Differentiable Optimization

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6 papers in the last four weeks, against 2 the four weeks before. 0.1% of all new papers.

Jul 13Week of Sep 28

Latest papers 58

Oct 6, 2026cs.LG

Decision-Focused Learning in MDPs: An Occupancy Measure Approach

In this work, we consider decision-focused learning (DFL) for a Markov decision process (MDP), where existing methods differentiate through the KKT conditions of the Bellman equation and require solving a linear system over all state-action pairs, limiting its scalability. We address this by reformulating the MDP as an occupancy measure-based linear program (LP), whose feasible region is induced by predicted dynamics, and we derive a closed-form gradient by identifying the active constraints in the feasible polyhedron via the pivoting algorithm. This occupancy measure-based LP layer raises two challenges: (1) LP's solution gradient is discontinuous when active constraints change, and (2) the LP backward cost still scales with the state size, which is costly for large or continuous state spaces. We address the challenges with an augmented Lagrangian surrogate and smooth the boundary jumps by random row sketching of the constraints, and a learnable soft state-aggregation layer and its function-approximation generalization that scales the LP to large finite and continuous-state MDPs. Across multiple tasks, our methods reach lower regret than KKT-based DFL and two-stage baselines with significantly lower computation cost. The source code for all experiments is available at https://github.com/A-Eshragh/State_Aggregation_Project.

Zihao Zhao, Ashwath K. Karunakaram, Ali Eshragh +2

Georgia Institute of Technology, Atlanta, GA · Johns Hopkins Carey Business School, Washington, DC · International Computer Science Institute, Berkeley, CA

Markov Decision Processes Decision-Focused Learning Differentiable Optimization

Sep 30, 2026cs.LG

Same Loss, Different Gradients

Differentiable learning typically assumes that the scalar objective evaluated in the forward pass and the gradient supplied to the optimizer in the backward pass describe the same mathematical object. We show that this correspondence can fail when probabilistic objectives rely on finite special-function recurrences, custom backward rules, and numerical clipping. In high-dimensional von Mises-Fisher learning, real numerical implementations can produce identical forward scores and losses at the same learning state while supplying different gradients and following different optimization trajectories. We characterize the structure of this mismatch in finite-start Bessel recurrence and show that classwise radial mismatch can compose through probabilities into a locally nonconservative update field. Evaluating the accuracy of special-function values and derivatives separately is therefore insufficient to characterize the realized learning objective. Motivated by this observation, we introduce AR/FR, a fixed-depth analytic realization that constructs a potential and its derivative jointly, ensuring forward-backward coherence by construction. We establish a uniform cubic-order error bound relative to the exact Bessel ratio over the entire nonnegative concentration axis and propagate this guarantee to learning scores and objectives. As representation dimension increases, the original finite recurrence becomes sequentially deeper, whereas the worst-case AR/FR error guarantee tightens cubically, jointly providing coherence, certified fidelity, and fixed-depth computation. These results suggest that a differentiable numerical primitive is defined by both the values it realizes and the derivatives it actually supplies to the optimizer; together, they constitute the numerical realization of the learning algorithm.

Ningkang Peng, Xiaoqian Peng, Yifan He +4

Nanjing Normal University · Nanjing University of Chinese Medicine

Automatic Differentiation Differentiable Optimization

Sep 28, 2026cs.RO

On the Numerical Reliability of Differentiable Physics-Based Optimization for Robotic Material Manipulation

Differentiable physics is increasingly used in robotic material manipulation for system identification, trajectory or skill optimization, demonstration generation, and robot or end-effector design. These applications depend on gradients propagated through long, contact-rich simulation rollouts. We study the numerical reliability of those gradients using two Material Point Method (MPM) system-identification benchmarks derived from elastoplastic and granular manipulation. The benchmarks provide controlled cases for three effects that also arise in broader differentiable physics-based optimization. GPU many-to-one sums whose order depends on thread scheduling changed long-horizon gradients and reversed the sign of one parameter gradient relative to a deterministic reference. Finite-difference checks became less reliable for longer rollouts because repeated-run loss variation grew much faster than the loss change produced by the tested parameter perturbations. Observation and loss definitions changed optimization behaviour and the solution preferred by an independent metric. These results motivate reproducible accumulation, finite-difference validation that compares perturbation-induced loss changes with repeated-run variation, and explicit reporting of objective construction when differentiable simulation is used for robotic optimization.

Xintong Yang, Minglun Wei, Yu-Kun Lai +1

School of Engineering, Cardiff University, Cardiff, United Kingdom · School of Computer Science and Informatics, Cardiff University, Cardiff, United Kingdom · College of Mechanical and Electrical Engineering, Hohai University, China

Robot Dynamics Identification Material Point Method Robotic Manipulation Differentiable Physics Differentiable Optimization

Sep 27, 2026cs.LG

dOPT: Differentiating Conic Optimization via Geometric Reduction

Optimization layers enable the incorporation of structured constraints and decision problems into learning systems. Training such systems requires differentiating through the embedded optimization problem, which can be challenging for general conic programs. We introduce dOPT, a solver-agnostic framework that, rather than differentiating the full conic formulation, reduces it at a computed primal-dual solution to an equality-constrained quadratic program that preserves the reference solution and its first-order sensitivity. The reduction captures the local first- and second-order conic geometry relevant to differentiation and remains well defined at singular configurations. Computing solution derivatives then requires a single symmetric linear solve, independently of the forward solver. We derive explicit reductions for convex NLPs, QPs, SOCPs, and SDPs. Numerical experiments validate the computed gradients and show favorable backward-pass scalability, with substantial speedups over existing differentiable conic optimization methods as problem size increases.

