Differentiable Optimization

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Differentiable optimization studies how to compute gradients through iterative algorithms and constrained decisions so their parameters or inputs can be learned end to end. Common approaches unroll solver iterations, use implicit differentiation or adjoints, or build differentiable layers from projections, quadratic programs, and conic solvers. These methods support tasks such as hyperparameter tuning, decision-focused prediction, robotics control, and physics-based design, where improving the final decision matters as much as improving a prediction. A central challenge is balancing scalable, reliable gradients with exact feasibility and discrete or nonsmooth decisions, especially in large or numerically sensitive problems.

Latest papers

84 papers10 papers in the last four weeks, up 25% on the four weeks before. 0.1% of all new papers.

May 2, 2026

Decision-Focused Learning via Tangent-Space Projection of Prediction Error

Decision-Focused Learning (DFL) trains predictors to improve downstream decision quality, but computing regret gradients typically requires differentiating through solvers or relying on surrogate losses, which can be computationally expensive or deviate from the true objective. We show that, under standard regularity with locally stable active constraints, the regret gradient admits a closed-form geometric characterization, equivalent to the prediction error projected onto the tangent space of active constraints, scaled by local curvature. This reveals that regret gradients can be obtained by filtering decision-irrelevant components from the MSE gradient, providing a simpler and more direct alternative to existing approaches. Based on this, we propose PEAR (Projected Error As Regret-gradient), which computes regret gradients via a reduced linear system over active constraints, avoiding differentiation through solver iterations or additional optimization solves. Experiments on LP benchmarks and a real-world QP task show that PEAR achieves the best decision quality among all baselines while being the most computationally efficient, with gains that persist under constraint shifts.

Junhyeong Lee, Sangjin Jin, Yongjae Lee

Department of Industrial Engineering, Ulsan National Institute of Science and Technology, Ulsan, South Korea.

Decision-Focused Learning Differentiable Optimization

May 1, 2026

Differentiable Multiphysics Co-Optimization via Implicit Neural Representations: A Transient Hamburger-Cooking Benchmark

The co-optimization of geometry and physical parameters remains challenging in transient multiphysics systems involving moving boundaries, nonlinear material response, phase transitions, and competing objectives. Existing methods often optimize geometry and physical variables separately, rely on simplified steady-state physics, or require offline data generation and reduced design spaces. Here, we present an end-to-end differentiable co-optimization framework that couples an implicit neural representation of geometry with a JAX-compiled Eulerian multiphysics solver. Geometry is represented as a signed distance field using Fourier-feature-encoded spatial coordinates, while boundary conditions, initial conditions, process controls, and material parameters are optimized within the same differentiable loop. Continuous relaxations represent non-smooth physical transitions while preserving compatibility with reverse-mode automatic differentiation and backpropagation through time. We demonstrate the framework using a transient hamburger-cooking benchmark, selected as an interpretable multiphysics problem rather than a culinary optimization exercise. The benchmark combines conductive and convective heat transfer, latent energy effects, moisture and fat transport, shrinkage-induced geometry evolution, evolving contact boundary conditions, flipping-induced boundary-condition changes, and competing quality objectives. Results show that geometry-only optimization modifies shape to relieve thermal bottlenecks, while joint co-optimization distributes the design response across geometry, material state, process variables, and boundary conditions through gradients propagated over the full transient rollout.

Navid Zobeiry

Department of Materials Science and Engineering, University of Washington · William E. Boeing Department of Aeronautics & Astronautics, University of Washington, Seattle, WA 98195, USA

Differentiable Optimization Multi-Objective Optimization Multiphysics Simulation

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 (

k

-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 Optimization Learning to Optimize Constrained Optimization Differentiable Optimization

Apr 21, 2026

Differentiable Satellite Constellation Configuration via Relaxed Coverage and Revisit Objectives

