搜索结果：convergence

We investigate the convergence of symmetric stochastic differential games with interactions via control, where the volatility terms of both idiosyncratic and common noises are controlled. We apply the stochastic maximum principle, following the approach of Laurière and Tangpi, to reduce the convergence analysis to the study of forward-backward propagation of chaos. Under the standard monotonicity conditions, we derive quantitative convergence rates for open-loop Nash equilibria of $N$-player stochastic differential games toward the corresponding mean field equilibrium. As a prerequisite, we also establish the well-posedness of the conditional McKean--Vlasov forward-backward stochastic differential equations by the method of continuation. Moreover, we analyze a specific class of linear-quadratic settings to demonstrate the applicability of our main result.

A depth-recurrent transformer applies a weight-tied core a variable number of times, and prior work has shown that training with a randomized recursion count yields one checkpoint usable across a range of inference depths. We ask what such a model actually computes per token, and measure it directly. On a 135M-class model trained on FineWeb-Edu, the recurrent state converges to a per-token fixed point: mean successive-output KL divergence falls from 3.9e-1 at the second loop to 8.5e-6 by the sixteenth, and per-token state change decays in step. Crucially, this convergence is not uniform across tokens. The median token converges by loop six, while approximately 10 percent of tokens continue to update at the training-mean depth of eight, and mean convergence depth is ordered by token type (whitespace shallowest, content words deepest). This per-token variation is the central object of the paper. We show it is directly readable and that reading it outperforms learning to predict it: a training-free rule that halts each token once its output stabilizes attains uniform depth-8 quality at 4.94 average loops (a 38 percent reduction in average depth) and matches uniform depth across the av

Classical optimisation theory guarantees monotonic objective decrease for gradient descent (GD) when employed in a small step size, or ``stable", regime. In contrast, gradient descent on neural networks is frequently performed in a large step size regime called the ``edge of stability", in which the objective decreases non-monotonically with an observed implicit bias towards flat minima. In this paper, we take a step toward quantifying this phenomenon by providing convergence rates for gradient descent with large learning rates in an overparametrised least squares setting. The key insight behind our analysis is that, as a consequence of overparametrisation, the set of global minimisers forms a Riemannian manifold $M$, which enables the decomposition of the GD dynamics into components parallel and orthogonal to $M$. The parallel component corresponds to Riemannian gradient descent on the objective sharpness, while the orthogonal component is a bifurcating dynamical system. This insight allows us to derive convergence rates in three regimes characterised by the learning rate size: (a) the subcritical regime, in which transient instability is overcome in finite time before linear conv

In this paper we investigate the convergence of the Policy Iteration Algorithm (PIA) for a class of general continuous-time entropy-regularized stochastic control problems. In particular, instead of employing sophisticated PDE estimates for the iterative PDEs involved in the algorithm (see, e.g., Huang-Wang-Zhou(2025)), we shall provide a simple proof from scratch for the convergence of the PIA. Our approach builds on probabilistic representation formulae for solutions of PDEs and their derivatives. Moreover, in the finite horizon model and in the infinite horizon model with large discount factor, the similar arguments lead to a super-exponential rate of convergence without tear. Finally, with some extra efforts we show that our approach can be extended to the diffusion control case in the one dimensional setting, also with a super-exponential rate of convergence.

Continual learning, which aims to learn multiple tasks sequentially, has gained extensive attention. However, most existing work focuses on empirical studies, and the theoretical aspect remains under-explored. Recently, a few investigations have considered the theory of continual learning only for linear regressions, establishes the results based on the strict independent and identically distributed (i.i.d.) assumption and the persistent excitation on the feature data that may be difficult to verify or guarantee in practice. To overcome this fundamental limitation, in this paper, we provide a general and comprehensive theoretical analysis for continual learning of regression models. By utilizing the stochastic Lyapunov function and martingale estimation techniques, we establish the almost sure convergence results of continual learning under a general data condition for the first time. Additionally, without any excitation condition imposed on the data, the convergence rates for the forgetting and regret metrics are provided.

In the framework of real Hilbert spaces, we investigate first-order dynamical systems governed by monotone and continuous operators. We demonstrate that when the monotone operator flow is augmented with a Tikhonov regularization term, the resulting trajectory converges strongly to the element of the set of zeros with minimal norm. In addition, rates of convergence in norm for the trajectory's velocity and the operator along the trajectory can be derived in terms of the regularization function. In some particular cases, these rates of convergence can outperform the ones of the coercive operator flows and can be as fast as $O(\frac{1}{t})$ as $t \rightarrow +\infty$. In this way, we emphasize a surprising acceleration feature of the Tikhonov regularization. Additionally, we explore these properties for monotone operator flows that incorporate time rescaling and an anchor point and show that they are closely linked to second-order dynamics with a vanishing damping term. The convergence and convergence rate results we achieve for these systems complement recent findings for the Fast Optimistic Gradient Descent Ascent (OGDA) dynamics. When the monotone operator is defined as the identit

