arXiv subjects

Vinit Ranjan

Publications and source records attributed to Vinit Ranjan.

5 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↗

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

Verification of First-Order Methods for Parametric Quadratic Optimization

We introduce a numerical framework to verify the finite step convergence of first-order methods for parametric convex quadratic optimization. We formulate the verification problem as a mathematical optimization problem where we maximize a performance metric (e.g., fixed-point residual at the last iteration) subject to constraints representing proximal algorithm steps (e.g., linear system solutions, projections, or gradient steps). Our framework is highly modular because we encode a wide range of proximal algorithms as variations of two primitive steps: affine steps and element-wise maximum steps. Compared to standard convergence analysis and performance estimation techniques, we can explicitly quantify the effects of warm-starting by directly representing the sets where the initial iterates and parameters live. We show that the verification problem is NP-hard, and we construct strong semidefinite programming relaxations using various constraint tightening techniques. Numerical examples in nonnegative least squares, network utility maximization, Lasso, and optimal control show a significant reduction in pessimism of our framework compared to standard worst-case convergence analysis techniques.

math.OC↗

arXiv2018

Block size in Geometric(p)-biased permutations

Fix a probability distribution $\mathbf p = (p_1, p_2, \cdots)$ on the positive integers. The first block in a $\mathbf p$-biased permutation can be visualized in terms of raindrops that land at each positive integer $j$ with probability $p_j$. It is the first point $K$ so that all sites in $[1,K]$ are wet and all sites in $(K,\infty)$ are dry. For the geometric distribution $p_j= p(1-p)^{j-1}$ we show that $p \log K$ converges in probability to an explicit constant as $p$ tends to 0. Additionally, we prove that if $\mathbf p$ has a stretch exponential distribution, then $K$ is infinite with positive probability.

math.PR↗