A Mean-Field Analysis of Neural Gradient Descent-Ascent: Applications to Functional Conditional Moment Equations

We study minimax optimization problems defined over infinite-dimensional function classes. In particular, we restrict the functions to the class of overparameterized two-layer neural networks and study (i) the convergence of the gradient descent-ascent algorithm and (ii) the representation learning of the neural network. As an initial step, we consider the minimax optimization problem stemming from estimating a functional equation defined by conditional expectations via adversarial estimation, where the objective function is quadratic in the functional space. For this problem, we establish convergence under the mean-field regime by considering the continuous-time and infinite-width limit of the optimization dynamics. Under this regime, gradient descent-ascent corresponds to a Wasserstein gradient flow over the space of probability measures defined over the space of neural network parameters. We prove that the Wasserstein gradient flow converges globally to a stationary point of the minimax objective at a $\mathcal{O}(T^{-1} + \alpha^{-1} ) $ sublinear rate, and additionally finds the solution to the functional equation when the regularizer of the minimax objective is strongly convex. Here $T$ denotes the time and $\alpha$ is a scaling parameter of the neural network. In terms of representation learning, our results show that the feature representation induced by the neural networks is allowed to deviate from the initial one by the magnitude of $\mathcal{O}(\alpha^{-1})$, measured in terms of the Wasserstein distance. Finally, we apply our general results to concrete examples including policy evaluation, nonparametric instrumental variable regression, and asset pricing.

Advancing Object Goal Navigation Through LLM-enhanced Object Affinities Transfer

In object goal navigation, agents navigate towards objects identified by category labels using visual and spatial information. Previously, solely network-based methods typically rely on historical data for object affinities estimation, lacking adaptability to new environments and unseen targets. Simultaneously, employing Large Language Models (LLMs) for navigation as either planners or agents, though offering a broad knowledge base, is cost-inefficient and lacks targeted historical experience. Addressing these challenges, we present the LLM-enhanced Object Affinities Transfer (LOAT) framework, integrating LLM-derived object semantics with network-based approaches to leverage experiential object affinities, thus improving adaptability in unfamiliar settings. LOAT employs a dual-module strategy: a generalized affinities module for accessing LLMs' vast knowledge and an experiential affinities module for applying learned object semantic relationships, complemented by a dynamic fusion module harmonizing these information sources based on temporal context. The resulting scores activate semantic maps before feeding into downstream policies, enhancing navigation systems with context-aware inputs. Our evaluations in AI2-THOR and Habitat simulators demonstrate improvements in both navigation success rates and efficiency, validating the LOAT's efficacy in integrating LLM insights for improved object goal navigation.

Can Large Language Models Play Games? A Case Study of A Self-Play Approach

Large Language Models (LLMs) harness extensive data from the Internet, storing a broad spectrum of prior knowledge. While LLMs have proven beneficial as decision-making aids, their reliability is hampered by limitations in reasoning, hallucination phenomenon, and so on. On the other hand, Monte-Carlo Tree Search (MCTS) is a heuristic search algorithm that provides reliable decision-making solutions, achieved through recursive rollouts and self-play. However, the effectiveness of MCTS relies heavily on heuristic pruning and external value functions, particularly in complex decision scenarios. This work introduces an innovative approach that bolsters LLMs with MCTS self-play to efficiently resolve deterministic turn-based zero-sum games (DTZG), such as chess and go, without the need for additional training. Specifically, we utilize LLMs as both action pruners and proxies for value functions without the need for additional training. We theoretically prove that the suboptimality of the estimated value in our proposed method scales with $\tilde{\mathcal O}\Bigl(\frac{|\tilde {\mathcal A}|}{\sqrt{N}} + \epsilon_\mathrm{pruner} + \epsilon_\mathrm{critic}\Bigr)$, where \(N\) is the number of simulations, $|\tilde {\mathcal A}|$ is the cardinality of the pruned action space by LLM, and $\epsilon_\mathrm{pruner}$ and $\epsilon_\mathrm{critic}$ quantify the errors incurred by adopting LLMs as action space pruner and value function proxy, respectively. Our experiments in chess and go demonstrate the capability of our method to address challenges beyond the scope of MCTS and improve the performance of the directly application of LLMs.

