CVFeb 9Code
MOVA: Towards Scalable and Synchronized Video-Audio GenerationSII-OpenMOSS Team, Donghua Yu, Mingshu Chen et al.
Audio is indispensable for real-world video, yet generation models have largely overlooked audio components. Current approaches to producing audio-visual content often rely on cascaded pipelines, which increase cost, accumulate errors, and degrade overall quality. While systems such as Veo 3 and Sora 2 emphasize the value of simultaneous generation, joint multimodal modeling introduces unique challenges in architecture, data, and training. Moreover, the closed-source nature of existing systems limits progress in the field. In this work, we introduce MOVA (MOSS Video and Audio), an open-source model capable of generating high-quality, synchronized audio-visual content, including realistic lip-synced speech, environment-aware sound effects, and content-aligned music. MOVA employs a Mixture-of-Experts (MoE) architecture, with a total of 32B parameters, of which 18B are active during inference. It supports IT2VA (Image-Text to Video-Audio) generation task. By releasing the model weights and code, we aim to advance research and foster a vibrant community of creators. The released codebase features comprehensive support for efficient inference, LoRA fine-tuning, and prompt enhancement.
CLFeb 23, 2023
Empathetic Response Generation via Emotion Cause Transition GraphYushan Qian, Bo Wang, Ting-En Lin et al.
Empathetic dialogue is a human-like behavior that requires the perception of both affective factors (e.g., emotion status) and cognitive factors (e.g., cause of the emotion). Besides concerning emotion status in early work, the latest approaches study emotion causes in empathetic dialogue. These approaches focus on understanding and duplicating emotion causes in the context to show empathy for the speaker. However, instead of only repeating the contextual causes, the real empathic response often demonstrate a logical and emotion-centered transition from the causes in the context to those in the responses. In this work, we propose an emotion cause transition graph to explicitly model the natural transition of emotion causes between two adjacent turns in empathetic dialogue. With this graph, the concept words of the emotion causes in the next turn can be predicted and used by a specifically designed concept-aware decoder to generate the empathic response. Automatic and human experimental results on the benchmark dataset demonstrate that our method produces more empathetic, coherent, informative, and specific responses than existing models.
SDMay 28, 2022
SuperVoice: Text-Independent Speaker Verification Using Ultrasound Energy in Human SpeechHanqing Guo, Qiben Yan, Nikolay Ivanov et al.
Voice-activated systems are integrated into a variety of desktop, mobile, and Internet-of-Things (IoT) devices. However, voice spoofing attacks, such as impersonation and replay attacks, in which malicious attackers synthesize the voice of a victim or simply replay it, have brought growing security concerns. Existing speaker verification techniques distinguish individual speakers via the spectrographic features extracted from an audible frequency range of voice commands. However, they often have high error rates and/or long delays. In this paper, we explore a new direction of human voice research by scrutinizing the unique characteristics of human speech at the ultrasound frequency band. Our research indicates that the high-frequency ultrasound components (e.g. speech fricatives) from 20 to 48 kHz can significantly enhance the security and accuracy of speaker verification. We propose a speaker verification system, SUPERVOICE that uses a two-stream DNN architecture with a feature fusion mechanism to generate distinctive speaker models. To test the system, we create a speech dataset with 12 hours of audio (8,950 voice samples) from 127 participants. In addition, we create a second spoofed voice dataset to evaluate its security. In order to balance between controlled recordings and real-world applications, the audio recordings are collected from two quiet rooms by 8 different recording devices, including 7 smartphones and an ultrasound microphone. Our evaluation shows that SUPERVOICE achieves 0.58% equal error rate in the speaker verification task, it only takes 120 ms for testing an incoming utterance, outperforming all existing speaker verification systems. Moreover, within 91 ms processing time, SUPERVOICE achieves 0% equal error rate in detecting replay attacks launched by 5 different loudspeakers.
CVAug 19, 2023
Breast Lesion Diagnosis Using Static Images and Dynamic VideoYunwen Huang, Hongyu Hu, Ying Zhu et al.
