arXiv subjects

Madeleine Udell

Publications and source records attributed to Madeleine Udell.

At least 19 recordsLinked to original sources

arXiv2026

Stochastic Gradient Methods with Online Scaling

This paper introduces Stochastic Online Scaled Gradient Methods (SOSGM), a generalization of the recently developed adaptive preconditioning framework in arXiv:2505.23081 and arXiv:2509.11007 to stochastic optimization. Under standard assumptions, we establish convergence guarantees for SOSGM using large batchsize or variance reduction. SOSGM is compatible with popular diagonal and/or low-rank preconditioners as well as heavy-ball momentum, while maintaining memory and computation cost comparable to Adam. Extensive numerical experiments demonstrate the strong empirical performance of SOSGM. Using a diagonal preconditioner, SOSGM and its variants substantially outperform existing adaptive first-order methods across a range of statistical learning tasks.

math.OC↗

arXiv2026

Operator Splitting Methods with Online Scaling

This paper develops a principled framework for automatically tuning preconditioners in operator splitting methods, including forward-backward splitting, Douglas-Rachford splitting, and the alternating direction method of multipliers. We interpret these splitting methods as preconditioned gradient descent applied to their corresponding envelope functions and use online learning to adaptively learn the preconditioner during optimization. Our framework achieves asymptotically faster convergence rates and accelerates the solution of important optimization problems, including Lasso, $\ell_1$-regularized logistic regression, and quadratic programming. Extensive numerical experiments validate our theoretical results and demonstrate the practical effectiveness of the proposed approach.

math.OC↗

arXiv2026

GPU-Enabled Large-Scale Optimization Using Randomized Linear Algebra

This paper introduces rlaopt, a PyTorch-based package for large-scale optimization and scientific computing using randomized numerical linear algebra (RandNLA). Despite substantial progress in RandNLA-based algorithms, few implementations combine GPU acceleration with a simple interface for specifying optimization problems. rlaopt addresses this gap by providing GPU-enabled solvers for positive-definite linear systems and convex empirical risk minimization with constraints and regularizers. These solvers use RandNLA to accelerate conjugate gradient (NystromPCG), operator splitting (NysADMM), and stochastic gradient methods (SAPPHIRE). Moreover, rlaopt includes a modeling language that lets users specify problems using natural mathematical syntax. rlaopt automatically checks compatibility with the selected solver and performs the required problem decomposition. The solvers also support differentiation through their iterations, enabling applications such as hyperparameter tuning. Experiments on ridge regression, bounded multinomial logistic regression, and bounded elastic net identify when randomized preconditioning improves performance and demonstrate substantial speedups from GPU execution. The package is open-source under an Apache license, with source code at https://github.com/udellgroup/rlaopt and version 0.1.0 available on PyPI.

cs.LG↗

arXiv2026

Convex Modeling of Price Cross-Impact over Time

Transaction costs can make or break a trading strategy, particularly in relative-value trading of commodity and macro markets, where edges are a few basis points. Price impact is a central component of transaction cost. Price impact models usually include self-impact (a trade in a contract moves that contract's price) but omit two well-documented effects: cross-impact (a trade in one contract also moves the prices of related contracts) and transient impact (price impact decays over time, so an unwind recovers part of the entry cost). A model without these effects overprices the impact of relative-value trades, whose correlated legs are built and unwound over days, and so forgoes potentially profitable trades. This paper models both effects with a convex quadratic cost. In each period, a positive semidefinite matrix built from volatility, volume, and correlation forecasts couples trades across contracts. A power-law decay kernel then couples trades across periods. The resulting cost admits no price manipulation even when liquidity varies over the planning horizon. The model is demonstrated empirically on calendar spread trading of crude oil futures around the commodity index roll.

math.OC↗

arXiv2026

FLARE: Verifying MILP Reformulations with LLM-Based Theorem Proving

Mixed-Integer Linear Programming (MILP) is a fundamental tool for combinatorial optimization with extensive real-world applications. A central challenge is designing computationally efficient MILP formulations. Large Language Models (LLMs) offer new opportunities to automate the modeling process, from deriving formulations to strengthening them. Reliable automation requires robust methods for verifying that proposed formulations preserve the underlying optimization problem. However, existing approaches evaluate formulations numerically and fail to reason about general problem instances. We resolve this limitation by introducing a constructive definition of MILP reformulation that can be formalized in Lean and machine-checked. We develop FLARE (Formulation-Level Automated Reformulation Evaluation), a method that uses an LLM-based agent and the Lean proof assistant to verify proposed reformulations against a reference formulation. To evaluate our approach, we introduce FormulationBench, a challenging dataset of 20 problems and 109 formulations. FLARE outperforms existing methods, with 100% accuracy on the NP-hard subset of FormulationBench. Furthermore, FLARE produces a machine-checkable certificate for every reformulation it accepts. For cases where formal guarantees are not necessary, we introduce FLARE-NL, a fast and cheap LLM proxy that matches FLARE's accuracy but produces no certificate. These methods enable reliable verification in automated optimization modeling.

