In this work, we tackle the following question: Can neural networks trained with gradient-based methods achieve the optimal statistical-computational tradeoff in learning Gaussian single-index models? Prior research has shown that any polynomial-time algorithm under the statistical query (SQ) framework requires $\Omega(d^{s^\star/2}\lor d)$ samples, where $s^\star$ is the generative exponent representing the intrinsic difficulty of learning the underlying model. However, it remains unknown whether neural networks can achieve this sample complexity. Inspired by prior techniques such as label transformation and landscape smoothing for learning single-index models, we propose a unified gradient-based algorithm for training a two-layer neural network in polynomial time. Our method is adaptable to a variety of loss and activation functions, covering a broad class of existing approaches. We show that our algorithm learns a feature representation that strongly aligns with the unknown signal $\theta^\star$, with sample complexity $\tilde O (d^{s^\star/2} \lor d)$, matching the SQ lower bound up to a polylogarithmic factor for all generative exponents $s^\star\geq 1$. Furthermore, we extend our approach to the setting where $\theta^\star$ is $k$-sparse for $k = o(\sqrt{d})$ by introducing a novel weight perturbation technique that leverages the sparsity structure. We derive a corresponding SQ lower bound of order $\tilde\Omega(k^{s^\star})$, matched by our method up to a polylogarithmic factor. Our framework, especially the weight perturbation technique, is of independent interest, and suggests potential gradient-based solutions to other problems such as sparse tensor PCA.

We study the dynamics of gradient flow for training a multi-head softmax attention model for in-context learning of multi-task linear regression. We establish the global convergence of gradient flow under suitable choices of initialization. In addition, we prove that an interesting “task allocation" phenomenon emerges during the gradient flow dynamics, where each attention head focuses on solving a single task of the multi-task model. Specifically, we prove that the gradient flow dynamics can be split into three phases — a warm-up phase where the loss decreases rather slowly and the attention heads gradually build up their inclination towards individual tasks, an emergence phase where each head selects a single task and the loss rapidly decreases, and a convergence phase where the attention parameters converge to a limit. Furthermore, we prove the optimality of gradient flow in the sense that the limiting model learned by gradient flow is on par with the best possible multi-head softmax attention model up to a constant factor. Our analysis also delineates a strict separation in terms of the prediction accuracy of ICL between single-head and multi-head attention models. The key technique for our convergence analysis is to map the gradient flow dynamics in the parameter space to a set of ordinary differential equations in the spectral domain, where the relative magnitudes of the semi-singular values of the attention weights determines task allocation. To our best knowledge, our work provides the first convergence result for the multi-head softmax attention model.

In-context learning (ICL) is a cornerstone of large language model (LLM) functionality, yet its theoretical foundations remain elusive due to the complexity of transformer architectures. In particular, most existing work only theoretically explains how the attention mechanism facilitates ICL under certain data models. It remains unclear how the other building blocks of the transformer contribute to ICL. To address this question, we study how a two-attention-layer transformer is trained to perform ICL on $n$-gram Markov chain data, where each token in the Markov chain statistically depends on the previous n tokens. We analyze a sophisticated transformer model featuring relative positional embedding, multi-head softmax attention, and a feed-forward layer with normalization. We prove that the gradient flow with respect to a cross-entropy ICL loss converges to a limiting model that performs a generalized version of the "induction head" mechanism with a learned feature, resulting from the congruous contribution of all the building blocks. Specifically, the first attention layer acts as a copier, copying past tokens within a given window to each position, and the feed-forward network with normalization acts as a selector that generates a feature vector by only looking at informationally relevant parents from the window. Finally, the second attention layer is a classifier that compares these features with the feature at the output position, and uses the resulting similarity scores to generate the desired output. Our theory is further validated by simulation experiments.