Deep learning based Computer Aided Diagnosis (CAD) systems have been developed to treat breast ultrasound. Most of them focus on a single ultrasound imaging modality, either using representative static images or the dynamic video of a real-time scan. In fact, these two image modalities are complementary for lesion diagnosis. Dynamic videos provide detailed three-dimensional information about the lesion, while static images capture the typical sections of the lesion. In this work, we propose a multi-modality breast tumor diagnosis model to imitate the diagnosing process of radiologists, which learns the features of both static images and dynamic video and explores the potential relationship between the two modalities. Considering that static images are carefully selected by professional radiologists, we propose to aggregate dynamic video features under the guidance of domain knowledge from static images before fusing multi-modality features. Our work is validated on a breast ultrasound dataset composed of 897 sets of ultrasound images and videos. Experimental results show that our model boosts the performance of Benign/Malignant classification, achieving 90.0% in AUC and 81.7% in accuracy.
LGMay 19
Complementing reinforcement learning with SFT through logit averaging in the post training of LLMsXingwei Gan, Ying Zhu
We introduce a novel method that averages the logits of a frozen reference policy (e.g., SFT) and a trainable policy, and incorporate the method into Group Relative Policy Optimization (GRPO). In contrast to Reinforcement Learning with Verifiable Rewards (RLVR) methods, our proposal does not involve a Kullback Leibler (KL) regularization or critic; the trainable policy and the reference anchor are coupled through the logit averaging structure to leverage the reasoning expertise of the trainable policy while maintaining the formatting advantage of SFT. Our method is evaluated on MATH, cn-k12, and MMLU, and the results show a higher accuracy or at least comparable accuracy relative to the canonical KL-regularized GRPO.
GTApr 27, 2022
On the limitations of data-based price discriminationHaitian Xie, Ying Zhu, Denis Shishkin
The classic third degree price discrimination (3PD) model requires the knowledge of the distribution of buyer valuations and the covariate to set the price conditioned on the covariate. In terms of generating revenue, the classic result shows that 3PD is at least as good as uniform pricing. What if the seller has to set a price based only on a sample of observations from the underlying distribution? Is it still obvious that the seller should engage in 3PD? This paper sheds light on these fundamental questions. In particular, the comparison of the revenue performance between 3PD and uniform pricing is ambiguous overall when prices are set based on samples. This finding is in the nature of statistical learning under uncertainty: a curse of dimensionality, but also other small sample complications.
CLJan 30
FourierSampler: Unlocking Non-Autoregressive Potential in Diffusion Language Models via Frequency-Guided GenerationSiyang He, Qiqi Wang, Xiaoran Liu et al.
Despite the non-autoregressive potential of diffusion language models (dLLMs), existing decoding strategies demonstrate positional bias, failing to fully unlock the potential of arbitrary generation. In this work, we delve into the inherent spectral characteristics of dLLMs and present the first frequency-domain analysis showing that low-frequency components in hidden states primarily encode global structural information and long-range dependencies, while high-frequency components are responsible for characterizing local details. Based on this observation, we propose FourierSampler, which leverages a frequency-domain sliding window mechanism to dynamically guide the model to achieve a "structure-to-detail" generation. FourierSampler outperforms other inference enhancement strategies on LLADA and SDAR, achieving relative improvements of 20.4% on LLaDA1.5-8B and 16.0% on LLaDA-8B-Instruct. It notably surpasses similarly sized autoregressive models like Llama3.1-8B-Instruct.
DBJul 28, 2024
Evaluating LLMs for Text-to-SQL Generation With Complex SQL WorkloadLimin Ma, Ken Pu, Ying Zhu
This study presents a comparative analysis of the a complex SQL benchmark, TPC-DS, with two existing text-to-SQL benchmarks, BIRD and Spider. Our findings reveal that TPC-DS queries exhibit a significantly higher level of structural complexity compared to the other two benchmarks. This underscores the need for more intricate benchmarks to simulate realistic scenarios effectively. To facilitate this comparison, we devised several measures of structural complexity and applied them across all three benchmarks. The results of this study can guide future research in the development of more sophisticated text-to-SQL benchmarks. We utilized 11 distinct Language Models (LLMs) to generate SQL queries based on the query descriptions provided by the TPC-DS benchmark. The prompt engineering process incorporated both the query description as outlined in the TPC-DS specification and the database schema of TPC-DS. Our findings indicate that the current state-of-the-art generative AI models fall short in generating accurate decision-making queries. We conducted a comparison of the generated queries with the TPC-DS gold standard queries using a series of fuzzy structure matching techniques based on query features. The results demonstrated that the accuracy of the generated queries is insufficient for practical real-world application.
CVApr 24, 2024
A Survey on Visual MambaHanwei Zhang, Ying Zhu, Dan Wang et al.
