Then-OptimizeSequential Decision MakingLearning to OptimizeDecision-Focused LearningDecision-Making under Uncertainty
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Apr 10, 2026

A Self-Attentive Meta-Optimizer with Group-Adaptive Learning Rates and Weight Decay

Adaptive optimizers like AdamW apply uniform hyperparameters across all parameter groups, ignoring heterogeneous optimization dynamics across layers and modules. We address this limitation by proposing MetaAdamW - a new optimizer that integrates a self-attention mechanism to dynamically modulate per-group learning rates and weight decay. The modulation factors are produced by a lightweight Transformer encoder that operates on statistical features (gradient norms, momentum norms, correlations) extracted from each parameter group. To train the attention module, we introduce a meta-learning objective that combines gradient alignment, loss decrease, and generalization gap. A key novel contribution is the extension of homoscedastic uncertainty weighting (HUW) with task-specific priorities that directly scale the regularization terms - enabling domain knowledge to guide automatic loss balancing. Extensive experiments on five diverse tasks-time series forecasting (ETT), language modeling (WikiText-2), machine translation (Multi30k), image classification (CIFAR-10), and sentiment analysis (IMDB) - demonstrate that MetaAdamW consistently outperforms the standard AdamW baseline in terms of validation loss, accuracy, or perplexity. Depending on the task, MetaAdamW either reduces overall training time (by up to 17.11%) or improves performance (by up to 11.08%) while introducing only moderate overhead; in some cases, it can also mitigate issues of insufficient convergence caused by premature early stopping. Ablation studies validate the effectiveness of each component, including feature versions, grouping strategies, and the proposed priority-injected uncertainty weighting.

JiangBo Zhao, ZhaoXin Liu

Deep Learning OptimizationLearning to OptimizeLayer-Wise Learning Rate AdaptationAdaptive Loss Weighting
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Apr 4, 2026

One-Shot Localisation of the Global Minimum of a Noisy One-Dimensional Function: A Controlled Comparison of an Iterative Neural Minimizer, Set Transformers and Classical Estimators

We study passive global optimisation: from twenty noisy samples of an unknown one-dimensional function, with no further queries, estimate where its global minimum lies. We build a benchmark and a self-checking harness and use them for a controlled comparison. The Neural Function Minimizer (NFM), which we introduce, moves a position across the domain for fifteen steps and re-reads the samples from it at every step; a same-size Set Transformer answers in one pass. In a 2x2 design each has one output head or an added mixture head, with the same data and inputs and three seeds each. Within the training families the two are tied on location error, and the mixture loss lowers it by about 0.45 points of the domain in both. The NFM's answers have lower regret, by 2.90 and 3.12 points of the range with one and two heads (significant after correction, and again on a block of fresh cases), mainly because its right-basin answers lie closer to the floor; a Set Transformer looped to the NFM's computation does not reproduce this within the training families, though it has the lower error on unseen functions (exploratory). The NFM also differs from the Set Transformers in encoder, local read, activation, losses and training time. At near-tied minima the form of the answer decides: point answers hedge in about a quarter of exact ties, mixture modes almost never, mixture means in more than 80%. Gaussian-process estimators, with no training, have the lowest regret and are the most accurate on unseen functions. We will release the benchmark, the models and the harness.

Qusay Muzaffar, David Levin, Michael Werman

Department of Computer Science, The Hebrew University of Jerusalem, Jerusalem, Israel · Department of Applied Mathematics, Tel Aviv University, Tel Aviv, Israel

Black-Box OptimizationZeroth-Order OptimizationLearning to OptimizeGlobal Optimization
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Apr 4, 2026

Portfolio Optimization Proxies under Label Scarcity and Regime Shifts via Bayesian and Deterministic Students under Semi-Supervised Sandwich Training