Blurring mean shift (BMS) algorithm, a variant of the mean shift algorithm, is a kernel-based iterative method for data clustering, where data points are clustered according to their convergent points via iterative blurring. In this paper, we analyze convergence properties of the BMS algorithm by leveraging its interpretation as an optimization procedure, which is known but has been underutilized in existing convergence studies. Whereas existing results on convergence properties applicable to multi-dimensional data only cover the case where all the blurred data point sequences converge to a single point, this study provides a convergence guarantee even when those sequences can converge to multiple points, yielding multiple clusters. This study also shows that the convergence of the BMS algorithm is fast by further leveraging geometrical characterization of the convergent points.

It has been shown that many first-order methods satisfy the perturbed Fenchel duality inequality, which yields a unified derivation of convergence. More first-order methods are discussed in this paper, e.g., dual averaging and bundle method. We show primal-dual convergence of them on convex optimization by proving the perturbed Fenchel duality property. We also propose a single-cut bundle method for saddle problem, and prove its convergence in a similar manner.

We give conditions under which weak convergence of a stochastic process indexed in the class of $d$-dimensional hyperrectangles is sufficient to ensure convergence in the larger class of functions of uniformly bounded Hardy-Krause variation. When applied to the empirical process, this can further be extended to derive weak convergence of V-processes indexed in the class of kernel functions which are coordinate-wise of uniformly bounded Hardy-Krause variation. Our proofs use a generalisation of the Koksma-Hlawka inequality for linear operators, allowing us to establish our results without any continuity assumptions on the functions involved. Our theory is complemented by two separate applications: First, we establish asymptotic normality of Chatterjee's rank correlation in the fully general setting. Second, we present new limit theorems for U- and V-processes of strongly mixing data.

Regret matching (RM) -- and its modern variants -- is a foundational online algorithm that has been at the heart of many AI breakthrough results in solving benchmark zero-sum games, such as poker. Yet, surprisingly little is known so far in theory about its convergence beyond two-player zero-sum games. For example, whether regret matching converges to Nash equilibria in potential games has been an open problem for two decades. Even beyond games, one could try to use RM variants for general constrained optimization problems. Recent empirical evidence suggests that they -- particularly regret matching$^+$ (RM$^+$) -- attain strong performance on benchmark constrained optimization problems, outperforming traditional gradient descent-type algorithms. We show that RM$^+$ converges to an $ε$-KKT point after $O_ε(1/ε^4)$ iterations, establishing for the first time that it is a sound and fast first-order optimizer. Our argument relates the KKT gap to the accumulated regret, two quantities that are entirely disparate in general but interact in an intriguing way in our setting, so much so that when regrets are bounded, our complexity bound improves all the way to $O_ε(1/ε^2)$. From a technic

We study convergence rates of the Trotter splitting $e^{A+L} = \lim_{n \to \infty} (e^{L/n} e^{A/n})^n$ in the strong operator topology. In the first part, we use complex interpolation theory to treat generators $L$ and $A$ of contraction semigroups on Banach spaces, with $L$ relatively $A$-bounded. In the second part, we study unitary dynamics on Hilbert spaces and develop a new technique based on the concept of energy constraints. Our results provide a complete picture of the convergence rates for the Trotter splitting for all common types of Schrödinger and Dirac operators, including singular, confining and magnetic vector potentials, as well as molecular many-body Hamiltonians in dimension $d=3$. Using the Brezis-Mironescu inequality, we derive convergence rates for the Schrödinger operator with $V(x)=\pm |x|^{-a}$ potential. In each case, our conditions are fully explicit.

We investigate the joint convergence of independent random Toeplitz matrices with complex input entries that have a pair-correlation structure, along with deterministic Toeplitz matrices and the backward identity permutation matrix. Further, we study the joint convergence of independent generalized Toeplitz matrices along with other related matrices. The limits depend only on the correlation structure but are universal otherwise, in that they do not depend on the underlying distributions of the entries. In particular, these results provide the joint convergence of asymmetric Hankel matrices. Earlier results in the literature on the joint convergence of random symmetric Toeplitz and symmetric Hankel matrices with real entries follow as special cases.

Let $Φ'$ denote the strong dual of a nuclear space $Φ$ and let $C_{\infty}(Φ')$ be the collection of all continuous mappings $x:[0,\infty) \rightarrow Φ'$ equipped with the topology of local uniform convergence. In this paper we prove sufficient conditions for tightness of probability measures on $C_{\infty}(Φ')$ and for weak convergence in $C_{\infty}(Φ')$ for a sequence of $Φ'$-valued processes. We illustrate our results with two applications. First, we show the central limit theorem for local martingales taking values in the dual of an ultrabornological nuclear space. Second, we prove sufficient conditions for the weak convergence in $C_{\infty}(Φ')$ for a sequence of solutions to stochastic partial differential equations driven by semimartingale noise.

