EPA: Neural Collapse Inspired Robust Out-of-Distribution Detector

Out-of-distribution (OOD) detection plays a crucial role in ensuring the security of neural networks. Existing works have leveraged the fact that In-distribution (ID) samples form a subspace in the feature space, achieving state-of-the-art (SOTA) performance. However, the comprehensive characteristics of the ID subspace still leave under-explored. Recently, the discovery of Neural Collapse ($\mathcal{NC}$) sheds light on novel properties of the ID subspace. Leveraging insight from $\mathcal{NC}$, we observe that the Principal Angle between the features and the ID feature subspace forms a superior representation for measuring the likelihood of OOD. Building upon this observation, we propose a novel $\mathcal{NC}$-inspired OOD scoring function, named Entropy-enhanced Principal Angle (EPA), which integrates both the global characteristic of the ID subspace and its inner property. We experimentally compare EPA with various SOTA approaches, validating its superior performance and robustness across different network architectures and OOD datasets.

Unravel Anomalies: An End-to-end Seasonal-Trend Decomposition Approach for Time Series Anomaly Detection

Traditional Time-series Anomaly Detection (TAD) methods often struggle with the composite nature of complex time-series data and a diverse array of anomalies. We introduce TADNet, an end-to-end TAD model that leverages Seasonal-Trend Decomposition to link various types of anomalies to specific decomposition components, thereby simplifying the analysis of complex time-series and enhancing detection performance. Our training methodology, which includes pre-training on a synthetic dataset followed by fine-tuning, strikes a balance between effective decomposition and precise anomaly detection. Experimental validation on real-world datasets confirms TADNet's state-of-the-art performance across a diverse range of anomalies.

Linear Speedup of Incremental Aggregated Gradient Methods on Streaming Data

This paper considers a type of incremental aggregated gradient (IAG) method for large-scale distributed optimization. The IAG method is well suited for the parameter server architecture as the latter can easily aggregate potentially staled gradients contributed by workers. Although the convergence of IAG in the case of deterministic gradient is well known, there are only a few results for the case of its stochastic variant based on streaming data. Considering strongly convex optimization, this paper shows that the streaming IAG method achieves linear speedup when the workers are updating frequently enough, even if the data sample distribution across workers are heterogeneous. We show that the expected squared distance to optimal solution decays at O((1+T)/(nt)), where $n$ is the number of workers, t is the iteration number, and T/n is the update frequency of workers. Our analysis involves careful treatments of the conditional expectations with staled gradients and a recursive system with both delayed and noise terms, which are new to the analysis of IAG-type algorithms. Numerical results are presented to verify our findings.

SageFormer: Series-Aware Graph-Enhanced Transformers for Multivariate Time Series Forecasting

Multivariate time series forecasting plays a critical role in diverse domains. While recent advancements in deep learning methods, especially Transformers, have shown promise, there remains a gap in addressing the significance of inter-series dependencies. This paper introduces SageFormer, a Series-aware Graph-enhanced Transformer model designed to effectively capture and model dependencies between series using graph structures. SageFormer tackles two key challenges: effectively representing diverse temporal patterns across series and mitigating redundant information among series. Importantly, the proposed series-aware framework seamlessly integrates with existing Transformer-based models, augmenting their ability to model inter-series dependencies. Through extensive experiments on real-world and synthetic datasets, we showcase the superior performance of SageFormer compared to previous state-of-the-art approaches.

Bridge the Performance Gap in Peak-hour Series Forecasting: The Seq2Peak Framework

Peak-Hour Series Forecasting (PHSF) is a crucial yet underexplored task in various domains. While state-of-the-art deep learning models excel in regular Time Series Forecasting (TSF), they struggle to achieve comparable results in PHSF. This can be attributed to the challenges posed by the high degree of non-stationarity in peak-hour series, which makes direct forecasting more difficult than standard TSF. Additionally, manually extracting the maximum value from regular forecasting results leads to suboptimal performance due to models minimizing the mean deficit. To address these issues, this paper presents Seq2Peak, a novel framework designed specifically for PHSF tasks, bridging the performance gap observed in TSF models. Seq2Peak offers two key components: the CyclicNorm pipeline to mitigate the non-stationarity issue, and a simple yet effective trainable-parameter-free peak-hour decoder with a hybrid loss function that utilizes both the original series and peak-hour series as supervised signals. Extensive experimentation on publicly available time series datasets demonstrates the effectiveness of the proposed framework, yielding a remarkable average relative improvement of 37.7\% across four real-world datasets for both transformer- and non-transformer-based TSF models.

