Asymptotically Optimal Regret for Reinforcement Learning without Horizon Dependence

We study horizon-free regret minimization for finite-horizon time-homogeneous tabular Markov decision processes with $S$ states, $A$ actions, horizon $H$, and per-trajectory total reward bounded by $1$. We propose a new algorithm and prove a regret upper bound \[\tilde O(\sqrt{SAK}+S^8A^3)\] with failure probability $δ$, where $K$ is the number of episodes and $\tilde O(\cdot)$ hides $\mathsf{poly}\log(S,A,K,1/δ)$. Thus, the regret is $H$-free and asymptotically optimal, matching the contextual-bandit lower bound $Ω(\sqrt{SAK})$ up to logarithmic factors. This completely removes the $\log H$ dependence from the previous $\tilde O(\sqrt{SAK\log H}+S^2A\log H)$ guarantee of Zhang et al. (2021), and drastically improves the prior best horizon-free regret $\tilde O(\sqrt{S^9A^3K})$ of Zhang et al. (2022) asymptotically. The main technical difficulty is that the optimal value functions $\{V_h^*\}_{h=1}^H$ are time-inhomogeneous even though the transition kernel is time-homogeneous. A direct union bound over all value functions typically incurs an additional $\min\{\log H,S\}$ factor. We avoid this factor by (i) exploiting the monotonicity of $V_h^*$ in $h$ and (ii) non-trivially projecting the value functions onto an $S$-dimensional grid. Our analysis relies on three additional ingredients. First, we introduce a horizon-truncation argument that enables reward-based exploration and removes the cost of a separate reward-free exploration phase. Second, we design a cutting bonus that preserves both optimism and the monotonicity needed for planning. Third, we prove a new bound on total deviation for time-homogeneous MDPs, which controls the clipped variance terms in the cutting bonus with adjustable polynomial dependence on $S$ and without any dependence on $H$. Together, these tools yield an asymptotically optimal horizon-free regret guarantee.

Two-Layer Linear Auto-Regressive Models Estimate Latent States

Auto-regressive models have emerged as powerful tools for sequential data, from language to video. Understanding how and why these models learn latent representations remains an open theoretical question. In this work, we demonstrate that when trained by empirical risk minimization on data from partially observed linear dynamical systems, two-layer linear auto-regressive models naturally learn to approximate Kalman filtering. In particular, we show that the learned hidden representation coincides, up to a similarity transformation, with the state estimates produced by the optimal (Kalman) filter, even though the model has no explicit knowledge of the underlying dynamics or state. The result follows from three main insights. First, we establish that the Kalman filter is well approximated by an auto-regressive model with bounded truncation error. Second, we show that despite non-convexity, the two-layer optimization landscape is benign, i.e., all stationary points are either strict saddles or global minima. Finally, as our main contributions, we provide finite-sample guarantees on prediction error, parameter estimation error, and latent state recovery. Numerical simulations support the theoretical results and demonstrate that the latent representations of auto-regressive models recover state estimates.

Local linear convergence of gradient methods for overparameterized Gaussian mixtures

We study the problem of learning Gaussian mixture models under overparameterization. Prior work has shown that while overparameterization is essential for avoiding spurious local optima and enables global recovery of the ground-truth model using the gradient-EM (expectation-maximization) algorithm, it can dramatically slow down the local rate of convergence. Under certain assumptions on the mixture weights, we show that a standard divergence measure minimized by statistical learning procedures possesses a manifold of slow growth on which the well-known Polyak stepsize reduces the loss geometrically, and design a gradient-based method that converges to minimizers at a locally linear rate. Additionally, we show that our method converges to nearly optimal solutions -- up to a natural misspecification threshold -- for mixtures with arbitrary weights. At a high level, the method alternates between several "short" gradient descent steps that approach the manifold and "long" Polyak steps that contract the distance to minimizers. Our results suggest that slow convergence is not an intrinsic challenge of overparameterization, but can be overcome by exploiting the favorable structure of the loss landscape.

