When training deep neural networks, a model's generalization error is often observed to follow a power scaling law dependent both on the model size and the data size. Perhaps the best known example of such scaling laws are for transformer-based large language models, where networks with billions of parameters are trained on trillions of tokens of text. Yet, despite sustained widespread interest, a rigorous understanding of why transformer scaling laws exist is still missing. To answer this question, we establish novel statistical estimation and mathematical approximation theories for transformers when the input data are concentrated on a low-dimensional manifold. Our theory predicts a power law between the generalization error and both the training data size and the network size for transformers, where the power depends on the intrinsic dimension $d$ of the training data. Notably, the constructed model architecture is shallow, requiring only logarithmic depth in $d$. By leveraging low-dimensional data structures under a manifold hypothesis, we are able to explain transformer scaling laws in a way which respects the data geometry. Moreover, we test our theory with empirical observation by training LLMs on natural language datasets. We find the observed empirical data scaling laws closely agree with our theoretical predictions. Taken together, these results rigorously show the intrinsic dimension of data to be a crucial quantity affecting transformer scaling laws in both theory and practice.

Gaussian processes (GPs) are a ubiquitous tool for geostatistical modeling with high levels of flexibility and interpretability, and the ability to make predictions at unseen spatial locations through a process called Kriging. Estimation of Kriging weights relies on the inversion of the process' covariance matrix, creating a computational bottleneck for large spatial datasets. In this paper, we propose an Amortized Bayesian Local Interpolation NetworK (A-BLINK) for fast covariance parameter estimation, which uses two pre-trained deep neural networks to learn a mapping from spatial location coordinates and covariance function parameters to Kriging weights and the spatial variance, respectively. The fast prediction time of these networks allows us to bypass the matrix inversion step, creating large computational speedups over competing methods in both frequentist and Bayesian settings, and also provides full posterior inference and predictions using Markov chain Monte Carlo sampling methods. We show significant increases in computational efficiency over comparable scalable GP methodology in an extensive simulation study with lower parameter estimation error. The efficacy of our approach is also demonstrated using a temperature dataset of US climate normals for 1991--2020 based on over 7,000 weather stations.

In reinforcement learning with image-based inputs, it is crucial to establish a robust and generalizable state representation. Recent advancements in metric learning, such as deep bisimulation metric approaches, have shown promising results in learning structured low-dimensional representation space from pixel observations, where the distance between states is measured based on task-relevant features. However, these approaches face challenges in demanding generalization tasks and scenarios with non-informative rewards. This is because they fail to capture sufficient long-term information in the learned representations. To address these challenges, we propose a novel State Chrono Representation (SCR) approach. SCR augments state metric-based representations by incorporating extensive temporal information into the update step of bisimulation metric learning. It learns state distances within a temporal framework that considers both future dynamics and cumulative rewards over current and long-term future states. Our learning strategy effectively incorporates future behavioral information into the representation space without introducing a significant number of additional parameters for modeling dynamics. Extensive experiments conducted in DeepMind Control and Meta-World environments demonstrate that SCR achieves better performance comparing to other recent metric-based methods in demanding generalization tasks.

Portfolio construction is the science of balancing reward and risk; it is at the core of modern finance. In this paper, we tackle the question of optimal decision-making within a Bayesian paradigm, starting from a decision-theoretic formulation. Despite the inherent intractability of the optimal decision in any interesting scenarios, we manage to rewrite it as a saddle-point problem. Leveraging the literature on variational Bayes (VB), we propose a relaxation of the original problem. This novel methodology results in an efficient algorithm that not only performs well but is also provably convergent. Furthermore, we provide theoretical results on the statistical consistency of the resulting decision with the optimal Bayesian decision. Using real data, our proposal significantly enhances the speed and scalability of portfolio selection problems. We benchmark our results against state-of-the-art algorithms, as well as a Monte Carlo algorithm targeting the optimal decision.

This paper explores the application of Positive-Unlabeled (PU) learning for enhanced Distributed Denial-of-Service (DDoS) detection in cloud environments. Utilizing the $\texttt{BCCC-cPacket-Cloud-DDoS-2024}$ dataset, we implement PU learning with four machine learning algorithms: XGBoost, Random Forest, Support Vector Machine, and Na\"{i}ve Bayes. Our results demonstrate the superior performance of ensemble methods, with XGBoost and Random Forest achieving $F_{1}$ scores exceeding 98%. We quantify the efficacy of each approach using metrics including $F_{1}$ score, ROC AUC, Recall, and Precision. This study bridges the gap between PU learning and cloud-based anomaly detection, providing a foundation for addressing Context-Aware DDoS Detection in multi-cloud environments. Our findings highlight the potential of PU learning in scenarios with limited labeled data, offering valuable insights for developing more robust and adaptive cloud security mechanisms.

The driver warning system that alerts the human driver about potential risks during driving is a key feature of an advanced driver assistance system. Existing driver warning technologies, mainly the forward collision warning and unsafe lane change warning, can reduce the risk of collision caused by human errors. However, the current design methods have several major limitations. Firstly, the warnings are mainly generated in a one-shot manner without modeling the ego driver's reactions and surrounding objects, which reduces the flexibility and generality of the system over different scenarios. Additionally, the triggering conditions of warning are mostly rule-based threshold-checking given the current state, which lacks the prediction of the potential risk in a sufficiently long future horizon. In this work, we study the problem of optimally generating driver warnings by considering the interactions among the generated warning, the driver behavior, and the states of ego and surrounding vehicles on a long horizon. The warning generation problem is formulated as a partially observed Markov decision process (POMDP). An optimal warning generation framework is proposed as a solution to the proposed POMDP. The simulation experiments demonstrate the superiority of the proposed solution to the existing warning generation methods.

