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Ruijia Zhang

Publications and source records attributed to Ruijia Zhang.

At least 19 recordsLinked to original sources
arXiv2026

Entropy and round-point convergence for the $σ_k$-curvature flow

We prove that every smooth closed strictly convex hypersurface in $\mathbb R^{n+1}$ contracts to a round point under the $σ_k$-curvature flow for $2\leq k\leq n-1$, without additional pinching or symmetry assumptions. After rescaling about the extinction point to mean width two, the hypersurfaces converge smoothly and exponentially in normalized time to the unit sphere. The proof is based on an entropy which is invariant under translations and dilations and admits a representation by weighted boundary areas of orthogonal projections. A strengthened Codazzi inequality yields its monotonicity. An upper bound for the entropy controls the inradius relative to the mean width, while its dissipation controls the spatial variation of the normalized speed. We also obtain two-sided curvature bounds at the natural extinction scale.

math.DG↗
arXiv2026

Width and Depth Limits Commute During Training in Linear Residual Networks

Recent work established that the infinite-width and infinite-depth limits of residual networks commute at initialization when residual branches are scaled by $1/\sqrt{depth}$. In particular, the network output converges to the same limit whether width and depth tend to infinity sequentially or jointly. We extend this commutativity result to training in linear residual networks. Under the same $1/\sqrt{depth}$ residual-branch scaling (Depth-$μ$P) and the $μ$P width parameterization, we prove that, for any fixed number of gradient-descent steps with gradient clipping, the network output converges in $L^2$ along every joint width--depth growth path as both width and depth tend to infinity. Consequently, the infinite-width and infinite-depth limits commute throughout any finite training horizon. Without gradient clipping, the corresponding $L^2$ convergence does not hold in general, but we establish the weaker convergence in probability.

cs.LG↗
arXiv2026

Curvature quotient flows and Alexandrov--Fenchel inequalities in hyperbolic space

We study quermassintegral-preserving curvature flows in $\mathbb H^{n+1}$, $n\geq3$, driven by the curvature quotient $(E_k/E_\ell)^{1/(k-\ell)}$. For $1\leq\ell<k$ with $k+\ell\leq n$, we prove that positive sectional curvature is preserved. Starting from a smooth closed strictly convex hypersurface with positive sectional curvature, the flow exists for all time and converges smoothly and exponentially to a geodesic sphere. As an application, we establish sharp Alexandrov--Fenchel inequalities between $W_k$ and $W_\ell$ for smooth bounded convex domains whose boundaries have nonnegative sectional curvature, whenever $0\leq\ell<k\leq n$ and $k+\ell\leq n$. Equality holds if and only if the domain is a geodesic ball. Together with previously known cases, these inequalities give the full family of Alexandrov--Fenchel inequalities for this class of domains in $\mathbb H^6$. We also construct counterexamples showing that $k+\ell\leq n$ is the optimal range for preservation of positive sectional curvature by these flows.

math.DG↗
arXiv2026

Artificial intelligences and human scientists exhibit complementary strengths in theory building

We investigate the effectiveness of artificial intelligences (AI)-specifically large language models (LLMs)-relative to human scientists at high-level cognitive tasks in social science such as theory formulation, predictions of novel empirical results, and theory revision in response to new evidence. The research domain was academic discourse regarding gender and race inequality. Our findings, comparing 25 LLMs with 13 senior researchers and 60 doctoral scholars, reveal that the AIs outperformed most humans individually on most of the present tasks, while human theories were more diverse and exhibited greater gains in predictive accuracy from aggregation. AI-generated theories were more extensively elaborated, involving additional theoretical paths and latent variables, and were rated as higher quality than human theories by independent raters blinded to source. However, this theoretical complexity was in part ornamental, in that it was not associated with more accurate predictions about empirical patterns in data; in contrast, human scientists achieved greater predictive efficiency with simpler theories. The AIs were significantly more likely than human scientists to revise their theories to incorporate new evidence; human scientists updated their beliefs in a selective way that is sensitive to prior prediction errors. We speculate that the superior processing capacity of artificial intelligences makes them especially well-suited to tasks requiring grappling with complexity, but that the greater diversity of human ideas is essential to wise crowds and collective creativity.