State space models (SSMs) with selection mechanisms and hardware-aware architectures, namely Mamba, have recently demonstrated significant promise in long-sequence modeling. Since the self-attention mechanism in transformers has quadratic complexity with image size and increasing computational demands, the researchers are now exploring how to adapt Mamba for computer vision tasks. This paper is the first comprehensive survey aiming to provide an in-depth analysis of Mamba models in the field of computer vision. It begins by exploring the foundational concepts contributing to Mamba's success, including the state space model framework, selection mechanisms, and hardware-aware design. Next, we review these vision mamba models by categorizing them into foundational ones and enhancing them with techniques such as convolution, recurrence, and attention to improve their sophistication. We further delve into the widespread applications of Mamba in vision tasks, which include their use as a backbone in various levels of vision processing. This encompasses general visual tasks, Medical visual tasks (e.g., 2D / 3D segmentation, classification, and image registration, etc.), and Remote Sensing visual tasks. We specially introduce general visual tasks from two levels: High/Mid-level vision (e.g., Object detection, Segmentation, Video classification, etc.) and Low-level vision (e.g., Image super-resolution, Image restoration, Visual generation, etc.). We hope this endeavor will spark additional interest within the community to address current challenges and further apply Mamba models in computer vision.
CVJun 12, 2024Code
Controllable Dance Generation with Style-Guided Motion DiffusionHongsong Wang, Ying Zhu, Xin Geng et al.
Dance plays an important role as an artistic form and expression in human culture, yet automatically generating dance sequences is a significant yet challenging endeavor. Existing approaches often neglect the critical aspect of controllability in dance generation. Additionally, they inadequately model the nuanced impact of music styles, resulting in dances that lack alignment with the expressive characteristics inherent in the conditioned music. To address this gap, we propose Style-Guided Motion Diffusion (SGMD), which integrates the Transformer-based architecture with a Style Modulation module. By incorporating music features with user-provided style prompts, the SGMD ensures that the generated dances not only match the musical content but also reflect the desired stylistic characteristics. To enable flexible control over the generated dances, we introduce a spatial-temporal masking mechanism. As controllable dance generation has not been fully studied, we construct corresponding experimental setups and benchmarks for tasks such as trajectory-based dance generation, dance in-betweening, and dance inpainting. Extensive experiments demonstrate that our approach can generate realistic and stylistically consistent dances, while also empowering users to create dances tailored to diverse artistic and practical needs. Code is available on Github: https://github.com/mucunzhuzhu/DGSDP
CVNov 7, 2024Code
D$^3$epth: Self-Supervised Depth Estimation with Dynamic Mask in Dynamic ScenesSiyu Chen, Hong Liu, Wenhao Li et al.
Depth estimation is a crucial technology in robotics. Recently, self-supervised depth estimation methods have demonstrated great potential as they can efficiently leverage large amounts of unlabelled real-world data. However, most existing methods are designed under the assumption of static scenes, which hinders their adaptability in dynamic environments. To address this issue, we present D$^3$epth, a novel method for self-supervised depth estimation in dynamic scenes. It tackles the challenge of dynamic objects from two key perspectives. First, within the self-supervised framework, we design a reprojection constraint to identify regions likely to contain dynamic objects, allowing the construction of a dynamic mask that mitigates their impact at the loss level. Second, for multi-frame depth estimation, we introduce a cost volume auto-masking strategy that leverages adjacent frames to identify regions associated with dynamic objects and generate corresponding masks. This provides guidance for subsequent processes. Furthermore, we propose a spectral entropy uncertainty module that incorporates spectral entropy to guide uncertainty estimation during depth fusion, effectively addressing issues arising from cost volume computation in dynamic environments. Extensive experiments on KITTI and Cityscapes datasets demonstrate that the proposed method consistently outperforms existing self-supervised monocular depth estimation baselines. Code is available at \url{https://github.com/Csyunling/D3epth}.
AIAug 22, 2024
GRATR: Zero-Shot Evidence Graph Retrieval-Augmented Trustworthiness ReasoningYing Zhu, Shengchang Li, Ziqian Kong et al.
