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The multi-stage phenomenon in the training loss curves of neural networks has been widely observed, reflecting the non-linearity and complexity inherent in the training process. In this work, we investigate the training dynamics of neural networks (NNs), with particular emphasis on the small initialization regime and identify three distinct stages observed in the loss curve during training: initial plateau stage, initial descent stage, and secondary plateau stage. Through rigorous analysis, we reveal the underlying challenges causing slow training during the plateau stages. Building on existing work, we provide a more detailed proof for the initial plateau. This is followed by a comprehensive analysis of the dynamics in the descent stage. Furthermore, we explore the mechanisms that enable the network to overcome the prolonged secondary plateau stage, supported by both experimental evidence and heuristic reasoning. Finally, to better understand the relationship between global training trends and local parameter adjustments, we employ the Wasserstein distance to capture the microscopic evolution of weight amplitude distribution.

Large language models (LLMs) have garnered substantial attention due to their promising applications in diverse domains. Nevertheless, the increasing size of LLMs comes with a significant surge in the computational requirements for training and deployment. Memristor crossbars have emerged as a promising solution, which demonstrated a small footprint and remarkably high energy efficiency in computer vision (CV) models. Memristors possess higher density compared to conventional memory technologies, making them highly suitable for effectively managing the extreme model size associated with LLMs. However, deploying LLMs on memristor crossbars faces three major challenges. Firstly, the size of LLMs increases rapidly, already surpassing the capabilities of state-of-the-art memristor chips. Secondly, LLMs often incorporate multi-head attention blocks, which involve non-weight stationary multiplications that traditional memristor crossbars cannot support. Third, while memristor crossbars excel at performing linear operations, they are not capable of executing complex nonlinear operations in LLM such as softmax and layer normalization. To address these challenges, we present a novel architecture for the memristor crossbar that enables the deployment of state-of-the-art LLM on a single chip or package, eliminating the energy and time inefficiencies associated with off-chip communication. Our testing on BERT_Large showed negligible accuracy loss. Compared to traditional memristor crossbars, our architecture achieves enhancements of up to 39X in area overhead and 18X in energy consumption. Compared to modern TPU/GPU systems, our architecture demonstrates at least a 68X reduction in the area-delay product and a significant 69% energy consumption reduction.

In this paper, a multi-modal data based semi-supervised learning (SSL) framework that jointly use channel state information (CSI) data and RGB images for vehicle positioning is designed. In particular, an outdoor positioning system where the vehicle locations are determined by a base station (BS) is considered. The BS equipped with several cameras can collect a large amount of unlabeled CSI data and a small number of labeled CSI data of vehicles, and the images taken by cameras. Although the collected images contain partial information of vehicles (i.e. azimuth angles of vehicles), the relationship between the unlabeled CSI data and its azimuth angle, and the distances between the BS and the vehicles captured by images are both unknown. Therefore, the images cannot be directly used as the labels of unlabeled CSI data to train a positioning model. To exploit unlabeled CSI data and images, a SSL framework that consists of a pretraining stage and a downstream training stage is proposed. In the pretraining stage, the azimuth angles obtained from the images are considered as the labels of unlabeled CSI data to pretrain the positioning model. In the downstream training stage, a small sized labeled dataset in which the accurate vehicle positions are considered as labels is used to retrain the model. Simulation results show that the proposed method can reduce the positioning error by up to 30% compared to a baseline where the model is not pretrained.

In this paper, a multi-modal vehicle positioning framework that jointly localizes vehicles with channel state information (CSI) and images is designed. In particular, we consider an outdoor scenario where each vehicle can communicate with only one BS, and hence, it can upload its estimated CSI to only its associated BS. Each BS is equipped with a set of cameras, such that it can collect a small number of labeled CSI, a large number of unlabeled CSI, and the images taken by cameras. To exploit the unlabeled CSI data and position labels obtained from images, we design an meta-learning based hard expectation-maximization (EM) algorithm. Specifically, since we do not know the corresponding relationship between unlabeled CSI and the multiple vehicle locations in images, we formulate the calculation of the training objective as a minimum matching problem. To reduce the impact of label noises caused by incorrect matching between unlabeled CSI and vehicle locations obtained from images and achieve better convergence, we introduce a weighted loss function on the unlabeled datasets, and study the use of a meta-learning algorithm for computing the weighted loss. Subsequently, the model parameters are updated according to the weighted loss function of unlabeled CSI samples and their matched position labels obtained from images. Simulation results show that the proposed method can reduce the positioning error by up to 61% compared to a baseline that does not use images and uses only CSI fingerprint for vehicle positioning.