Average Gradient Outer Product in kernel regression provably recovers the central subspace for multi-index models

We study a prototypical situation when a learned predictor can discover useful low-dimensional structure in data, while using fewer samples than are needed for accurate prediction. Specifically, we consider the problem of recovering a multi-index polynomial $f^*(x)=h(Ux)$, with $U\in\mathbb{R}^{r\times d}$ and $r\ll d$, from finitely many data/label pairs. Importantly, the target function depends on input $x$ only through the projection onto an unknown $r$-dimensional central subspace. The algorithm we analyze is appealingly simple: fit kernel ridge regression (KRR) to the data and compute the Average Gradient Outer Product (AGOP) from the fitted predictor. Our main results show that under reasonable assumptions the top $r$-dimensional eigenspace of AGOP provably recovers the central subspace, even in regimes when the prediction error remains large. Specifically, if the target function $f^*$ has degree $p^*$, it is known that $n\asymp d^{p^*}$ samples are necessary for KRR to achieve accurate prediction. In contrast, we show that if a low degree $p$ component of $f^*$ already carries all relevant directions for prediction, subspace recovery occurs in the much lower sample regime $n\asymp d^{p+δ}$ for any $δ\in(0,1)$. Our results thus demonstrate a separation between prediction and representation, and provide an explanation for why iterative kernel methods such as Recursive Feature Machines (RFM) can be sample-efficient in practice.

Dynamics of Learning under User Choice: Overspecialization and Peer-Model Probing

In many economically relevant contexts where machine learning is deployed, multiple platforms obtain data from the same pool of users, each of whom selects the platform that best serves them. Prior work in this setting focuses exclusively on the "local" losses of learners on the distribution of data that they observe. We find that there exist instances where learners who use existing algorithms almost surely converge to models with arbitrarily poor global performance, even when models with low full-population loss exist. This happens through a feedback-induced mechanism, which we call the overspecialization trap: as learners optimize for users who already prefer them, they become less attractive to users outside this base, which further restricts the data they observe. Inspired by the recent use of knowledge distillation in modern ML, we propose an algorithm that allows learners to "probe" the predictions of peer models, enabling them to learn about users who do not select them. Our analysis characterizes when probing succeeds: this procedure converges almost surely to a stationary point with bounded full-population risk when probing sources are sufficiently informative, e.g., a known market leader or a majority of peers with good global performance. We verify our findings with semi-synthetic experiments on the MovieLens, Census, and Amazon Sentiment datasets.

Cold-Start Personalization via Training-Free Priors from Structured World Models

Cold-start personalization requires inferring user preferences through interaction when no user-specific historical data is available. The core challenge is a routing problem: each task admits dozens of preference dimensions, yet individual users care about only a few, and which ones matter depends on who is asking. With a limited question budget, asking without structure will miss the dimensions that matter. Reinforcement learning is the natural formulation, but in multi-turn settings its terminal reward fails to exploit the factored, per-criterion structure of preference data, and in practice learned policies collapse to static question sequences that ignore user responses. We propose decomposing cold-start elicitation into offline structure learning and online Bayesian inference. Pep (Preference Elicitation with Priors) learns a structured world model of preference correlations offline from complete profiles, then performs training-free Bayesian inference online to select informative questions and predict complete preference profiles, including dimensions never asked about. The framework is modular across downstream solvers and requires only simple belief models. Across medical, mathematical, social, and commonsense reasoning, Pep achieves 80.8% alignment between generated responses and users' stated preferences versus 68.5% for RL, with 3-5x fewer interactions. When two users give different answers to the same question, Pep changes its follow-up 39-62% of the time versus 0-28% for RL. It does so with ~10K parameters versus 8B for RL, showing that the bottleneck in cold-start elicitation is the capability to exploit the factored structure of preference data.

Unregularized Linear Convergence in Zero-Sum Game from Preference Feedback

Aligning large language models (LLMs) with human preferences has proven effective for enhancing model capabilities, yet standard preference modeling using the Bradley-Terry model assumes transitivity, overlooking the inherent complexity of human population preferences. Nash learning from human feedback (NLHF) addresses this by framing non-transitive preferences as a two-player zero-sum game, where alignment reduces to finding the Nash equilibrium (NE). However, existing algorithms typically rely on regularization, incurring unavoidable bias when computing the duality gap in the original game. In this work, we provide the first convergence guarantee for Optimistic Multiplicative Weights Update ($\mathtt{OMWU}$) in NLHF, showing that it achieves last-iterate linear convergence after a burn-in phase whenever an NE with full support exists, with an instance-dependent linear convergence rate to the original NE, measured by duality gaps. Compared to prior results in Wei et al. (2020), we do not require the assumption of NE uniqueness. Our analysis identifies a novel marginal convergence behavior, where the probability of rarely played actions grows exponentially from exponentially small values, enabling exponentially better dependence on instance-dependent constants than prior results. Experiments corroborate the theoretical strengths of $\mathtt{OMWU}$ in both tabular and neural policy classes, demonstrating its potential for LLM applications.