This paper proposes a machine learning assisted portfolio optimization framework designed for low data environments and regime uncertainty. We construct a teacher student learning pipeline in which a Conditional Value at Risk (CVaR) optimizer generates supervisory labels, and neural models (Bayesian and deterministic) are trained using both real and synthetically augmented data. The synthetic data is generated using a factor based model with t copula residuals, enabling training beyond the limited real sample of 104 labeled observations. We evaluate four student models under a structured experimental framework comprising (i) controlled synthetic experiments (3 x 5 seed grid), (ii) in-distribution real market evaluation (C2A) and (iii) cross-universe generalization (D2A). In real-market settings, models are deployed using a rolling evaluation protocol where a frozen pretrained model is periodically fine tuned on recent observations and reset to its base state, ensuring stability while allowing limited adaptation. Results show that student models can match or outperform the CVaR teacher in several settings, while achieving improved robustness under regime shifts and reduced turnover. These findings suggest that hybrid optimization learning approaches can enhance portfolio construction in data constrained environments

Adhiraj Chattopadhyay

Department of Management Studies, Indian Institute of Technology Roorkee, India

Portfolio OptimizationLearning to OptimizeSemi-Supervised Learning
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Apr 2, 2026

Sparse Bayesian Learning Algorithms Revisited: From Learning Majorizers to Structured Algorithmic Learning using Neural Networks

Sparse Bayesian Learning is one of the most popular sparse signal recovery methods, and various algorithms exist under the SBL paradigm. However, given a performance metric and a sparse recovery problem, it is difficult to know a-priori the best algorithm to choose. This difficulty is in part due to a lack of a unified framework to derive SBL algorithms. We address this issue by first showing that the most popular SBL algorithms can be derived using the majorization-minimization (MM) principle, providing hitherto unknown convergence guarantees to this class of SBL methods. Moreover, we show that the two most popular SBL update rules not only fall under the MM framework but are both valid descent steps for a common majorizer, revealing a deeper analytical compatibility between these algorithms. Using this insight and properties from MM theory we expand the class of SBL algorithms, and address finding the best SBL algorithm via data within the MM framework. Second, we go beyond the MM framework by introducing the powerful modeling capabilities of deep learning to further expand the class of SBL algorithms, aiming to learn a superior SBL update rule from data. We propose a novel deep learning architecture that can outperform the classical MM based ones across different sparse recovery problems. Our architecture's complexity does not scale with the measurement matrix dimension, hence providing a unique opportunity to test generalization capability across different matrices. For parameterized dictionaries, this invariance allows us to train and test the model across different parameter ranges. We also showcase our model's ability to learn a functional mapping by its zero-shot performance on unseen measurement matrices. Finally, we test our model's performance across different numbers of snapshots, signal-to-noise ratios, and sparsity levels.

Rushabha Balaji, Kuan-Lin Chen, Danijela Cabric +1

Department of Electrical and Computer Engineering, University of California, Los Angeles, CA 90095, USA · Department of Electrical and Computer Engineering, University of California, San Diego, La Jolla, CA 92093, USA

Sparse RecoveryLearning to OptimizeBayesian Inference
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Apr 2, 2026

When does learning pay off? A study on DRL-based dynamic algorithm configuration for carbon-aware scheduling

Deep reinforcement learning (DRL) has recently emerged as a promising tool for Dynamic Algorithm Configuration (DAC), enabling evolutionary algorithms to adapt their parameters online rather than relying on static tuned configurations. While DRL can learn effective control policies, training is computationally expensive. This cost may be justified if learned policies generalize, allowing the training effort to transfer across instance types and problem scales. Yet, for real-world optimization problems, it remains unclear whether this promise holds in practice and under which conditions the investment in learning pays off. In this work, we investigate this question in the context of the carbon-aware permutation flow-shop scheduling problem. We develop a DRL-based DAC framework and train it exclusively on small, simple instances. We then deploy the learned policy on both similar and more complex unseen instances and compare its performance against a static tuned baseline, which provides a fair point of comparison. Our findings show that the proposed method provides a strong dynamic algorithm control policy that can be effectively transferred to different unseen problem instances. Notably, on simple and cheap to compute instances, similar to those observed during training and tuning, DRL performs comparably with the statically tuned baseline. However, as instance characteristics diverge and computational complexities increase, the DRL-learned policy continuously outperforms static tuning. These results confirm that DRL can acquire robust and generalizable control policies which are effective beyond the training instance distributions. This ability to generalize across instance types makes the initial computational investment worthwhile, particularly in settings where static tuning struggles to adapt to changing problem scenarios.