We study a heterogeneous agent macroeconomic model with an infinite number of households and firms competing in a labor market. Each household earns income and engages in consumption at each time step while aiming to maximize a concave utility subject to the underlying market conditions. The households aim to find the optimal saving strategy that maximizes their discounted cumulative utility given the market condition, while the firms determine the market conditions through maximizing corporate profit based on the household population behavior. The model captures a wide range of applications in macroeconomic studies, and we propose a data-driven reinforcement learning framework that finds the regularized competitive equilibrium of the model. The proposed algorithm enjoys theoretical guarantees in converging to the equilibrium of the market at a sub-linear rate.

We study multi-agent reinforcement learning in the setting of episodic Markov decision processes, where many agents cooperate via communication through a central server. We propose a provably efficient algorithm based on value iteration that can simultaneously allow asynchronous communication and guarantee the benefit of cooperation with low communication complexity. Under linear function approximation, we prove that our algorithm enjoys a $\tilde{\mathcal{O}}(d^{3/2}H^2\sqrt{K})$ regret upper bound with $\tilde{\mathcal{O}}(dHM^2)$ communication complexity, where $d$ is the feature dimension, $H$ is the horizon length, $M$ is the total number of agents, and $K$ is the total number of episodes. We also provide a lower bound showing that an $\Omega(dM)$ communication complexity is necessary to improve the performance through collaboration.

Linear contextual bandits represent a fundamental class of models with numerous real-world applications, and it is critical to develop algorithms that can effectively manage noise with unknown variance, ensuring provable guarantees for both worst-case constant-variance noise and deterministic reward scenarios. In this paper, we study linear contextual bandits with heteroscedastic noise and propose the first noise-adaptive Thompson sampling-style algorithm that achieves a variance-dependent regret upper bound of $\widetilde O\Big(d^{3/2} + d^{3/2} \sqrt{\sum_{t=1}^T \sigma_t^2}\Big)$, where $d$ is the dimension of the context vectors and $\sigma_t^2$ is the variance of the reward in round $t$. This recovers the existing $\widetilde O(d^{3/2}\sqrt{T})$ regret guarantee in the constant-variance regime and further improves to $\widetilde O(d^{3/2})$ in the deterministic regime, thus achieving a smooth interpolation in between. Our approach utilizes a stratified sampling procedure to overcome the too-conservative optimism in the linear Thompson sampling algorithm for linear contextual bandits.

We study the stochastic shortest path (SSP) problem in reinforcement learning with linear function approximation, where the transition kernel is represented as a linear mixture of unknown models. We call this class of SSP problems as linear mixture SSPs. We propose a novel algorithm with Hoeffding-type confidence sets for learning the linear mixture SSP, which can attain an $\tilde{\mathcal{O}}(d B_{\star}^{1.5}\sqrt{K/c_{\min}})$ regret. Here $K$ is the number of episodes, $d$ is the dimension of the feature mapping in the mixture model, $B_{\star}$ bounds the expected cumulative cost of the optimal policy, and $c_{\min}>0$ is the lower bound of the cost function. Our algorithm also applies to the case when $c_{\min} = 0$, and an $\tilde{\mathcal{O}}(K^{2/3})$ regret is guaranteed. To the best of our knowledge, this is the first algorithm with a sublinear regret guarantee for learning linear mixture SSP. Moreover, we design a refined Bernstein-type confidence set and propose an improved algorithm, which provably achieves an $\tilde{\mathcal{O}}(d B_{\star}\sqrt{K/c_{\min}})$ regret. In complement to the regret upper bounds, we also prove a lower bound of $\Omega(dB_{\star} \sqrt{K})$. Hence, our improved algorithm matches the lower bound up to a $1/\sqrt{c_{\min}}$ factor and poly-logarithmic factors, achieving a near-optimal regret guarantee.

We study federated contextual linear bandits, where $M$ agents cooperate with each other to solve a global contextual linear bandit problem with the help of a central server. We consider the asynchronous setting, where all agents work independently and the communication between one agent and the server will not trigger other agents' communication. We propose a simple algorithm named FedLinUCB based on the principle of optimism. We prove that the regret of FedLinUCB is bounded by $\widetilde{\mathcal{O}}(d\sqrt{\sum_{m=1}^M T_m})$ and the communication complexity is $\widetilde{O}(dM^2)$, where $d$ is the dimension of the contextual vector and $T_m$ is the total number of interactions with the environment by agent $m$. To the best of our knowledge, this is the first provably efficient algorithm that allows fully asynchronous communication for federated linear bandits, while achieving the same regret guarantee as in the single-agent setting.