Fengyu Yang, Connor W. Magoon, Tyler Watts +1

Department of Mathematics University of North Carolina at Chapel Hill

Differentiable Programming Constrained Optimization Differentiable Optimization

Sep 27, 2026math.OC

lapanda: A Matrix-Free Differentiable Solver for Nonconvex Constrained Optimization Layers

Differentiable optimization brings the structural guarantees of mathematical optimization to network pipelines, allowing them to be trained end-to-end. However, its application remains challenging for nonconvex constrained problems, as existing differentiable solvers often suffer from limited modeling expressiveness due to their reliance on specialized problem structures, while also incurring substantial computation time and memory overhead in both the forward and backward passes. To address these challenges, we propose lapanda, a matrix-free differentiable solver for nonconvex optimization with general constraints. It reformulates the problem to a sequence of augmented Lagrangian subproblems, each handled by a first-order inner solver through a proximal averaged quasi-Newton algorithm with adaptive linesearch, thus enabling efficient forward optimization. We establish local well-posedness of the solution map and convergence of the outer iterations, and further derive a sensitivity alignment between the original problem and the final subproblem in the backward pass, demonstrating that the subproblem sensitivity, which can be computed efficiently in a matrix-free manner, provides a principled approximation to the exact optimizer sensitivity. We evaluate lapanda on nonconvex constrained Rosenbrock benchmarks, imitation learning with several representative constrained optimal control problems, and embedded robotic obstacle-avoidance tasks. Compared with state-of-the-art differentiable solvers, lapanda delivers substantial reductions in computation time and memory footprint while maintaining reliable constraint satisfaction and learning performance.

Yuankun Chen, Zifei Nie, Kangyu Lin +2

School of Artificial Intelligence, Jilin University, China · Graduate School of Informatics, Kyoto University, JPN · Department of Civil and Systems Engineering, Johns Hopkins University, USA

Differentiable Programming Nonconvex Optimization Quasi-Newton Methods Augmented Lagrangian Method Differentiable Optimization

Sep 17, 2026cs.AI

Customizable and Jointly Optimized Route Planning: A Deep Architecture Enabling Differentiable Shortest-Path Search

With the widespread use of online navigation and ride-hailing services, achieving optimal route planning for diverse user preferences has recently attracted increasing attention. Classic graph algorithms for pathfinding use heuristic cost functions to define edge weight, thus providing no optimality guarantee of route quality. Prior data-driven approaches equating ground truth of the optimal route with user trajectory, which is however moderately influenced by the navigation service, suffers from the feedback loop problem. To address these issues, we propose a deep architecture that is able to jointly optimize cost functions and route-ranking model towards any route preference. First, we run a multi-objective Dijkstra algorithm offline to collect the set of Pareto optimal routes, deeming it as the complete candidate set. Exploiting the property of such a set, we design a neural network structure that emulates shortest-path search and route ranking in an end-to-end differentiable manner. Second, we define route preference as a task of constrained optimization of route attributes, and propose a novel loss function that optimizes a single-objective variable, with other variables strictly under constraints. We conduct extensive experiments on real-world datasets. The results show that our architecture significantly outperforms state-of-the-art methods in route quality and customizability.

Rui Zhao, Chao Chen, Longfei Xu +4

Alibaba Group Beijing, China · Alibaba Group Hangzhou, China

Graph-Based Planning Differentiable Optimization Multi-Objective Optimization

Sep 14, 2026cs.LG

HGTO: A Unified Graph-Based Physics-Informed Formulation for Structural Topology Optimization

Density-based topology optimization is typically structured as a nested sequence of material updates, structural analyses, and sensitivity assessments. While neural density parameterization and dual-field physics-informed approaches provide data-free alternatives, most existing methods represent density and displacement as coordinate fields and make limited use of the discrete relationships inherent in the finite element mesh. The present study introduces HGTO, a unified graph-based formulation that extends complete neural topology optimization from coordinate space to finite-element graph space. Element densities are parameterized on the element graph derived from the mesh, and the structural state is determined on the corresponding node--element hypergraph. Finite element kinematics, numerical quadrature, constitutive response, and force assembly remain explicitly defined operations within the differentiable computation. The material field and equilibrium state are therefore coupled through a common finite-element incidence structure. Numerical studies show compliance comparable to conventional density-based optimization at substantially lower computational cost than a representative coordinate-based dual-field neural method. The same coupled formulation accommodates high-resolution and irregular meshes, three-dimensional structures, finite deformation, and elastoplastic response.

Kangzheng Liu, Uday Kumar Punna, Leixin Ma

School for Engineering of Matter, Transport and Energy, Arizona State University, Tempe, 85287, AZ, USA

Topology Optimization Structural Mechanics Differentiable Optimization

Sep 3, 2026cs.CE

Differentiable Hybrid Modelling for Learning and Optimising Chemical Transport Processes from Experimental Data

Reliable transport models are essential when modelling and optimising many chemical engineering processes, yet, most models assume hand-picked constitutive laws which may not reflect reality, and often assume initial conditions are known exactly. Both restrictions can significantly bias model predictions and lead to systematic error when used in predictive and control settings. Black-box neural surrogate alternatives for modelling can better match real example data, but are confined to the task they were trained on and cannot be interrogated for physical consistency. Here we introduce a general-purpose differentiable hybrid modelling framework for transport processes, specifically for the case of population balance equations. Our framework integrates a JAX finite volume population balance solver with learnable neural network components which are trained to both discover constitutive laws and fit initial conditions from real experimental data, allowing us to better model real experimental transport systems. Furthermore, we use our framework for process optimisation, using its differentiability to allow us to direct optimising experimental settings for quantities of interest. This work highlights the huge potential of such differentiable hybrid modelling frameworks for learning and optimising any given chemical separation which involves mass, energy, and/or momentum transport.