Andrea Mencaroni, Robbert Reijnen, Yingqian Zhang +1

Department of Industrial Systems Engineering and Product Design, Ghent University, Ghent, Belgium · Industrial Systems Engineering (ISyE), Flanders Make, Kortrijk, Belgium · Eindhoven Artificial Intelligence Systems Institute, Eindhoven University of Technology, The Netherlands +1

RL ControlLearning to OptimizeCombinatorial Optimization
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Apr 1, 2026

Learning to Model Optimization Problems from Solver Outcomes

Automating optimization modeling with LLMs is a promising path toward scalable decision intelligence, but existing approaches either rely on agentic pipelines built on closed-source LLMs with high inference latency or fine-tune smaller LLMs using costly process supervision that often overfits to a single solver API. We study how reinforcement learning from inexpensive outcome labels can improve optimization modeling and support reuse across heterogeneous solver environments. To this end, we propose Optimization Modeling from Solver Outcomes (OMSO), which uses reinforcement learning (RL) with a mathematical programming solver as a deterministic, interactive environment. Given a natural-language problem and a target solver, OMSO generates solver-specific code, executes it in a sandboxed harness, and converts execution outcomes into scalar rewards in a closed-loop generate-execute-feedback-update process. This RL formulation learns optimization modeling without process-level supervision and supports cross-solver generalization by switching the solver environment rather than reconstructing solver-specific datasets. Experiments on NL4OPT, MAMO, IndustryOR, and OptiBench across Gurobi, OR-Tools, and COPT show that OMSO matches or outperforms process-supervised SFT, supports zero-shot training-free transfer, direct RL training, and SFT-cold-started RL, and improves modeling reasoning and semantic correctness beyond solver API usage.

Runda Guan, Xiangqing Shen, Jiajun Zhang +3

School of Computer Science and Engineering, Nanjing University of Science and Technology · School of Intelligence Science and Technology, Nanjing University · Institute of Automation, Chinese Academy of Sciences

Natural Language to Optimization ModelingLearning to OptimizeLLM-Based Program SynthesisReinforcement Learning with Verifiable Rewards
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Mar 31, 2026

Speeding Up Mixed-Integer Programming Solvers with Sparse Learning for Branching

Machine learning is increasingly used to improve decisions within branch-and-bound algorithms for mixed-integer programming. Many existing approaches rely on deep learning, which often requires very large training datasets and substantial computational resources for both training and deployment, typically with GPU parallelization. In this work, we take a different path by developing interpretable models that are simple but effective. We focus on approximating strong branching (SB) scores, a highly effective yet computationally expensive branching rule. Using sparse learning methods, we build models with fewer than 4% of the parameters of a state-of-the-art graph neural network (GNN) while achieving competitive accuracy. Relative to SCIP's built-in branching rules and the GNN-based model, our CPU-only models are faster than the default solver and the GPU-accelerated GNN. The models are simple to train and deploy, and they remain effective with small training sets, which makes them practical in low-resource settings. Extensive experiments across diverse problem classes demonstrate the efficiency of this approach.