Satellite constellation design requires optimizing orbital parameters across multiple satellites to maximize mission specific metrics. For many types of mission, it is desirable to maximize coverage and minimize revisit gaps over ground targets. Existing approaches to constellation design either restrict the design space to symmetric parametric families such as Walker constellations, or rely on metaheuristic methods that require significant compute and many iterations. Gradient-based optimization has been considered intractable due to the non-differentiability of coverage and revisit metrics, which involve binary visibility indicators and discrete max operations. We introduce four continuous relaxations: soft sigmoid visibility, noisy-OR multi-satellite aggregation, leaky integrator revisit gap tracking, and LogSumExp soft-maximum, which when composed with the

\partial

SGP4 differentiable orbit propagator, yield a fully differentiable pipeline from orbital elements to mission-level objectives. We show that this scheme can recover Walker-Delta geometry from irregular initializations, and discovers elliptical Molniya-like orbits with apogee dwell over extreme latitudes from only gradients. Compared to simulated annealing (SA), genetic algorithm (GA), and differential evolution (DE) baselines, our gradient-based method recovers Walker-equivalent geometry within

{\sim}750

evaluations, whereas the three black-box baselines plateau at with significantly worse revisit even with roughly four times the evaluation budget.

Shreeyam Kacker, Kerri Cahoy

Department of Aeronautics and Astronautics MIT

Differentiable Programming Differentiable Optimization

Apr 8, 2026

VertAX: a differentiable vertex model for learning epithelial tissue mechanics

Epithelial tissues dynamically reshape through local mechanical interactions among cells, a process well captured by vertex models. Yet their many tunable parameters make inference and optimization challenging, motivating computational frameworks that flexibly model and learn tissue mechanics. We introduce VertAX, a differentiable JAX-based framework for vertex-modeling of confluent epithelia. VertAX provides automatic differentiation, GPU acceleration, and end-to-end bilevel optimization for forward simulation, parameter inference, and inverse mechanical design. Users can define arbitrary energy and cost functions in pure Python, enabling seamless integration with machine-learning pipelines. We demonstrate VertAX on three representative tasks: (i) forward modeling of tissue morphogenesis, (ii) mechanical parameter inference, and (iii) inverse design of tissue-scale behaviors. We benchmark three differentiation strategies-automatic differentiation, implicit differentiation, and equilibrium propagation-showing that the latter can approximate gradients using repeated forward, adjoint-free simulations alone, offering a simple route for extending inverse biophysical problems to non-differentiable simulators with limited additional engineering effort.

Alessandro Pasqui, Jim Martin Catacora Ocana, Anshuman Sinha +7

Center for Interdisciplinary Research in Biology (CIRB), Collège de France, Université PSL, CNRS, INSERM, 75005 Paris, France. · Current address: Center for Integrative Biology (CBI), University of Toulouse, CNRS, Toulouse, France. · Center for Life Nano & Neuro Science, IIT, 00161, Rome, Italy. +3

Differentiable Physics Differentiable Optimization Inverse Design Automatic Differentiation

Apr 7, 2026

LMI-Net: Linear Matrix Inequality--Constrained Neural Networks via Differentiable Projection Layers

Linear matrix inequalities (LMIs) have played a central role in certifying stability, robustness, and forward invariance of dynamical systems. Despite rapid development in learning-based methods for control design and certificate synthesis, existing approaches often fail to preserve the hard matrix inequality constraints required for formal guarantees. We propose LMI-Net, an efficient and modular differentiable projection layer that enforces LMI constraints by construction. Our approach lifts the set defined by LMI constraints into the intersection of an affine equality constraint and the positive semidefinite cone, performs the forward pass via Douglas-Rachford splitting, and supports efficient backward propagation through implicit differentiation. We establish theoretical guarantees that the projection layer converges to a feasible point, certifying that LMI-Net transforms a generic neural network into a reliable model satisfying LMI constraints. Evaluated on experiments including invariant ellipsoid synthesis and joint controller-and-certificate design for a family of disturbed linear systems, LMI-Net substantially improves feasibility over soft-constrained models under distribution shift while retaining fast inference speed, bridging semidefinite-program-based certification and modern learning techniques.