Neural network-based methods have emerged as powerful tools for solving partial differential equations (PDEs) in scientific and engineering applications, particularly when handling complex domains or incorporating empirical data. These methods leverage neural networks as basis functions to approximate PDE solutions. However, training such networks can be challenging, often resulting in limited accuracy. In this paper, we investigate the training dynamics of neural network-based PDE solvers with a focus on the impact of initialization techniques. We assess training difficulty by analyzing the eigenvalue distribution of the kernel and apply the concept of effective rank to quantify this difficulty, where a larger effective rank correlates with faster convergence of the training error. Building upon this, we discover through theoretical analysis and numerical experiments that two initialization techniques, partition of unity (PoU) and variance scaling (VS), enhance the effective rank, thereby accelerating the convergence of training error. Furthermore, comprehensive experiments using popular PDE-solving frameworks, such as PINN, Deep Ritz, and the operator learning framework DeepOnet, confirm that these initialization techniques consistently speed up convergence, in line with our theoretical findings.

As the world enters the journey toward the 6th generation (6G) of wireless technology, the promises of ultra-high data rates, unprecedented low latency, and a massive surge in connected devices require crucial exploration of network energy saving (NES) solutions to minimize the carbon footprint and overall energy usage of future cellular networks. On the other hand, network-controlled repeaters (NCRs) have been introduced by 3rd generation partnership project (3GPP) as a cost-effective solution to improve network coverage. However, their impact on network power consumption and energy efficiency has not been thoroughly investigated. This paper studies NES schemes for next-generation 6G networks aided by NCRs and proposes optimal NES strategies aiming at maximizing the overall energy efficiency of the network. Repeaters are shown to allow for power savings at next-generation nodeB (gNB), and offer higher overall energy efficiency (EE) and spectral efficiency (SE), thus providing an energy-efficient and cost-efficient alternative to increase the performance of future 6G networks, Comment: 7 pages, conference

Road segmentation is a critical task for autonomous driving systems, requiring accurate and robust methods to classify road surfaces from various environmental data. Our work introduces an innovative approach that integrates LiDAR point cloud data, visual image, and relative depth maps derived from images. The integration of multiple data sources in road segmentation presents both opportunities and challenges. One of the primary challenges is the scarcity of large-scale, accurately labeled datasets that are necessary for training robust deep learning models. To address this, we have developed the [UdeerLID+] framework under a semi-supervised learning paradigm. Experiments results on KITTI datasets validate the superior performance.

This paper is mainly concerned with the free boundary problem for an approximate model (for example, arising from the study of sonoluminescence) of a gas bubble of finite mass enclosed within a bounded incompressible viscous liquid, accounting for surface tensions at both the gas-liquid interface and the external free surface of the entire gas-liquid region. It is found that any regular spherically symmetric steady-state solution is characterized by a positive root of a ninth-degree polynomial for which the existence and uniqueness are proved and a one-to-one correspondence between equilibria and pairs of gas mass and liquid volume is established. We prove that these equilibria exhibit nonlinear and exponential asymptotic stability under small perturbations that conserve gas mass and liquid volume, and an equilibrium solution acts as a local minimizer of the energy functional, even under relatively large perturbations, with the proportionality constant determined by the adiabatic constant. Moreover, we construct a global center manifold to apply the center manifold theory. Our results apply to gases and liquids of all sizes. Furthermore, we derive the optimal exponential decay rate for small liquid volumes by analyzing the spectrum bounds of the associated linear operator and show that decreasing the gas mass or increasing the temperature can accelerate the convergence rate, a behavior not seen in unbounded liquid scenarios., Comment: 38 pages, changed the title and inserted the funding information

This paper studies performance of various types of relay nodes in a 5G wireless network: conventional amplify-forward repeaters, (semi-)smart/smart amplify-forward repeaters with different levels of side information, and half-duplex/full-duplex decode-forward relay nodes with and without spatial reuse. End-to-end effective signal to interference and noise ratios (SINRs) and achievable rates are derived for these different types of relay nodes. Performance and complexity tradeoffs are discussed with a simulation over a Manhattan topology setting. Over-the-air (OTA) test results corroborates the findings in this paper., Comment: Presented in IEEE ICC 2024 Industry workshop