Selin Bayramoğlu, George L Nemhauser, Nikolaos V Sahinidis

H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology · School of Chemical & Biomolecular Engineering, Georgia Institute of Technology

Learning to OptimizeBranch and BoundMixed-Integer Linear Programming
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Mar 31, 2026

End-to-End Learning-based Operation of Integrated Energy Systems for Buildings and Data Centers

Buildings and data centers (DCs) are energy-intensive sectors, playing a critical role to achieve the low-carbon and sustainable energy transition targets. To this end, integrated energy system (IES) that incorporates diverse renewables, energy generation, conversion, and storage technologies to enable coordinated multi-energy supply have been widely investigated for both buildings and DCs. However, few works consider the two sectors jointly within IES to exploit their substantial synergistic benefits. Meanwhile, the operational optimization of IES remains challenging due to the difficulty to predict the multi-energy demand and supply accurately. To address these gaps, this paper investigates IES for coordinated multi-energy supply of buildings and DC, where the waste heat from DCs is recovered and reused to enhance energy efficiency. Moreover, an end-to-end learning-based method is proposed for the operational optimization of IES under uncertainty. Unlike conventional predict-then-optimize approaches, the proposed method integrates the training of prediction models for uncertain variables with the constrained optimization of IES into a unified learning framework, guiding the training of prediction models to improve operational performance, rather than prediction accuracy, thereby mitigating the impacts of predictions errors. Case studies based on real-world datasets show that the proposed methods improves the operational performance of IES by about 7-9% compared to existing predict-then-optimize methods. In addition, coordinating buildings and DCs within IES shows substantial economic benefits. In particular, the waste heat recovery from DCs leads to approximately 10% of total energy cost reduction of the IES.

Zhenyu Pu, Yu Yang, Liang Yu +1

School of Automation Science and Engineering, Xi’an Jiaotong University, Shaanxi, China · College of Automation, Nanjing University of Posts and Telecommunications, Nanjing 210023, China · Center for Intelligent and Networked Systems, Department of Automation, Tsinghua University, Beijing 100084, China +1

Data Center Energy ManagementLearning to OptimizeBuilding Energy ManagementDecision-Focused Learning
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Mar 30, 2026

Machine Learning-Assisted High-Dimensional Matrix Estimation

Efficient estimation of high-dimensional matrices-including covariance and precision matrices-is a cornerstone of modern multivariate statistics. Most existing studies have focused primarily on the theoretical properties of the estimators (e.g., consistency and sparsity), while largely overlooking the computational challenges inherent in high-dimensional settings. Motivated by recent advances in learning-based optimization method-which integrate data-driven structures with classical optimization algorithms-we explore high-dimensional matrix estimation assisted by machine learning. Specifically, for the optimization problem of high-dimensional matrix estimation, we first present a solution procedure based on the Linearized Alternating Direction Method of Multipliers (LADMM). We then introduce learnable parameters and model the proximal operators in the iterative scheme with neural networks, thereby improving estimation accuracy and accelerating convergence. Theoretically, we first prove the convergence of LADMM, and then establish the convergence, convergence rate, and monotonicity of its reparameterized counterpart; importantly, we show that the reparameterized LADMM enjoys a faster convergence rate. Notably, the proposed reparameterization theory and methodology are applicable to the estimation of both high-dimensional covariance and precision matrices. We validate the effectiveness of our method by comparing it with several classical optimization algorithms across different structures and dimensions of high-dimensional matrices.

Wan Tian, Hui Yang, Zhouhui Lian +2

Advanced Institute of Information Technology, Peking University · Wangxuan Institute of Computer Technology, Peking University, China, 100871 · Guanghua School of Management, Peking University, Beijing, China, 100871 +3

Learning to OptimizeAlternating Direction Method of MultipliersAlgorithm UnrollingCovariance Estimation
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Mar 30, 2026

A Learning-Based Cooperative Coevolution Framework for Heterogeneous Large-Scale Global Optimization