Sunbochen Tang, Andrea Goertzen, Navid Azizan

Laboratory for Information and Decision Systems (LIDS), Massachusetts Institute of Technology, Cambridge, MA 02139, USA

Constrained Optimization Differentiable Optimization Semidefinite Programming

Apr 3, 2026

Differentiable Stroke Planning with Dual Parameterization for Efficient and High-Fidelity Painting Creation

In stroke-based rendering, search methods often get trapped in local minima due to discrete stroke placement, while differentiable optimizers lack structural awareness and produce unstructured layouts. To bridge this gap, we propose a dual representation that couples discrete polylines with continuous Bézier control points via a bidirectional mapping mechanism. This enables collaborative optimization: local gradients refine global stroke structures, while content-aware stroke proposals help escape poor local optima. Our representation further supports Gaussian-splatting-inspired initialization, enabling highly parallel stroke optimization across the image. Experiments show that our approach reduces the number of strokes by 30-50%, achieves more structurally coherent layouts, and improves reconstruction quality, while cutting optimization time by 30-40% compared to existing differentiable vectorization methods.

Jinfan Liu, Wuze Zhang, Zhangli Hu +3

Shanghai Jiao Tong University

Differentiable Rendering Differentiable Optimization

Apr 3, 2026

Differentiable SpaTiaL: Symbolic Learning and Reasoning with Geometric Temporal Logic for Manipulation Tasks

Executing complex manipulation in cluttered environments requires satisfying coupled geometric and temporal constraints. Although Spatio-Temporal Logic (SpaTiaL) offers a principled specification framework, its use in gradient-based optimization is limited by non-differentiable geometric operations. Existing differentiable temporal logics focus on the robot's internal state and neglect interactive object-environment relations, while spatial logic approaches that capture such interactions rely on discrete geometry engines that break the computational graph and preclude exact gradient propagation. To overcome this limitation, we propose Differentiable SpaTiaL, a fully tensorized toolbox that constructs smooth, autograd-compatible geometric primitives directly over polygonal sets. To the best of our knowledge, this is the first end-to-end differentiable symbolic spatio-temporal logic toolbox. By analytically deriving differentiable relaxations of key spatial predicates--including signed distance, intersection, containment, and directional relations--we enable an end-to-end differentiable mapping from high-level semantic specifications to low-level geometric configurations, without invoking external discrete solvers. This fully differentiable formulation unlocks two core capabilities: (i) massively parallel trajectory optimization under rigorous spatio-temporal constraints, and (ii) direct learning of spatial logic parameters from demonstrations via backpropagation. Experimental results validate the effectiveness and scalability of the proposed framework.

Licheng Luo, Kaier Liang, Cristian-Ioan Vasile +1

UC Riverside · Lehigh University

Differentiable Logic Spatiotemporal Reasoning Robotic Manipulation Differentiable Optimization Trajectory Optimization

Mar 31, 2026

Efficient Bilevel Optimization with KFAC-Based Hypergradients

Bilevel optimization (BO) is widely applicable to many machine learning problems. Scaling BO, however, requires repeatedly computing hypergradients, which involves solving inverse Hessian-vector products (IHVPs). In practice, these operations are often approximated using crude surrogates such as one-step gradient unrolling or identity/short Neumann expansions, which discard curvature information. We build on implicit function theorem-based algorithms and propose to incorporate Kronecker-factored approximate curvature (KFAC), yielding curvature-aware hypergradients with a better performance efficiency trade-off than Conjugate Gradient (CG) or Neumann methods and consistently outperforming unrolling. We evaluate this approach across diverse tasks, including meta-learning and AI safety problems. On models up to BERT, we show that curvature information is valuable at scale, and KFAC can provide it with only modest memory and runtime overhead. Our implementation is available at https://github.com/liaodisen/NeuralBo.