Trustworthiness reasoning aims to enable agents in multiplayer games with incomplete information to identify potential allies and adversaries, thereby enhancing decision-making. In this paper, we introduce the graph retrieval-augmented trustworthiness reasoning (GRATR) framework, which retrieves observable evidence from the game environment to inform decision-making by large language models (LLMs) without requiring additional training, making it a zero-shot approach. Within the GRATR framework, agents first observe the actions of other players and evaluate the resulting shifts in inter-player trust, constructing a corresponding trustworthiness graph. During decision-making, the agent performs multi-hop retrieval to evaluate trustworthiness toward a specific target, where evidence chains are retrieved from multiple trusted sources to form a comprehensive assessment. Experiments in the multiplayer game \emph{Werewolf} demonstrate that GRATR outperforms the alternatives, improving reasoning accuracy by 50.5\% and reducing hallucination by 30.6\% compared to the baseline method. Additionally, when tested on a dataset of Twitter tweets during the U.S. election period, GRATR surpasses the baseline method by 10.4\% in accuracy, highlighting its potential in real-world applications such as intent analysis.
LGFeb 26
Predicting Tennis Serve directions with Machine LearningYing Zhu, Ruthuparna Naikar
Serves, especially first serves, are very important in professional tennis. Servers choose their serve directions strategically to maximize their winning chances while trying to be unpredictable. On the other hand, returners try to predict serve directions to make good returns. The mind game between servers and returners is an important part of decision-making in professional tennis matches. To help understand the players' serve decisions, we have developed a machine learning method for predicting professional tennis players' first serve directions. Through feature engineering, our method achieves an average prediction accuracy of around 49\% for male players and 44\% for female players. Our analysis provides some evidence that top professional players use a mixed-strategy model in serving decisions and that fatigue might be a factor in choosing serve directions. Our analysis also suggests that contextual information is perhaps more important for returners' anticipatory reactions than previously thought.
CLMar 3
Evaluating Prompting Strategies for Chart Question Answering with Large Language ModelsRuthuparna Naikar, Ying Zhu
Prompting strategies affect LLM reasoning performance, but their role in chart-based QA remains underexplored. We present a systematic evaluation of four widely used prompting paradigms (Zero-Shot, Few-Shot, Zero-Shot Chain-of-Thought, and Few-Shot Chain-of-Thought) across GPT-3.5, GPT-4, and GPT-4o on the ChartQA dataset. Our framework operates exclusively on structured chart data, isolating prompt structure as the only experimental variable, and evaluates performance using two metrics: Accuracy and Exact Match. Results from 1,200 diverse ChartQA samples show that Few-Shot Chain-of-Thought prompting consistently yields the highest accuracy (up to 78.2\%), particularly on reasoning-intensive questions, while Few-Shot prompting improves format adherence. Zero-Shot performs well only with high-capacity models on simpler tasks. These findings provide actionable guidance for selecting prompting strategies in structured data reasoning tasks, with implications for both efficiency and accuracy in real-world applications.
CLMay 28, 2025
RISE: Reasoning Enhancement via Iterative Self-Exploration in Multi-hop Question AnsweringBolei He, Xinran He, Mengke Chen et al.
Large Language Models (LLMs) excel in many areas but continue to face challenges with complex reasoning tasks, such as Multi-Hop Question Answering (MHQA). MHQA requires integrating evidence from diverse sources while managing intricate logical dependencies, often leads to errors in reasoning. Retrieval-Augmented Generation (RAG), widely employed in MHQA tasks, faces challenges in effectively filtering noisy data and retrieving all necessary evidence, thereby limiting its effectiveness in addressing MHQA challenges. To address these challenges, we propose RISE:Reasoning Enhancement via Iterative Self-Exploration, a novel framework designed to enhance models' reasoning capability through iterative self-exploration. Specifically, RISE involves three key steps in addressing MHQA tasks: question decomposition, retrieve-then-read, and self-critique. By leveraging continuous self-exploration, RISE identifies accurate reasoning paths, iteratively self-improving the model's capability to integrate evidence, maintain logical consistency, and enhance performance in MHQA tasks. Extensive experiments on multiple MHQA benchmarks demonstrate that RISE significantly improves reasoning accuracy and task performance.
MAMay 21, 2025
Swarm Intelligence Enhanced Reasoning: A Density-Driven Framework for LLM-Based Multi-Agent OptimizationYing Zhu, Heng Zhou, Rui Su et al.
