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Let $(M, g, f)$ be a $5$-dimensional complete noncompact gradient shrinking Ricci soliton with the equation $Ric+\nabla^2f= \lambda g$, where $\text{Ric}$ is the Ricci tensor and $\nabla^2f$ is the Hessian of the potential function $f$. We prove that it is a finite quotient of $\mathbb{R}^2\times \mathbb{S}^3$ if $M$ has constant scalar curvature $R=3 \lambda$.

Suppose $(M^n, g, f)$ is a complete shrinking gradient Ricci soliton. We give several rigidity results under some natural conditions, generalizing the results in \cite{Petersen-Wylie,Guan-Lu-Xu}. Using maximum principle, we prove that shrinking gradient Ricci soliton with constant scalar curvature $R=1$ is isometric to a finite quotient of $\mathbb{R}^2\times \mathbb{S}^2$, giving a new proof of the main results of Cheng-Zhou \cite{Cheng-Zhou}.

Unlike professional Business-to-Consumer (B2C) e-commerce platforms (e.g., Amazon), Consumer-to-Consumer (C2C) platforms (e.g., Facebook marketplace) are mainly targeting individual sellers who usually lack sufficient experience in e-commerce. Individual sellers often struggle to compose proper descriptions for selling products. With the recent advancement of Multimodal Large Language Models (MLLMs), we attempt to integrate such state-of-the-art generative AI technologies into the product listing process. To this end, we develop IPL, an Intelligent Product Listing tool tailored to generate descriptions using various product attributes such as category, brand, color, condition, etc. IPL enables users to compose product descriptions by merely uploading photos of the selling product. More importantly, it can imitate the content style of our C2C platform Xianyu. This is achieved by employing domain-specific instruction tuning on MLLMs and adopting the multi-modal Retrieval-Augmented Generation (RAG) process. A comprehensive empirical evaluation demonstrates that the underlying model of IPL significantly outperforms the base model in domain-specific tasks while producing less hallucination. IPL has been successfully deployed in our production system, where 72% of users have their published product listings based on the generated content, and those product listings are shown to have a quality score 5.6% higher than those without AI assistance.

Extreme cornering in racing often induces large side-slip angles, presenting a formidable challenge in vehicle control. To tackle this issue, this paper introduces an Active Exploration with Double GPR (AEDGPR) system. The system initiates by planning a minimum-time trajectory with a Gaussian Process Regression(GPR) compensated model. The planning results show that in the cornering section, the yaw angular velocity and side-slip angle are in opposite directions, indicating that the vehicle is drifting. In response, we develop a drift controller based on Model Predictive Control (MPC) and incorporate Gaussian Process Regression to correct discrepancies in the vehicle dynamics model. Moreover, the covariance from the GPR is employed to actively explore various cornering states, aiming to minimize trajectory tracking errors. The proposed algorithm is validated through simulations on the Simulink-Carsim platform and experiments using a 1/10 scale RC vehicle.

Autoregressively trained transformers have brought a profound revolution to the world, especially with their in-context learning (ICL) ability to address downstream tasks. Recently, several studies suggest that transformers learn a mesa-optimizer during autoregressive (AR) pretraining to implement ICL. Namely, the forward pass of the trained transformer is equivalent to optimizing an inner objective function in-context. However, whether the practical non-convex training dynamics will converge to the ideal mesa-optimizer is still unclear. Towards filling this gap, we investigate the non-convex dynamics of a one-layer linear causal self-attention model autoregressively trained by gradient flow, where the sequences are generated by an AR process $x_{t+1} = W x_t$. First, under a certain condition of data distribution, we prove that an autoregressively trained transformer learns $W$ by implementing one step of gradient descent to minimize an ordinary least squares (OLS) problem in-context. It then applies the learned $\widehat{W}$ for next-token prediction, thereby verifying the mesa-optimization hypothesis. Next, under the same data conditions, we explore the capability limitations of the obtained mesa-optimizer. We show that a stronger assumption related to the moments of data is the sufficient and necessary condition that the learned mesa-optimizer recovers the distribution. Besides, we conduct exploratory analyses beyond the first data condition and prove that generally, the trained transformer will not perform vanilla gradient descent for the OLS problem. Finally, our simulation results verify the theoretical results., Comment: Accepted by NeurIPS2024, 45 pages. The final version of the previous preprint

In this paper, we prove some Willmore-type inequalities for closed hypersurfaces in weighted manifolds with nonnegative Bakry-\'Emery Ricci curvature. In particular, we get a sharp Willmore type inequality steady gradient Ricci solitons. We also prove a sharp Willmore-like inequality in shrinking gradient Ricci solitons. Moreover, we characterize the equality cases of our Willmore-type inequalities. These results can be regarded as generalizations of Agostiniani-Fogagnolo-Mazzieri's Willmore-type inequality to weighted manifolds. As applications, we derive some isoperimetric type inequalities under certain existence assumptions of isoperimetric regions in weighted manifolds., Comment: Mini corrected, Corollary 1.3 added