Global Convergence of Four-Layer Matrix Factorization under Random Initialization

Gradient descent dynamics on the deep matrix factorization problem is extensively studied as a simplified theoretical model for deep neural networks. Although the convergence theory for two-layer matrix factorization is well-established, no global convergence guarantee for general deep matrix factorization under random initialization has been established to date. To address this gap, we provide a polynomial-time global convergence guarantee for randomly initialized gradient descent on four-layer matrix factorization, given certain conditions on the target matrix and a standard balanced regularization term. Our analysis employs new techniques to show saddle-avoidance properties of gradient decent dynamics, and extends previous theories to characterize the change in eigenvalues of layer weights.

Global Convergence of Gradient EM for Over-Parameterized Gaussian Mixtures

Learning Gaussian Mixture Models (GMMs) is a fundamental problem in machine learning, with the Expectation-Maximization (EM) algorithm and its popular variant gradient EM being arguably the most widely used algorithms in practice. In the exact-parameterized setting, where both the ground truth GMM and the learning model have the same number of components $m$, a vast line of work has aimed to establish rigorous recovery guarantees for EM. However, global convergence has only been proven for the case of $m=2$, and EM is known to fail to recover the ground truth when $m\geq 3$. In this paper, we consider the $\textit{over-parameterized}$ setting, where the learning model uses $n>m$ components to fit an $m$-component ground truth GMM. In contrast to the exact-parameterized case, we provide a rigorous global convergence guarantee for gradient EM. Specifically, for any well separated GMMs in general position, we prove that with only mild over-parameterization $n = \Omega(m\log m)$, randomly initialized gradient EM converges globally to the ground truth at a polynomial rate with polynomial samples. Our analysis proceeds in two stages and introduces a suite of novel tools for Gaussian Mixture analysis. We use Hermite polynomials to study the dynamics of gradient EM and employ tensor decomposition to characterize the geometric landscape of the likelihood loss. This is the first global convergence and recovery result for EM or Gradient EM beyond the special case of $m=2$.

Sharp Gap-Dependent Variance-Aware Regret Bounds for Tabular MDPs

We consider the gap-dependent regret bounds for episodic MDPs. We show that the Monotonic Value Propagation (MVP) algorithm achieves a variance-aware gap-dependent regret bound of $$\tilde{O}\left(\left(\sum_{\Delta_h(s,a)>0} \frac{H^2 \log K \land \mathtt{Var}_{\max}^{\text{c}}}{\Delta_h(s,a)} +\sum_{\Delta_h(s,a)=0}\frac{ H^2 \land \mathtt{Var}_{\max}^{\text{c}}}{\Delta_{\mathrm{min}}} + SAH^4 (S \lor H) \right) \log K\right),$$ where $H$ is the planning horizon, $S$ is the number of states, $A$ is the number of actions, and $K$ is the number of episodes. Here, $\Delta_h(s,a) =V_h^* (a) - Q_h^* (s, a)$ represents the suboptimality gap and $\Delta_{\mathrm{min}} := \min_{\Delta_h (s,a) > 0} \Delta_h(s,a)$. The term $\mathtt{Var}_{\max}^{\text{c}}$ denotes the maximum conditional total variance, calculated as the maximum over all $(\pi, h, s)$ tuples of the expected total variance under policy $\pi$ conditioned on trajectories visiting state $s$ at step $h$. $\mathtt{Var}_{\max}^{\text{c}}$ characterizes the maximum randomness encountered when learning any $(h, s)$ pair. Our result stems from a novel analysis of the weighted sum of the suboptimality gap and can be potentially adapted for other algorithms. To complement the study, we establish a lower bound of $$\Omega \left( \sum_{\Delta_h(s,a)>0} \frac{H^2 \land \mathtt{Var}_{\max}^{\text{c}}}{\Delta_h(s,a)}\cdot \log K\right),$$ demonstrating the necessity of dependence on $\mathtt{Var}_{\max}^{\text{c}}$ even when the maximum unconditional total variance (without conditioning on $(h, s)$) approaches zero.