Cooperative Coevolution (CC) effectively addresses Large-Scale Global Optimization (LSGO) via decomposition but struggles with the emerging class of Heterogeneous LSGO (H-LSGO) problems arising from real-world applications, where subproblems exhibit diverse dimensions and distinct landscapes. The prevailing CC paradigm, relying on a fixed low-dimensional optimizer, often fails to navigate this heterogeneity. To address this limitation, we propose the Learning-Based Heterogeneous Cooperative Coevolution Framework (LH-CC). By formulating the optimization process as a Markov Decision Process, LH-CC employs a meta-agent to adaptively select the most suitable optimizer for each subproblem. We also introduce a flexible benchmark suite to generate diverse H-LSGO problem instances. Extensive experiments on 3000-dimensional problems with complex coupling relationships demonstrate that LH-CC achieves superior solution quality and computational efficiency compared to state-of-the-art baselines. Furthermore, the framework exhibits robust generalization across varying problem instances, optimization horizons, and optimizers. Our findings reveal that dynamic optimizer selection is a pivotal strategy for solving complex H-LSGO problems.

Wenjie Qiu, Zixin Wang, Hongyu Fang +2

South China University of Technology, Guangzhou, Guangdong, China

Global OptimizationEvolutionary OptimizationLearning to Optimize
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Mar 12, 2026

Inverse Neural Operator for ODE Parameter Optimization

We propose the Inverse Neural Operator (INO), a two-stage framework for recovering hidden ODE parameters from sparse, partial observations. In Stage 1, a Conditional Fourier Neural Operator (C-FNO) with cross-attention learns a differentiable surrogate that reconstructs full ODE trajectories from arbitrary sparse inputs, suppressing high-frequency artifacts via spectral regularization. In Stage 2, an Amortized Drifting Model (ADM) learns a kernel-weighted velocity field in parameter space, transporting random parameter initializations toward the ground truth without backpropagating through the surrogate, avoiding the Jacobian instabilities that afflict gradient-based inversion in stiff regimes. Experiments on a real-world stiff atmospheric chemistry benchmark (POLLU, 25 parameters) and a synthetic Gene Regulatory Network (GRN, 40 parameters) show that INO outperforms gradient-based and amortized baselines in parameter recovery accuracy while requiring only 0.23s inference time, a 487x speedup over iterative gradient descent.

Zhi-Song Liu, Wenqing Peng, Helmi Toropainen +5

Department of Computational Engineering, Lappeenranta-Lahti University of Technology, Lahti, Finland · Atmospheric Modelling Center Lahti (AMC-Lahti), Finland · University of Helsinki, Helsinki, Finland +3

Parameter EstimationOrdinary Differential EquationsNeural OperatorsLearning to OptimizeNeural Surrogate Modeling
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Mar 11, 2026

Learning to Score: Tuning Cluster Schedulers through Reinforcement Learning

Efficiently allocating incoming jobs to nodes in large-scale clusters can lead to substantial improvements in both cluster utilization and job performance. In order to allocate incoming jobs, cluster schedulers usually rely on a set of scoring functions to rank feasible nodes. Results from individual scoring functions are usually weighted equally, which could lead to sub-optimal deployments as the one-size-fits-all solution does not take into account the characteristics of each workload. Tuning the weights of scoring functions, however, requires expert knowledge and is computationally expensive. This paper proposes a reinforcement learning approach for learning the weights in scheduler scoring algorithms with the overall objective of improving the end-to-end performance of jobs for a given cluster. Our approach is based on percentage improvement reward, frame-stacking, and limiting domain information. We propose a percentage improvement reward to address the objective of multi-step parameter tuning. The inclusion of frame-stacking allows for carrying information across an optimization experiment. Limiting domain information prevents overfitting and improves performance in unseen clusters and workloads. The policy is trained on different combinations of workloads and cluster setups. We demonstrate the proposed approach improves performance on average by 33% compared to fixed weights and 12% compared to the best-performing baseline in a lab-based serverless scenario.