Imitation learning aims to solve the problem of defining reward functions in real-world decision-making tasks. The current popular approach is the Adversarial Imitation Learning (AIL) framework, which matches expert state-action occupancy measures to obtain a surrogate reward for forward reinforcement learning. However, the traditional discriminator is a simple binary classifier and doesn't learn an accurate distribution, which may result in failing to identify expert-level state-action pairs induced by the policy interacting with the environment. To address this issue, we propose a method named diffusion adversarial imitation learning (DiffAIL), which introduces the diffusion model into the AIL framework. Specifically, DiffAIL models the state-action pairs as unconditional diffusion models and uses diffusion loss as part of the discriminator's learning objective, which enables the discriminator to capture better expert demonstrations and improve generalization. Experimentally, the results show that our method achieves state-of-the-art performance and significantly surpasses expert demonstration on two benchmark tasks, including the standard state-action setting and state-only settings. Our code can be available at the link https://github.com/ML-Group-SDU/DiffAIL., Comment: Accepted at AAAI 2024

We obtain some rigidity results for metrics whose Schouten tensor is bounded from below after conformal transformations. Liang Cheng recently proved that a complete, nonflat, locally conformally flat manifold with Ricci pinching condition ($Ric-\epsilon Rg\geq 0$) must be compact. This answers higher dimensional Hamilton's pinching conjecture on locally conformally flat manifolds affirmatively. Since (modified) Schouten tensor being nonnegative is equivalent to a Ricci pinching condition, our main result yields a simple proof of Cheng's theorem.

Generative data augmentation, which scales datasets by obtaining fake labeled examples from a trained conditional generative model, boosts classification performance in various learning tasks including (semi-)supervised learning, few-shot learning, and adversarially robust learning. However, little work has theoretically investigated the effect of generative data augmentation. To fill this gap, we establish a general stability bound in this not independently and identically distributed (non-i.i.d.) setting, where the learned distribution is dependent on the original train set and generally not the same as the true distribution. Our theoretical result includes the divergence between the learned distribution and the true distribution. It shows that generative data augmentation can enjoy a faster learning rate when the order of divergence term is $o(\max\left( \log(m)\beta_m, 1 / \sqrt{m})\right)$, where $m$ is the train set size and $\beta_m$ is the corresponding stability constant. We further specify the learning setup to the Gaussian mixture model and generative adversarial nets. We prove that in both cases, though generative data augmentation does not enjoy a faster learning rate, it can improve the learning guarantees at a constant level when the train set is small, which is significant when the awful overfitting occurs. Simulation results on the Gaussian mixture model and empirical results on generative adversarial nets support our theoretical conclusions. Our code is available at https://github.com/ML-GSAI/Understanding-GDA., Comment: 39 pages

Macro-AUC is the arithmetic mean of the class-wise AUCs in multi-label learning and is commonly used in practice. However, its theoretical understanding is far lacking. Toward solving it, we characterize the generalization properties of various learning algorithms based on the corresponding surrogate losses w.r.t. Macro-AUC. We theoretically identify a critical factor of the dataset affecting the generalization bounds: \emph{the label-wise class imbalance}. Our results on the imbalance-aware error bounds show that the widely-used univariate loss-based algorithm is more sensitive to the label-wise class imbalance than the proposed pairwise and reweighted loss-based ones, which probably implies its worse performance. Moreover, empirical results on various datasets corroborate our theory findings. To establish it, technically, we propose a new (and more general) McDiarmid-type concentration inequality, which may be of independent interest., Comment: ICML 2023

In this paper, we give various curvature pinching conditions such that shrinkers are compact. On one hand, we prove that shrinkers with positive Ricci curvature are compact when they have bounded curvature and certain curvature pinching conditions. On the other hand, we prove that shrinkers with certain asymptotically nonnegative sectional curvature are compact. As applications, some related classifications of shrinkers are provided., Comment: Reference [14] and some comments added

A large-scale deep model pre-trained on massive labeled or unlabeled data transfers well to downstream tasks. Linear evaluation freezes parameters in the pre-trained model and trains a linear classifier separately, which is efficient and attractive for transfer. However, little work has investigated the classifier in linear evaluation except for the default logistic regression. Inspired by the statistical efficiency of naive Bayes, the paper revisits the classical topic on discriminative vs. generative classifiers. Theoretically, the paper considers the surrogate loss instead of the zero-one loss in analyses and generalizes the classical results from binary cases to multiclass ones. We show that, under mild assumptions, multiclass naive Bayes requires $O(\log n)$ samples to approach its asymptotic error while the corresponding multiclass logistic regression requires $O(n)$ samples, where $n$ is the feature dimension. To establish it, we present a multiclass $\mathcal{H}$-consistency bound framework and an explicit bound for logistic loss, which are of independent interests. Simulation results on a mixture of Gaussian validate our theoretical findings. Experiments on various pre-trained deep vision models show that naive Bayes consistently converges faster as the number of data increases. Besides, naive Bayes shows promise in few-shot cases and we observe the "two regimes" phenomenon in pre-trained supervised models. Our code is available at https://github.com/ML-GSAI/Revisiting-Dis-vs-Gen-Classifiers., Comment: Accepted by ICML 2023, 58 pages

