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.

We propose a scalable method for training prediction (machine learning) models in the predict-then-optimize paradigm, where model outputs serve as coefficients for a subsequent linear optimization task. Directly minimizing the empirical decision regret is intractable for linear programming and combinatorial optimization since the decision mapping is piecewise constant, and the gradients are zero almost everywhere. While existing methods address this by smoothing the differentiation process, they suffer from scalability issues, since a computationally expensive solver call is required for every gradient evaluation. To address this, we propose a decision-focused learning pipeline based on a measure transformation principle, which yields a new surrogate loss that is completely optimization-solver-free during training. We establish theoretical guarantees, including Fisher consistency and excess risk bounds. Empirically, our method achieves decision quality competitive with state-of-the-art methods while reducing training time by orders of magnitude.