Martin Asenov, Qiwen Deng, Gingfung Yeung +1

Edinburgh Research Centre, Central Software Institute, Huawei · School of Computer Science, University of St Andrews

Learning to OptimizeReinforcement LearningReward Design for RL
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Mar 6, 2026

Learning to Solve Orienteering Problem with Time Windows and Variable Profits

The orienteering problem with time windows and variable profits (OPTWVP) is common in many real-world applications and involves continuous time variables. Current approaches fail to develop an efficient solver for this orienteering problem variant with discrete and continuous variables. In this paper, we propose a learning-based two-stage DEcoupled discrete-Continuous optimization with Service-time-guided Trajectory (DeCoST), which aims to effectively decouple the discrete and continuous decision variables in the OPTWVP problem, while enabling efficient and learnable coordination between them. In the first stage, a parallel decoding structure is employed to predict the path and the initial service time allocation. The second stage optimizes the service times through a linear programming (LP) formulation and provides a long-horizon learning of structure estimation. We rigorously prove the global optimality of the second-stage solution. Experiments on OPTWVP instances demonstrate that DeCoST outperforms both state-of-the-art constructive solvers and the latest meta-heuristic algorithms in terms of solution quality and computational efficiency, achieving up to 6.6x inference speedup on instances with fewer than 500 nodes. Moreover, the proposed framework is compatible with various constructive solvers and consistently enhances the solution quality for OPTWVP.

Songqun Gao, Zanxi Ruan, Patrick Floor +3

Department of Industrial Engineering, Università di Trento, Trento, Italy · Interdepartmental Robotics Labs (IDRA), Trento, Italy · Department of Engineering for Innovation Medicine, Università di Verona, Verona, Italy +2

Learning to OptimizeCombinatorial OptimizationNeural Combinatorial Optimization
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Mar 5, 2026

Cheap Thrills: Effective Amortized Optimization Using Inexpensive Labels

To scale optimization and simulation, prior work has explored training machine-learning surrogates that map problem parameters to solutions inexpensively at inference time. Unfortunately, commonly used approaches, including supervised and self-supervised learning with either soft or hard feasibility enforcement, face inherent challenges such as reliance on expensive high-quality labels or difficult optimization landscapes. To address their trade-offs, we propose a novel framework that collects "cheap" imperfect labels, performs supervised model pretraining with a merit loss-based termination scheme, and finally refines the model through self-supervised learning to improve final performance. Empirical validation across challenging domains -- including nonconvex constrained optimization, power-grid operation, and stiff dynamical systems -- shows that this three-stage strategy yields faster convergence; improved accuracy, feasibility, and optimality; and up to 59x reductions in total offline computational cost. We further analyze why and when our framework improves surrogate model training, finding that (i) merit loss is an informative signal and (ii) only small numbers of cheap, inexact labels are needed to place the model in a favorable regime for self-supervised learning.

Khai Nguyen, Petros Ellinas, Anvita Bhagavathula +1

Massachusetts Institute of Technology, Cambridge, MA, USA · Technical University of Denmark, Kongens Lyngby, Denmark

Learning to OptimizeNeural Surrogate ModelingConstrained Optimization
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Mar 3, 2026

Optimal trajectory-guided stochastic co-optimization for e-fuel system design and real-time operation

E-fuels are promising long-term energy carriers supporting the net-zero transition. However, the large combinatorial design-operation spaces under renewable uncertainty make the use of mathematical programming impractical for co-optimizing e-fuel production systems. Here, we present MasCOR, a machine-learning-assisted co-optimization framework that learns from global operational trajectories. By encoding system design and renewable trends, a single MasCOR agent generalizes dynamic operation across diverse configurations and scenarios, substantially simplifying design-operation co-optimization under uncertainty. Benchmark comparisons against state-of-the-art reinforcement learning baselines demonstrate near-optimal performance, while computational costs are substantially lower than those of mathematical programming, enabling rapid parallel evaluation of designs within the co-optimization loop. This framework enables rapid screening of feasible design spaces together with corresponding operational policies. When applied to four potential European sites targeting e-methanol production, MasCOR shows that most locations benefit from reducing system load below 50 MW to achieve carbon-neutral methanol production, with production costs of 1.0-1.2 USD per kg. In contrast, Dunkirk (France), with limited renewable availability and high grid prices, favors system loads above 200 MW and expanded storage to exploit dynamic grid exchange and hydrogen sales to the market. These results underscore the value of the MasCOR framework for site-specific guidance from system design to real-time operation.

