A Theoretical Study of (Hyper) Self-Attention through the Lens of Interactions: Representation, Training, Generalization

Self-attention has emerged as a core component of modern neural architectures, yet its theoretical underpinnings remain elusive. In this paper, we study self-attention through the lens of interacting entities, ranging from agents in multi-agent reinforcement learning to alleles in genetic sequences, and show that a single layer linear self-attention can efficiently represent, learn, and generalize functions capturing pairwise interactions, including out-of-distribution scenarios. Our analysis reveals that self-attention acts as a mutual interaction learner under minimal assumptions on the diversity of interaction patterns observed during training, thereby encompassing a wide variety of real-world domains. In addition, we validate our theoretical insights through experiments demonstrating that self-attention learns interaction functions and generalizes across both population distributions and out-of-distribution scenarios. Building on our theories, we introduce HyperFeatureAttention, a novel neural network module designed to learn couplings of different feature-level interactions between entities. Furthermore, we propose HyperAttention, a new module that extends beyond pairwise interactions to capture multi-entity dependencies, such as three-way, four-way, or general n-way interactions.

Thinking Beyond Visibility: A Near-Optimal Policy Framework for Locally Interdependent Multi-Agent MDPs

Decentralized Partially Observable Markov Decision Processes (Dec-POMDPs) are known to be NEXP-Complete and intractable to solve. However, for problems such as cooperative navigation, obstacle avoidance, and formation control, basic assumptions can be made about local visibility and local dependencies. The work DeWeese and Qu 2024 formalized these assumptions in the construction of the Locally Interdependent Multi-Agent MDP. In this setting, it establishes three closed-form policies that are tractable to compute in various situations and are exponentially close to optimal with respect to visibility. However, it is also shown that these solutions can have poor performance when the visibility is small and fixed, often getting stuck during simulations due to the so called "Penalty Jittering" phenomenon. In this work, we establish the Extended Cutoff Policy Class which is, to the best of our knowledge, the first non-trivial class of near optimal closed-form partially observable policies that are exponentially close to optimal with respect to the visibility for any Locally Interdependent Multi-Agent MDP. These policies are able to remember agents beyond their visibilities which allows them to perform significantly better in many small and fixed visibility settings, resolve Penalty Jittering occurrences, and under certain circumstances guarantee fully observable joint optimal behavior despite the partial observability. We also propose a generalized form of the Locally Interdependent Multi-Agent MDP that allows for transition dependence and extended reward dependence, then replicate our theoretical results in this setting.

Natural Policy Gradient for Average Reward Non-Stationary RL

We consider the problem of non-stationary reinforcement learning (RL) in the infinite-horizon average-reward setting. We model it by a Markov Decision Process with time-varying rewards and transition probabilities, with a variation budget of $\Delta_T$. Existing non-stationary RL algorithms focus on model-based and model-free value-based methods. Policy-based methods despite their flexibility in practice are not theoretically well understood in non-stationary RL. We propose and analyze the first model-free policy-based algorithm, Non-Stationary Natural Actor-Critic (NS-NAC), a policy gradient method with a restart based exploration for change and a novel interpretation of learning rates as adapting factors. Further, we present a bandit-over-RL based parameter-free algorithm BORL-NS-NAC that does not require prior knowledge of the variation budget $\Delta_T$. We present a dynamic regret of $\tilde{\mathscr O}(|S|^{1/2}|A|^{1/2}\Delta_T^{1/6}T^{5/6})$ for both algorithms, where $T$ is the time horizon, and $|S|$, $|A|$ are the sizes of the state and action spaces. The regret analysis leverages a novel adaptation of the Lyapunov function analysis of NAC to dynamic environments and characterizes the effects of simultaneous updates in policy, value function estimate and changes in the environment.

Whole-Body Model-Predictive Control of Legged Robots with MuJoCo

We demonstrate the surprising real-world effectiveness of a very simple approach to whole-body model-predictive control (MPC) of quadruped and humanoid robots: the iterative LQR (iLQR) algorithm with MuJoCo dynamics and finite-difference approximated derivatives. Building upon the previous success of model-based behavior synthesis and control of locomotion and manipulation tasks with MuJoCo in simulation, we show that these policies can easily generalize to the real world with few sim-to-real considerations. Our baseline method achieves real-time whole-body MPC on a variety of hardware experiments, including dynamic quadruped locomotion, quadruped walking on two legs, and full-sized humanoid bipedal locomotion. We hope this easy-to-reproduce hardware baseline lowers the barrier to entry for real-world whole-body MPC research and contributes to accelerating research velocity in the community. Our code and experiment videos will be available online at:https://johnzhang3.github.io/mujoco_ilqr

ASAP: Aligning Simulation and Real-World Physics for Learning Agile Humanoid Whole-Body Skills

Humanoid robots hold the potential for unparalleled versatility in performing human-like, whole-body skills. However, achieving agile and coordinated whole-body motions remains a significant challenge due to the dynamics mismatch between simulation and the real world. Existing approaches, such as system identification (SysID) and domain randomization (DR) methods, often rely on labor-intensive parameter tuning or result in overly conservative policies that sacrifice agility. In this paper, we present ASAP (Aligning Simulation and Real-World Physics), a two-stage framework designed to tackle the dynamics mismatch and enable agile humanoid whole-body skills. In the first stage, we pre-train motion tracking policies in simulation using retargeted human motion data. In the second stage, we deploy the policies in the real world and collect real-world data to train a delta (residual) action model that compensates for the dynamics mismatch. Then, ASAP fine-tunes pre-trained policies with the delta action model integrated into the simulator to align effectively with real-world dynamics. We evaluate ASAP across three transfer scenarios: IsaacGym to IsaacSim, IsaacGym to Genesis, and IsaacGym to the real-world Unitree G1 humanoid robot. Our approach significantly improves agility and whole-body coordination across various dynamic motions, reducing tracking error compared to SysID, DR, and delta dynamics learning baselines. ASAP enables highly agile motions that were previously difficult to achieve, demonstrating the potential of delta action learning in bridging simulation and real-world dynamics. These results suggest a promising sim-to-real direction for developing more expressive and agile humanoids.

