Larkin Liu

Larkin Liu

We investigate Nash equilibrium learning in a competitive Markov Game (MG) environment, where multiple agents compete, and multiple Nash equilibria can exist. In particular, for an oligopolistic dynamic pricing environment, exact Nash equilibria are difficult to obtain due to the curse-of-dimensionality. We develop a new model-free method to find approximate Nash equilibria. Gradient-free black box optimization is then applied to estimate $ε$, the maximum reward advantage of an agent unilaterally deviating from any joint policy, and to also estimate the $ε$-minimizing policy for any given state. The policy-$ε$ correspondence and the state to $ε$-minimizing policy are represented by neural networks, the latter being the Nash Policy Net. During batch update, we perform Nash Q learning on the system, by adjusting the action probabilities using the Nash Policy Net. We demonstrate that an approximate Nash equilibrium can be learned, particularly in the dynamic pricing domain where exact solutions are often intractable.

Larkin Liu, Jalal Etesami

We explore the use of expert-guided bandit learning, which we refer to as online mixture-of-experts (OMoE). In this setting, given a context, a candidate committee of experts must determine how to aggregate their outputs to achieve optimal results in terms of aggregate accuracy. We propose two algorithms to address this problem. The first algorithm combines aggregate voting with UCB-driven successive elimination, efficiently pruning suboptimal exploration actions. The second algorithm employs an online weighted-majority-voting mechanism, leveraging the respective voting power of each expert proportional to their predictive power. We derive theoretical guarantees for the regret properties in the bandit setting under ideal circumstances, and empirical results are provided accordingly. As a modern study on applications, these methods are applied to the online fine-tuning of a set of expert large language models (LLMs), where after each response, the generative LLM dynamically reweighs its set of experts and/or selects the optimal committee of experts to generate the most accurate response. Our results introduce new methodologies and no-regret guarantees for combining multiple experts to improve on the performance of the an aggregate model overall.

Larkin Liu, Kashif Rasul, Yutong Chao et al.

We present a novel framework for online learning in Stackelberg general-sum games, where two agents, the leader and follower, engage in sequential turn-based interactions. At the core of this approach is a learned diffeomorphism that maps the joint action space to a smooth spherical Riemannian manifold, referred to as the Stackelberg manifold. This mapping, facilitated by neural normalizing flows, ensures the formation of tractable isoplanar subspaces, enabling efficient techniques for online learning. Leveraging the linearity of the agents' reward functions on the Stackelberg manifold, our construct allows the application of linear bandit algorithms. We then provide a rigorous theoretical basis for regret minimization on the learned manifold and establish bounds on the simple regret for learning Stackelberg equilibrium. This integration of manifold learning into game theory uncovers a previously unrecognized potential for neural normalizing flows as an effective tool for multi-agent learning. We present empirical results demonstrating the effectiveness of our approach compared to standard baselines, with applications spanning domains such as cybersecurity and economic supply chain optimization.

Larkin Liu, Shiqi Liu, Yinruo Hua et al.

Markov Decision Processes (MDPs), as a general-purpose framework, often overlook the benefits of incorporating the causal structure of the transition and reward dynamics. For a subclass of resource allocation problems, we introduce the Structurally Decomposed MDP (SD-MDP), which leverages causal disentanglement to partition an MDP's temporal causal graph into independent components. By exploiting this disentanglement, SD-MDP enables dimensionality reduction and computational efficiency gains in optimal value function estimation. We reduce the sequential optimization problem to a fractional knapsack problem with log-linear complexity $O(T \log T)$, outperforming traditional stochastic programming methods that exhibit polynomial complexity with respect to the time horizon $T$. Additionally, SD-MDP's computational advantages are independent of state-action space size, making it viable for high-dimensional spaces. Furthermore, our approach integrates seamlessly with Monte Carlo Tree Search (MCTS), achieving higher expected rewards under constrained simulation budgets while providing a vanishing simple regret bound. Empirical results demonstrate superior policy performance over benchmarks across various logistics and finance domains.

Larkin Liu, Jun Tao Luo

Flexible implementations of Monte Carlo Tree Search (MCTS), combined with domain specific knowledge and hybridization with other search algorithms, can be powerful for finding the solutions to problems in complex planning. We introduce mctreesearch4j, an MCTS implementation written as a standard JVM library following key design principles of object oriented programming. We define key class abstractions allowing the MCTS library to flexibly adapt to any well defined Markov Decision Process or turn-based adversarial game. Furthermore, our library is designed to be modular and extensible, utilizing class inheritance and generic typing to standardize custom algorithm definitions. We demonstrate that the design of the MCTS implementation provides ease of adaptation for unique heuristics and customization across varying Markov Decision Process (MDP) domains. In addition, the implementation is reasonably performant and accurate for standard MDP's. In addition, via the implementation of mctreesearch4j, the nuances of different types of MCTS algorithms are discussed.

Larkin Liu, Yu-Chung Lin, Joshua Reid

Language models based on deep neural networks and traditional stochastic modelling have become both highly functional and effective in recent times. In this work, a general survey into the two types of language modelling is conducted. We investigate the effectiveness of the Hidden Markov Model (HMM), and the Long Short-Term Memory Model (LSTM). We analyze the hidden state structures common to both models, and present an analysis on structural similarity of the hidden states, common to both HMM's and LSTM's. We compare the LSTM's predictive accuracy and hidden state output with respect to the HMM for a varying number of hidden states. In this work, we justify that the less complex HMM can serve as an appropriate approximation of the LSTM model.

Larkin Liu, Richard Downe, Joshua Reid

A survey is performed of various Multi-Armed Bandit (MAB) strategies in order to examine their performance in circumstances exhibiting non-stationary stochastic reward functions in conjunction with delayed feedback. We run several MAB simulations to simulate an online eCommerce platform for grocery pick up, optimizing for product availability. In this work, we evaluate several popular MAB strategies, such as $ε$-greedy, UCB1, and Thompson Sampling. We compare the respective performances of each MAB strategy in the context of regret minimization. We run the analysis in the scenario where the reward function is non-stationary. Furthermore, the process experiences delayed feedback, where the reward function is not immediately responsive to the arm played. We devise a new adaptive technique (AG1) tailored for non-stationary reward functions in the delayed feedback scenario. The results of the simulation show show superior performance in the context of regret minimization compared to traditional MAB strategies.