Arthur Jessop, Mohammed Alsubeihi, Ben Moseley +1

Department of Chemical Engineering, The University of Manchester, Manchester, M13 9PL, United Kingdom. · Department of Earth Science and Engineering, Imperial College London, South Kensington, SW7 2BX, United Kingdom. · I-X Centre for AI in Science, Imperial College London, South Kensington, SW7 2BX, United Kingdom.

Differentiable Physics Differentiable Optimization

Sep 2, 2026cs.LG

Differentiable Electricity-Market Clearing for Gradient-Based Planning

Planning a large data center is difficult because a facility big enough to matter changes the electricity prices it will pay. Those prices are set by market clearing, a constrained optimization problem solved anew in every operating condition. However, simulating the market tells a planner how a candidate plan performs but not how to improve it. Here we treat market clearing as a differentiable optimization layer: each forward pass solves the market, and reverse-mode automatic differentiation propagates the planning cost back through the cleared prices to the plan. After validating these gradients against finite differences, we apply them to a concrete problem: allocating 50 MW of data-center load across six candidate buses in two synthetic networks, under a fixed cost per active site, evaluated over 36 operating states. Judged against exhaustive enumeration of all site combinations, gradient optimization recovers the continuous allocations almost exactly, with worst-case objective gaps of 2.3% and 8.5% of the cost difference between the best and worst single site. Its one systematic error is instructive: near the costs at which a site should close, the smooth relaxation of the discrete site count shrinks the site rather than closing it, so discrete transitions arrive late. Differentiable market clearing thus turns market-aware planning into a problem gradients can search.

Luca Mungo, Maarten P. Scholl, Arnau Quera-Bofarull

Macrocosm Inc. · Institute for New Economic Thinking, University of Oxford

Differentiable Optimization

Aug 7, 2026cs.RO

Hölder Signed Distance: A Differentiable, Signed, Parallelizable Metric for Robotics

Computing distances between sets is essential in robotic motion planning and control, where differentiable gradients enable real-time optimization. The Euclidean Signed Distance Function (SDF), however, is not differentiable everywhere, and existing alternatives often sacrifice differentiability, sign information, or computational efficiency. In this letter, we introduce a novel differentiable signed distance between convex polyhedra. To this end, we first propose differentiable versions of the minimum and maximum operators, termed the Hölder minimum and Hölder maximum. We then replace the original min-max operators in the classical SDF formulation, yielding the Hölder signed distance. Unlike prior differentiable distance formulations that rely on iterative algorithms, our approach is computed in closed form, eliminating convergence issues while remaining naturally amenable to GPU parallelization. We validate the practical advantages and computational performance of the proposed distance through runtime comparisons with existing approaches. We also present a robotic manipulator experiment, demonstrating its suitability for applications in control.

Felipe Bartelt, Ali Umut Kaypak, Anthony Tzes +3

Signed Distance Functions Differentiable Optimization Robot Motion Planning

Aug 7, 2026cs.AI

Fast LapSum: Exact Differentiable Top- $k$ at Million Scale

Selecting the top-

k

elements is a fundamental operation for inducing sparsity in large-scale models and optimization problems, enabling robust expert activation, token routing or attention pruning. However, hard top-

k

is non-differentiable, while existing differentiable alternatives become increasingly expensive as the number of coordinates grows. We introduce Fast LapSum, a scalable solver for the LapSum soft top-

k

formulation that preserves an exact selection mass of

k

, while supporting end-to-end differentiation. In Fast LapSum, we reduce sorting cost using probabilistic bracketing, which restricts sorting to a narrow band of scores around the threshold using a binomial order-statistic from kernel-noised samples. A certification pass upgrades the probabilistic localization to a verified one at the cost of one additional linear pass, with a full-sort fallback that covers the worst-case scenario. Our certified GPU implementation processes

10^6

,

10^7

, and

10^8

scores in median times of

0.92

,

1.39

, and

7.24

ms, respectively, making exact-budget soft top-

k

practical within million-scale optimization loops. We demonstrate this capability in two applications: megapixel sparse adversarial examples with a small fraction of initial image pixels, where Fast LapSum achieves an order-of-magnitude speedup over state-of-the-art methods, and 3D Gaussian splatting. In the latter, we use Fast LapSum to reduce the number of Gaussians produced by Adaptive Density Control to a substantially smaller number while retaining nearly the same rendering quality and massively reducing computation. These results demonstrate that exact-budget differentiable top-

k

can be incorporated into practical million-scale optimization pipelines.

Jakub Antczak, Joanna Wojciechowicz, Kamil Książek +3

Wrocław University of Science and Technology, Wrocław, Poland. · Faculty of Mathematics and Computer Science, Jagiellonian University, Kraków, Poland.

Differentiable Programming Differentiable Optimization Consistent Adversarial Attacks

Aug 5, 2026cs.LG

Differentiating Through Dual Prices: End-to-End Policy Learning Under Capacity Constraints

Many social services assign scarce resources, such as housing assistance or hospital interventions, to people who arrive one at a time: each arrival must receive a decision immediately, and the long-run usage of every resource must stay within its capacity. We study how to learn such an assignment policy from logged observational data. The standard pipeline is decision-blind: fit one outcome model per arm by regression, price each capacitated resource from the fitted models, and assign each arrival the arm whose predicted outcome minus price is largest. We instead train the outcome models end-to-end, differentiating an off-policy estimate of the deployed policy's value through the dual prices themselves. We study two formulations: an exact nonconvex one, and a convex relaxation whose optimum always satisfies the capacity constraints in expectation and which is suboptimal by at most a term linear in the smoothing temperature and logarithmic in the number of arms. Every method is evaluated in a queueing simulation with resources replenished at their capacity rates. Across six datasets, the two end-to-end variants take the top slots on a deployment-adjusted value index at every delay cost, including zero; when capacities are binding, decision-blind baselines frequently violate them and incur much longer queueing delays. On the largest dataset, a hospital cohort of seventy thousand patients, end-to-end training also achieves significantly higher policy value, a margin that survives a capacity-matched neural baseline. Flexible decision-blind regression remains the stronger pure predictor where ground truth is measurable; end-to-end training is best suited to settings where resources are genuinely scarce and feasibility matters.

