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Additionally, it is evident that the results in (ii) do not implytheconvergenceof f(xk) to0.

Minimization (SAM), a recently proposed gradient-based optimization method [Foret et al., 2021] that significantly improves the generalization of deep neural

Nonconvex and nonsmooth problems have recently received considerable attention in signal/image processing, statistics and machine learning. However, solving the nonconvex and nonsmooth optimization problems remains a big challenge. Accelerated proximal gradient (APG) is an excellent method for convex programming. However, it is still unknown whether the usual APG can ensure the convergence to a critical point in nonconvex programming. In this paper, we extend APG for general nonconvex and nonsmooth programs by introducing a monitor that satisfies the sufficient descent property. Accordingly, we propose a monotone APG and a nonmonotone APG. The latter waives the requirement on monotonic reduction of the objective function and needs less computation in each iteration.

Understanding the convergence property of EM is highly nontrivial due to the non-convexity of the negative log-likelihood function.

This article provides a rigorous analysis of convergence and stability of Episodic Upside-Down Reinforcement Learning, Goal-Conditioned Supervised Learning and Online Decision Transformers. These algorithms performed competitively across various benchmarks, from games to robotic tasks, but their theoretical understanding is limited to specific environmental conditions. This work initiates a theoretical foundation for algorithms that build on the broad paradigm of approaching reinforcement learning through supervised learning or sequence modeling. At the core of this investigation lies the analysis of conditions on the underlying environment, under which the algorithms can identify optimal solutions. We also assess whether emerging solutions remain stable in situations where the environment is subject to tiny levels of noise. Specifically, we study the continuity and asymptotic convergence of command-conditioned policies, values and the goal-reaching objective depending on the transition kernel of the underlying Markov Decision Process. We demonstrate that near-optimal behavior is achieved if the transition kernel is located in a sufficiently small neighborhood of a deterministic kernel. The mentioned quantities are continuous (with respect to a specific topology) at deterministic kernels, both asymptotically and after a finite number of learning cycles. The developed methods allow us to present the first explicit estimates on the convergence and stability of policies and values in terms of the underlying transition kernels. On the theoretical side we introduce a number of new concepts to reinforcement learning, like working in segment spaces, studying continuity in quotient topologies and the application of the fixed-point theory of dynamical systems. The theoretical study is accompanied by a detailed investigation of example environments and numerical experiments.

This work proposes a mathematical approach that (re)defines a property of Machine Learning models named stability and determines sufficient conditions to validate it. Machine Learning models are represented as functions, and the characteristics in scope depend upon the domain of the function, what allows us to adopt topological and metric spaces theory as a basis. Finally, this work provides some equivalences useful to prove and test stability in Machine Learning models. The results suggest that whenever stability is aligned with the notion of function smoothness, then the stability of Machine Learning models primarily depends upon certain topological, measurable properties of the classification sets within the ML model domain.

Despite the number of successful applications of the Extreme Learning Machine (ELM), we show that its underlying foundational principles do not have a rigorous mathematical justification. Specifically, we refute the proofs of two main statements, and we also create a dataset that provides a counterexample to the ELM learning algorithm and explain its design, which leads to many such counterexamples. Finally, we provide alternative statements of the foundations, which justify the efficiency of ELM in some theoretical cases.

Motivated by the widespread use of approximate derivatives in machine learning and optimization, we study inexact subgradient methods with non-vanishing additive errors and step sizes. In the nonconvex semialgebraic setting, under boundedness assumptions, we prove that the method provides points that eventually fluctuate close to the critical set at a distance proportional to $\epsilon^\rho$ where $\epsilon$ is the error in subgradient evaluation and $\rho$ relates to the geometry of the problem. In the convex setting, we provide complexity results for the averaged values. We also obtain byproducts of independent interest, such as descent-like lemmas for nonsmooth nonconvex problems and some results on the limit of affine interpolants of differential inclusions.

Nonconvex and nonsmooth problems have recently received considerable attention in signal/image processing, statistics and machine learning. However, solving the nonconvex and nonsmooth optimization problems remains a big challenge. Accelerated proximal gradient (APG) is an excellent method for convex programming. However, it is still unknown whether the usual APG can ensure the convergence to a critical point in nonconvex programming. In this paper, we extend APG for general nonconvex and nonsmooth programs by introducing a monitor that satisfies the sufficient descent property. Accordingly, we propose a monotone APG and a nonmonotone APG. The latter waives the requirement on monotonic reduction of the objective function and needs less computation in each iteration.