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Zhao Song

Publications and source records attributed to Zhao Song.

At least 37 records · Page 2Linked to original sources
arXiv2026

Social welfare optimisation under institutional reward and punishment

Institutional incentives are widely used to promote cooperation among autonomous, self-regarding agents, from human societies to multi-agent and AI systems. Existing work typically treats incentive design as a bi-objective problem: minimise institutional cost while achieving a high long-run frequency of cooperation. Whether such schemes also maximise social welfare - total population payoff net of institutional expenditure - has remained largely unexplored. We develop a welfare-centric framework for institutional incentives in finite, well-mixed populations playing a social dilemma (Donation Game and Public Goods Game), considering both rewards for cooperators and punishments for defectors. For each mechanism, we derive explicit expressions for expected social welfare and characterise how it depends on incentive efficiency and selection intensity. Analytically, we identify parameter regimes where social welfare has a single optimal incentive level and regimes with qualitative phase transitions, in which welfare becomes non-monotonic with multiple local optima. We prove that any welfare-maximising incentive is either zero or concentrated around a simple closed-form target, and we provide an efficient algorithm to compute these optima. Comparing reward and punishment, we further derive close-formed conditions under which reward outperform punishment in terms of social welfare for any given budget. Overall, our results reveal a systematic gap between incentives optimised for cost or cooperation frequency and those that maximise welfare.

cs.GT↗
arXiv2026

Strategic commitments shape collective cybersecurity under AI inequality

The growing integration of AI into cybersecurity is reshaping the balance between attackers and defenders. When access to advanced AI-enabled defence tools is uneven, resource-limited defenders may be unable to adopt effective protection, creating persistent system vulnerabilities. We study the impact of differential AI access using an evolutionary game-theoretic model in a finite population. We first show that when high-capability defence is costly, the population is driven toward low-cost, weak-defence behaviour, sustaining attacks and weakening long-run security. To address this problem, we introduce differential access to AI defence tools by allowing defenders to choose between low- and high-capability protection based on their resources. We then examine the role of a small group of committed defenders who always adopt strong defence and influence others through social learning. Although commitment increases the prevalence of strong defence, it alone cannot stabilise secure outcomes due to high defence costs. We therefore incorporate a targeted subsidy to remove the cost disadvantage from committed defenders. Our analysis shows that subsidised commitment significantly increases strong defence adoption, suppresses successful attacks, and improves overall system resilience. Simulations across a broad parameter space confirm that subsidies consistently outperform commitment alone. In addition, social-welfare analysis shows improved defender outcomes while keeping attacker gains low. These findings suggest that targeted support for key defenders can be an effective mechanism for stabilising cybersecurity in AI-driven environments and provide a theoretical bridge between cybersecurity policy, AI governance, and strategic allocation of defensive AI capabilities.

cs.AI↗
arXiv2026

Computing Flows in Subquadratic Space

Space complexity is a critical factor in various computational models, including streaming, parallel/distributed computing, and communication complexity. We study the space complexity of the minimum-cost flow problem, a generalization of the st-max flow problem, focusing on computing flows in subquadratic space. In the general case with arbitrary capacities, minimum cost and $st$-maximum flows can use up to $Ω(n^2)$ edges, so computing the flow on each edge (rather than just the size/cost) seems impossible in subquadratic space. Indeed, there are lower bounds proving quadratic space is needed to store the flow on every edge, which has been used to prove lower bounds on streaming algorithms. However, we show that these lower bounds can be circumvented, opening up improvements for streaming and communication complexity. For a directed graph with integer capacities and costs bounded by $W$, we provide a $\tilde O(n^{1.5}\log (W/ε))$-space $\tilde O(\sqrt{n} \log(W/ε))$-pass streaming algorithm, which during the last pass returns the flow on each edge up to an additive error of $ε$. Crucially, the algorithm does not return the flow at the end of the last pass but returns the flow on an edge, as the edge is read in the stream. This allows us to circumvent existing $Ω(n^2)$ space lower bounds. In the 2-party communication model, our algorithm implies $\tilde O(n^{1.5}\log^2 W)$ bits of communication.

