Related papers: Perov's Contraction Principle and Dynamic Programm…
While numerous extensions of Banach's fixed point theorem typically offer only sufficient conditions for the existence and uniqueness of a fixed point and the convergence of iterative sequences, this study introduces a generalization…
A constraint-reduced Mehrotra-Predictor-Corrector algorithm for convex quadratic programming is proposed. (At each iteration, such algorithms use only a subset of the inequality constraints in constructing the search direction, resulting in…
Recently, the concept of k-contraction has been introduced as a promising generalization of contraction for dynamical systems. However, the study of k-contraction properties has faced significant challenges due to the reliance on complex…
This paper considers the analysis of continuous time gradient-based optimization algorithms through the lens of nonlinear contraction theory. It demonstrates that in the case of a time-invariant objective, most elementary results on…
We consider the problem of optimising the expected value of a loss functional over a nonlinear model class of functions, assuming that we have only access to realisations of the gradient of the loss. This is a classical task in statistics,…
This paper investigates theoretical and methodological foundations for stochastic optimal control (SOC) in discrete time. We start formulating the control problem in a general dynamic programming framework, introducing the mathematical…
Sufficient dimension reduction is a powerful tool to extract core information hidden in the high-dimensional data and has potentially many important applications in machine learning tasks. However, the existing nonlinear sufficient…
The key element of the approach to the theory of necessary conditions in optimal control discussed in the paper is reduction of the original constrained problem to unconstrained minimization with subsequent application of a suitable…
In many experimental settings, one is tasked with obtaining information about certain relationships by applying perturbations to a set of independent variables and noting the changes in the set of dependent ones. While traditional…
In Bayesian statistics, posterior contraction rates (PCRs) quantify the speed at which the posterior distribution concentrates on arbitrarily small neighborhoods of a true model, in a suitable way, as the sample size goes to infinity. In…
In the present note, the Banach contraction principle is proved in complete modular spaces via an order theoretic approach.
The last decade witnessed a rise in the importance of supervised learning applications involving {\em big data} and {\em big models}. Big data refers to situations where the amounts of training data available and needed causes difficulties…
This paper proposes a hierarchical modeling approach to perform stochastic model specification in Markov switching vector error correction models. We assume that a common distribution gives rise to the regime-specific regression…
Dropout is a simple yet effective algorithm for regularizing neural networks by randomly dropping out units through Bernoulli multiplicative noise, and for some restricted problem classes, such as linear or logistic regression, several…
New approaches to the theory of dynamic programming view dynamic programs as families of policy operators acting on partially ordered sets. In this paper, we extend these ideas by shifting from arbitrary partially ordered sets to ordered…
We consider convex optimization problems formulated using dynamic programming equations. Such problems can be solved using the Dual Dynamic Programming algorithm combined with the Level 1 cut selection strategy or the Territory algorithm to…
Preference optimization is widely used to align large language models (LLMs) with human preferences. However, many margin-based methods also suppress the chosen response when they try to suppress the rejected one, and there is no general…
We consider the problem of learning the optimal policy for infinite-horizon Markov decision processes (MDPs). For this purpose, some variant of Stochastic Mirror Descent is proposed for convex programming problems with Lipschitz-continuous…
In this work, we leverage the 2-contraction theory, which extends the capabilities of classical contraction theory, to develop a global stability framework. Coupled with powerful geometric tools such as the Poincare index theory, the…
This paper, based on the compactness-continuity and finite value conditions, establishes the sufficiency of the class of stationary policies out of the general class of history-dependent ones for a constrained continuous-time Markov…