In deep reinforcement learning applications, maximizing discounted reward is often employed instead of maximizing total reward to ensure the convergence and stability of algorithms, even though the performance metric for evaluating the policy remains the total reward. However, the optimal policies corresponding to these two objectives may not always be consistent. To address this issue, we analyzed the suboptimality of the policy obtained through maximizing discounted reward in relation to the policy that maximizes total reward and identified the influence of hyperparameters. Additionally, we proposed sufficient conditions for aligning the optimal policies of these two objectives under various settings. The primary contributions are as follows: We theoretically analyzed the factors influencing performance when using discounted reward as a proxy for total reward, thereby enhancing the theoretical understanding of this scenario. Furthermore, we developed methods to align the optimal policies of the two objectives in certain situations, which can improve the performance of reinforcement learning algorithms.

Semi-gradient Q-learning is applied in many fields, but due to the absence of an explicit loss function, studying its dynamics and implicit bias in the parameter space is challenging. This paper introduces the Fokker--Planck equation and employs partial data obtained through sampling to construct and visualize the effective loss landscape within a two-dimensional parameter space. This visualization reveals how the global minima in the loss landscape can transform into saddle points in the effective loss landscape, as well as the implicit bias of the semi-gradient method. Additionally, we demonstrate that saddle points, originating from the global minima in loss landscape, still exist in the effective loss landscape under high-dimensional parameter spaces and neural network settings. This paper develop a novel approach for probing implicit bias in semi-gradient Q-learning.

Recent works have shown the remarkable superiority of transformer models in reinforcement learning (RL), where the decision-making problem is formulated as sequential generation. Transformer-based agents could emerge with self-improvement in online environments by providing task contexts, such as multiple trajectories, called in-context RL. However, due to the quadratic computation complexity of attention in transformers, current in-context RL methods suffer from huge computational costs as the task horizon increases. In contrast, the Mamba model is renowned for its efficient ability to process long-term dependencies, which provides an opportunity for in-context RL to solve tasks that require long-term memory. To this end, we first implement Decision Mamba (DM) by replacing the backbone of Decision Transformer (DT). Then, we propose a Decision Mamba-Hybrid (DM-H) with the merits of transformers and Mamba in high-quality prediction and long-term memory. Specifically, DM-H first generates high-value sub-goals from long-term memory through the Mamba model. Then, we use sub-goals to prompt the transformer, establishing high-quality predictions. Experimental results demonstrate that DM-H achieves state-of-the-art in long and short-term tasks, such as D4RL, Grid World, and Tmaze benchmarks. Regarding efficiency, the online testing of DM-H in the long-term task is 28$\times$ times faster than the transformer-based baselines., Comment: arXiv admin note: text overlap with arXiv:2405.20692. arXiv admin note: text overlap with arXiv:2405.20692; text overlap with arXiv:2305.16554, arXiv:2210.14215 by other authors

This paper presents a comprehensive analysis of critical point sets in two-layer neural networks. To study such complex entities, we introduce the critical embedding operator and critical reduction operator as our tools. Given a critical point, we use these operators to uncover the whole underlying critical set representing the same output function, which exhibits a hierarchical structure. Furthermore, we prove existence of saddle branches for any critical set whose output function can be represented by a narrower network. Our results provide a solid foundation to the further study of optimization and training behavior of neural networks.

For the initial boundary problem of the incompressible MHD equations in a bounded domain with general curved boundary in 3D with the general Navier-slip boundary conditions for the velocity field and the perfect conducting condition for the magnetic field, we establish the uniform regularity of conormal Sobolev norms and Lipschitz norms to addressing the anisotropic regularity of tangential and normal directions, which enable us to prove the vanishing dissipation limit as the viscosity and the magnetic diffusion coefficients tend to zero. We overcome the difficulties caused by the intricate interaction of boundary curvature, velocity field, and magnetic fields and resolve the issue caused by the problem that the viscosity and the magnetic diffusion coefficients are not required to be equal.