cs.DS↗
arXiv2026

Mathematical Modelling of Ethical AI Use in Higher Education: A Coordination Game Framework for Future-Facing Learning

The rapid uptake of generative artificial intelligence (AI) in higher education is reshaping assessment practices and intensifying concerns around academic integrity, fairness, and learning quality. While institutional responses increasingly emphasise policy guidance and ethical principles, there remains limited formal understanding of how collective norms of responsible or opportunistic AI use emerge and stabilise within student cohorts. This paper reframes student AI use in assessment as a coordination problem shaped by peer expectations and assessment design rather than individual compliance alone. We develop a coordination-based evolutionary game-theoretic framework that captures learning value, effort, perceived fairness, and transparency, with institutional AI governance modelled implicitly through reflective assessment incentives. We use analytical results and finite-population simulations to reveal threshold-driven behavioural transitions in student AI use: small, well-calibrated changes in reflective assessment incentives can trigger rapid shifts towards responsible, learning-oriented AI-use norms, whereas weak or misaligned incentives allow opportunistic practices to persist. These non-linear dynamics explain why policy statements alone often fail to change behaviour, while modest assessment redesigns can have disproportionate effects. By providing a mechanism-level account of how assessment structures shape collective AI-use practices, this work offers higher education institutions an analytically grounded tool for Future Facing Learning, supporting proportionate, pedagogy-led AI governance without reliance on surveillance or punitive enforcement.

cs.CY↗
arXiv2026

Tracking High-order Evolutions via Cascading Low-rank Fitting

Diffusion models have become the de facto standard for modern visual generation, including well-established frameworks such as latent diffusion and flow matching. Recently, modeling high-order dynamics has emerged as a promising frontier in generative modeling. Rather than only learning the first-order velocity field that transports random noise to a target data distribution, these approaches simultaneously learn higher-order derivatives, such as acceleration and jerk, yielding a diverse family of higher-order diffusion variants. To represent higher-order derivatives, naive approaches instantiate separate neural networks for each order, which scales the parameter space linearly with the derivative order. To overcome this computational bottleneck, we introduce cascading low-rank fitting, an ordinary differential equation inspired method that approximates successive derivatives by applying a shared base function augmented with sequentially accumulated low-rank components. Theoretically, we analyze the rank dynamics of these successive matrix differences. We prove that if the initial difference is linearly decomposable, the generic ranks of high-order derivatives are guaranteed to be monotonically non-increasing. Conversely, we demonstrate that without this structural assumption, the General Leibniz Rule allows ranks to strictly increase. Furthermore, we establish that under specific conditions, the sequence of derivative ranks can be designed to form any arbitrary permutation. Finally, we present a straightforward algorithm to efficiently compute the proposed cascading low-rank fitting.

cs.LG↗
arXiv2026

Sima 1.0: A Collaborative Multi-Agent Framework for Documentary Video Production

Content creation for major video-sharing platforms demands significant manual labor, particularly for long-form documentary videos spanning one to two hours. In this work, we introduce Sima 1.0, a multi-agent system designed to optimize the weekly production pipeline for high-quality video generation. The framework partitions the production process into an 11-step pipeline distributed across a hybrid workforce. While foundational creative tasks and physical recording are executed by a human operator, time-intensive editing, caption refinement, and supplementary asset integration are delegated to specialized junior and senior-level AI agents. By systematizing tasks from script annotation to final asset exportation, Sima 1.0 significantly reduces the production workload, empowering a single creator to efficiently sustain a rigorous weekly publishing schedule.