Recently, many approaches, such as Chain-of-Thought (CoT) prompting and Multi-Agent Debate (MAD), have been proposed to further enrich Large Language Models' (LLMs) complex problem-solving capacities in reasoning scenarios. However, these methods may fail to solve complex problems due to the lack of ability to find optimal solutions. Swarm Intelligence has been serving as a powerful tool for finding optima in the field of traditional optimization problems. To this end, we propose integrating swarm intelligence into the reasoning process by introducing a novel Agent-based Swarm Intelligence (ASI) paradigm. In this paradigm, we formulate LLM reasoning as an optimization problem and use a swarm intelligence scheme to guide a group of LLM-based agents in collaboratively searching for optimal solutions. To avoid swarm intelligence getting trapped in local optima, we further develop a Swarm Intelligence Enhancing Reasoning (SIER) framework, which develops a density-driven strategy to enhance the reasoning ability. To be specific, we propose to perform kernel density estimation and non-dominated sorting to optimize both solution quality and diversity simultaneously. In this case, SIER efficiently enhances solution space exploration through expanding the diversity of the reasoning path. Besides, a step-level quality evaluation is used to help agents improve solution quality by correcting low-quality intermediate steps. Then, we use quality thresholds to dynamically control the termination of exploration and the selection of candidate steps, enabling a more flexible and efficient reasoning process. Extensive experiments are ...
LGOct 13, 2025
Redundancy as a Structural Information Principle for Learning and GeneralizationYuda Bi, Ying Zhu, Vince D Calhoun
We present a theoretical framework that extends classical information theory to finite and structured systems by redefining redundancy as a fundamental property of information organization rather than inefficiency. In this framework, redundancy is expressed as a general family of informational divergences that unifies multiple classical measures, such as mutual information, chi-squared dependence, and spectral redundancy, under a single geometric principle. This reveals that these traditional quantities are not isolated heuristics but projections of a shared redundancy geometry. The theory further predicts that redundancy is bounded both above and below, giving rise to an optimal equilibrium that balances over-compression (loss of structure) and over-coupling (collapse). While classical communication theory favors minimal redundancy for transmission efficiency, finite and structured systems, such as those underlying real-world learning, achieve maximal stability and generalization near this equilibrium. Experiments with masked autoencoders are used to illustrate and verify this principle: the model exhibits a stable redundancy level where generalization peaks. Together, these results establish redundancy as a measurable and tunable quantity that bridges the asymptotic world of communication and the finite world of learning.
AIFeb 20, 2025
Narrative-Driven Travel Planning: Geoculturally-Grounded Script Generation with Evolutionary Itinerary OptimizationZiyu Zhang, Ran Ding, Ying Zhu et al.
To enhance tourists' experiences and immersion, this paper proposes a narrative-driven travel planning framework called NarrativeGuide, which generates a geoculturally-grounded narrative script for travelers, offering a novel, role-playing experience for their journey. In the initial stage, NarrativeGuide constructs a knowledge graph for attractions within a city, then configures the worldview, character setting, and exposition based on the knowledge graph. Using this foundation, the knowledge graph is combined to generate an independent scene unit for each attraction. During the itinerary planning stage, NarrativeGuide models narrative-driven travel planning as an optimization problem, utilizing a genetic algorithm (GA) to refine the itinerary. Before evaluating the candidate itinerary, transition scripts are generated for each pair of adjacent attractions, which, along with the scene units, form a complete script. The weighted sum of script coherence, travel time, and attraction scores is then used as the fitness value to update the candidate solution set. In our experiments, we incorporated the TravelPlanner benchmark to systematically evaluate the planning capability of NarrativeGuide under complex constraints. In addition, we assessed its performance in terms of narrative coherence and cultural fit. The results show that NarrativeGuide demonstrates strong capabilities in both itinerary planning and script generation.
LGMar 4, 2024
Approximating invariant functions with the sorting trick is theoretically justifiedWee Chaimanowong, Ying Zhu
Many machine learning models leverage group invariance which is enjoyed with a wide-range of applications. For exploiting an invariance structure, one common approach is known as \emph{frame averaging}. One popular example of frame averaging is the \emph{group averaging}, where the entire group is used to symmetrize a function. Another example is the \emph{canonicalization}, where a frame at each point consists of a single group element which transforms the point to its orbit representative, for example, sorting. Compared to group averaging, canonicalization is more efficient computationally. However, it results in non-differentiablity or discontinuity of the canonicalized function. As a result, the theoretical performance of canonicalization has not been given much attention. In this work, we establish an approximation theory for canonicalization. Specifically, we bound the point-wise and $L^2(\mathbb{P})$ approximation errors as well as the eigenvalue decay rates associated with a canonicalization trick applied to reproducing kernels. We discuss two key insights from our theoretical analyses and why they point to an interesting future research direction on how one can choose a design to fully leverage canonicalization in practice.