We show that variational learning can significantly improve the accuracy and calibration of Low-Rank Adaptation (LoRA) without a substantial increase in the cost. We replace AdamW by the Improved Variational Online Newton (IVON) algorithm to finetune large language models. For Llama-2 with 7 billion parameters, IVON improves the accuracy over AdamW by 2.8% and expected calibration error by 4.6%. The accuracy is also better than the other Bayesian alternatives, yet the cost is lower and the implementation is easier. Our work provides additional evidence for the effectiveness of IVON for large language models.

Wearable laser scanning (WLS) system has the advantages of flexibility and portability. It can be used for determining the user's path within a prior map, which is a huge demand for applications in pedestrian navigation, collaborative mapping, augmented reality, and emergency rescue. However, existing LiDAR-based global localization methods suffer from insufficient robustness, especially in complex large-scale outdoor scenes with insufficient features and incomplete coverage of the prior map. To address such challenges, we propose LiDAR-based reliable global localization (Reliable-loc) exploiting the verifiable cues in the sequential LiDAR data. First, we propose a Monte Carlo Localization (MCL) based on spatially verifiable cues, utilizing the rich information embedded in local features to adjust the particles' weights hence avoiding the particles converging to erroneous regions. Second, we propose a localization status monitoring mechanism guided by the sequential pose uncertainties and adaptively switching the localization mode using the temporal verifiable cues to avoid the crash of the localization system. To validate the proposed Reliable-loc, comprehensive experiments have been conducted on a large-scale heterogeneous point cloud dataset consisting of high-precision vehicle-mounted mobile laser scanning (MLS) point clouds and helmet-mounted WLS point clouds, which cover various street scenes with a length of over 20km. The experimental results indicate that Reliable-loc exhibits high robustness, accuracy, and efficiency in large-scale, complex street scenes, with a position accuracy of 1.66m, yaw accuracy of 3.09 degrees, and achieves real-time performance. For the code and detailed experimental results, please refer to https://github.com/zouxianghong/Reliable-loc.

In standard RL, a learner attempts to learn an optimal policy for a Markov Decision Process whose structure (e.g. state space) is known. In online model selection, a learner attempts to learn an optimal policy for an MDP knowing only that it belongs to one of $M >1$ model classes of varying complexity. Recent results have shown that this can be feasibly accomplished in episodic online RL. In this work, we propose $\mathsf{MRBEAR}$, an online model selection algorithm for the average reward RL setting. The regret of the algorithm is in $\tilde O(M C_{m^*}^2 \mathsf{B}_{m^*}(T,\delta))$ where $C_{m^*}$ represents the complexity of the simplest well-specified model class and $\mathsf{B}_{m^*}(T,\delta)$ is its corresponding regret bound. This result shows that in average reward RL, like the episodic online RL, the additional cost of model selection scales only linearly in $M$, the number of model classes. We apply $\mathsf{MRBEAR}$ to the interaction between a learner and an opponent in a two-player simultaneous general-sum repeated game, where the opponent follows a fixed unknown limited memory strategy. The learner's goal is to maximize its utility without knowing the opponent's utility function. The interaction is over $T$ rounds with no episode or discounting which leads us to measure the learner's performance by average reward regret. In this application, our algorithm enjoys an opponent-complexity-dependent regret in $\tilde O(M(\mathsf{sp}(h^*) B^{m^*} A^{m^*+1})^{\frac{3}{2}} \sqrt{T})$, where $m^*\le M$ is the unknown memory limit of the opponent, $\mathsf{sp}(h^*)$ is the unknown span of optimal bias induced by the opponent, and $A$ and $B$ are the number of actions for the learner and opponent respectively. We also show that the exponential dependency on $m^*$ is inevitable by proving a lower bound on the learner's regret.

As a key task in machine learning, data classification is essentially to find a suitable coordinate system to represent data features of different classes of samples. This paper proposes the mutual-energy inner product optimization method for constructing a feature coordinate system. First, by analyzing the solution space and eigenfunctions of partial differential equations describing a non-uniform membrane, the mutual-energy inner product is defined. Second, by expressing the mutual-energy inner product as a series of eigenfunctions, it shows a significant advantage of enhancing low-frequency features and suppressing high-frequency noise, compared with the Euclidean inner product. And then, a mutual-energy inner product optimization model is built to extract data features, and convexity and concavity properties of its objective function are discussed. Next, by combining the finite element method, a stable and efficient sequential linearization algorithm is constructed to solve the optimization model. This algorithm only solves equations including positive definite symmetric matrix and linear programming with a few constraints, and its vectorized implementation is discussed. Finally, the mutual-energy inner product optimization method is used to construct feature coordinates, and multi-class Gaussian classifiers are trained on the MINST training set. Good prediction results of Gaussian classifiers are achieved on the MINST test set.