Disen Liao, Felix Dangel, Yaoliang Yu

University of Waterloo · Vector Institute

Kronecker-Factored Approximation Bilevel Optimization Hypergradient Optimization Differentiable Optimization Curvature-Aware Optimization Second-Order Optimization

Mar 30, 2026

Differentiable Initialization-Accelerated CPU-GPU Hybrid Combinatorial Scheduling

This paper presents a hybrid CPU-GPU framework for solving combinatorial scheduling problems formulated as Integer Linear Programming (ILP). While scheduling underpins many optimization tasks in computing systems, solving these problems optimally at scale remains a long-standing challenge due to their NP-hard nature. We introduce a novel approach that combines differentiable optimization with classical ILP solving. Specifically, we utilize differentiable presolving to rapidly generate high-quality partial solutions, which serve as warm-starts for commercial ILP solvers (CPLEX, Gurobi) and rising open-source solver HiGHS. This method enables significantly improved early pruning compared to state-of-the-art standalone solvers. Empirical results across industry-scale benchmarks demonstrate up to a

10\times

performance gain over baselines, narrowing the optimality gap to

<0.1\%

. This work represents the first demonstration of utilizing differentiable optimization to initialize exact ILP solvers for combinatorial scheduling, opening new opportunities to integrate machine learning infrastructure with classical exact optimization methods across broader domains.

Mingju Liu, Jiaqi Yin, Alvaro Velasquez +1

University of Maryland, College Park, Maryland, USA · University of Colorado, Boulder, Colorado, USA

Combinatorial Optimization Integer Linear Programming Differentiable Optimization

Mar 28, 2026

MetaTune: Adjoint-based Meta-tuning via Robotic Differentiable Dynamics

Disturbance observer-based control has shown promise in robustifying robotic systems against uncertainties. However, tuning such systems remains challenging due to the strong coupling between controller gains and observer parameters. In this work, we propose MetaTune, a unified framework for joint auto-tuning of feedback controllers and disturbance observers through differentiable closed-loop meta-learning. MetaTune integrates a portable neural policy with physics-informed gradients derived from differentiable system dynamics, enabling adaptive gains across tasks and operating conditions. We develop an adjoint method that efficiently computes the meta-gradients with respect to adaptive gains backward in time to directly minimize the cost-to-go. Compared to existing forward methods, our approach reduces the computational complexity to be linear in the data horizon. On quadrotor control tasks, MetaTune achieves competitive or improved tracking performance while reducing gradient computation time by more than 50%. In PX4-Gazebo hardware-in-the-loop simulation, the learned policy transfers zero-shot and reduces tracking RMSE by about 15--20% in aggressive flight and up to 40% under strong disturbances.

Xiexin Peng, Bingheng Wang, Tao Zhang +2

School of Artificial Intelligence and Automation, Huazhong University of Science and Technology, Wuhan 430074, China · Department of Electrical and Computer Engineering, National University of Singapore, Singapore 117583, Singapore · Wuhan Zhenyou Technology Co., Ltd., Wuhan 430205, China

Robust Control Feedback Control Quadrotor Control Differentiable Optimization Adaptive Control Disturbance Rejection

Mar 16, 2026

Sample-Efficient Hypergradient Estimation for Decentralized Bi-Level Reinforcement Learning

Many strategic decision-making problems, such as environment design for warehouse robots, can be naturally formulated as bi-level reinforcement learning (RL), where a leader agent optimizes its objective while a follower solves a Markov decision process (MDP) conditioned on the leader's decisions. In many situations, a fundamental challenge arises when the leader cannot intervene in the follower's optimization process; it can only observe the optimization outcome. We address this decentralized setting by deriving the hypergradient of the leader's objective, i.e., the gradient of the leader's strategy that accounts for changes in the follower's optimal policy. Unlike prior hypergradient-based methods that require extensive data for repeated state visits or rely on gradient estimators whose complexity can increase substantially with the high-dimensional leader's decision space, we leverage the Boltzmann covariance trick to derive an alternative hypergradient formulation. This enables efficient hypergradient estimation solely from interaction samples, even when the leader's decision space is high-dimensional. Additionally, to our knowledge, this is the first method that enables hypergradient-based optimization for 2-player Markov games in decentralized settings. Experiments highlight the impact of hypergradient updates and demonstrate our method's effectiveness in both discrete and continuous state tasks.