How Can LLM Guide RL? A Value-Based Approach

Reinforcement learning (RL) has become the de facto standard practice for sequential decision-making problems by improving future acting policies with feedback. However, RL algorithms may require extensive trial-and-error interactions to collect useful feedback for improvement. On the other hand, recent developments in large language models (LLMs) have showcased impressive capabilities in language understanding and generation, yet they fall short in exploration and self-improvement capabilities for planning tasks, lacking the ability to autonomously refine their responses based on feedback. Therefore, in this paper, we study how the policy prior provided by the LLM can enhance the sample efficiency of RL algorithms. Specifically, we develop an algorithm named LINVIT that incorporates LLM guidance as a regularization factor in value-based RL, leading to significant reductions in the amount of data needed for learning, particularly when the difference between the ideal policy and the LLM-informed policy is small, which suggests that the initial policy is close to optimal, reducing the need for further exploration. Additionally, we present a practical algorithm SLINVIT that simplifies the construction of the value function and employs subgoals to reduce the search complexity. Our experiments across three interactive environments ALFWorld, InterCode, and BlocksWorld demonstrate that our method achieves state-of-the-art success rates and also surpasses previous RL and LLM approaches in terms of sample efficiency. Our code is available at https://github.com/agentification/Language-Integrated-VI.

Double Duality: Variational Primal-Dual Policy Optimization for Constrained Reinforcement Learning

We study the Constrained Convex Markov Decision Process (MDP), where the goal is to minimize a convex functional of the visitation measure, subject to a convex constraint. Designing algorithms for a constrained convex MDP faces several challenges, including (1) handling the large state space, (2) managing the exploration/exploitation tradeoff, and (3) solving the constrained optimization where the objective and the constraint are both nonlinear functions of the visitation measure. In this work, we present a model-based algorithm, Variational Primal-Dual Policy Optimization (VPDPO), in which Lagrangian and Fenchel duality are implemented to reformulate the original constrained problem into an unconstrained primal-dual optimization. Moreover, the primal variables are updated by model-based value iteration following the principle of Optimism in the Face of Uncertainty (OFU), while the dual variables are updated by gradient ascent. Moreover, by embedding the visitation measure into a finite-dimensional space, we can handle large state spaces by incorporating function approximation. Two notable examples are (1) Kernelized Nonlinear Regulators and (2) Low-rank MDPs. We prove that with an optimistic planning oracle, our algorithm achieves sublinear regret and constraint violation in both cases and can attain the globally optimal policy of the original constrained problem.

Human-Instruction-Free LLM Self-Alignment with Limited Samples

Aligning large language models (LLMs) with human values is a vital task for LLM practitioners. Current alignment techniques have several limitations: (1) requiring a large amount of annotated data; (2) demanding heavy human involvement; (3) lacking a systematic mechanism to continuously improve. In this work, we study aligning LLMs to a new domain with limited samples (e.g. < 100). We propose an algorithm that can self-align LLMs iteratively without active human involvement. Unlike existing works, our algorithm relies on neither human-crafted instructions nor labeled rewards, significantly reducing human involvement. In addition, our algorithm can self-improve the alignment continuously. The key idea is to first retrieve high-quality samples related to the target domain and use them as In-context Learning examples to generate more samples. Then we use the self-generated samples to finetune the LLM iteratively. We show that our method can unlock the LLMs' self-generalization ability to perform alignment with near-zero human supervision. We test our algorithm on three benchmarks in safety, truthfulness, and instruction-following, and show good performance in alignment, domain adaptability, and scalability.