cs.AI↗

arXiv2026

Twin: Playing an Unknown Game with a Test-Time Digital Twin

We present a Test-time World-model Inference (Twin) system, in which a frontier coding agent writes an executable world model for completing continual learning tasks, such as ARC-AGI-3 games. Traditional approaches hand-engineer such models, one custom design per task. Each game hides its rules and goal, and our system constructs them from simulation and interaction alone. Its inductive prior over grid games is strong enough to recover the true transitions of the game and the goal on nearly all levels. Replay validation happens in a twin world model. The harness enforces that an action is not made until the program reproduces every previous observed game transition. Each mismatch between a world model prediction and the actual action result becomes a counterexample that is used to repair the world model. Twin clears 179 out of 183 levels (97.8%), and does so more efficiently than humans in 158 out of 179 levels (88.3%). The system infers the goal before any reward on 156 of the levels it clears (87.2%), and in the remaining levels automatically discovers the goal by search. The benchmark scores completion and action efficiency, between 0 and 100, against humans playing each game for the first time. Played directly, the base model scores only 7.8%; an off-the-shelf harness increases it to 61.1%, whereas our twin world model increases the same base model to 93.3%, clearing 23 out of 25 games. Building a usable world model is simpler than anticipated, whereas the harder problem is inferring the right goal.

cs.AI↗

arXiv2026

Tight Nonasymptotic Local Convergence of Sinkhorn-Knopp

We revisit the Sinkhorn-Knopp (SK) algorithm for the matrix scaling problem. Despite extensive literature on the global convergence of SK and its variants, its local linear convergence behavior remains less understood. We address this gap by providing the first nonasymptotic local analysis of SK that matches the rate obtained from existing asymptotic Jacobian-based arguments. We show that under certain connectivity conditions, SK is a polynomial-time algorithm for doubly stochastic matrix scaling. With the developed tools, we showcase the local suboptimality of SK and provide accelerated variants. Finally, for dense matrices, we improve the complexity of existing first-order matrix scaling algorithms from $O(\tfrac{n^{7/3}}{\varepsilon^{2/3}})$ to $O(\tfrac{n^{9/4}}{\sqrt{\varepsilon}})$.

math.OC↗

arXiv2026

Battery Bidding under Price Uncertainty in Wholesale Electricity Markets

Grid-scale batteries increasingly influence outcomes in wholesale electricity markets, but their observed bid patterns remain difficult to interpret. In particular, bids that appear to reflect strategic withholding may instead arise from rational operations under price uncertainty and risk management. We develop an asset-level model of a price-taking battery that submits stepwise buy and sell bid curves in the day-ahead market under a finite set of price scenarios. The battery chooses quantity--price pairs to maximize a mean--CVaR objective subject to physical and market constraints. A direct formulation is a mixed-integer linear program, but we show that its integer decisions can be removed, yielding an exact linear programming reformulation suitable for empirical analysis. Our empirical results deliver three insights. First, withholding behavior can arise even without market power, because scarce stored energy and uncertain future prices increase the value of holding energy. Second, the effect of uncertainty depends on the state of charge: when stored energy is scarce, greater uncertainty raises sell bid prices, whereas when stored energy is abundant it can lower them. Third, risk management reshapes bid curves into layered structures that secure profitable execution across a broad set of scenarios while preserving some exposure to rare but valuable price spikes.

math.OC↗

arXiv2026

SubsurfaceGen: Procedural Generation of Field-Scale Earth Models and Seismic Data

Full waveform inversion (FWI) is the gold standard for subsurface imaging, with applications from carbon sequestration to energy and mineral exploration to earthquake hazard assessment. Machine learning approaches to FWI need field-scale, geologically diverse, and physically realistic training data, but existing resources such as Marmousi, SEAM, and OpenFWI fall short on spatial extent, temporal extent, geological diversity, and physical realism. We address these limitations with SubsurfaceGen, a GPU-accelerated generator for 3D velocity models and seismic data. Along with SubsurfaceGen, we release a paired dataset of 4,276 2D velocity slices, 5 s wavefields, and 8 s shot gathers drawn from 42 realistic, field-scale 3D velocity models, each spanning 10 km x 10 km laterally and 6.19 km deep at 10 m resolution. The dataset spans six geological settings -- four built with SubsurfaceGen and two drawn from prior sources -- relevant for carbon sequestration and hydrocarbon exploration. We use this dataset to evaluate neural operators on wavefield prediction and encoder-decoders on end-to-end velocity inversion, holding out one geological setting for out-of-distribution testing. These experiments surface failure modes at field-scale and demonstrate how SubsurfaceGen and the associated dataset can impact ML-based FWI.