Mikoto Kudo, Takumi Tanabe, Akifumi Wachi +1

University of Tsukuba · RIKEN AIP · LY Corporation

Hypergradient Optimization Decentralized MARL Sample-Efficient RL Differentiable Optimization

Mar 2, 2026

FAST-DIPS: Adjoint-Free Analytic Steps and Hard-Constrained Likelihood Correction for Diffusion-Prior Inverse Problems

Training-free diffusion priors enable inverse-problem solvers without retraining, but for nonlinear forward operators data consistency often relies on repeated derivatives or inner optimization/MCMC loops with conservative step sizes, incurring many iterations and denoiser/score evaluations. We propose a training-free solver that replaces these inner loops with a hard measurement-space feasibility constraint (closed-form projection) and an analytic, model-optimal step size, enabling a small, fixed compute budget per noise level. Anchored at the denoiser prediction, the correction is approximated via an adjoint-free, ADMM-style splitting with projection and a few steepest-descent updates, using one VJP and either one JVP or a forward-difference probe, followed by backtracking and decoupled re-annealing. We prove local model optimality and descent under backtracking for the step-size rule, and derive an explicit KL bound for mode-substitution re-annealing under a local Gaussian conditional surrogate. We also develop a latent variant and a one-parameter pixel

\rightarrow

latent hybrid schedule. Experiments achieve competitive PSNR/SSIM/LPIPS with up to 19.5

\times

speedup, without hand-coded adjoints or inner MCMC.

Minwoo Kim, Seunghyeok Shin, Hongki Lim

Department of Electrical and Computer Engineering, Inha University Incheon, 22212, South Korea

Diffusion-Based Inverse Problems Inverse Problems Diffusion Model Inference Acceleration Constrained Optimization Differentiable Optimization

Feb 27, 2026

End-to-end Differentiable Calibration and Reconstruction for Optical Particle Detectors

Large-scale homogeneous detectors with optical readouts are widely used in particle detection, with Cherenkov and scintillator neutrino detectors as prominent examples. Analyses in experimental physics rely on high-fidelity simulators to translate sensor-level information into physical quantities of interest. This task critically depends on accurate calibration, which aligns simulation behavior with real detector data, and on tracking, which infers particle properties from optical signals. We present the first end-to-end differentiable optical particle detector simulator, enabling simultaneous calibration and reconstruction through gradient-based optimization. Our approach unifies simulation, calibration, and tracking, which are traditionally treated as separate problems, within a single differentiable framework. We demonstrate that it achieves smooth and physically meaningful gradients across all key stages of light generation, propagation, and detection while maintaining computational efficiency. We show that gradient-based calibration and reconstruction greatly simplify existing analysis pipelines while matching or surpassing the performance of conventional non-differentiable methods in both accuracy and speed. Moreover, the framework's modularity allows straightforward adaptation to diverse detector geometries and target materials, providing a flexible foundation for experiment design and optimization. The results demonstrate the readiness of this technique for adoption in current and future optical detector experiments, establishing a new paradigm for simulation and reconstruction in particle physics.

Omar Alterkait, César Jesús-Valls, Ryo Matsumoto +2

Department of Physics and Astronomy, Tufts University, 574 Boston Ave, Medford, MA 02155, USA · The NSF AI Institute for Fundamental Interactions, MIT, Cambridge, MA 02139, USA · European Organization for Nuclear Research (CERN), 1211 Geneva 23, Switzerland +3

Differentiable Physics Differentiable Optimization

Feb 26, 2026

Differentiable Zero-One Loss via Hypersimplex Projections

Recent advances in machine learning have emphasized the integration of structured optimization components into end-to-end differentiable models, enabling richer inductive biases and tighter alignment with task-specific objectives. In this work, we introduce a novel differentiable approximation to the zero-one loss-long considered the gold standard for classification performance, yet incompatible with gradient-based optimization due to its non-differentiability. Our method constructs a smooth, order-preserving projection onto the n,k-dimensional hypersimplex through a constrained optimization framework, leading to a new operator we term Soft-Binary-Argmax. After deriving its mathematical properties, we show how its Jacobian can be efficiently computed and integrated into binary and multiclass learning systems. Empirically, our approach achieves significant improvements in generalization under large-batch training by imposing geometric consistency constraints on the output logits, thereby narrowing the performance gap traditionally observed in large-batch training.