cs.MA↗
arXiv2026

Lazy Kronecker Product

In this paper, we show how to generalize the lazy update regime from dynamic matrix product [Cohen, Lee, Song STOC 2019, JACM 2021] to dynamic kronecker product. We provide an algorithm that uses $n^{ω( \lceil k/2 \rceil, \lfloor k/2 \rfloor, a )-a}$ amortized update time and $ n^{ω( \lceil(k-s)/2 \rceil, \lfloor (k-s)/2 \rfloor,a )}$ worst case query time for dynamic kronecker product problem. Unless tensor MV conjecture is false, there is no algorithm that can use both $n^{ω( \lceil k/2 \rceil, \lfloor k/2 \rfloor, a )-a-Ω(1)}$ amortized update time, and $ n^{ω( \lceil(k-s)/2 \rceil, \lfloor (k-s)/2 \rfloor,a )-Ω(1)}$ worst case query time.

cs.DS↗
arXiv2026

Why Softmax Attention Outperforms Linear Attention

Large transformer models have achieved state-of-the-art results in numerous natural language processing tasks. Among the pivotal components of the transformer architecture, the attention mechanism plays a crucial role in capturing token interactions within sequences through the utilization of softmax function. Conversely, linear attention presents a more computationally efficient alternative by approximating the softmax operation with linear complexity. However, it exhibits substantial performance degradation when compared to the traditional softmax attention mechanism. In this paper, we bridge the gap in our theoretical understanding of the reasons behind the practical performance gap between softmax and linear attention. By conducting a comprehensive comparative analysis of these two attention mechanisms, we shed light on the underlying reasons for why softmax attention outperforms linear attention in most scenarios.

cs.CL↗
arXiv2026

Dynamic Kernel Graph Sparsifiers

A geometric graph associated with a set of points $P= \{x_1, x_2, \cdots, x_n \} \subset \mathbb{R}^d$ and a fixed kernel function $\mathsf{K}:\mathbb{R}^d\times \mathbb{R}^d\to\mathbb{R}_{\geq 0}$ is a complete graph on $P$ such that the weight of edge $(x_i, x_j)$ is $\mathsf{K}(x_i, x_j)$. We present a fully-dynamic data structure that maintains a spectral sparsifier of a geometric graph under updates that change the locations of points in $P$ one at a time. The update time of our data structure is $n^{o(1)}$ with high probability, and the initialization time is $n^{1+o(1)}$. Under certain assumption, our data structure can be made robust against adaptive adversaries, which makes our sparsifier applicable in iterative optimization algorithms. We further show that the Laplacian matrices corresponding to geometric graphs admit a randomized sketch for maintaining matrix-vector multiplication and projection in $n^{o(1)}$ time, under sparse updates to the query vectors, or under modification of points in $P$.

cs.DS↗
arXiv2026

Exploration enhances cooperation in the multi-agent communication system

Designing protocols enhancing cooperation for multi-agent systems remains a grand challenge. Cheap talk, defined as costless, non-binding communication before formal action, serves as a pivotal solution. However, existing theoretical frameworks often exclude random exploration, or noise, for analytical tractability, leaving its functional impact on system performance largely unexplored. To bridge this gap, we propose a two-stage evolutionary game-theoretical model, integrating signalling with a donation game, with exploration explicitly incorporated into the decision-making. Our agent-based simulations across topologies reveal a universal optimal exploration rate that maximises system-wide cooperation. Mechanistically, moderate exploration undermines the stability of defection and catalyses the self-organised cooperative alliances, facilitating their cyclic success. Moreover, the cooperation peak is enabled by the delicate balance between oscillation period and amplification. Our findings suggest that rather than pursuing deterministic rigidity, embracing strategic exploration, as a form of engineered randomness, is essential to sustain cooperation and realise optimal performance in communication-based intelligent systems.