cs.LG↗

arXiv2026

ShapeBench: A Scalable Benchmark and Diagnostic Suite for Standardized Evaluation in Aerodynamic Shape Optimization

Rapid progress in aerodynamic shape optimization (ASO) has outpaced currently-available standardized evaluation frameworks. Fair comparison requires a unified benchmark spanning diverse shape classes, objective formulations, and matched-budget state-of-the-art baselines. We introduce ShapeBench, an open-source ASO benchmark with a unified API spanning 103 tasks across eight shape categories and multiple optimization regimes. Each ShapeBench task includes a validated surrogate for fast search; when feasible, a high-fidelity Computational Fluid Dynamics (CFD) pipeline for final verification is available, enabling systematic fidelity-gap analysis. ShapeBench provides a reproducible protocol with well-configured baselines to compare fairly using a consistent budget metric, allowing for comparison among both classical and LLM-driven methods, including general-purpose optimizers and a new domain-specialized evolutionary LLM baseline, ShapeEvolve. Results on ShapeBench demonstrate substantial variance in optimizer rankings across shape categories and problem formulations, with mean pairwise Spearman $\rho = 0.013$, so single-task conclusions do not reliably generalize across problem classes. The benchmark is also far from saturation; classical methods are rarely applicable across all shape categories and tasks, further highlighting the need for more general-purpose approaches.

cs.LG↗

arXiv2026

SnareNet: Flexible Repair Layers for Neural Networks with Hard Constraints

Neural networks are increasingly used as fast surrogate models across various domains, but unconstrained predictions can violate physical, operational, or safety requirements. We propose SnareNet, a feasibility-controlled architecture to learn mappings whose outputs must satisfy input-dependent constraints. SnareNet appends a differentiable repair layer that navigates in the constraint map's range space, steering iterates toward feasibility and producing a repaired output that satisfies constraints to a user-specified tolerance. We stabilize end-to-end training by adaptive relaxation, a new training paradigm that snares the neural network at initialization and shrinks it into the feasible set, enabling early exploration and strict feasibility later in training. On optimization learning and trajectory planning benchmarks, SnareNet consistently attains improved objective quality while satisfying constraints more reliably than prior work, and it is the first to enforce non-convex constraints at medium-to-high precision robustly across instances.

cs.LG↗

arXiv2026

Small Gradient Norm Regret for Online Convex Optimization

This paper introduces a new problem-dependent regret measure for online convex optimization with smooth losses. The notion, which we call the $G^\star$ regret, depends on the cumulative squared gradient norm evaluated at the decision in hindsight. We show that the $G^\star$ regret strictly refines the existing $L^\star$ (small loss) regret, and that it can be arbitrarily sharper when the losses have vanishing curvature around the hindsight decision. We establish upper and lower bounds on the $G^\star$ regret and extend our results to dynamic regret and bandit settings. As a byproduct, we refine the existing convergence analysis of stochastic optimization algorithms in the interpolation regime. Some experiments validate our theoretical findings.

stat.ML↗

arXiv2025

New Results on the Polyak Stepsize: Tight Convergence Analysis and Universal Function Classes

In this paper, we revisit a classical adaptive stepsize strategy for gradient descent: the Polyak stepsize (PolyakGD), originally proposed in Polyak (1969). We study the convergence behavior of PolyakGD from two perspectives: tight worst-case analysis and universality across function classes. As our first main result, we establish the tightness of the known convergence rates of PolyakGD by explicitly constructing worst-case functions. In particular, we show that the $O((1-\frac{1}{\kappa})^K)$ rate for smooth strongly convex functions and the $O(1/K)$ rate for smooth convex functions are both tight. Moreover, we theoretically show that PolyakGD automatically exploits floating-point errors to escape the worst-case behavior. Our second main result provides new convergence guarantees for PolyakGD under both H\"older smoothness and H\"older growth conditions. These findings show that the Polyak stepsize is universal, automatically adapting to various function classes without requiring prior knowledge of problem parameters.