Camilo Gomez, Pengyang Wang, Liansheng Tang

School of Data, Mathematical, and Statistical Sciences, University of Central Florida, Orlando, USA · Department of CIS, University of Macau, Macao, China

Classification Deep Learning Optimization Differentiable Optimization

Feb 23, 2026

GauS: Differentiable Scheduling Optimization via Gaussian Reparameterization

Efficient operator scheduling is a fundamental challenge in software compilation and hardware synthesis. While recent differentiable approaches have sought to replace traditional ones like exact solvers or heuristics with gradient-based search, they typically rely on categorical distributions that fail to capture the ordinal nature of time and suffer from a parameter space that scales poorly. In this paper, we propose a novel differentiable framework, GauS, that models operator scheduling as a stochastic relaxation using Gaussian distributions, which fully utilize modern parallel computing devices like GPUs. By representing schedules as continuous Gaussian variables, we successfully capture the ordinal nature of time and reduce the optimization space by orders of magnitude. Our method is highly flexible to represent various objectives and constraints, which provides the first differentiable formulation for the complex pipelined scheduling problem. We evaluate our method on a range of benchmarks, demonstrating that Gaus achieves Pareto-optimal results.

Yaohui Cai, Vesal Bakhtazad, Cunxi Yu +1

Cornell University · University of Maryland, College Park

Differentiable Optimization

Feb 23, 2026

Understanding the Curse of Unrolling

Algorithm unrolling is ubiquitous in machine learning, particularly in hyperparameter optimization and meta-learning, where Jacobians of solution mappings are computed by differentiating through iterative algorithms. Although unrolling is known to yield asymptotically correct Jacobians under suitable conditions, recent work has shown that the derivative iterates may initially diverge from the true Jacobian, a phenomenon known as the curse of unrolling. In this work, we provide a non-asymptotic analysis that explains the origin of this behavior and identifies the algorithmic factors that govern it. We show that truncating early iterations of the derivative computation mitigates the curse while simultaneously reducing memory requirements. Finally, we demonstrate that warm-starting in bilevel optimization naturally induces an implicit form of truncation, providing a practical remedy. Our theoretical findings are supported by numerical experiments on representative examples.

Sheheryar Mehmood, Florian Knoll, Peter Ochs

Saarland University, Saarbrücken, Germany · Friedrich-Alexander University, Erlangen, Germany

Algorithm Unrolling Differentiable Optimization

Feb 20, 2026

Understanding the Generalization of Bilevel Programming in Hyperparameter Optimization: A Tale of Bias-Variance Decomposition

Gradient-based hyperparameter optimization (HPO) have emerged recently, leveraging bilevel programming techniques to optimize hyperparameter by estimating hypergradient w.r.t. validation loss. Nevertheless, previous theoretical works mainly focus on reducing the gap between the estimation and ground-truth (i.e., the bias), while ignoring the error due to data distribution (i.e., the variance), which degrades performance. To address this issue, we conduct a bias-variance decomposition for hypergradient estimation error and provide a supplemental detailed analysis of the variance term ignored by previous works. We also present a comprehensive analysis of the error bounds for hypergradient estimation. This facilitates an easy explanation of some phenomena commonly observed in practice, like overfitting to the validation set. Inspired by the derived theories, we propose an ensemble hypergradient strategy to reduce the variance in HPO algorithms effectively. Experimental results on tasks including regularization hyperparameter learning, data hyper-cleaning, and few-shot learning demonstrate that our variance reduction strategy improves hypergradient estimation. To explain the improved performance, we establish a connection between excess error and hypergradient estimation, offering some understanding of empirical observations.

Yubo Zhou, Jun Shu, Junmin Liu +1

School of Mathematics and Statistics and Ministry of Education Key Lab of Intelligent Networks and Network Security, Xi’an Jiaotong University, Xi’an, 100190, Shaan’xi Province, P. R. China. · Pazhou Lab (Huangpu), Street, Guangzhou, Guangdong Province, P. R. China.