cs.MA↗
arXiv2026

Evolution of fairness in hybrid populations with specialised AI agents

Fairness in hybrid societies hinges on a simple choice: should AI be a generous host or a strict gatekeeper? Moving beyond symmetric models, we show that asymmetric social structures--like those in hiring, regulation, and negotiation--AI that guards fairness outperforms AI that gifts it. We bridge this gap with a bipartite hybrid population model of the Ultimatum Game, separating humans and AI into distinct proposer and receiver groups. We first introduce Samaritan AI agents, which act as either unconditional fair proposers or strict receivers. Our results reveal a striking asymmetry: Samaritan AI receivers drive population-wide fairness far more effectively than Samaritan AI proposers. To overcome the limitations of the Samaritan AI proposer, we design the Discriminatory AI proposer, which predicts co-players' expectations and only offers fair portions to those with high acceptance thresholds. Our results demonstrate that this Discriminatory AI outperforms both types of Samaritan AI, especially in strong selection scenarios. It not only sustains fairness across both populations but also significantly lowers the critical mass of agents required to reach an equitable steady state. By transitioning from unconditional modelling to strategic enforcement, our work provides a pivotal framework for deploying asymmetric AIs in the increasingly hybrid society.

cs.MA↗
arXiv2026

Tensor Hinted Mv Conjectures

Brand, Nanongkai, and Saranurak introduced a conjecture known as the Hinted Mv Conjecture. Although it was originally formulated for the matrix case, we generalize it here to the tensor setting.

cs.CC↗
arXiv2026

Sublinear Time Quantum Algorithm for Attention Approximation

Given the query, key and value matrices $Q, K, V\in \mathbb{R}^{n\times d}$, the attention module is defined as $\mathrm{Att}(Q, K, V)=D^{-1}AV$ where $A=\exp(QK^\top/\sqrt{d})$ with $\exp(\cdot)$ applied entrywise, $D=\mathrm{diag}(A{\bf 1}_n)$. The attention module is the backbone of modern transformers and large language models, but explicitly forming the softmax matrix $D^{-1}A$ incurs $Ω(n^2)$ time, motivating numerous approximation schemes that reduce runtime to $\widetilde O(nd)$ via sparsity or low-rank factorization. We propose a quantum data structure that approximates any row of $\mathrm{Att}(Q, K, V)$ using only row queries to $Q, K, V$. Our algorithm preprocesses these matrices in $\widetilde{O}\left( ε^{-1} n^{0.5} \left( s_λ^{2.5} + s_λ^{1.5} d + α^{0.5} d \right) \right)$ time, where $ε$ is the target accuracy, $s_λ$ is the $λ$-statistical dimension of the exponential kernel defined by $Q$ and $K$, and $α$ measures the row distortion of $V$ that is at most $d/{\rm srank}(V)$, the stable rank of $V$. Each row query can be answered in $\widetilde{O}(s_λ^2 + s_λd)$ time. To our knowledge, this is the first quantum data structure that approximates rows of the attention matrix in sublinear time with respect to $n$. Our approach relies on a quantum Nyström approximation of the exponential kernel, quantum multivariate mean estimation for computing $D$, and quantum leverage score sampling for the multiplication with $V$.

quant-ph↗
arXiv2026

Toward Highly Efficient and Private Submodular Maximization via Matrix-Based Acceleration

Submodular function maximization is a critical building block for diverse tasks, such as document summarization, sensor placement, and image segmentation. Yet its practical utility is often limit by the $O(knd^2)$ computational bottleneck. In this paper, we propose an integrated framework that addresses efficiency and privacy simultaneously. First, we introduce a novel matrix-based computation paradigm that accelerates function evaluations. Second, we develop approximate data structures that further streamline the optimization process, achieving a theoretical complexity of $O(ε^{-2}(nd+kn+kd^2)\log(k/δ))$. Third, we integrate ($ε, δ$)-DP guaranties to address the privacy concerns inherent in sensitive optimization tasks.