math.OC↗

arXiv2025

PDE-SHARP: PDE Solver Hybrids through Analysis and Refinement Passes

Current LLM-driven approaches using test-time computing to generate PDE solvers execute a large number of solver samples to identify high-accuracy solvers. These paradigms are especially costly for complex PDEs requiring substantial computational resources for numerical evaluation. We introduce PDE-SHARP, a framework to reduce computational costs by replacing expensive scientific computation by cheaper LLM inference that achieves superior solver accuracy with 60-75% fewer computational evaluations. PDE-SHARP employs three stages: (1) Analysis: mathematical chain-of-thought analysis including PDE classification, solution type detection, and stability analysis; (2) Genesis: solver generation based on mathematical insights from the previous stage; and (3) Synthesis: collaborative selection-hybridization tournaments in which LLM judges iteratively refine implementations through flexible performance feedback. To generate high-quality solvers, PDE-SHARP requires fewer than 13 solver evaluations on average compared to 30+ for baseline methods, improving accuracy uniformly across tested PDEs by $4\times$ on average, and demonstrates robust performance across LLM architectures, from general-purpose to specialized reasoning models.

cs.LG↗

arXiv2025

"It Was a Magical Box": Understanding Practitioner Workflows and Needs in Optimization

Optimization underpins decision-making in domains from healthcare to logistics, yet for many practitioners it remains a "magical box": powerful but opaque, difficult to use, and reliant on specialized expertise. While prior work has extensively studied machine learning workflows, the everyday practices of optimization model developers (OMDs) have received little attention. We conducted semi-structured interviews with 15 OMDs across diverse domains to examine how optimization is done in practice. Our findings reveal a highly iterative workflow spanning six stages: problem elicitation, data processing, model development, implementation, validation, and deployment. Importantly, we find that optimization practice is not only about algorithms that deliver better decisions, but is equally shaped by data and dialogue - the ongoing communication with stakeholders that enables problem framing, trust, and adoption. We discuss opportunities for future tooling that foregrounds data and dialogue alongside decision-making, opening new directions for human-centered optimization.

cs.HC↗

arXiv2025

Gradient Methods with Online Scaling Part II. Practical Aspects

Part I of this work [Gao25] establishes online scaled gradient methods (OSGM), a framework that utilizes online convex optimization to adapt stepsizes in gradient methods. This paper focuses on the practical aspects of OSGM. We leverage the OSGM framework to design new adaptive first-order methods and provide insights into their empirical behavior. The resulting method, OSGM-Best, matches the performance of quasi-Newton variants while requiring less memory and cheaper iterations. We also extend OSGM to nonconvex optimization and outline directions that connect OSGM to existing branches of optimization theory and practice.

math.OC↗

arXiv2025

Gradient Methods with Online Scaling Part I. Theoretical Foundations

This paper establishes the theoretical foundations of the online scaled gradient methods (OSGM), a framework that utilizes online learning to adapt stepsizes and provably accelerate first-order methods. OSGM quantifies the effectiveness of a stepsize by a feedback function motivated from a convergence measure and uses the feedback to adjust the stepsize through an online learning algorithm. Consequently, instantiations of OSGM achieve convergence rates that are asymptotically no worse than the optimal stepsize. OSGM yields desirable convergence guarantees on smooth convex problems, including 1) trajectory-dependent global convergence on smooth convex objectives; 2) an improved complexity result on smooth strongly convex problems, and 3) local superlinear convergence. Notably, OSGM constitutes a new family of first-order methods with non-asymptotic superlinear convergence, joining the celebrated quasi-Newton methods. Finally, OSGM explains the empirical success of the popular hypergradient-descent heuristic in optimization for machine learning.

math.OC↗

arXiv2025

Turbocharging Gaussian Process Inference with Approximate Sketch-and-Project

Gaussian processes (GPs) play an essential role in biostatistics, scientific machine learning, and Bayesian optimization for their ability to provide probabilistic predictions and model uncertainty. However, GP inference struggles to scale to large datasets (which are common in modern applications), since it requires the solution of a linear system whose size scales quadratically with the number of samples in the dataset. We propose an approximate, distributed, accelerated sketch-and-project algorithm ($\texttt{ADASAP}$) for solving these linear systems, which improves scalability. We use the theory of determinantal point processes to show that the posterior mean induced by sketch-and-project rapidly converges to the true posterior mean. In particular, this yields the first efficient, condition number-free algorithm for estimating the posterior mean along the top spectral basis functions, showing that our approach is principled for GP inference. $\texttt{ADASAP}$ outperforms state-of-the-art solvers based on conjugate gradient and coordinate descent across several benchmark datasets and a large-scale Bayesian optimization task. Moreover, $\texttt{ADASAP}$ scales to a dataset with $> 3 \cdot 10^8$ samples, a feat which has not been accomplished in the literature.

cs.LG↗