Hypergradient Optimization Hyperparameter Optimization Bilevel Optimization Variance Reduction Differentiable Optimization

Feb 18, 2026

RoboLayout: Differentiable 3D Scene Generation for Embodied Agents

Recent advances in vision language models (VLMs) have shown strong potential for spatial reasoning and 3D scene layout generation from open-ended language instructions. However, generating layouts that are not only semantically coherent but also feasible for interaction by embodied agents remains challenging, particularly in physically constrained indoor environments. In this paper, RoboLayout is introduced as an extension of LayoutVLM that augments the original framework with agent-aware reasoning and improved optimization stability. RoboLayout integrates explicit reachability constraints into a differentiable layout optimization process, enabling the generation of layouts that are navigable and actionable by embodied agents. Importantly, the agent abstraction is not limited to a specific robot platform and can represent diverse entities with distinct physical capabilities, such as service robots, warehouse robots, humans of different age groups, or animals, allowing environment design to be tailored to the intended agent. In addition, a local refinement stage is proposed that selectively reoptimizes problematic object placements while keeping the remainder of the scene fixed, improving convergence efficiency without increasing global optimization iterations. Overall, RoboLayout preserves the strong semantic alignment and physical plausibility of LayoutVLM while enhancing applicability to agent-centric indoor scene generation, as demonstrated by experimental results across diverse scene configurations.

Ali Shamsaddinlou

3D Scene Layout Generation 3D Scene Generation Indoor Scene Generation Functional 3D Scene Generation Differentiable Optimization

Feb 5, 2026

Shiva-DiT: Residual-Based Differentiable Top- $k$ Selection for Efficient Diffusion Transformers

Diffusion Transformers (DiTs) are costly at high resolution because self-attention scales quadratically with token sequence length. Existing pruning methods do not jointly provide end-to-end learnability, low training overhead, and deterministic token counts for predictable token-dependent computation. We propose Shiva-DiT, based on Residual-Based Differentiable Top-k Selection. Its forward pass executes hard top-k selection, while a residual-aware straight-through estimator propagates gradients to both token scores and the budget k without evaluating a second backbone path. A Context-Aware Router and Adaptive Ratio Policy learn layer- and timestep-dependent retention schedules under a target average budget. Experiments on SD3-Medium, Flux.1-dev, and PixArt-Σ show consistent reductions in FLOPs and measured latency. On SD3-Medium, Shiva-DiT provides four fidelity-latency operating points and reaches a 1.54x wall-clock speedup with competitive fidelity.

Jiaji Zhang, Hailiang Zhao, Jiaju Wu +3

College of Computer Science and Technology, Zhejiang University · 3Nanyang Technological University

Diffusion Model Acceleration Diffusion Transformer Token Pruning Differentiable Optimization

Jan 23, 2026

Learning to Optimize by Differentiable Programming

Solving massive-scale optimization problems requires scalable first-order methods with low per-iteration cost. This tutorial highlights a shift in optimization: using differentiable programming not only to execute algorithms but to learn how to design them. Modern frameworks such as PyTorch, TensorFlow, and JAX enable this paradigm through efficient automatic differentiation. Embedding first-order methods within these systems allows end-to-end training that improves convergence and solution quality. Guided by Fenchel-Rockafellar duality, the tutorial demonstrates how duality-informed iterative schemes such as the alternating direction method of multipliers, and the primal-dual hybrid gradient can be learned and adapted through representative case studies.

Liping Tao, Xindi Tong, Chee Wei Tan

Nanyang Technological University, Singapore · Nanyang Technological University, Singapore, Singapore

Primal-Dual Optimization Learning to Optimize Alternating Direction Method of Multipliers Differentiable Programming Constrained Optimization Differentiable Optimization

Jan 12, 2026

Learning to bin: differentiable and Bayesian optimization for multi-dimensional discriminants in high-energy physics