cs.LG↗
arXiv2026

Beyond Classical Attention: Quantum Attention for Scalable Computation

As large language models (LLMs) demonstrate outstanding performance across various tasks, attention-driven models have profoundly transformed the field of machine learning. Since attention computations account for the primary computational overhead in both model inference and training, efficiently computing attention matrices has become one of the core challenges in accelerating large language models. It is well-known that quantum machines possess computational advantages over classical machines, and the role of quantum computing in LLMs remains largely unexplored. In this work, we focus on leveraging the Grover search algorithm to efficiently compute a sparse attention matrix. Through comparisons with classical algorithms, we demonstrate that our method achieves quantum acceleration in polynomial time. Additionally, we observe that the generated quantum attention matrices naturally exhibit low-rank structures, providing further theoretical support for efficient modeling. Moreover, within the specific context of attention matrix computation, we conduct a systematic and detailed analysis of the error and time complexity of the proposed algorithm.

quant-ph↗
arXiv2026

Exploring the Frontiers of Softmax: Provable Optimization, Applications in Diffusion Model, and Beyond

The softmax activation function plays a crucial role in the success of large language models (LLMs), particularly in the self-attention mechanism of the widely adopted Transformer architecture. However, the underlying learning dynamics that contribute to the effectiveness of softmax remain largely unexplored. As a step towards better understanding, this paper provides a theoretical study of the optimization and generalization properties of two-layer softmax neural networks, providing theoretical insights into their superior performance as other activation functions, such as ReLU and exponential. Leveraging the Neural Tangent Kernel (NTK) framework, our analysis reveals that the normalization effect of the softmax function leads to a good perturbation property of the induced NTK matrix, resulting in a good convex region of the loss landscape. Consequently, softmax neural networks can learn the target function in the over-parametrization regime. To demonstrate the broad applicability of our theoretical findings, we apply them to the task of learning score estimation functions in diffusion models, a promising approach for generative modeling. Our analysis shows that gradient-based algorithms can learn the score function with a provable accuracy. Our work provides a deeper understanding of the effectiveness of softmax neural networks and their potential in various domains, paving the way for further advancements in natural language processing and beyond.

cs.LG↗
arXiv2026

Training Tensor Attention Efficiently: From Cubic to Almost Linear Time

Tensor Attention, a multi-view attention that is able to capture high-order correlations among multiple modalities, can overcome the representational limitations of classical matrix attention. However, the $O(n^3)$ time complexity of tensor attention poses a significant obstacle to its utilization in transformers, where $n$ is the input sequence length. In this work, we prove that the backward gradient of tensor attention training can be computed in almost linear time $n^{1+o(1)}$, the same complexity as its forward computation under the bounded entries assumption. We provide a closed-form solution for the gradient and propose a fast computation method utilizing polynomial approximation methods and tensor algebraic techniques. Furthermore, we prove the necessity and tightness of our assumption through hardness analysis, showing that slightly weakening it renders the gradient problem unsolvable in truly subcubic time. Our theoretical results establish the feasibility of efficient higher-order transformer training and may facilitate practical applications of tensor attention architectures.

cs.LG↗
arXiv2026

RoPE Attention Can Be Trained in Almost Linear Time

The Rotary Position Embedding (RoPE) mechanism has become a powerful enhancement to the Transformer architecture, which enables models to capture token relationships when encoding positional information. However, the RoPE mechanisms make the computations of attention mechanisms more complicated, which makes efficient algorithms challenging. Earlier research introduced almost linear time algorithms for the forward computation under specific parameter settings of bounded entries (i.e., in time $n^{1+o(1)}$ where $n$ is the number of input tokens), but has not addressed backward computation. In this work, we develop the first almost linear time algorithm for backward computations in the RoPE-based attention under bounded entries. Our approach builds on recent advancements in fast RoPE attention computations, utilizing a novel combination of the polynomial method and the Fast Fourier Transform. Furthermore, we show that with lower bounds derived from the Strong Exponential Time Hypothesis (SETH), the bounded entry condition is necessary for subquadratic performance.

cs.LG↗