Sparse PCA with Oracle Property

In this paper, we study the estimation of the $k$-dimensional sparse principal subspace of covariance matrix $\Sigma$ in the high-dimensional setting. We aim to recover the oracle principal subspace solution, i.e., the principal subspace estimator obtained assuming the true support is known a priori. To this end, we propose a family of estimators based on the semidefinite relaxation of sparse PCA with novel regularizations. In particular, under a weak assumption on the magnitude of the population projection matrix, one estimator within this family exactly recovers the true support with high probability, has exact rank-$k$, and attains a $\sqrt{s/n}$ statistical rate of convergence with $s$ being the subspace sparsity level and $n$ the sample size. Compared to existing support recovery results for sparse PCA, our approach does not hinge on the spiked covariance model or the limited correlation condition. As a complement to the first estimator that enjoys the oracle property, we prove that, another estimator within the family achieves a sharper statistical rate of convergence than the standard semidefinite relaxation of sparse PCA, even when the previous assumption on the magnitude of the projection matrix is violated. We validate the theoretical results by numerical experiments on synthetic datasets.

Empowering Autonomous Driving with Large Language Models: A Safety Perspective

Autonomous Driving (AD) faces crucial hurdles for commercial launch, notably in the form of diminished public trust and safety concerns from long-tail unforeseen driving scenarios. This predicament is due to the limitation of deep neural networks in AD software, which struggle with interpretability and exhibit poor generalization capabilities in out-of-distribution and uncertain scenarios. To this end, this paper advocates for the integration of Large Language Models (LLMs) into the AD system, leveraging their robust common-sense knowledge, reasoning abilities, and human-interaction capabilities. The proposed approach deploys the LLM as an intelligent decision-maker in planning, incorporating safety verifiers for contextual safety learning to enhance overall AD performance and safety. We present results from two case studies that affirm the efficacy of our approach. We further discuss the potential integration of LLM for other AD software components including perception, prediction, and simulation. Despite the observed challenges in the case studies, the integration of LLMs is promising and beneficial for reinforcing both safety and performance in AD.

Provably Efficient High-Dimensional Bandit Learning with Batched Feedbacks

We study high-dimensional multi-armed contextual bandits with batched feedback where the $T$ steps of online interactions are divided into $L$ batches. In specific, each batch collects data according to a policy that depends on previous batches and the rewards are revealed only at the end of the batch. Such a feedback structure is popular in applications such as personalized medicine and online advertisement, where the online data often do not arrive in a fully serial manner. We consider high-dimensional and linear settings where the reward function of the bandit model admits either a sparse or low-rank structure and ask how small a number of batches are needed for a comparable performance with fully dynamic data in which $L = T$. For these settings, we design a provably sample-efficient algorithm which achieves a $ \mathcal{\tilde O}(s_0^2 \log^2 T)$ regret in the sparse case and $ \mathcal{\tilde O} ( r ^2 \log^2 T)$ regret in the low-rank case, using only $L = \mathcal{O}( \log T)$ batches. Here $s_0$ and $r$ are the sparsity and rank of the reward parameter in sparse and low-rank cases, respectively, and $ \mathcal{\tilde O}(\cdot)$ omits logarithmic factors involving the feature dimensions. In other words, our algorithm achieves regret bounds comparable to those in fully sequential setting with only $\mathcal{O}( \log T)$ batches. Our algorithm features a novel batch allocation method that adjusts the batch sizes according to the estimation accuracy within each batch and cumulative regret. Furthermore, we also conduct experiments with synthetic and real-world data to validate our theory.

A Principled Framework for Knowledge-enhanced Large Language Model

Large Language Models (LLMs) are versatile, yet they often falter in tasks requiring deep and reliable reasoning due to issues like hallucinations, limiting their applicability in critical scenarios. This paper introduces a rigorously designed framework for creating LLMs that effectively anchor knowledge and employ a closed-loop reasoning process, enhancing their capability for in-depth analysis. We dissect the framework to illustrate the contribution of each component to the LLMs' performance, offering a theoretical assurance of improved reasoning under well-defined assumptions.