Mohammadsaeed Haghi, Mahdi Salmani, Nima Kelidari

University of Southern California

Constrained Optimization Differentiable Optimization Online Resource Allocation

Aug 3, 2026cs.AI

Hard Constraints, Smooth Gradients: Learning Feasible Inventory Policies via Differentiable Projection

Many operational problems are constrained sequential decision processes with large, combinatorial action spaces and interdependent feasibility constraints. Mixed-integer linear programs (MILPs) handle such constraints flexibly but scale poorly in stochastic environments. Deep reinforcement learning (DRL) promises scalable decision rules, but existing methods either penalize constraints rather than enforce them, or rely on feasibility mechanisms that break down once constraints interact. We bridge this gap by embedding a differentiable convex optimization module inside the policy: a neural network proposes continuous action targets, a quadratic program projects them onto the relaxed feasible set, and a dual-informed integer mapping restores integrality while preserving feasibility. Given a differentiable simulator, the policy trains end to end from sampled trajectories using pathwise gradients, while handling hard constraints with similar flexibility to MILPs. We show that our feasibility enforcement has bounded error relative to an exact integer projection and ensures the entire feasible action space is reachable. We apply the method to multi-echelon production-inventory planning under shared resource and material constraints. Our policy attains an average optimality gap below 1% on small instances. It further outperforms state-of-the-art echelon base-stock policies by up to 9.75% and a rolling-horizon multi-stage stochastic program by at least 7.7% in larger networks. On an industry-scale case study from ASML, it reduces average cost by up to 3.22% relative to the best-known benchmark policy. The savings are largest where planning is hardest: in tightly capacitated systems with high demand variability. More broadly, our work shows that DRL can deliver economically significant savings in sequential decision problems with interdependent hard constraints, which are widespread in practice.

Patrick Helm, Jan-Niklas Doerr, Joren Gijsbrechts +1

TUM School of Management, Technical University of Munich · Esade, Ramon Llull University · Munich Data Science Institute (MDSI), Technical University of Munich

Inventory Control Constrained RL Supply Chain Management Differentiable Optimization RL for Combinatorial Optimization

Jul 27, 2026cs.RO

Amortising Trajectory Optimisation for Residual MPC via Implicit Contact Differentiation

Differentiable simulation can accelerate contact-rich trajectory optimisation by exposing local sensitivities of task outcomes to controls. Existing approaches either use finite differences, which are expensive and step-size sensitive; differentiate iterative contact solvers by unrolling automatic differentiation (AD), which stores a growing computation trace; or require intricate, solver-specific KKT sensitivity derivations. We introduce an AD-assisted implicit derivative for regularised smooth contacts and apply it to Mujoco MJX, based on the Implicit Function Theorem (IFT). The method differentiates the stationarity residual at the tolerance-converged solution, avoiding both solver unrolling and hand-assembled KKT systems. IFT keeps compiled temporary memory nearly constant with solver effort, changing by less than 4

\%

from one to ten iterations versus 10.6

\times

growth for unrolled AD. IFT memory grows slower with active contacts and model dimension, using 20

\times

less memory at 256 contacts and 6

\times

less at 16 contacts and 96 DoF. We further introduce optimiser distillation for residual MPC, amortising batched full-horizon iLQR into a policy that guides short-horizon residual iLQR. Across Finger, Franka, and Unitree, this raises six-step success by 28-98 percentage points over standard iLQR.

Daniel Layeghi, Thomas Corbères, Calum Arnott +4

School of Informatics, University of Edinburgh, Scotland.

Trajectory Optimization Automatic Differentiation Nonlinear MPC Model Predictive Control Differentiable Optimization

Jul 27, 2026math.OC

Smooth Learning with Hard Constraints via Legendre-Regularized Policies

We revisit contextual optimization from the perspective of policy class design. A desirable policy class should be expressive enough to learn rich context-decision relationships, should enforce hard feasibility constraints rather than soft penalty terms, and should remain smooth enough for gradient-based training on downstream decision losses. Existing approaches usually emphasize only part of these requirements. We propose Legendre-regularized policies, which parameterize decisions as solutions of regularized optimization problems over the original feasible region. This construction yields policies that are feasible by construction and differentiable with respect to learned latent parameters. We prove that the associated optimizer map is single-valued, maps onto the relative interior of the feasible set, admits an explicit Jacobian, is Lipschitz continuous, and can be made arbitrarily smooth. We also establish a universal approximation result showing that the proposed class can approximate any continuous feasible policy on compact context sets. The framework unifies explicitly regularized optimizers and implicit perturbation-based smooth optimizers. Experiments on contextual newsvendor and resource allocation problems show that our approach improves prescriptive performance relative to the benchmark methods.

Zikun Lin, Rui Chen, Yijie Wang

Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Hong Kong, China. · School of Data Science, The Chinese University of Hong Kong, Shenzhen, Guangdong, China. · School of Economics and Management, Tongji University, Shanghai, China.

Decision-Focused Learning Constrained Optimization Differentiable Optimization

Jul 22, 2026cs.LG

End-to-End Learning of Safe Optimal Feedback Control in High Dimensions with Control Barrier Function Layers

We consider the problem of learning high-dimensional semi-global feedback controllers under hard safety constraints enforced by control barrier functions (CBFs). Incorporating CBFs into end-to-end policy training requires embedding a quadratic-program-based safety filter as an optimization layer, but computational and differentiation bottlenecks have largely restricted prior approaches to low-dimensional systems, typically with at most 16 state dimensions. We address this limitation by combining operator splitting with the recently developed Jacobian-Free Backpropagation (JFB) method to enable scalable end-to-end training while preserving hard safety guarantees through the CBF safety filter. We justify this training methodology theoretically using nonsmooth analysis techniques and demonstrate its effectiveness on high-dimensional multi-agent nonlinear control problems with state and control dimensions up to 1200 and 400, respectively.

