arXiv subjects
Publications and source records attributed to Shengda Liu.
We define and study ordinary differential equations (ODEs) for functions valued in a Banach module $V$ over a finite-dimensional $\Bbbk$-algebra $\mathit{\Lambda}$ by using the tensor of Banach modules. Furthermore, we show that the solution space of a homogeneous linear ODE as above is shown to be a finitely generated $\mathit{\Lambda}$-submodule.
High-quality information set abstraction remains a core challenge in solving large-scale imperfect-information extensive-form games (IIEFGs)--such as no-limit Texas Hold'em--where the finite nature of spatial resources hinders solving strategies for the full game. State-of-the-art AI methods rely on pre-trained discrete clustering for abstraction, yet their hard classification irreversibly discards critical information: specifically, the quantifiable subtle differences between information sets--vital for strategy solving--thus compromising the quality of such solving. Inspired by the word embedding paradigm in natural language processing, this paper proposes the Embedding CFR algorithm, a novel approach for solving strategies in IIEFGs within an embedding space. The algorithm pre-trains and embeds the features of individual information sets into an interconnected low-dimensional continuous space, where the resulting vectors more precisely capture both the distinctions and connections between information sets. Embedding CFR introduces a strategy-solving process driven by regret accumulation and strategy updates in this embedding space, with supporting theoretical analysis verifying its ability to reduce cumulative regret. Experiments on poker show that with the same spatial overhead, Embedding CFR achieves significantly faster exploitability convergence compared to cluster-based abstraction algorithms, confirming its effectiveness. Furthermore, to our knowledge, it is the first algorithm in poker AI that pre-trains information set abstractions via low-dimensional embedding for strategy solving.
Excessive abstraction is a critical challenge in hand abstraction-a task specific to games like Texas hold'em-when solving large-scale imperfect-information games, as it impairs AI performance. This issue arises from extreme implementations of imperfect-recall abstraction, which entirely discard historical information. This paper presents KrwEmd, the first practical algorithm designed to address this problem. We first introduce the k-recall winrate feature, which not only qualitatively distinguishes signal observation infosets by leveraging both future and, crucially, historical game information, but also quantitatively captures their similarity. We then develop the KrwEmd algorithm, which clusters signal observation infosets using earth mover's distance to measure discrepancies between their features. Experimental results demonstrate that KrwEmd significantly improves AI gameplay performance compared to existing algorithms.
Hand abstraction is crucial for scaling imperfect-information games (IIGs) such as Texas Hold'em, yet progress is limited by the lack of a formal task model and by evaluations that require resource-intensive strategy solving. We introduce signal observation ordered games (SOOGs), a subclass of IIGs tailored to hold'em-style games that cleanly separates signal from player action sequences, providing a precise mathematical foundation for hand abstraction. Within this framework, we define a resolution bound-an information-theoretic upper bound on achievable performance under a given signal abstraction. Using the bound, we show that mainstream outcome-based imperfect-recall algorithms suffer substantial losses by arbitrarily discarding historical information; we formalize this behavior via potential-aware outcome Isomorphism (PAOI) and prove that PAOI characterizes their resolution bound. To overcome this limitation, we propose full-recall outcome isomorphism (FROI), which integrates historical information to raise the bound and improve policy quality. Experiments on hold'em-style benchmarks confirm that FROI consistently outperforms outcome-based imperfect-recall baselines. Our results provide a unified formal treatment of hand abstraction and practical guidance for designing higher-resolution abstractions in IIGs.
Recent breakthroughs in Large Language Models (LLMs) have led to a qualitative leap in artificial intelligence' s performance on reasoning tasks, particularly demonstrating remarkable capabilities in mathematical, symbolic, and commonsense reasoning. However, as a critical component of advanced human cognition, strategic reasoning, i.e., the ability to assess multi-agent behaviors in dynamic environments, formulate action plans, and adapt strategies, has yet to be systematically evaluated or modeled. To address this gap, this paper introduces WGSR-Bench, the first strategy reasoning benchmark for LLMs using wargame as its evaluation environment. Wargame, a quintessential high-complexity strategic scenario, integrates environmental uncertainty, adversarial dynamics, and non-unique strategic choices, making it an effective testbed for assessing LLMs' capabilities in multi-agent decision-making, intent inference, and counterfactual reasoning. WGSR-Bench designs test samples around three core tasks, i.e., Environmental situation awareness, Opponent risk modeling and Policy generation, which serve as the core S-POE architecture, to systematically assess main abilities of strategic reasoning. Finally, an LLM-based wargame agent is designed to integrate these parts for a comprehensive strategy reasoning assessment. With WGSR-Bench, we hope to assess the strengths and limitations of state-of-the-art LLMs in game-theoretic strategic reasoning and to advance research in large model-driven strategic intelligence.
This paper provides a new categorification of the Lebesgue integral with variable upper limits by using normed modules over finite-dimensional $\Bbbk$-algebras $\mathit{\Lambda}$ and the category $\mathscr{A}^p_{\mathit{\Lambda}}$ associated with $\mathit{\Lambda}$. The integration process is redefined through the introduction of an integral partially ordered set and an abstract integral with variable upper limits. Finally, we present two important applications: (1) the categorification of basic elementary functions, including (anti-)trigonometric and logarithmic functions, and (2) a new approach for characterizing the global dimensions of gentle algebras.
We define integrals for functions on finite-dimensional algebras, adapting methods from Leinster's research. This paper discusses the relationships between the integrals of functions defined on subsets $\mathbb{I}_1 \subseteq {\mathit{\Lambda}}_1$ and $\mathbb{I}_2 \subseteq {\mathit{\Lambda}}_2$ of two finite-dimensional algebras, under the influence of a mapping $\omega$, which can be an injection or a bijection. We explore four specific cases: $\bullet$ $\omega$ as a monotone non-decreasing and right-continuous function; $\bullet$ $\omega$ as an injective, absolutely continuous function; $\bullet$ $\omega$ as a bijection; $\bullet$ and $\omega$ as the identity on $\mathbb{R}$. These scenarios correspond to the frameworks of Lebesgue-Stieltjes integration, Riemann-Stieltjes integration, substitution rules for Lebesgue integrals, and traditional Lebesgue or Riemann integration, respectively.
We explore the assignment of norms to $\mathit{\Lambda}$-modules over a finite-dimensional algebra $\mathit{\Lambda}$, resulting in the establishment of normed $\mathit{\Lambda}$-modules. Our primary contribution lies in constructing two new categories $\mathscr{N}\!\!or^p$ and $\mathscr{A}^p$, where each object in $\mathscr{N}\!\!or^p$ is a normed $\mathit{\Lambda}$-module $N$ limited by a special element $v_N\in N$ and a special $\mathit{\Lambda}$-homomorphism $\delta_N: N^{\oplus 2^{\dim\mathit{\Lambda}}} \to N$, the morphism in $\mathscr{N}\!\!or^p$ is a $\mathit{\Lambda}$-homomorphism $\theta: N\to M$ such that $\theta(v_N) = v_M$ and $\theta\delta_N = \delta_M\theta^{\oplus 2^{\dim\mathit{\Lambda}}}$, and $\mathscr{A}^p$ is a full subcategory of $\mathscr{N}\!\!or^p$ generated by all Banach modules. By examining the objects and morphisms in these categories. We establish a framework for understanding the categorification of integration, series expansions, and derivatives. Furthermore, we obtain the Stone--Weierstrass approximation theorem in the sense of $\mathscr{A}^p$.