Q-learning with linear function approximation can be unstable because an arbitrary approximation architecture need not preserve the Bellman contraction. We develop a stable infinite-dimensional linear function approximation framework for Q-learning from a single Markovian behavior-policy trajectory. The learning variable is a coefficient field $θ\in C(\mathbb L)$ on a compact latent metric space $(\mathbb L,ρ)$. The framework uses a reconstruction operator that maps $θ$ to a continuous Q-function and a compression operator that maps Bellman updates back to latent coordinates. Nonexpansiveness of both operators induces a contractive latent Bellman map on $C(\mathbb L)$, with a unique fixed point $θ^*$ whose reconstruction approximates the optimal Q-function up to representation error. We propose two stochastic approximation (SA) algorithms and establish their sup-norm convergence bounds with a leading term of order $\widetilde O(n^{-1/2})$. The infinite-dimensional formulation provides a powerful abstraction for identifying the structures that govern statistical difficulty. Smoothness of the compression map in $ρ$ is inherited by $θ^*$ and the SA iterates, allowing uniform estimation errors to be controlled through covering numbers of $(\mathbb L,ρ)$ rather than the dimension of $C(\mathbb L)$. Remarkably, the SA algorithms we propose are agnostic to the choice of $ρ$, and thus can automatically adapt to both the smoothness and the geometry. We further illustrate the framework through Q-measure-learning with linear density approximation and output-layer neural weight training under a frozen pretrained network.
Ege C. Kaya, Arda Fazla, M. Berk Sahin +1cs.LG math.OC
We study stochastic approximation of fixed points of a non-expansive operator when the oracle samples originate from a continuing Markovian trajectory. A direct block-minibatch implementation of Halpern iteration attains an expected last-iterate residual of order $O(\log N/N)$, but accrues a substantive complexity of $\tilde O(ε^{-5})$ Markovian samples. We therefore introduce a variance-reduced Markovian PAGE-Halpern method whose refresh and same-state difference blocks are analyzed through the Poisson equation. In Hilbert spaces, the cocoercivity of $I-T$ results in an $O(ε^{-3})$ sample complexity. Our main result extends this construction to a general finite-dimensional Banach space. A displacement-level Halpern bound replaces the Hilbert-space potential and yields $\tilde O(ε^{-3})$ sample complexity in the original non-expansiveness norm. We also establish a high-probability guarantee with the same leading accuracy dependence by measuring the estimator in an auxiliary smooth norm. Non-smooth sup and block-sup geometries are covered through norm smoothing.
We study the implementable finite-batch particle algorithm for mean-field variational inference as a fully discrete stochastic approximation of the projected Wasserstein dynamics. The target potential is globally smooth but need not be strongly convex. The departure from contractivity is quantified by the curvature defect \[ \mathfrak d_α(x,y) = \bigl[α\|x-y\|^2- \langle\nabla V(x)-\nabla V(y),x-y\rangle\bigr]_+, \] which is the additive loss in the one-step Euler contraction estimate. We prove a non-asymptotic Wasserstein stability bound that separates initialization, product-empirical approximation, finite-batch drift error, time discretization, and the defects accumulated along the coupled trajectories. Under the uniform bound $\mathfrak d_α\leqβ$, the particle iterates remain within $O(\sqrt{β/α})$ of any MFVI minimizer, up to explicit errors in the particle number, batch size, and step size. The proof uses a stationary comparison array whose population law is an MFVI minimizer but whose particle-level law is a random product empirical measure, and it controls the resulting projected-drift discrepancy explicitly. We also give coordinatewise defect estimates and structural conditions for dimension-independent projected-drift sensitivity, construct an arbitrary-dimensional smooth nonconvex benchmark with a closed-form MFVI minimizer, and explain why polynomially growing drifts require a modification of the untamed explicit scheme.
Popularity bias in recommendation systems arises when a majority user class generates disproportionate interaction data, causing the system to increasingly favour it while degrading recommendation quality for niche users. While extensive empirical evidence of popularity bias exists, the dynamics leading to its emergence are not well understood. In this work, we study the coupled evolution of recommender model updates and user engagement through the lens of dynamical systems. We formulate a stochastic process and analyse its asymptotic behaviour through an ordinary differential equation (ODE) framework grounded in two-time-scale stochastic approximation. We characterise the equilibrium points of this dynamical system, and derive conditions under which popularity bias is provably emergent, as well as conditions under which symmetric retention of all user classes is possible. We conduct experiments on synthetic data and real-world production logs derived from a large-scale commercial music recommendation platform to validate our theoretical results.
