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The stochastic approximation (SA) method stands as the foundational mathematical tool for modern large-scale optimization and machine learning. Therefore, gaining a theoretical understanding of SA algorithms is of fundamental interest. In their paper titled "A Lyapunov Theory for Finite-Sample Guarantees of Markovian Stochastic Approximation," Chen et al. present a unified Lyapunov framework for the finite-sample analysis of a Markovian SA algorithm under a contractive operator with respect to an arbitrary norm. The key novelty lies in the construction of a smooth Lyapunov function called the generalized Moreau envelope. The authors demonstrate the effectiveness of their SA results in the context of reinforcement learning (RL), specifically through popular algorithms such as variants of temporal difference (TD) learning and Q-learning. As byproducts, the results provide theoretical insights into the efficiency of bootstrapping in TD learning with eligibility traces and the bias-variance tradeoff in off-policy learning. This paper develops a unified Lyapunov framework for finite-sample analysis of a Markovian stochastic approximation (SA) algorithm under a contraction operator with respect to an arbitrary norm. The main novelty lies in the construction of a valid Lyapunov function called the generalized Moreau envelope. The smoothness and an approximation property of the generalized Moreau envelope enable us to derive a one-step Lyapunov drift inequality, which is the key to establishing the finite-sample bounds. Our SA result has wide applications, especially in the context of reinforcement learning (RL). Specifically, we show that a large class of value-based RL algorithms can be modeled in the exact form of our Markovian SA algorithm. Therefore, our SA results immediately imply finite-sample guarantees for popular RL algorithms such as n-step temporal difference (TD) learning, TD (λ) , off-policy V-trace, and Q-learning. As byproducts, by analyzing the convergence bounds of n-step TD and TD (λ) , we provide theoretical insight into the problem about the efficiency of bootstrapping. Moreover, our finite-sample bounds of off-policy V-trace explicitly capture the tradeoff between the variance of the stochastic iterates and the bias in the limit. Funding: This work was supported by RTX, the National Science Foundation [Grants 2019844, 2107037, 211247, 2112533, 2144316, and 2240982], and the Machine Learning Laboratory at University of Texas at Austin. Supplemental Material: The online appendix is available at https://doi.org/10.1287/opre.2022.0249. [ABSTRACT FROM AUTHOR]

Two of the most well-known belief contraction operators are partial meet contractions (PMCs) and kernel contractions (KCs). In this paper we propose two new classes of contraction operators, namely the class of generalized partial meet contractions (GPMC) and the class of generalized kernel contractions (GKC), which strictly contain the classes of PMCs and of KCs, respectively. We identify some extra conditions that can be added to the definitions of GPMCs and of GKCs, which give rise to some interesting subclasses of those classes of functions, namely the classes of extensional and of uniform GPMCs/GKCs. In the context of contractions on belief sets the classes of partial meet contractions, uniform GPMCs and extensional GPMCs are all identical. Nevertheless, when considered as operations on belief bases, the class of uniform GPMCs coincides with the class of partial meet contractions, but the extensional GPMCs constitute a new kind of belief base contraction functions whose characterizing postulate of irrelevance of syntax is extensionality—the same postulate of irrelevance of syntax which occurs in the classical axiomatic characterization of partial meet contractions for belief sets—rather than the postulate of uniformity—which is the irrelevance of syntax postulate used in the axiomatic characterization for partial meet contractions on belief bases. Analogous results are obtained regarding the classes of extensional and of uniform GKCs. We present the interrelations in the sense of inclusion among all the new classes of operators presented in this paper and several well known classes of PMCs and of KCs. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new mapping namely L-contraction for fixed function and we also prove a novel fixed function theorem using L-contraction mapping. Moreover, we apply our result to solve integral equation. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to introduce a novel fixed point iterative scheme based on Green's function, called the Picard–Ishikawa–Green's iterative scheme and use it in approximating the solution of boundary value problems (BVPs). It is proved that Picard–Ishikawa–Green's scheme converges strongly for an integral operator which represents the solution of BVP and the scheme is stable. Moreover, we prove that the integral operator is a contraction. Furthermore, it is shown that the novel scheme converges faster than all of Ishikawa–Green's, Khan–Green's, and Mann–Green's schemes. Finally, numerical examples are given to substantiate the validity of our results for third‐order BVPs. Our results extend and generalize several other results in literature. [ABSTRACT FROM AUTHOR]