In this paper, we advance the understanding of neural network training dynamics by examining the intricate interplay of various factors introduced by weight parameters in the initialization process. Motivated by the foundational work of Luo et al. (J. Mach. Learn. Res., Vol. 22, Iss. 1, No. 71, pp 3327-3373), we explore the gradient descent dynamics of neural networks through the lens of macroscopic limits, where we analyze its behavior as width $m$ tends to infinity. Our study presents a unified approach with refined techniques designed for multi-layer fully connected neural networks, which can be readily extended to other neural network architectures. Our investigation reveals that gradient descent can rapidly drive deep neural networks to zero training loss, irrespective of the specific initialization schemes employed by weight parameters, provided that the initial scale of the output function $\kappa$ surpasses a certain threshold. This regime, characterized as the theta-lazy area, accentuates the predominant influence of the initial scale $\kappa$ over other factors on the training behavior of neural networks. Furthermore, our approach draws inspiration from the Neural Tangent Kernel (NTK) paradigm, and we expand its applicability. While NTK typically assumes that $\lim_{m\to\infty}\frac{\log \kappa}{\log m}=\frac{1}{2}$, and imposes each weight parameters to scale by the factor $\frac{1}{\sqrt{m}}$, in our theta-lazy regime, we discard the factor and relax the conditions to $\lim_{m\to\infty}\frac{\log \kappa}{\log m}>0$. Similar to NTK, the behavior of overparameterized neural networks within the theta-lazy regime trained by gradient descent can be effectively described by a specific kernel. Through rigorous analysis, our investigation illuminates the pivotal role of $\kappa$ in governing the training dynamics of neural networks.

Recent advancements in neural network quantisation have yielded remarkable outcomes, with three-bit networks reaching state-of-the-art full-precision accuracy in complex tasks. These achievements present valuable opportunities for accelerating neural networks by computing in reduced precision. Implementing it on FPGAs can take advantage of bit-level reconfigurability, which is not available on conventional CPUs and GPUs. Simultaneously, the high data intensity of neural network processing has inspired computing-in-memory paradigms, including on FPGA platforms. By programming the effects of trained model weights as lookup operations in soft logic, the transfer of weight data from memory units can be avoided, alleviating the memory bottleneck. However, previous methods face poor scalability - the high logic utilisation limiting them to small networks/sub-networks of binary models with low accuracy. In this paper, we introduce Table Lookup Multiply-Accumulate (TLMAC) as a framework to compile and optimise quantised neural networks for scalable lookup-based processing. TLMAC clusters and maps unique groups of weights to lookup-based processing elements, enabling highly parallel computation while taking advantage of parameter redundancy. Further place and route algorithms are proposed to reduce LUT utilisation and routing congestion. We demonstrate that TLMAC significantly improves the scalability of previous related works. Our efficient logic mapping and high degree of reuse enables entire ImageNet-scale quantised models with full-precision accuracy to be implemented using lookup-based computing on one commercially available FPGA.

Empirical and theoretical works show that the input weights of two-layer neural networks, when initialized with small values, converge towards isolated orientations. This phenomenon, referred to as condensation, indicates that the gradient descent methods tend to spontaneously reduce the complexity of neural networks during the training process. In this work, we elucidate the mechanisms behind the condensation phenomena occurring in the training of three-layer neural networks and distinguish it from the training of two-layer neural networks. Through rigorous theoretical analysis, we establish the blow-up property of effective dynamics and present a sufficient condition for the occurrence of condensation, findings that are substantiated by experimental results. Additionally, we explore the association between condensation and the low-rank bias observed in deep matrix factorization.

Deep reinforcement learning excels in numerous large-scale practical applications. However, existing performance analyses ignores the unique characteristics of continuous-time control problems, is unable to directly estimate the generalization error of the Bellman optimal loss and require a boundedness assumption. Our work focuses on continuous-time control problems and proposes a method that is applicable to all such problems where the transition function satisfies semi-group and Lipschitz properties. Under this method, we can directly analyze the \emph{a priori} generalization error of the Bellman optimal loss. The core of this method lies in two transformations of the loss function. To complete the transformation, we propose a decomposition method for the maximum operator. Additionally, this analysis method does not require a boundedness assumption. Finally, we obtain an \emph{a priori} generalization error without the curse of dimensionality.

