Simpler Methods Work Better for L1 Penalized Logistic Models and Large Datasets

The paper considers the computation of L1 regularization paths in a state space setting, which includes L1 regularized Kalman smoothing, linear SVM, LASSO, and more. The paper proposes two new algorithms, which are duals of each other; the first algorithm applies to L1 regularization of independent variables while the second applies to L1 regularization of dependent variables. The heart of the proposed algorithms is parametric Gaussian message passing (i.e., Kalman-type forward-backward recursions) in the pertinent factor graphs. The proposed methods are broadly applicable, they (usually) require only matrix multiplications, and their complexity can be competitive with prior methods in some cases.

Optimization problems are pervasive in sectors from manufacturing and distribution to healthcare. However, most such problems are still solved heuristically by hand rather than optimally by state-of-the-art solvers because the expertise required to formulate and solve these problems limits the widespread adoption of optimization tools and techniques. We introduce a Large Language Model (LLM)-based system designed to formulate and solve (mixed integer) linear programming problems from their natural language descriptions. Our system can develop mathematical models, write and debug solver code, evaluate the generated solutions, and improve efficiency and correctness of its model and code based on these evaluations. OptiMUS is designed as a productivity tool for optimization practitioners who understand the problem domain and can describe it precisely, but seek to accelerate the modeling and implementation workflow. OptiMUS-0.3 utilizes a modular structure to process problems, allowing it to handle problems with long descriptions and complex data without long prompts. Experiments demonstrate that OptiMUS-0.3 outperforms direct-prompting baselines by over 43% on easy and 18% on hard instances. It remains competitive with fine-tuned specialist models on benchmark problems, and outperforms them on real-world case studies (28.6% vs. 0%) where fine-tuned models fail to generalize. Ablation studies show that modular architecture with error correction is central to these gains. A key finding is that system architecture is a stronger driver of performance than model capability. Structured decomposition with targeted error correction enables weaker models to match stronger models under naive prompting, and remains competitive with fine-tuned specialist models without retraining costs.

The growth of large language model (LLM) inference and search services increases the scale of online linear programming problems, motivating computationally efficient algorithms. We develop resource-adaptive stochastic gradient descent (RASGD) for stochastic online linear programming. The algorithm uses one request and current inventory to update resource prices, requiring O(m) operations for m resources and memory per arrival and no LP or sample-average optimization. The central idea is to express the current-resource pricing logic of re-solving through a first-order SGD update: each arrival refreshes the remaining-inventory allowance in the dual objective, while the stepsize decreases for early learning and increases later to match the speed of inventory adjustment. Under standard non-degeneracy conditions, our algorithm is feasible on every sample path and achieves O(\log T) expected regret against the realized fractional hindsight optimum, which matches the lower bound, even for policies that know the distribution and have unrestricted computation. The analysis converts curvature around the fixed reference price into inventory stability without tracking optimal prices at changing resource levels. Numerical experiments show that RASGD achieves regret competitive with per-arrival LP re-solving and improves upon the tested first-order baselines, while retaining the computational efficiency of first-order methods. These results establish RASGD as a computationally efficient approach to achieving high allocation quality in large-scale OLP.