Xingjian Li, Kelvin Kan, Deepanshu Verma +3

UT Austin · UCLA · Clemson University +1

Safety-Critical Control Control Barrier Functions Feedback Control Differentiable Optimization Optimal Control

Jul 8, 2026cs.CV

Widest-Path Reachability Fields for Connectivity-Preserving Slender Structure Segmentation

Segmenting slender curvilinear structures such as retinal vessels, cracks, and roads demands topological correctness, as even a single-pixel discontinuity can fragment a continuous network and invalidate downstream analysis. Under standard binary-mask supervision, models optimized for pixel-level overlap frequently produce topologically broken predictions. We trace this to a fundamental mismatch: pixel-wise losses distribute gradients uniformly, yet connectivity hinges on a sparse set of bottleneck pixels. These pixels are vastly outnumbered by thick structures and background, rendering their aggregate gradient contribution negligible. We term this phenomenon topological gradient starvation (TGS). To address it, we propose Widest-Path Reachability Fields (WPRF), a differentiable Max-Min reachability objective that redirects gradient flow to connectivity bottlenecks. The module is plug-and-play, backbone-agnostic, and incurs no inference overhead. WPRF implements a differentiable Max-Min objective via dynamic programming on a domain-restricted graph, coupled with a bottleneck-aware observation term that balances gradient contributions across varying structures. Compared to prior topology-aware losses that rely on post-hoc skeletonization or homology computation, WPRF directly optimizes end-to-end reachability via differentiable Max-Min algebra, enabling gradient flow to concentrate on connectivity bottlenecks without auxiliary structures. We introduce OMVIS, a new oral microvessel segmentation dataset. Experiments across nine architectures and six datasets validate the bottleneck-focused gradient routing mechanism. WPRF improves 87% of experiments with fixed hyperparameters and achieves clDice gains of 7.2 percentage points on structurally fragile datasets.

Youcheng Zong, Runda Jia, Minxuan Hu +2

College of Information Science and Engineering, Northeastern University, Shenyang 110004, China · Changsha Stomatological Hospital, Changsha 410005, China

Image Segmentation Topology-Preserving Image Segmentation Medical Image Segmentation Differentiable Optimization

Jul 6, 2026cs.LG

Directly Optimizing Mean Demographic Parity for Nonlinear Regression

We focus on regression settings where the fairness goal is to equalize average predictions across values of a sensitive attribute, a criterion known as mean demographic parity. Directly optimizing this criterion is difficult because it depends on a conditional mean that is unknown and changes during training. Common dependence penalties and adversarial methods do not estimate this conditional mean; instead, they push predictions toward full independence. This stronger constraint can reduce accuracy even when average predictions are already equal. Existing conditional-mean methods are limited to linear predictors or low-dimensional sensitive attributes. We enable direct optimization of mean demographic parity using DPVar, a fairness measure defined as the variance of the conditional mean prediction. Because the conditional mean must be estimated as the predictor changes, optimizing DPVar leads to a functional bilevel problem. We develop two solvers: FBO, which uses a closed-form hypergradient, and an iterative-differentiation (ITD) solver that differentiates through updates of the conditional-mean estimator. Unlike previous conditional-mean methods, our approach applies to nonlinear predictors and high-dimensional continuous sensitive attributes. Across a semi-synthetic benchmark built from 21 tabular regression datasets and Communities & Crime data, FBO and ITD recover competitive or better accuracy-DPVar trade-offs than existing methods.

Ieva Petrulionyte, Julien Mairal, Michael Arbel

Univ. Grenoble Alpes, Inria, CNRS, Grenoble INP, LJK, 38000 Grenoble, France

Algorithmic Fairness Group Fairness Bilevel Optimization Differentiable Optimization

Jul 4, 2026q-bio.QM

Smooth $\%$ MinMax: A Differentiable Relaxation for Codon Harmonization

Codon harmonization aims to adapt the coding sequences for heterologous expression while preserving the native-like patterns of frequent and rare codons that may influence local translation dynamics and co-translational protein folding. However, widely used harmonization metrics, such as

\%

MinMax, are defined on discrete codon sequences and are, therefore, not readily compatible with gradient-based neural codon design. Here, we introduce Smooth

\%

MinMax, denoted as

\%{\rm MinMax}_{(s)}

, a differentiable relaxation of the conventional hard

\%

MinMax metric, denoted as

\%{\rm MinMax}_{(h)}

.

\%{\rm MinMax}_{(s)}

replaces the discrete codon-usage values with probability-weighted synonymous-codon usage values and replaces the hard

\%

Max/

\%

Min branch with a sigmoid-gated interpolation. This formulation preserves the signed interpretation of

\%{\rm MinMax}_{(h)}

, while enabling optimization with respect to the synonymous-codon probabilities and learnable parameters. In human-to-Escherichia coli codon harmonization experiments,

\%{\rm MinMax}_{(s)}

closely approximates

\%{\rm MinMax}_{(h)}

and supports gradient-based profile matching in synonymous-codon probability space. These results suggest

\%{\rm MinMax}_{(s)}

as a practical bridge between profile-based codon harmonization and neural synonymous-sequence design.

Yoonho Jeong, Hyunwoo Choi, Ryan Fernandez Medina Hariri +3

Department of Chemistry, KAIST, Daejeon 34141, Korea. · School of Chemistry and Chemical Engineering, Highfield Campus, University of Southampton, Southampton SO17 1BJ, United Kingdom.