Bilevel optimization (BLO) is fundamental to hierarchical decision-making but suffers from critical instability under heavy-tailed stochastic noise. Existing variance-reduction techniques typically rely on myopic magnitude checks, which fail to distinguish informative geometric signals from impulsive outliers. To resolve this, we propose \textbf{RQ-TTSA} (Robust Quantile-guided TTSA), a distribution-aware framework that leverages historical gradient buffers to estimate rolling quantiles for adaptive Huber-style clipping, effectively preserving local optimization geometry while strictly bounding effective variance. Theoretically, we provide a convergence analysis for quantile-guided TTSA under nonconvex-strongly convex assumptions with infinite-variance noise ($p \in (1,2]$), deriving a rate of $\mathcal{O}(T^{-\frac{p-1}{3p-2}})$ that recovers optimal dependence on the heavy-tailed parameter. Empirically, across six diverse tasks, spanning heterogeneous vision benchmarks, dynamic games under momentum poisoning, and offline reinforcement learning, RQ-TTSA consistently outperforms state-of-the-art baselines by eliminating divergence spikes and ensuring stable convergence. Our method demonstrates significant robustness to hyperparameter variations and incurs negligible computational overhead ($\approx 2.7\%$ increase), validating distribution-aware gradient control as a practical and necessary component for reliable bilevel learning.
Many central machine learning tasks, from entropy tuning in reinforcement learning to equilibrating generative adversarial networks, are fundamentally stochastic root-finding problems rather than loss minimization. Yet, they are frequently forced into a minimization framework via squared residuals, introducing a critical flaw we identify as the Variance Trap. Standard bilevel minimization algorithms require estimating hypergradients involving implicit Jacobians; in stochastic settings, these terms act as noise amplifiers, destabilizing convergence. We formalize Root-Finding Bilevel Optimization (RF-BO) as a distinct problem class that bypasses this pathology. We propose a Jacobian-free solution using Two-Time-Scale Stochastic Approximation (TTSA) that updates directly along the root error, structurally avoiding variance amplification. We provide the first non-asymptotic convergence guarantees for TTSA in this setting under Markovian noise. Extensive experiments demonstrate the decisive advantage of this paradigm: compared to squared-residual and implicit-gradient baselines, our framework achieves a 2.6\% top-1 accuracy gain in SimCLR, 17$\times$ faster convergence in non-linear ODE control where baselines fail, significantly improved entropy stability in reinforcement learning, and an 11.1\% quality improvement in generative modeling.
Attention mechanisms have demonstrated remarkable empirical success in identifying relevant information from large collections of tokens, yet the theoretical principles underlying this behavior remain poorly understood. We study a stylized softmax-attention model in which a query vector is learned by stochastic gradient ascent from a collection of informative and nuisance tokens. Exploiting the symmetry of the model, we derive a population objective and characterize the limiting ordinary differential equation governing the learning dynamics. Using tools from stochastic approximation and dynamical systems theory, we establish a rigorous connection between the stochastic learning algorithm and its deterministic limit. Our main result shows that, under suitable high-dimensional scaling assumptions and standard step-size conditions, the learned query converges almost surely to the one-dimensional signal subspace spanned by the latent informative direction. Equivalently, the query asymptotically recovers the latent signal up to the intrinsic sign ambiguity. These results provide a rigorous theoretical foundation for understanding attention mechanisms as signal extraction procedures in high-dimensional noisy environments and offer a dynamical-systems perspective on how attention discovers relevant information in the presence of substantial noise.
In this article, we analyse convergence of the averaged Adam optimizer to an attracting zero of the Adam vector field. We provide a central limit theorem that, in particular, quantifies exactly the speed of convergence. The order of convergence is $n^{-1/2}$ in the number of steps of the algorithm which coincides with the order observed for classical stochastic approximation algorithms. The covariance in the central limit theorem is given in terms of properties of the Adam algorithm in the state of the attractor.
M. Forzo, E. Monzio Compagnoni, A. Russo +1stat.ML cs.LG math.PR
Temporal difference (TD) learning with linear function approximation is a core method for policy evaluation. Its classical continuous-time description is an ordinary differential equation (ODE), which captures the asymptotic mean dynamics but neglects stochastic fluctuations determining the error floor. We introduce a stochastic differential equation (SDE) approximation for linear TD(0) under Markovian noise. The resulting model distinguishes the contraction dynamics governed by the projected Bellman operator from the influence of Markovian sampling. As a consequence, the model explains the constant-stepsize error floor through the interaction between Markovian long-run covariance and the contraction geometry of the projected Bellman operator.
Adaptive optimizers combining preconditioning, momentum, and weight decay (Adam and AdamW) are, under Polyak-Ruppert averaging, candidate engines for one-pass inference. Does the averaged iterate keep the classical Polyak-Ruppert central limit theorem (CLT), with sandwich covariance $H^{-1}SH^{-1}$ (Hessian $H$, gradient covariance $S$), under momentum and non-convergent preconditioning? The preconditioner-only analysis does not carry over: with momentum the canonical decomposition collapses to a tautology. Treating the augmented state (iterate, momentum buffer) as a time-varying linear stochastic approximation (SA), we prove (under local stabilization) positive drift stability, a non-autonomous Polyak-Ruppert CLT, and a projection identity. The upshot: the iterate-marginal covariance is exactly the plain stochastic gradient descent (SGD) sandwich $H^{-1}SH^{-1}$, so the adaptivity is asymptotically invisible. This holds for SA-Adam (sub-linearly vanishing momentum gain, $γ\in(α,1)$; the sub-linear regime is essential), not constant-$β$ deployed Adam. Coupled $L_2$ weight decay yields the ridge-penalized sandwich, extending one-pass inference to regularized problems.