European Institutions and scholars are reflecting upon the nature of the requirements that are or can be used to implement sustainability through Public Procurement. As far as public organisations are concerned, the Portuguese government's actions are in the same wavelength as the European guidelines on the matter. Public procurement involves economic operators. Thus, it is important to try to understand if the European Union proposal for a directive on "Corporate Sustainability Due Diligence" is an interesting tool in the discussion. Recently, the Portuguese Council of Ministers issued Resolution no. 132/2023, of 25 October, defining the ecological criteria applicable to certain kinds of contracts and products, goods and services concerning some administrative bodies, a Resolution which, nevertheless, is prone to criticism. May the shift of the European Union Public Procurement paradigm help implement Sustainable Public Procurement through mandatory requirements? Can the future Directive help economic operators align with contracting authorities to implement Sustainable Public Procurement? Might the obligation to include environmental factors help fulfil the Directive's due diligence obligations? And, lastly, has Portugal taken the right step with the issuance of Resolution no. 132/2023? [ABSTRACT FROM AUTHOR]

In this paper, we establish the quantitative mean ergodic theorems for two subclasses of power bounded operators on a fixed noncommutative L p space with 1 < p < ∞ , which mainly concerns power bounded invertible operators and Lamperti contractions. Our approach to the quantitative ergodic theorems relies on noncommutative square function inequalities. The establishment of the latter involves several new ingredients such as the almost orthogonality and Calderón-Zygmund arguments for non-smooth kernels from semi-commutative harmonic analysis, the extension properties of the operators under consideration from operator theory, and a noncommutative version of the classical transference method due to Coifman and Weiss. [ABSTRACT FROM AUTHOR]

Background: Tumor immune microenvironment (TiME) is prognostically instructive in Pancreatic adenocarcinoma (PAAD). However, the potential value of TiME-related genes in the individualized immunotherapy of PAAD has not been clarified. Methods: Correlation between Immune-Related Genes (IRGs) and immunerelated transcription factors (TFs) was performed to prove the immune correlation of selected genes. Immune-related molecular subtypes were identified by consensus clustering. The TiME-score, an immune microenvironment-related prognostic signature for PAAD, was constructed using minimum absolute contraction and selection operator regression (Lasso-Cox). The International Cancer Genome Consortium (ICGC) dataset validated the reliability of TiME-score as external validation. Single-cell samples from GSE197177 confirmed microenvironment differences of TiME-score hub genes between tumor and its paracancer tissues. Then, RARRES3, a hub gene in TiMEscore, was further analyzed about its upstream TP53 mutation and the specific immune landscape of itself in transcriptome and Single-cell level. Eventually, TiME-score were validated in different therapeutic cohorts of PAAD mice models. Results: A 14-genes PAAD immune-related risk signature, TiME-score, was constructed based on IRGs. The differences of TiME-score hub genes in single-cell samples of PAAD cancer tissues and adjacent tissues were consistent with the transcriptome. Single-cell samples of cancer tissues showed more pronounced immune cell infiltration. The upstream mutation factor TP53 of RARRES3 was significantly enriched in immune-related biological processes. High RARRES3 expression was correlated with a worse prognosis and high macrophages M1 infiltration. Additionally, the immunohistochemistry of hub genes AGT, DEFB1, GH1, IL20RB, and TRAF3 in different treatment cohorts of mice PAAD models were consistent with the predicted results. The combination of immunotherapy, chemotherapy and targeted therapy has shown significantly better therapeutic effects than single drug therapy in PAAD. Conclusion: TiME-score, as a prognostic signature related to PAAD-specific immune microenvironment constructed based on RARRES3, has predictive value for prognosis and the potential to guide individualized immunotherapy for PAAD patients. [ABSTRACT FROM AUTHOR]