Differentiable Optimization

Jul 3, 2026cs.LG

Differentiate the Evaluator, Not the Program: An Efficient Runtime Representation for Neuro-Symbolic Learning

AI systems increasingly propose executable scientific models whose value depends on both their symbolic structure and their fitted continuous parameters. This makes parameter calibration the bottleneck of program-and-parameter co-search: an outer loop can generate thousands of candidate programs, but each needs an inner gradient-based optimization before it can be assessed. Staging each candidate into its own differentiable graph makes individual models fast but sacrifices the program-as-data property that keeps search fluid; interpreter-based approaches preserve programs as runtime data but pay interpreter overhead that dominates the numerical work. We present the Native Differentiable Virtual Machine (NDVM), a runtime representation that differentiates executable programs without compiling each candidate into a separate graph. NDVM separates symbolic structure from differentiable numeric state: tags, symbols, environments, and control remain native runtime data, while numeric payloads live in dense batched buffers with exact reverse-mode gradients recorded along the realized execution trace, so one evaluator walk is amortized across large populations of parameter vectors. A locked cost model of a real differentiable self-hosted Scheme interpreter motivates the design. We realize NDVM as a native runtime with forward and gradient equivalence to the reference backend, about 60x per-lane batch amortization, near-linear multicore scaling, and two independent front ends. In fixed-budget co-search over LLM-proposed programs, NDVM reaches high-quality solutions about 24x sooner in wall-clock time, suggesting runtime differentiation as a practical systems foundation for scientific discovery workflows.

Lucas Sheneman

Institute for Interdisciplinary Data Sciences University of Idaho

Automatic Differentiation Differentiable Programming Differentiable Optimization Neurosymbolic Learning

Jul 2, 2026cs.RO

Saturation-Aware Robust Trajectory Optimization for Reusable Launch Vehicles via Differentiable Physics

The high-angle-of-attack flip maneuver of reusable launch vehicles presents significant challenges for robust trajectory optimization due to the combined effects of highly nonlinear dynamics, aerodynamic uncertainties, and actuator saturation. This paper presents a differentiable physics framework for saturation-aware robust trajectory optimization. At its core, a Differentiable Particle Tube Control (DPTC) scheme is developed to optimize uncertainty evolution through an ensemble-based distribution shaping strategy. State uncertainty is represented by a Lagrangian particle ensemble, while hard actuator projection operators are embedded directly into the computational graph, enabling the joint optimization of the nominal feedforward trajectory and a time-varying feedback policy via end-to-end backpropagation. The proposed framework is evaluated against an automatic differentiation-based Successive Convexification (AD-SCvx) baseline combined with a conventional covariance steering feedback strategy. Six-degree-of-freedom Monte Carlo simulations demonstrate that, although the baseline achieves nominal fuel-optimal solutions, its unconstrained feedback formulation becomes susceptible to actuator saturation under aerodynamic disturbances, leading to degraded closed-loop robustness. In contrast, the proposed DPTC framework proactively performs a constraint-aware performance trade-off by relaxing spatial tracking to preserve critical control authority. These results demonstrate that integrating differentiable physics with ensemble-based optimization provides an effective and practical framework for robust guidance in highly constrained aerospace flight systems.

Liwei Chen, Tong Qin

Beijing Institute of Aeronautical Systems Engineering, Beijing 100076, China

Trajectory Optimization Robust Control Feedback Control Differentiable Optimization

Jul 1, 2026cs.LG

Decision-focused Sparse Tangent Portfolio Optimization

Sparse tangent portfolio optimization aims to learn an interpretable, low-cardinality portfolio in the tangency direction of the mean-variance frontier. However, the associated cardinality-constrained formulation is NP-hard, and standard predict-then-optimize pipelines often misalign forecasting accuracy with downstream portfolio quality. We propose an end-to-end decision-focused learning framework that reformulates Sharpe ratio maximization as a Disciplined Parametrized Programming (DPP)-compliant convex programming layer and replaces discrete selection with a smooth top-

k

operator enforcing an exact cardinality

k

. This enables gradient flow through prediction, asset selection, and re-optimization, allowing the predictive model to directly optimize portfolio performance. Across four major equity markets, our method achieves competitive and often superior out-of-sample Sharpe ratios compared with historical and prediction-focused baselines, with particularly strong gains in larger asset universes. Our code is publicly available.

Haeun Jeon, Seunghoon Choi, Hyunglip Bae +2

Department of Industrial and Systems Engineering, Korea Advanced Institute of Science and Technology, Daejeon, Republic of Korea · Department of Big Data Convergence, and Bigdata Research Lab, Chonnam National University, Gwangju, Republic of Korea · Department of Industrial Engineering, Ulsan National Institute of Science and Technology, Ulsan, Republic of Korea

Decision-Focused Learning Portfolio Optimization Differentiable Optimization Finance

Jun 15, 2026cs.RO

Elastic ODYN: Differentiable Optimization for Infeasible Control and Learning in Robotics

Robotic systems routinely encounter conflicting objectives, modeling errors, and degenerate contact conditions that render quadratic programs (QPs) infeasible. Yet most optimization solvers and differentiable QP layers assume feasibility, leading to numerical failures, unstable gradients, or solver breakdown when constraints cannot be simultaneously satisfied. We present Elastic ODYN, a primal-dual non-interior-point QP solver that handles infeasibility through smooth squared-

\ell_2

elastic relaxations. The formulation remains well posed under ill-conditioning and degeneracy, supports warm starting, and converges to closest-to-feasible solutions, with lightweight refinement recovering physically meaningful dual variables. Building on this framework, we develop Elastic ODYNLayer, a differentiable QP layer with stable gradients under infeasibility, and Elastic OdynSQP, an SQP method that resolves inconsistent subproblems and intrinsically infeasible optimal control tasks through selective constraint elasticity. Across benchmark QPs, singular contact mechanics, differentiable parameter identification, and quadrupedal and humanoid trajectory optimization, Elastic ODYN outperforms state-of-the-art elastic QP solvers in robustness, warm-start performance, and convergence reliability, enabling optimization, simulation, control, and learning beyond standard feasibility assumptions.