cs.AI↗
arXiv2026

Pogorelov interior estimates for sum-of-Hessians equations

In this paper, we develop a new approach to sum-of-Hessians equations involving multiple $k$-Hessian operators of distinct orders. By exploiting the concavity of sums of Hessian operators, we derive Pogorelov estimates for the corresponding equations under the dynamic semi-convexity condition. Moreover, when the highest order is $n-1$ or $n$, we establish such estimates for admissible solutions without relying on this condition, and they are therefore optimal. As an application, when the right-hand side is identically $1$, we prove that every entire admissible solution in $\mathbb{R}^n$ with quadratic growth is necessarily a quadratic polynomial.

math.AP↗
arXiv2026

Interior Hessian estimates for Hessian quotient equations

In this paper, we establish interior $C^2$ estimates for admissible semiconvex solutions to the general Hessian quotient equation $\frac{σ_k}{σ_l}(D^2u)=f(x,u),$ for the cases $l=k-1$ and $l=k-2$, where $f$ is a positive $C^2$ function. Such estimates are known to fail in general for $k-l\geq 3$, even for convex solutions, as shown by counterexamples due to Lu \cite{LuGeneral}. The main ingredient is a quantitative concavity inequality for the Hessian quotient operator under the semiconvex condition. Our result provides a unified argument to such general Hessian quotient equations for $2\leq k\leq n-1$ in arbitrary dimensions.

math.AP↗
arXiv2026

Geometry-Preserving Orthonormal Initialization for Low-Rank Adaptation in RLVR

Low-rank adaptation (LoRA) and its variants enable parameter-efficient fine-tuning of large language models under the supervised fine-tuning (SFT) paradigm. However, their efficacy and behavior under Reinforcement learning with verifiable rewards (RLVR) are less well understood. In particular, two structurally initialized LoRA variants, PiSSA and MiLoRA, which outperform standard LoRA under SFT, can underperform standard LoRA under RLVR and may even exhibit training instability. These observations suggest that how to initialize the low-rank matrices in RLVR remains unclear. In this work, we develop a theoretical analysis of LoRA in RLVR, showing that orthonormal initialization achieves the minimal gap between LoRA outcome and that of full fine-tuning. Guided by this insight, we propose geometry-preserving orthonormal initialization for low-rank adaptation in RLVR, leading to two new variants, RLPO and RLMO. Experiments on mathematical reasoning benchmarks show that the proposed orthonormal initialization stabilizes RLVR training and outperforms standard LoRA, contrasting with PiSSA and MiLoRA. Finally, our unified analysis for LoRA initialization also explains why PiSSA and MiLoRA can underperform in RLVR, which may be of independent interest. Code and checkpoints are publicly available at https://github.com/Richard-ZZZ/geometry-preserving-orthonormal-init-rlvr.

cs.LG↗
arXiv2026

On the Scaling of PEFT: Towards Million Personal Models of Trillion Parameters

Parameter-efficient fine-tuning (PEFT) is usually treated as a cheaper alternative to full fine-tuning. We study a broader role: small trainable adapters as persistent local state on top of strong shared foundation models. In this framing, the base model provides shared competence while adapters carry instance-specific behavior such as preferences, skills, tool habits, and memory-like updates. We organize the problem around three scaling axes: Scale Up, where stronger shared priors make small local updates more useful; Scale Down, where we study how small adapters can be while remaining reliable; and Scale Out, where many persistent adapted instances coexist. MinT provides one infrastructure example for managing adapter identity, revision, provenance, evaluation, and serving residency. Together, the results suggest that PEFT can be a compact substrate for persistent personal models rather than only a budget substitute for full fine-tuning.