Deep Ensemble (DE) is an effective alternative to Bayesian neural networks for uncertainty quantification in deep learning. The uncertainty of DE is usually conveyed by the functional inconsistency among the ensemble members, say, the disagreement among their predictions. Yet, the functional inconsistency stems from unmanageable randomness and may easily collapse in specific cases. To render the uncertainty of DE reliable, we propose a refinement of DE where the functional inconsistency is explicitly characterized, and further tuned w.r.t. the training data and certain priori beliefs. Specifically, we describe the functional inconsistency with the empirical covariance of the functions dictated by ensemble members, which, along with the mean, define a Gaussian process (GP). Then, with specific priori uncertainty imposed, we maximize functional evidence lower bound to make the GP specified by DE approximate the Bayesian posterior. In this way, we relate DE to Bayesian inference to enjoy reliable Bayesian uncertainty. Moreover, we provide strategies to make the training efficient. Our approach consumes only marginally added training cost than the standard DE, but achieves better uncertainty quantification than DE and its variants across diverse scenarios.

Zeroth-order (ZO) optimization is widely used to handle challenging tasks, such as query-based black-box adversarial attacks and reinforcement learning. Various attempts have been made to integrate prior information into the gradient estimation procedure based on finite differences, with promising empirical results. However, their convergence properties are not well understood. This paper makes an attempt to fill up this gap by analyzing the convergence of prior-guided ZO algorithms under a greedy descent framework with various gradient estimators. We provide a convergence guarantee for the prior-guided random gradient-free (PRGF) algorithms. Moreover, to further accelerate over greedy descent methods, we present a new accelerated random search (ARS) algorithm that incorporates prior information, together with a convergence analysis. Finally, our theoretical results are confirmed by experiments on several numerical benchmarks as well as adversarial attacks., Comment: NeurIPS 2021; code available at https://github.com/csy530216/pg-zoo

The (gradient-based) bilevel programming framework is widely used in hyperparameter optimization and has achieved excellent performance empirically. Previous theoretical work mainly focuses on its optimization properties, while leaving the analysis on generalization largely open. This paper attempts to address the issue by presenting an expectation bound w.r.t. the validation set based on uniform stability. Our results can explain some mysterious behaviours of the bilevel programming in practice, for instance, overfitting to the validation set. We also present an expectation bound for the classical cross-validation algorithm. Our results suggest that gradient-based algorithms can be better than cross-validation under certain conditions in a theoretical perspective. Furthermore, we prove that regularization terms in both the outer and inner levels can relieve the overfitting problem in gradient-based algorithms. In experiments on feature learning and data reweighting for noisy labels, we corroborate our theoretical findings.

(Partial) ranking loss is a commonly used evaluation measure for multi-label classification, which is usually optimized with convex surrogates for computational efficiency. Prior theoretical work on multi-label ranking mainly focuses on (Fisher) consistency analyses. However, there is a gap between existing theory and practice -- some pairwise losses can lead to promising performance but lack consistency, while some univariate losses are consistent but usually have no clear superiority in practice. In this paper, we attempt to fill this gap through a systematic study from two complementary perspectives of consistency and generalization error bounds of learning algorithms. Our results show that learning algorithms with the consistent univariate loss have an error bound of $O(c)$ ($c$ is the number of labels), while algorithms with the inconsistent pairwise loss depend on $O(\sqrt{c})$ as shown in prior work. This explains that the latter can achieve better performance than the former in practice. Moreover, we present an inconsistent reweighted univariate loss-based learning algorithm that enjoys an error bound of $O(\sqrt{c})$ for promising performance as well as the computational efficiency of univariate losses. Finally, experimental results validate our theoretical analyses.

Various evaluation measures have been developed for multi-label classification, including Hamming Loss (HL), Subset Accuracy (SA) and Ranking Loss (RL). However, there is a gap between empirical results and the existing theories: 1) an algorithm often empirically performs well on some measure(s) while poorly on others, while a formal theoretical analysis is lacking; and 2) in small label space cases, the algorithms optimizing HL often have comparable or even better performance on the SA measure than those optimizing SA directly, while existing theoretical results show that SA and HL are conflicting measures. This paper provides an attempt to fill up this gap by analyzing the learning guarantees of the corresponding learning algorithms on both SA and HL measures. We show that when a learning algorithm optimizes HL with its surrogate loss, it enjoys an error bound for the HL measure independent of $c$ (the number of labels), while the bound for the SA measure depends on at most $O(c)$. On the other hand, when directly optimizing SA with its surrogate loss, it has learning guarantees that depend on $O(\sqrt{c})$ for both HL and SA measures. This explains the observation that when the label space is not large, optimizing HL with its surrogate loss can have promising performance for SA. We further show that our techniques are applicable to analyze the learning guarantees of algorithms on other measures, such as RL. Finally, the theoretical analyses are supported by experimental results., Comment: To Appear in NeurIPS 2020