Aristotelis Papatheodorou, Jose Rojas, Ioannis Havoutis +1

Department of Engineering Science, University of Oxford, Oxford, UK · Robot Motor Intelligence (RoMI) Lab, Heriot-Watt University, Edinburgh, UK

Robotic Control Primal-Dual Optimization Trajectory Optimization Constrained Optimization Quadratic Programming Differentiable Optimization

Jun 15, 2026cs.CV

Differentiable Packing of Irregular 3D Objects with Adaptive Container Estimation

Most existing approaches either fix the container in advance or optimize only a single container dimension through an outer search loop, leaving the remaining dimensions as a manual tuning problem. We present a differentiable packing framework that jointly optimizes all 6N object pose parameters and all three container side lengths inside a single gradient-based loop. The formulation combines six physics-inspired, differentiable loss terms computed directly on triangle meshes through axis-aligned bounding-box proxies. An adaptive squeezing mechanism periodically tightens the container whenever the overlap loss falls below a pair-count-scaled threshold, producing a large initial drop in container volume, followed by small refinements. All pairwise computations are written in tensor-broadcasting form, giving a 3.4 to 54 times speedup over a reference loop-based implementation. The pipeline is implemented in Python and PyTorch, with no physics engine, FFT library, or convex decomposition. On multiple object categories, the method produces containers that are 11 to 32 percent smaller than time-matched DBLF and simulated-annealing baselines at N =100, while running in under 4 minutes per instance on a single consumer GPU.

Palak Gupta, Shanmuganathan Raman

Indian Institute of Technology Gandhinagar, India

3D Spatial Reasoning Differentiable Optimization

Jun 11, 2026math.OC

Scalable Deep Unfolding of Conic Optimizers

Deep unfolding (DU) accelerates iterative optimizers by introducing learnable components and training them through unrolled iterations, but extending DU to the large-scale semidefinite programs (SDPs) common in robotics has remained limited. Unrolling a full-update conic solver such as COSMO exposes two obstacles that prior work on learned conic solvers has not: backpropagating through the per-iteration linear-system solve incurs memory quadratic in the problem size once the coefficient matrix is formed explicitly, and backpropagating through the positive semidefinite (PSD) cone projection becomes numerically unstable when eigenvalues coincide. We address the first obstacle with a matrix-free implicit differentiation rule that operates entirely through matrix-vector products, reducing memory from

O(n^2)

to

O(n)

and enabling backpropagation at scales where direct factorization runs out of memory. We address the second with a backward rule based on the Dalečkii--Krein representation of the Fréchet derivative, which remains well-defined under repeated eigenvalues. Together these make it possible to learn lightweight hyperparameter policies and warm-starts for a full-update conic solver. We evaluate on nonlinear covariance steering problems solved via sequential convex programming (SCP), as well as standalone SDPs and second-order cone programs ranging from max-cut and Lovász

\vartheta

SDPs to robust estimation and control problems. The learned policies outperform state-of-the-art solvers across all problems, and can provide up to a 50

\times

speedup depending on the class. When used as a subroutine in SCP, the learned approach delivers over a 30

\times

speedup compared to COSMO.

Alex Oshin, Rahul Vodeb Ghosh, Evangelos A. Theodorou

Georgia Institute of Technology

Semidefinite Programming Algorithm Unrolling Constrained Optimization Matrix Optimization Differentiable Optimization Deep Unfolding

Jun 7, 2026cs.PL

Compile Once, Differentiate Everywhere: A Differentiable Meta-Circular Interpreter

The boundary between program execution and gradient-based optimization has long limited the use of code itself as a learnable scientific model. We present a compiler that translates a self-hosting subset of Scheme into differentiable computation graphs for autograd backends. Because the subset can compile its own evaluator, this yields differentiable meta-circular interpretation (DMCI): a compiled Scheme interpreter executes programs supplied as data, while reverse-mode autodiff propagates gradients to continuous constants embedded in those programs. The interpreter is compiled once, so new programs inherit differentiability without recompilation or custom gradient machinery, while retaining closures, recursion, and data structures. We prove that gradients through the compiled interpreter are correct almost everywhere and show that they match direct compilation to numerical precision across 171 recursive and higher-order program-seed pairs. We then use DMCI for program-and-parameter co-search, where a large language model proposes Scheme programs and exact gradients calibrate their continuous parameters through a single frozen interpreter. This enables OpenEvolve-style program search in which an outer loop proposes discrete program structures and DMCI supplies exact gradient-based calibration of each candidate's continuous parameters. On battery capacity-fade data, the search recovers a knee-like degradation structure and improves held-out extrapolation over hand-crafted baselines on the harder early-extrapolation split, matching them on the later split. On a high-dimensional El Nino inverse problem, DMCI optimizes an interpreted Kalman-filter likelihood where gradient-free search fails. These results extend symbolic regression and neurosymbolic search from closed-form expressions to executable, stateful programs, making model-generated code directly optimizable against data.