STDec 7, 2021
Phase transitions in nonparametric regressionsYing Zhu
When the unknown regression function of a single variable is known to have derivatives up to the $(γ+1)$th order bounded in absolute values by a common constant everywhere or a.e. (i.e., $(γ+1)$th degree of smoothness), the minimax optimal rate of the mean integrated squared error (MISE) is stated as $\left(\frac{1}{n}\right)^{\frac{2γ+2}{2γ+3}}$ in the literature. This paper shows that: (i) if $n\leq\left(γ+1\right)^{2γ+3}$, the minimax optimal MISE rate is $\frac{\log n}{n\log(\log n)}$ and the optimal degree of smoothness to exploit is roughly $\max\left\{ \left\lfloor \frac{\log n}{2\log\left(\log n\right)}\right\rfloor ,\,1\right\} $; (ii) if $n>\left(γ+1\right)^{2γ+3}$, the minimax optimal MISE rate is $\left(\frac{1}{n}\right)^{\frac{2γ+2}{2γ+3}}$ and the optimal degree of smoothness to exploit is $γ+1$. The fundamental contribution of this paper is a set of metric entropy bounds we develop for smooth function classes. Some of our bounds are original, and some of them improve and/or generalize the ones in the literature (e.g., Kolmogorov and Tikhomirov, 1959). Our metric entropy bounds allow us to show phase transitions in the minimax optimal MISE rates associated with some commonly seen smoothness classes as well as non-standard smoothness classes, and can also be of independent interest outside the nonparametric regression problems.
HCMay 23, 2021
Visualization -- a vital decision driving tool for enterprisesRajath Chikkatur Srinivasa, Supriya Arun, Lauren James et al.
This report documents the results found through surveys and interviews on how visualizations help the employees in their workspace. The objectives of this study were to get in-depth knowledge on what prepares an employee to have the right skill set in constructing an informative visualization as well as the tools and techniques that they use on their daily basis for analysis and visualization purposes. Using the results gathered, we sorted the information in many different ways for analysis and came to conclusions ranging from corporation-based strategies to individualized employee and position preferences.
STNov 23, 2020
Classes of ODE solutions: smoothness, covering numbers, implications for noisy function fitting, and the curse of smoothness phenomenonYing Zhu, Mozhgan Mirzaei
Many numerical methods for recovering ODE solutions from data rely on approximating the solutions using basis functions or kernel functions under a least square criterion. The accuracy of this approach hinges on the smoothness of the solutions. This paper provides a theoretical foundation for these methods by establishing novel results on the smoothness and covering numbers of ODE solution classes (as a measure of their "size"). Our results provide answers to "how do the degree of smoothness and the "size" of a class of ODEs affect the "size" of the associated class of solutions?" We show that: (1) for $y^{'}=f\left(y\right)$ and $y^{'}=f\left(x,\,y\right)$, if the absolute values of all $k$th ($k\leqβ+1$) order derivatives of $f$ are bounded by $1$, then the solution can end up with the $(k+1)$th derivative whose magnitude grows factorially fast in $k$ -- "a curse of smoothness"; (2) our upper bounds for the covering numbers of the $(β+2)-$degree smooth solution classes are greater than those of the "standard" $(β+2)-$degree smooth class of univariate functions; (3) the mean squared error of least squares fitting for noisy recovery has a convergence rate no larger than $\left(\frac{1}{n}\right)^{\frac{2\left(β+2\right)}{2\left(β+2\right)+1}}$ if $n=Ω\left(\left(β\sqrt{\log\left(β\vee1\right)}\right)^{4β+10}\right)$, and under this condition, the rate $\left(\frac{1}{n}\right)^{\frac{2\left(β+2\right)}{2\left(β+2\right)+1}}$ is minimax optimal in the case of $y^{'}=f\left(x,\,y\right)$; (4) more generally, for the higher order Picard type ODEs, $y^{\left(m\right)}=f\left(x,\,y,\,y^{'},\,...,y^{\left(m-1\right)}\right)$, the covering number of the solution class is bounded from above by the product of the covering number of the class $\mathcal{F}$ that $f$ ranges over and the covering number of the set where initial values lie.