We study a Markov matching market involving a planner and a set of strategic agents on the two sides of the market.At each step, the agents are presented with a dynamical context, where the contexts determine the utilities. The planner controls the transition of the contexts to maximize the cumulative social welfare, while the agents aim to find a myopic stable matching at each step. Such a setting captures a range of applications including ridesharing platforms. We formalize the problem by proposing a reinforcement learning framework that integrates optimistic value iteration with maximum weight matching. The proposed algorithm addresses the coupled challenges of sequential exploration, matching stability, and function approximation. We prove that the algorithm achieves sublinear regret.

We study the off-policy evaluation (OPE) problem in reinforcement learning with linear function approximation, which aims to estimate the value function of a target policy based on the offline data collected by a behavior policy. We propose to incorporate the variance information of the value function to improve the sample efficiency of OPE. More specifically, for time-inhomogeneous episodic linear Markov decision processes (MDPs), we propose an algorithm, \texttt{VA-OPE}, which uses the estimated variance of the value function to reweight the Bellman residual in Fitted Q-Iteration. We show that our algorithm achieves a tighter error bound than the best-known result. We also provide a fine-grained characterization of the distribution shift between the behavior policy and the target policy. Extensive numerical experiments corroborate our theory.

We study reinforcement learning (RL) with linear function approximation under the adaptivity constraint. We consider two popular limited adaptivity models: the batch learning model and the rare policy switch model, and propose two efficient online RL algorithms for episodic linear Markov decision processes, where the transition probability and the reward function can be represented as a linear function of some known feature mapping. In specific, for the batch learning model, our proposed LSVI-UCB-Batch algorithm achieves an $\tilde O(\sqrt{d^3H^3T} + dHT/B)$ regret, where $d$ is the dimension of the feature mapping, $H$ is the episode length, $T$ is the number of interactions and $B$ is the number of batches. Our result suggests that it suffices to use only $\sqrt{T/dH}$ batches to obtain $\tilde O(\sqrt{d^3H^3T})$ regret. For the rare policy switch model, our proposed LSVI-UCB-RareSwitch algorithm enjoys an $\tilde O(\sqrt{d^3H^3T[1+T/(dH)]^{dH/B}})$ regret, which implies that $dH\log T$ policy switches suffice to obtain the $\tilde O(\sqrt{d^3H^3T})$ regret. Our algorithms achieve the same regret as the LSVI-UCB algorithm \citep{jin2020provably}, yet with a substantially smaller amount of adaptivity. We also establish a lower bound for the batch learning model, which suggests that the dependency on $B$ in our regret bound is tight.

We present a new framework to analyze accelerated stochastic mirror descent through the lens of continuous-time stochastic dynamic systems. It enables us to design new algorithms, and perform a unified and simple analysis of the convergence rates of these algorithms. More specifically, under this framework, we provide a Lyapunov function based analysis for the continuous-time stochastic dynamics, as well as several new discrete-time algorithms derived from the continuous-time dynamics. We show that for general convex objective functions, the derived discrete-time algorithms attain the optimal convergence rate. Empirical experiments corroborate our theory.

We provide a second-order stochastic differential equation (SDE), which characterizes the continuous-time dynamics of accelerated stochastic mirror descent (ASMD) for strongly convex functions. This SDE plays a central role in designing new discrete-time ASMD algorithms via numerical discretization, and providing neat analyses of their convergence rates based on Lyapunov functions. Our results suggest that the only existing ASMD algorithm, namely, AC-SA proposed in Ghadimi & Lan (2012) is one instance of its kind, and we can actually derive new instances of ASMD with fewer tuning parameters. This sheds light on revisiting accelerated stochastic optimization through the lens of SDEs, which can lead to a better understanding of acceleration in stochastic optimization, as well as new simpler algorithms. Numerical experiments on both synthetic and real data support our theory.