cs.LG↗
arXiv2026

MinT: Managed Infrastructure for Training and Serving Millions of LLMs

We present MindLab Toolkit (MinT), a managed infrastructure system for Low-Rank Adaptation (LoRA) post-training and online serving. MinT targets a setting where many trained policies are produced over a small number of expensive base-model deployments. Instead of materializing each policy as a merged full checkpoint, MinT keeps the base model resident and moves exported LoRA adapter revisions through rollout, update, export, evaluation, serving, and rollback, hiding distributed training, serving, scheduling, and data movement behind a service interface. MinT scales this path along three axes. Scale Up extends LoRA RL to frontier-scale dense and MoE architectures, including MLA and DSA attention paths, with training and serving validated beyond 1T total parameters. Scale Down moves only the exported LoRA adapter, which can be under 1% of base-model size in rank-1 settings; adapter-only handoff reduces the measured step by 18.3x on a 4B dense model and 2.85x on a 30B MoE, while concurrent multi-policy GRPO shortens wall time by 1.77x and 1.45x without raising peak memory. Scale Out separates durable policy addressability from CPU/GPU working sets: a tensor-parallel deployment supports 10^6-scale addressable catalogs (measured single-engine sweeps through 100K) and thousand-adapter active waves at cluster scale, with cold loading treated as scheduled service work and packed MoE LoRA tensors improving live engine loading by 8.5-8.7x. MinT thus manages million-scale LoRA policy catalogs while training and serving selected adapter revisions over shared 1T-class base models.

cs.LG↗
arXiv2026

SafeSieve: From Heuristics to Experience in Progressive Pruning for LLM-based Multi-Agent Communication

LLM-based multi-agent systems exhibit strong collaborative capabilities but often suffer from redundant communication and excessive token overhead. Existing methods typically enhance efficiency through pretrained GNNs or greedy algorithms, but often isolate pre- and post-task optimization, lacking a unified strategy. To this end, we present SafeSieve, a progressive and adaptive multi-agent pruning algorithm that dynamically refines the inter-agent communication through a novel dual-mechanism. SafeSieve integrates initial LLM-based semantic evaluation with accumulated performance feedback, enabling a smooth transition from heuristic initialization to experience-driven refinement. Unlike existing greedy Top-k pruning methods, SafeSieve employs 0-extension clustering to preserve structurally coherent agent groups while eliminating ineffective links. Experiments across benchmarks (SVAMP, HumanEval, etc.) showcase that SafeSieve achieves 94.01% average accuracy while reducing token usage by 12.4%-27.8%. Results further demonstrate robustness under prompt injection attacks (1.23% average accuracy drop). In heterogeneous settings, SafeSieve reduces deployment costs by 13.3% while maintaining performance. These results establish SafeSieve as an efficient, GPU-free, and scalable framework for practical multi-agent systems. Our code can be found here: https://github.com/csgen/SafeSieve

cs.MA↗
arXiv2026

POLAR: A Pessimistic Model-based Policy Learning Algorithm for Dynamic Treatment Regimes

Dynamic treatment regimes (DTRs) provide a principled framework for optimizing sequential decision-making in domains where decisions must adapt over time in response to individual trajectories, such as healthcare, education, and digital interventions. However, existing statistical methods often rely on strong positivity assumptions and lack robustness under partial data coverage, while offline reinforcement learning approaches typically focus on average training performance, lack statistical guarantees, and require solving complex optimization problems. To address these challenges, we propose POLAR, a novel pessimistic model-based policy learning algorithm for offline DTR optimization. POLAR estimates the transition dynamics from offline data and quantifies uncertainty for each history-action pair. A pessimistic penalty is then incorporated into the reward function to discourage actions with high uncertainty. Unlike many existing methods that focus on average training performance or provide guarantees only for an oracle policy, POLAR directly targets the suboptimality of the final learned policy and offers theoretical guarantees, without relying on computationally intensive minimax or constrained optimization procedures. To the best of our knowledge, POLAR is the first model-based DTR method to provide both statistical and computational guarantees, including finite-sample bounds on policy suboptimality. Empirical results on both synthetic data and the MIMIC-III dataset demonstrate that POLAR outperforms state-of-the-art methods and yields near-optimal, history-aware treatment strategies.

stat.ML↗
arXiv2026

LLM Agents in Law: Taxonomy, Applications, and Challenges

Large language models (LLMs) have precipitated a dramatic improvement in the legal domain, yet the deployment of standalone models faces significant limitations regarding hallucination, outdated information, and verifiability. Recently, LLM agents have attracted significant attention as a solution to these challenges, utilizing advanced capabilities such as planning, memory, and tool usage to meet the rigorous standards of legal practice. In this paper, we present a comprehensive survey of LLM agents for legal tasks, analyzing how these architectures bridge the gap between technical capabilities and domain-specific needs. Our major contributions include: (1) systematically analyzing the technical transition from standard legal LLMs to legal agents; (2) presenting a structured taxonomy of current agent applications across distinct legal practice areas; (3) discussing evaluation methodologies specifically for agentic performance in law; and (4) identifying open challenges and outlining future directions for developing robust and autonomous legal assistants.