Lucas Sheneman

Institute for Interdisciplinary Data Sciences University of Idaho

Automatic Differentiation Program Synthesis Differentiable Programming Differentiable Optimization Symbolic Regression

Jun 5, 2026physics.optics

Beyond the Thin-Layer Limit: Differentiable Volumetric Training for Visible-Range Diffractive Neural Networks

Diffractive deep neural networks (D2NNs) promise miniaturized, power-efficient, light-speed optical front-ends for machine vision, yet the most mature demonstrations remain in the terahertz regime, built from readily fabricated millimeter-scale neurons. Translating D2NNs to the visible range, where nearly all vision pipelines operate, was long blamed on the difficulty of fabricating nanoscale neurons; but even after recent advances removed that barrier, visible-range D2NNs matching their terahertz counterparts remain out of reach. We identify the true obstacle as the thin-layer approximation underlying nearly all D2NN training, which treats each diffractive layer as an infinitely thin mask. It fails not because of the short wavelength, as is commonly assumed, but because the low-refractive-index materials (n approximately 1.3-1.5) used at visible wavelengths require relief structures thick enough that intra-layer diffraction and phase accumulation become significant. To overcome this, we introduce a differentiable beam-propagation (

\partial

BPM) layer that models each element as a finite-thickness volume and propagates light through it during training, keeping the fabrication-compatible height map end-to-end trainable without full-wave simulation in the loop. Across MNIST, Fashion-MNIST, and CIFAR-100 classification and imaging,

\partial

BPM training substantially reduces the design-to-device mismatch, and full-wave FDTD validation raises classification accuracy from 50% to 90% without re-optimization. The

\partial

BPM layer thus offers a scalable, physics-aware bridge between efficient optical neural-network optimization and fabrication-consistent diffractive design.

Dineth Jayakody, Dushan N. Wadduwage

Department of Computer Science, Old Dominion University, Norfolk, VA 23529, USA · School of Data Science, Old Dominion University, Norfolk, VA 23529, USA · Department of Physics, Old Dominion University, Norfolk, VA 23529, USA

Differentiable Physics Differentiable Optimization

Jun 5, 2026cs.NI

DIFFRACT: Neuralized Utility Maximization for Wireless Networks by Differentiable Programming

Next-generation wireless networks, including satellite-to-Open RAN systems, demand agile and intelligent resource management capable of handling dynamic multi-user interference under stochastic quality of service constraints. This paper introduces DIFFRACT, a neuralized utility maximization framework that leverages differentiable programming to integrate deep learning with optimization in wireless networks. Central to our approach is the exploitation of the mathematical structure of standard interference functions, which are foundational in wireless power control. By developing a duality theory for these functions, we map iterative interference management algorithms into differentiable neural network architectures via algorithm unrolling. This enables distributed, end-to-end gradient-based learning at the network edge, supporting real-time adaptation to interference in both terrestrial and non-terrestrial environments. DIFFRACT allows for scalable and robust utility maximization by modeling complex channel dynamics and leveraging the expressiveness of differentiable models. Experimental results confirm the framework's theoretical soundness and practical effectiveness for next-generation wireless systems.

Chee Wei Tan, Siya Chen

Nanyang Technological University, Singapore, Nanyang Ave., Singapore · Department of Computer Science, City University of Hong Kong, Hong Kong

Algorithm Unrolling Wireless Resource Allocation Differentiable Optimization

Jun 3, 2026cs.LG

DiffSlack: Learning under Nonlinear Inequality Constraints via Learnable Slack Variables

Enforcing nonlinear inequality constraints in neural networks remains challenging, especially when the output is subject to many coupled constraints. Existing hard constraint methods often impose structural restrictions on the constraint set or introduce substantial computational overhead for large-scale nonlinear problems. Here, we propose DiffSlack, a differentiable projection layer for nonlinear inequality-constrained neural prediction. DiffSlack reformulates inequalities as equalities with learnable slack variables, which are predicted as part of the augmented network output and provide a data-driven warm start for damped Gauss-Newton projection. The projection layer maps raw predictions onto the augmented feasible manifold while preserving end-to-end differentiability. A two-stage curriculum further stabilizes training and improves constraint satisfaction. We evaluate DiffSlack on vehicle path planning with 200 nonlinear inequality constraints from collision avoidance, curvature limits, and waypoint spacing. Compared with existing learning-based baselines, DiffSlack achieves a higher planning success rate and stronger geometric constraint satisfaction under a comparable inference budget. Ablation studies further show that the hard projection layer reduces sensitivity to supervision quality. Closed-loop tracking in CARLA and real-world vehicle experiments confirms the executability of the generated trajectories. These results demonstrate that DiffSlack provides a practical and scalable approach to embedding hard inequality constraints into neural networks for engineering applications.

Ziqian Wang, Chenxi Fang, Zhen Zhang

State Key Laboratory of Tribology in Advanced Equipment, Tsinghua University, Beijing 100084, China · Beijing Key Laboratory of Transformative High-end Manufacturing Equipment and Technology, Department of Mechanical Engineering, Tsinghua University, Beijing 100084, China · Automotive Electronics Business Unit at Hirain Inc., China

Constrained Optimization Differentiable Optimization

Jun 1, 2026cs.LG

Regularized Large Neighborhood Search

Operations research practitioners typically tackle NP-hard combinatorial problems using large neighborhood search (LNS), a scalable heuristic that iteratively refines a current solution by locally re-optimizing subsets of its variables. In contrast, most existing approaches for integrating combinatorial optimization layers into neural networks still assume access to an exact global solution, which is computationally intractable. We bridge this gap by introducing regularized LNS (RLNS). By regularizing or perturbing local subproblems, we turn the LNS heuristic into an efficient MCMC sampler over the combinatorial set of feasible solutions, with associated Fenchel-Young losses. Under entropic regularization, we prove that RLNS performs exact block Gibbs sampling. Furthermore, adjusting the number of RLNS iterations allows us to interpolate between pseudolikelihood and exact maximum likelihood estimation, for end-to-end learning without global solvers. We demonstrate our approach on

k

-subset selection, generalized assignment, and stochastic vehicle scheduling problems.

Germain Vivier-Ardisson, Laurent Demonet, Axel Parmentier +1

Google DeepMind & CERMICS · Google Research · CERMICS, ENPC, CNRS, IPP +1

Markov Chain Monte Carlo Metaheuristic Optimization Combinatorial Optimization Differentiable Optimization Neural Combinatorial Optimization Large Neighborhood Search