Categorizing events using discriminant observables is central to many high-energy physics analyses. Yet, bin boundaries are often chosen manually. A simple, popular choice in multi-classification tasks is to assign events according to the largest per-class score ("argmax") and to apply equidistant binning to the resulting one-dimensional discriminants. We propose a binning optimization for signal significance directly in multi-dimensional discriminants. We use a Gaussian Mixture Model (GMM) to define flexible regions in the score space, which can be interpreted either as bins or as analysis categories. While this GMM-based strategy is applicable in both one and multiple dimensions, we also study a direct bin-boundary optimization in one dimension as a simpler alternative for binary discriminants. On this binning model, we study two optimization strategies: a differentiable and a Bayesian optimization approach. We study two toy setups: a binary classification and a three-class problem with two signals and backgrounds. In the one-dimensional case, both approaches achieve similar gains in signal sensitivity compared to equidistant binning for a given number of bins, while in the multi-dimensional case the differentiable approach performs best. We show that the GMM-based optimization can outperform argmax classification even after optimized binning is applied to the one-dimensional projections. We further study the performance of our methods on the FAIR Universe

H\rightarrowττ

dataset, where the GMM-based optimization gives the highest signal significance. Both methods are released as lightweight Python plugins intended for straightforward integration into existing analyses.

Johannes Erdmann, Nitish Kumar Kasaraguppe, Florian Mausolf

RWTH Aachen University, III. Physikalisches Institut A, Aachen, Germany

High-Energy Physics Differentiable Optimization

Sep 2, 2025

Differentiable Expectation-Maximisation and Applications to Gaussian Mixture Model Optimal Transport

The Expectation-Maximisation (EM) algorithm is a central tool in statistics and machine learning, widely used for latent-variable models such as Gaussian Mixture Models (GMMs). Despite its ubiquity, EM is typically treated as a non-differentiable black box, preventing its integration into modern learning pipelines where end-to-end gradient propagation is essential. In this work, we present and compare several differentiation strategies for EM, from full automatic differentiation to approximate methods, assessing their accuracy and computational efficiency. As a key application, we leverage this differentiable EM in the computation of the Mixture Wasserstein distance

\mathrm{MW}_2

between GMMs, allowing

\mathrm{MW}_2

to be used as a differentiable loss in imaging and machine learning tasks. To complement our practical use of

\mathrm{MW}_2

, we contribute a novel stability result which provides theoretical justification for the use of

\mathrm{MW}_2

with EM, and also introduce a novel unbalanced variant of

\mathrm{MW}_2

. Numerical experiments on barycentre computation, colour and style transfer, image generation, and texture synthesis illustrate the versatility of the proposed approach in different settings.

Samuel Boïté, Eloi Tanguy, Julie Delon +2

Université Paris Cité, CNRS, MAP5, F-75006 Paris, France · Centre Borelli, CNRS and ENS Paris-Saclay, F-91190 Gif-sur-Yvette, France · CMAP, CNRS, Ecole Polytechnique, Institut Polytechnique de Paris

Gaussian Mixture Models Expectation-Maximization Differentiable Optimization Optimal Transport Wasserstein Distance

May 28, 2025

Accelerating Optimization via Differentiable Stopping Time

Optimization is an important module of modern machine learning applications. Tremendous efforts have been made to accelerate optimization algorithms. A common formulation is achieving a lower loss at a given time. This enables a differentiable framework with respect to the algorithm hyperparameters. In contrast, its dual, minimizing the time to reach a target loss, is believed to be non-differentiable, as the time is not differentiable. As a result, it usually serves as a conceptual framework or is optimized using zeroth-order methods. To address this limitation, we propose a differentiable stopping time and theoretically justify it based on differential equations. An efficient algorithm is designed to backpropagate through it. As a result, the proposed differentiable stopping time enables a new differentiable formulation for accelerating algorithms. We further discuss its applications, such as online hyperparameter tuning and learning to optimize. Our proposed methods show superior performance in comprehensive experiments across various problems, which confirms their effectiveness.

Zhonglin Xie, Yiman Fong, Haoran Yuan +1

Beijing International Center for Mathematical Research, Peking University · Department of Industrial Engineering, Tsinghua University · School of Mathematics Science, Peking University

Differentiable Optimization Hyperparameter Optimization Learning to Optimize