Albeit progress has been made in Composed Image Retrieval (CIR), we empirically find that a certain percentage of failure retrieval results are not consistent with their relative captions. To address this issue, this work provides a Visual Question Answering (VQA) perspective to boost the performance of CIR. The resulting VQA4CIR is a post-processing approach and can be directly plugged into existing CIR methods. Given the top-C retrieved images by a CIR method, VQA4CIR aims to decrease the adverse effect of the failure retrieval results being inconsistent with the relative caption. To find the retrieved images inconsistent with the relative caption, we resort to the "QA generation to VQA" self-verification pipeline. For QA generation, we suggest fine-tuning LLM (e.g., LLaMA) to generate several pairs of questions and answers from each relative caption. We then fine-tune LVLM (e.g., LLaVA) to obtain the VQA model. By feeding the retrieved image and question to the VQA model, one can find the images inconsistent with relative caption when the answer by VQA is inconsistent with the answer in the QA pair. Consequently, the CIR performance can be boosted by modifying the ranks of inconsistently retrieved images. Experimental results show that our proposed method outperforms state-of-the-art CIR methods on the CIRR and Fashion-IQ datasets.

Transformer models trained on long sequences often achieve higher accuracy than short sequences. Unfortunately, conventional transformers struggle with long sequence training due to the overwhelming computation and memory requirements. Existing methods for long sequence training offer limited speedup and memory reduction, and may compromise accuracy. This paper presents a novel and efficient distributed training method, the Long Short-Sequence Transformer (LSS Transformer), for training transformer with long sequences. It distributes a long sequence into segments among GPUs, with each GPU computing a partial self-attention for its segment. Then, it uses a fused communication and a novel double gradient averaging technique to avoid the need to aggregate partial self-attention and minimize communication overhead. We evaluated the performance between LSS Transformer and the state-of-the-art Nvidia sequence parallelism on a Wikipedia enwik8 dataset. Results show that our proposed method lead to 5.6x faster and 10.2x more memory-efficient implementation compared to state-of-the-art sequence parallelism on 144 Nvidia V100 GPUs. Moreover, our algorithm scales to an extreme sequence length of 50,112 at 3,456 GPUs, achieving 161% super-linear parallel efficiency and a throughput of 32 petaflops.

For the problem of the non-isentropic compressible Euler Equations coupled with a nonlinear Poisson equation with the electric potential satisfying the Dirichlet boundary condition in three spatial dimensions with a general free boundary not restricting to a graph, we identify suitable stability conditions on the electric potential and the pressure under which we obtain a priori estimates on the Sobolev norms of the fluid and electric variables and bounds for geometric quantities of free surface. The stability conditions in this case for a general variable entropy are that the outer normal derivative of the electric potential is positive on the free surface, whereas that on the pressure is negative. In the isentropic case, the stability condition reduces to a single one, the outer normal derivative of the difference of the enthalpy and the electric potential is negative on the free surface. For the free boundary problem of the non-isentropic compressible Euler equations with variable entropy without coupling with the nonlinear Poisson equation, the corresponding higher-order estimates are also obtained under the Taylor sign condition. It is also found that one less derivative is needed to close the energy estimates for the problem for the non-isentropic compressible Euler Equations coupled with a nonlinear Poisson equation when the electric potential satisfies the Dirichlet boundary condition under the stability conditions on the electric potential and the pressure, compared with the problem of the non-isentropic compressible Euler equations.

As a popular and easy-to-implement machine learning method for solving differential equations, the physics-informed neural network (PINN) sometimes may fail and find poor solutions which bias against the exact ones. In this paper, we establish a framework of modified equation to explain the failure phenomenon and characterize the implicit bias of a general residual minimization (RM) method. We provide a simple way to derive the modified equation which models the numerical solution obtained by RM methods. Next, we show the modified solution deviates from the original exact solution. The proof uses a by-product of this paper, that is, a necessary and sufficient condition on characterizing the singularity of the coefficients. This equivalent condition can be extended to other types of equations in the future. Finally, we prove, as a complete characterization of the implicit bias, that RM method implicitly biases the numerical solution against the exact solution and towards a modified solution. In this work, we focus on elliptic equations with discontinuous coefficients, but our approach can be extended to other types of equations and our understanding of the implicit bias may shed light on further development of deep learning based methods for solving equations.