arXiv subjects

Jisun Park

Publications and source records attributed to Jisun Park.

9 recordsLinked to original sources

arXiv2026

Distributionally-Robust Learning to Optimize

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

cs.LG↗

arXiv2025

Data-Driven Analysis of First-Order Methods via Distributionally Robust Optimization

We consider the problem of analyzing the probabilistic performance of first-order methods when solving convex optimization problems drawn from an unknown distribution only accessible through samples. By combining performance estimation and Wasserstein distributionally robust optimization, we formulate the analysis as a tractable conic program. Our approach unifies worst-case and average-case analyses by incorporating data-driven information from the observed convergence of first-order methods on a limited number of problem instances. This yields probabilistic, data-driven performance guarantees in terms of the expectation or conditional value-at-risk of the selected performance metric. Our open-source implementation computes these guarantees directly from sampled algorithm trajectories using off-the-shelf conic solvers. Experiments on convex quadratic minimization, real-data logistic regression using a credit-scoring dataset, and Lasso show that our method significantly reduces the conservatism of classical worst-case bounds and narrows the gap between theoretical and empirical performance.

math.OC↗

arXiv2025

AlgoTune: Can Language Models Speed Up General-Purpose Numerical Programs?

Despite progress in language model (LM) capabilities, evaluations have thus far focused on models' performance on tasks that humans have previously solved, including in programming (Jimenez et al., 2024) and mathematics (Glazer et al., 2024). We therefore propose testing models' ability to design and implement algorithms in an open-ended benchmark: We task LMs with writing code that efficiently solves computationally challenging problems in computer science, physics, and mathematics. Our AlgoTune benchmark consists of 154 coding tasks collected from domain experts and a framework for validating and timing LM-synthesized solution code, which is compared to reference implementations from popular open-source packages. In addition, we develop a baseline LM agent, AlgoTuner, and evaluate its performance across a suite of frontier models. AlgoTuner uses a simple, budgeted loop that edits code, compiles and runs it, profiles performance, verifies correctness on tests, and selects the fastest valid version. AlgoTuner achieves an average 1.72x speedup against our reference solvers, which use libraries such as SciPy, sk-learn and CVXPY. However, we find that current models fail to discover algorithmic innovations, instead preferring surface-level optimizations. We hope that AlgoTune catalyzes the development of LM agents exhibiting creative problem solving beyond state-of-the-art human performance.

cs.SE↗

arXiv2024

Exact Verification of First-Order Methods via Mixed-Integer Linear Programming

We present exact mixed-integer linear programming formulations for verifying the performance of first-order methods for parametric quadratic optimization. We formulate the verification problem as a mixed-integer linear program where the objective is to maximize the infinity norm of the fixed-point residual after a given number of iterations. Our approach captures a wide range of gradient, projection, proximal iterations through affine or piecewise affine constraints. We derive tight polyhedral convex hull formulations of the constraints representing the algorithm iterations. To improve the scalability, we develop a custom bound tightening technique combining interval propagation, operator theory, and optimization-based bound tightening. Numerical examples, including linear and quadratic programs from network optimization, sparse coding using Lasso, and optimal control, show that our method provides several orders of magnitude reductions in the worst-case fixed-point residuals, closely matching the true worst-case performance.

math.OC↗

arXiv2024

Coordinate-Update Algorithms can Efficiently Detect Infeasible Optimization Problems

Coordinate update/descent algorithms are widely used in large-scale optimization due to their low per-iteration cost and scalability, but their behavior on infeasible or misspecified problems has not been much studied compared to the algorithms that use full updates. For coordinate-update methods to be as widely adopted to the extent so that they can be used as engines of general-purpose solvers, it is necessary to also understand their behavior under pathological problem instances. In this work, we show that the normalized iterates of randomized coordinate-update fixed-point iterations (RC-FPI) converge to the infimal displacement vector and use this result to design an efficient infeasibility detection method. We then extend the analysis to the setup where the coordinates are defined by non-orthonormal basis using the Friedrichs angle and then apply the machinery to decentralized optimization problems.

math.OC↗

arXiv2023

Continuous-time Analysis of Anchor Acceleration

Recently, the anchor acceleration, an acceleration mechanism distinct from Nesterov's, has been discovered for minimax optimization and fixed-point problems, but its mechanism is not understood well, much less so than Nesterov acceleration. In this work, we analyze continuous-time models of anchor acceleration. We provide tight, unified analyses for characterizing the convergence rate as a function of the anchor coefficient $β(t)$, thereby providing insight into the anchor acceleration mechanism and its accelerated $\mathcal{O}(1/k^2)$-convergence rate. Finally, we present an adaptive method inspired by the continuous-time analyses and establish its effectiveness through theoretical analyses and experiments.

math.OC↗

arXiv2023

Accelerated Infeasibility Detection of Constrained Optimization and Fixed-Point Iterations

As first-order optimization methods become the method of choice for solving large-scale optimization problems, optimization solvers based on first-order algorithms are being built. Such general-purpose solvers must robustly detect infeasible or misspecified problem instances, but the computational complexity of first-order methods for doing so has yet to be formally studied. In this work, we characterize the optimal accelerated rate of infeasibility detection. We show that the standard fixed-point iteration achieves a $\mathcal{O}(1/k^2)$ and $\mathcal{O}(1/k)$ rates, respectively, on the normalized iterates and the fixed-point residual converging to the infimal displacement vector, while the accelerated fixed-point iteration achieves $\mathcal{O}(1/k^2)$ and $\tilde{\mathcal{O}}(1/k^2)$ rates. We then provide a matching complexity lower bound to establish that $Θ(1/k^2)$ is indeed the optimal accelerated rate.

math.OC↗

arXiv2022

Exact Optimal Accelerated Complexity for Fixed-Point Iterations

Despite the broad use of fixed-point iterations throughout applied mathematics, the optimal convergence rate of general fixed-point problems with nonexpansive nonlinear operators has not been established. This work presents an acceleration mechanism for fixed-point iterations with nonexpansive operators, contractive operators, and nonexpansive operators satisfying a Hölder-type growth condition. We then provide matching complexity lower bounds to establish the exact optimality of the acceleration mechanisms in the nonexpansive and contractive setups. Finally, we provide experiments with CT imaging, optimal transport, and decentralized optimization to demonstrate the practical effectiveness of the acceleration mechanism.

math.OC↗

arXiv2021

Factor-$\sqrt{2}$ Acceleration of Accelerated Gradient Methods

The optimized gradient method (OGM) provides a factor-$\sqrt{2}$ speedup upon Nesterov's celebrated accelerated gradient method in the convex (but non-strongly convex) setup. However, this improved acceleration mechanism has not been well understood; prior analyses of OGM relied on a computer-assisted proof methodology, so the proofs were opaque for humans despite being verifiable and correct. In this work, we present a new analysis of OGM based on a Lyapunov function and linear coupling. These analyses are developed and presented without the assistance of computers and are understandable by humans. Furthermore, we generalize OGM's acceleration mechanism and obtain a factor-$\sqrt{2}$ speedup in other setups: acceleration with a simpler rational stepsize, the strongly convex setup, and the mirror descent setup.

math.OC↗