Search

This paper proposes a novel first-order algorithm that solves composite nonsmooth and stochastic convex optimization problem with function constraints. Most of the works in the literature provide convergence rate guarantees on the average-iterate solution. There is growing interest in the convergence guarantees of the last iterate solution due to its favorable structural properties, such as sparsity or privacy guarantees and good performance in practice. We provide the first method that obtains the best-known convergence rate guarantees on the last iterate for stochastic composite nonsmooth convex function-constrained optimization problems. Our novel and easy-to-implement algorithm is based on the augmented Lagrangian technique and uses a new linearized approximation of constraint functions, leading to its name, the Augmented Constraint Extrapolation (Aug-ConEx) method. We show that Aug-ConEx achieves $\mathcal{O}(1/\sqrt{K})$ convergence rate in the nonsmooth stochastic setting without any strong convexity assumption and $\mathcal{O}(1/K)$ for the same problem with strongly convex objective function. While optimal for nonsmooth and stochastic problems, the Aug-ConEx method also accelerates convergence in terms of Lipschitz smoothness constants to $\mathcal{O}(1/K)$ and $\mathcal{O}(1/K^2)$ in the aforementioned cases, respectively. To our best knowledge, this is the first method to obtain such differentiated convergence rate guarantees on the last iterate for a composite nonsmooth stochastic setting without additional $\log{K}$ factors. We validate the efficiency of our algorithm by comparing it with a state-of-the-art algorithm through numerical experiments., Comment: 26 Pages, 3 Figures

Background: This study aimed to evaluate and compare the performance of classical machine learning models (CMLs) and large language models (LLMs) in predicting mortality associated with COVID-19 by utilizing a high-dimensional tabular dataset. Materials and Methods: We analyzed data from 9,134 COVID-19 patients collected across four hospitals. Seven CML models, including XGBoost and random forest (RF), were trained and evaluated. The structured data was converted into text for zero-shot classification by eight LLMs, including GPT-4 and Mistral-7b. Additionally, Mistral-7b was fine-tuned using the QLoRA approach to enhance its predictive capabilities. Results: Among the CML models, XGBoost and RF achieved the highest accuracy, with F1 scores of 0.87 for internal validation and 0.83 for external validation. In the LLM category, GPT-4 was the top performer with an F1 score of 0.43. Fine-tuning Mistral-7b significantly improved its recall from 1% to 79%, resulting in an F1 score of 0.74, which was stable during external validation. Conclusion: While LLMs show moderate performance in zero-shot classification, fine-tuning can significantly enhance their effectiveness, potentially aligning them closer to CML models. However, CMLs still outperform LLMs in high-dimensional tabular data tasks., Comment: Code is available at: https://github.com/mohammad-gh009/Large-Language-Models-vs-Classical-Machine-learning and https://github.com/Sdamirsa/Tehran_COVID_Cohort. The datasets are available from the corresponding author on reasonable request (sdamirsa@ymail.com)

The monotone Variational Inequality (VI) is a general model with important applications in various engineering and scientific domains. In numerous instances, the VI problems are accompanied by function constraints that can be data-driven, making the usual projection operator challenging to compute. This paper presents novel first-order methods for the function-constrained Variational Inequality (FCVI) problem in smooth or nonsmooth settings with possibly stochastic operators and constraints. We introduce the AdOpEx method, which employs an operator extrapolation on the KKT operator of the FCVI in a smooth deterministic setting. Since this operator is not uniformly Lipschitz continuous in the Lagrange multipliers, we employ an adaptive two-timescale algorithm leading to bounded multipliers and achieving the optimal $O(1/T)$ convergence rate. For the nonsmooth and stochastic VIs, we introduce design changes to the AdOpEx method and propose a novel P-OpEx method that takes partial extrapolation. It converges at the rate of $O(1/\sqrt{T})$ when both the operator and constraints are stochastic or nonsmooth. This method has suboptimal dependence on the noise and Lipschitz constants of function constraints. We propose a constraint extrapolation approach leading to the OpConEx method that improves this dependence by an order of magnitude. All our algorithms easily extend to saddle point problems with function constraints that couple the primal and dual variables while maintaining the same complexity results. To the best of our knowledge, all our complexity results are new in the literature

We investigate a primal-dual (PD) method for the saddle point problem (SPP) that uses a linear approximation of the primal function instead of the standard proximal step, resulting in a linearized PD (LPD) method. For convex-strongly concave SPP, we observe that the LPD method has a suboptimal dependence on the Lipschitz constant of the primal function. To fix this issue, we combine features of Accelerated Gradient Descent with the LPD method resulting in a single-loop Accelerated Linearized Primal-Dual (ALPD) method. ALPD method achieves the optimal gradient complexity when the SPP has a semi-linear coupling function. We also present an inexact ALPD method for SPPs with a general nonlinear coupling function that maintains the optimal gradient evaluations of the primal parts and significantly improves the gradient evaluations of the coupling term compared to the ALPD method. We verify our findings with numerical experiments.