Jeongdong Kim, Minsu Kim, Jonggeol Na +1

Department of Chemical Engineering , Massachusetts Institute of Technology, 77 Massachusetts Ave, Cambridge, 02139, Massachusetts, United States. · Department of Chemical and Biomolecular Engineering, Yonsei University, 50 Yonsei-ro, Seoul, 03722, Republic of Korea. · Department of Chemical Engineering and Materials Science, Ewha Womans University, 52 Ewhayeodae-gil, Seoul, 03760, Republic of Korea. +2

Learning to OptimizeStochastic OptimizationStochastic Optimal Control
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Feb 23, 2026

Behavior Learning (BL): Learning Hierarchical Optimization Structures from Data

Inspired by behavioral science, we propose Behavior Learning (BL), a novel general-purpose machine learning framework that learns interpretable and identifiable optimization structures from data, ranging from single optimization problems to hierarchical compositions. It unifies predictive performance, intrinsic interpretability, and identifiability, with broad applicability to scientific domains involving optimization. BL parameterizes a compositional utility function built from intrinsically interpretable modular blocks, which induces a data distribution for prediction and generation. Each block represents and can be written in symbolic form as a utility maximization problem (UMP), a foundational paradigm in behavioral science and a universal framework of optimization. BL supports architectures ranging from a single UMP to hierarchical compositions, the latter modeling hierarchical optimization structures. Its smooth and monotone variant (IBL) guarantees identifiability. Theoretically, we establish the universal approximation property of BL, and analyze the M-estimation properties of IBL. Empirically, BL demonstrates strong predictive performance, intrinsic interpretability and scalability to high-dimensional data. Code: https://github.com/MoonYLiang/Behavior-Learning ; install via pip install blnetwork.

Zhenyao Ma, Yue Liang, Dongxu Li

Xiamen University · University of Tübingen · Xi’an Jiaotong University

Interpretable MLLearning to OptimizeScientific ML
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Feb 22, 2026

Celo2: Towards Learned Optimization Free Lunch

Learned optimizers are powerful alternatives to hand-designed update rules like Adam, yet they have seen limited practical adoption since they often fail to meta-generalize beyond their training distribution and incur high meta-training cost. For instance, prior work, VeLO, scaled meta-training to 4,000 TPU months (∼\sim10×\times GPT-3 compute) to meta-train a general-purpose optimizer but it failed to generalize beyond 600M parameters tasks. In this work, we present a surprising finding: by crafting a simple normalized optimizer architecture and augmenting meta-training, it becomes feasible to meta-train a performant general-purpose learned update rule on a tiny fraction of VeLO compute, 4.5 GPU hours to be precise. Our learned update rule scales stably to a billion-scale pretraining task (GPT-3 XL 1.3B) which is six orders of magnitude larger than its meta-training distribution. Furthermore, it shows strong performance across diverse out-of-distribution tasks and is compatible with modern optimization harness that includes orthogonalization, distinct update rules for input-output and hidden weights, and decoupled weight decay. In all, this work paves the way for practically applicable learnable optimization algorithms, unlocking exploration of richer meta-training and data curation recipes to further improve performance.

Abhinav Moudgil, Boris Knyazev, Eugene Belilovsky

Mila · Concordia University · Samsung AI Lab, Montreal +1

Learning to OptimizeDeep Learning OptimizationMeta-Learning
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