cs.CY↗
arXiv2025

Improved Rates of Differentially Private Nonconvex-Strongly-Concave Minimax Optimization

In this paper, we study the problem of (finite sum) minimax optimization in the Differential Privacy (DP) model. Unlike most of the previous studies on the (strongly) convex-concave settings or loss functions satisfying the Polyak-Lojasiewicz condition, here we mainly focus on the nonconvex-strongly-concave one, which encapsulates many models in deep learning such as deep AUC maximization. Specifically, we first analyze a DP version of Stochastic Gradient Descent Ascent (SGDA) and show that it is possible to get a DP estimator whose $l_2$-norm of the gradient for the empirical risk function is upper bounded by $\tilde{O}(\frac{d^{1/4}}{({nε})^{1/2}})$, where $d$ is the model dimension and $n$ is the sample size. We then propose a new method with less gradient noise variance and improve the upper bound to $\tilde{O}(\frac{d^{1/3}}{(nε)^{2/3}})$, which matches the best-known result for DP Empirical Risk Minimization with non-convex loss. We also discussed several lower bounds of private minimax optimization. Finally, experiments on AUC maximization, generative adversarial networks, and temporal difference learning with real-world data support our theoretical analysis.

cs.LG↗
arXiv2025

Understanding Inverse Reinforcement Learning under Overparameterization: Non-Asymptotic Analysis and Global Optimality

The goal of the Inverse reinforcement learning (IRL) task is to identify the underlying reward function and the corresponding optimal policy from a set of expert demonstrations. While most IRL algorithms' theoretical guarantees rely on a linear reward structure, we aim to extend the theoretical understanding of IRL to scenarios where the reward function is parameterized by neural networks. Meanwhile, conventional IRL algorithms usually adopt a nested structure, leading to computational inefficiency, especially in high-dimensional settings. To address this problem, we propose the first two-timescale single-loop IRL algorithm under neural network parameterized reward and provide a non-asymptotic convergence analysis under overparameterization. Although prior optimality results for linear rewards do not apply, we show that our algorithm can identify the globally optimal reward and policy under certain neural network structures. This is the first IRL algorithm with a non-asymptotic convergence guarantee that provably achieves global optimality in neural network settings.

stat.ML↗
arXiv2025

Memory-updated-based Framework for 100% Reliable Flexible Flat Cables Insertion

Automatic assembly lines have increasingly replaced human labor in various tasks; however, the automation of Flexible Flat Cable (FFC) insertion remains unrealized due to its high requirement for effective feedback and dynamic operation, limiting approximately 11% of global industrial capacity. Despite lots of approaches, like vision-based tactile sensors and reinforcement learning, having been proposed, the implementation of human-like high-reliable insertion (i.e., with a 100% success rate in completed insertion) remains a big challenge. Drawing inspiration from human behavior in FFC insertion, which involves sensing three-dimensional forces, translating them into physical concepts, and continuously improving estimates, we propose a novel framework. This framework includes a sensing module for collecting three-dimensional tactile data, a perception module for interpreting this data into meaningful physical signals, and a memory module based on Bayesian theory for reliability estimation and control. This strategy enables the robot to accurately assess its physical state and generate reliable status estimations and corrective actions. Experimental results demonstrate that the robot using this framework can detect alignment errors of 0.5 mm with an accuracy of 97.92% and then achieve a 100% success rate in all completed tests after a few iterations. This work addresses the challenges of unreliable perception and control in complex insertion tasks, highlighting the path toward the development of fully automated production lines.

cs.RO↗
arXiv2025

$C^2$ estimates for $k$-Hessian equations and a rigidity theorem

We derive a concavity inequality for $k$-Hessian operators under the semi-convexity condition. As an application, we establish interior estimates for semi-convex solutions of the $k$-Hessian equations with vanishing Dirichlet boundary and obtain a Liouville-type result. Additionally, we provide new and simple proofs of Guan-Ren-Wang's results on global curvature estimates for $k$-curvature equations.

math.AP↗