Mean-Field Sampling for Cooperative Multi-Agent Reinforcement Learning

Designing efficient algorithms for multi-agent reinforcement learning (MARL) is fundamentally challenging due to the fact that the size of the joint state and action spaces are exponentially large in the number of agents. These difficulties are exacerbated when balancing sequential global decision-making with local agent interactions. In this work, we propose a new algorithm \texttt{SUBSAMPLE-MFQ} (\textbf{Subsample}-\textbf{M}ean-\textbf{F}ield-\textbf{Q}-learning) and a decentralized randomized policy for a system with $n$ agents. For $k\leq n$, our algorithm system learns a policy for the system in time polynomial in $k$. We show that this learned policy converges to the optimal policy in the order of $\tilde{O}(1/\sqrt{k})$ as the number of subsampled agents $k$ increases. We validate our method empirically on Gaussian squeeze and global exploration settings.

Predictive Control and Regret Analysis of Non-Stationary MDP with Look-ahead Information

Policy design in non-stationary Markov Decision Processes (MDPs) is inherently challenging due to the complexities introduced by time-varying system transition and reward, which make it difficult for learners to determine the optimal actions for maximizing cumulative future rewards. Fortunately, in many practical applications, such as energy systems, look-ahead predictions are available, including forecasts for renewable energy generation and demand. In this paper, we leverage these look-ahead predictions and propose an algorithm designed to achieve low regret in non-stationary MDPs by incorporating such predictions. Our theoretical analysis demonstrates that, under certain assumptions, the regret decreases exponentially as the look-ahead window expands. When the system prediction is subject to error, the regret does not explode even if the prediction error grows sub-exponentially as a function of the prediction horizon. We validate our approach through simulations, confirming the efficacy of our algorithm in non-stationary environments.

Locally Interdependent Multi-Agent MDP: Theoretical Framework for Decentralized Agents with Dynamic Dependencies

Many multi-agent systems in practice are decentralized and have dynamically varying dependencies. There has been a lack of attempts in the literature to analyze these systems theoretically. In this paper, we propose and theoretically analyze a decentralized model with dynamically varying dependencies called the Locally Interdependent Multi-Agent MDP. This model can represent problems in many disparate domains such as cooperative navigation, obstacle avoidance, and formation control. Despite the intractability that general partially observable multi-agent systems suffer from, we propose three closed-form policies that are theoretically near-optimal in this setting and can be scalable to compute and store. Consequentially, we reveal a fundamental property of Locally Interdependent Multi-Agent MDP's that the partially observable decentralized solution is exponentially close to the fully observable solution with respect to the visibility radius. We then discuss extensions of our closed-form policies to further improve tractability. We conclude by providing simulations to investigate some long horizon behaviors of our closed-form policies.

Learning to Stabilize Unknown LTI Systems on a Single Trajectory under Stochastic Noise

We study the problem of learning to stabilize unknown noisy Linear Time-Invariant (LTI) systems on a single trajectory. It is well known in the literature that the learn-to-stabilize problem suffers from exponential blow-up in which the state norm blows up in the order of $\Theta(2^n)$ where $n$ is the state space dimension. This blow-up is due to the open-loop instability when exploring the $n$-dimensional state space. To address this issue, we develop a novel algorithm that decouples the unstable subspace of the LTI system from the stable subspace, based on which the algorithm only explores and stabilizes the unstable subspace, the dimension of which can be much smaller than $n$. With a new singular-value-decomposition(SVD)-based analytical framework, we prove that the system is stabilized before the state norm reaches $2^{O(k \log n)}$, where $k$ is the dimension of the unstable subspace. Critically, this bound avoids exponential blow-up in state dimension in the order of $\Theta(2^n)$ as in the previous works, and to the best of our knowledge, this is the first paper to avoid exponential blow-up in dimension for stabilizing LTI systems with noise.

Efficient Reinforcement Learning for Global Decision Making in the Presence of Local Agents at Scale

We study reinforcement learning for global decision-making in the presence of many local agents, where the global decision-maker makes decisions affecting all local agents, and the objective is to learn a policy that maximizes the rewards of both the global and the local agents. Such problems find many applications, e.g. demand response, EV charging, queueing, etc. In this setting, scalability has been a long-standing challenge due to the size of the state/action space which can be exponential in the number of agents. This work proposes the SUB-SAMPLE-Q algorithm where the global agent subsamples $k\leq n$ local agents to compute an optimal policy in time that is only exponential in $k$, providing an exponential speedup from standard methods that are exponential in $n$. We show that the learned policy converges to the optimal policy in the order of $\tilde{O}(1/\sqrt{k}+\epsilon_{k,m})$ as the number of sub-sampled agents $k$ increases, where $\epsilon_{k,m}$ is the Bellman noise. We also conduct numerical simulations in a demand-response setting and a queueing setting.