Very recently, the papers "Point Convergence of Nesterov's Accelerated Gradient Method: An AI-Assisted Proof" by Jang and Ryu, and "The Iterates of Nesterov's Accelerated Algorithm Converge in the Critical Regimes" by Bot, Fadili, and Nguyen simultaneously have resolved a long-standing open problem concerning Nesterov's accelerated gradient method. These works show that the iterates of the algorithm (known in its composite form as FISTA) indeed converge to an optimal solution. In this work, we extend these results and prove that, in infinite dimensional Hilbert spaces, the iterates of such an algorithm still converge (in the weak sense) even when the proximity operator and the gradient are computed inexactly, with the latter possibly stochastic.

Recently, there has been a great deal of attention in a class of controllers based on time-varying gains, called prescribed-time controllers, that steer the system's state to the origin in the desired time, a priori set by the user, regardless of the initial condition. Furthermore, such a class of controllers has been shown to maintain a prescribed-time convergence in the presence of disturbances even if the disturbance bound is unknown. However, such properties require a time-varying gain that becomes singular at the terminal time, which limits its application to scenarios under quantization or measurement noise. This chapter presents a methodology to design a broader class of controllers, called predefined-time controllers, with a prescribed convergence-time bound. Our approach allows designing robust predefined-time controllers based on time-varying gains while maintaining uniformly bounded time-varying gains. We analyze the condition for uniform Lyapunov stability under the proposed time-varying controllers.

Recently, much progress has been made on particle swarm optimization (PSO). A number of works have been devoted to analyzing the convergence of the underlying algorithms. Nevertheless, in most cases, rather simplified hypotheses are used. For example, it often assumes that the swarm has only one particle. In addition, more often than not, the variables and the points of attraction are assumed to remain constant throughout the optimization process. In reality, such assumptions are often violated. Moreover, there are no rigorous rates of convergence results available to date for the particle swarm, to the best of our knowledge. In this paper, we consider a general form of PSO algorithms, and analyze asymptotic properties of the algorithms using stochastic approximation methods. We introduce four coefficients and rewrite the PSO procedure as a stochastic approximation type iterative algorithm. Then we analyze its convergence using weak convergence method. It is proved that a suitably scaled sequence of swarms converge to the solution of an ordinary differential equation. We also establish certain stability results. Moreover, convergence rates are ascertained by using weak convergence

We consider two classes of natural stochastic processes on finite unlabeled graphs. These are Euclidean stochastic optimization algorithms on the adjacency matrix of weighted graphs and a modified version of the Metropolis MCMC algorithm on stochastic block models over unweighted graphs. In both cases we show that, as the size of the graph goes to infinity, the random trajectories of the stochastic processes converge to deterministic curves on the space of measure-valued graphons. Measure-valued graphons, introduced by Lovász and Szegedy in \cite{lovasz2010decorated}, are a refinement of the concept of graphons that can distinguish between two infinite exchangeable arrays that give rise to the same graphon limit. We introduce new metrics on this space which provide us with a natural notion of convergence for our limit theorems. This notion is equivalent to the convergence of infinite-exchangeable arrays. Under suitable assumptions and a specified time-scaling, the Metropolis chain admits a diffusion limit as the number of vertices go to infinity. We then demonstrate that, in an appropriately formulated zero-noise limit, the stochastic process of adjacency matrices of this diffusion

Let $Φ'$ denote the strong dual of a nuclear space $Φ$. In this paper we introduce sufficient conditions for the convergence uniform on compacts in probability for a sequence of $Φ'$-valued processes with continuous or càdlàg paths. We illustrate the usefulness of our results by considering two applications to stochastic analysis. First, we introduce a topology on the space of $Φ'$-valued semimartingales which are good integrators and show that this topology is complete and that the stochastic integral mapping is continuous on the integrators. Second, we introduce sufficient conditions for the convergence uniform on compacts in probability of the solutions to a sequence of linear stochastic evolution equations driven by semimartingale noise.

Under mild assumptions, we show the exact convergence rate in total variation is also exact in weaker Wasserstein distances for the Metropolis-Hastings independence sampler. We develop a new upper and lower bound on the worst-case Wasserstein distance when initialized from points. For an arbitrary point initialization, we show the convergence rate is the same and matches the convergence rate in total variation. We derive exact convergence expressions for more general Wasserstein distances when initialization is at a specific point. Using optimization, we construct a novel centered independent proposal to develop exact convergence rates in Bayesian quantile regression and many generalized linear model settings. We show the exact convergence rate can be upper bounded in Bayesian binary response regression (e.g. logistic and probit) when the sample size and dimension grow together.