Breaking the Sample Complexity Barrier to Regret-Optimal Model-Free Reinforcement Learning

Achieving sample efficiency in online episodic reinforcement learning (RL) requires optimally balancing exploration and exploitation. When it comes to a finite-horizon episodic Markov decision process with $S$ states, $A$ actions and horizon length $H$, substantial progress has been achieved towards characterizing the minimax-optimal regret, which scales on the order of $\sqrt{H^2SAT}$ (modulo log factors) with $T$ the total number of samples. While several competing solution paradigms have been proposed to minimize regret, they are either memory-inefficient, or fall short of optimality unless the sample size exceeds an enormous threshold (e.g., $S^6A^4 \,\mathrm{poly}(H)$ for existing model-free methods). To overcome such a large sample size barrier to efficient RL, we design a novel model-free algorithm, with space complexity $O(SAH)$, that achieves near-optimal regret as soon as the sample size exceeds the order of $SA\,\mathrm{poly}(H)$. In terms of this sample size requirement (also referred to the initial burn-in cost), our method improves -- by at least a factor of $S^5A^3$ -- upon any prior memory-efficient algorithm that is asymptotically regret-optimal. Leveraging the recently introduced variance reduction strategy (also called {\em reference-advantage decomposition}), the proposed algorithm employs an {\em early-settled} reference update rule, with the aid of two Q-learning sequences with upper and lower confidence bounds. The design principle of our early-settled variance reduction method might be of independent interest to other RL settings that involve intricate exploration-exploitation trade-offs.

The Rate of Convergence of Variation-Constrained Deep Neural Networks

Multi-layer feedforward networks have been used to approximate a wide range of nonlinear functions. An important and fundamental problem is to understand the learnability of a network model through its statistical risk, or the expected prediction error on future data. To the best of our knowledge, the rate of convergence of neural networks shown by existing works is bounded by at most the order of $n^{-1/4}$ for a sample size of $n$. In this paper, we show that a class of variation-constrained neural networks, with arbitrary width, can achieve near-parametric rate $n^{-1/2+\delta}$ for an arbitrarily small positive constant $\delta$. It is equivalent to $n^{-1 +2\delta}$ under the mean squared error. This rate is also observed by numerical experiments. The result indicates that the neural function space needed for approximating smooth functions may not be as large as what is often perceived. Our result also provides insight to the phenomena that deep neural networks do not easily suffer from overfitting when the number of neurons and learning parameters rapidly grow with $n$ or even surpass $n$. We also discuss the rate of convergence regarding other network parameters, including the input dimension, network layer, and coefficient norm.

Sample-Efficient Reinforcement Learning Is Feasible for Linearly Realizable MDPs with Limited Revisiting

Low-complexity models such as linear function representation play a pivotal role in enabling sample-efficient reinforcement learning (RL). The current paper pertains to a scenario with value-based linear representation, which postulates the linear realizability of the optimal Q-function (also called the "linear $Q^{\star}$ problem"). While linear realizability alone does not allow for sample-efficient solutions in general, the presence of a large sub-optimality gap is a potential game changer, depending on the sampling mechanism in use. Informally, sample efficiency is achievable with a large sub-optimality gap when a generative model is available but is unfortunately infeasible when we turn to standard online RL settings. In this paper, we make progress towards understanding this linear $Q^{\star}$ problem by investigating a new sampling protocol, which draws samples in an online/exploratory fashion but allows one to backtrack and revisit previous states in a controlled and infrequent manner. This protocol is more flexible than the standard online RL setting, while being practically relevant and far more restrictive than the generative model. We develop an algorithm tailored to this setting, achieving a sample complexity that scales polynomially with the feature dimension, the horizon, and the inverse sub-optimality gap, but not the size of the state/action space. Our findings underscore the fundamental interplay between sampling protocols and low-complexity structural representation in RL.

Minimax Estimation of Linear Functions of Eigenvectors in the Face of Small Eigen-Gaps

Eigenvector perturbation analysis plays a vital role in various statistical data science applications. A large body of prior works, however, focused on establishing $\ell_{2}$ eigenvector perturbation bounds, which are often highly inadequate in addressing tasks that rely on fine-grained behavior of an eigenvector. This paper makes progress on this by studying the perturbation of linear functions of an unknown eigenvector. Focusing on two fundamental problems -- matrix denoising and principal component analysis -- in the presence of Gaussian noise, we develop a suite of statistical theory that characterizes the perturbation of arbitrary linear functions of an unknown eigenvector. In order to mitigate a non-negligible bias issue inherent to the natural "plug-in" estimator, we develop de-biased estimators that (1) achieve minimax lower bounds for a family of scenarios (modulo some logarithmic factor), and (2) can be computed in a data-driven manner without sample splitting. Noteworthily, the proposed estimators are nearly minimax optimal even when the associated eigen-gap is substantially smaller than what is required in prior theory.

Is Q-Learning Minimax Optimal? A Tight Sample Complexity Analysis

Q-learning, which seeks to learn the optimal Q-function of a Markov decision process (MDP) in a model-free fashion, lies at the heart of reinforcement learning. When it comes to the synchronous setting (such that independent samples for all state-action pairs are drawn from a generative model in each iteration), substantial progress has been made recently towards understanding the sample efficiency of Q-learning. Take a $\gamma$-discounted infinite-horizon MDP with state space $\mathcal{S}$ and action space $\mathcal{A}$: to yield an entrywise $\varepsilon$-accurate estimate of the optimal Q-function, state-of-the-art theory for Q-learning proves that a sample size on the order of $\frac{|\mathcal{S}||\mathcal{A}|}{(1-\gamma)^5\varepsilon^{2}}$ is sufficient, which, however, fails to match with the existing minimax lower bound. This gives rise to natural questions: what is the sharp sample complexity of Q-learning? Is Q-learning provably sub-optimal? In this work, we settle these questions by (1) demonstrating that the sample complexity of Q-learning is at most on the order of $\frac{|\mathcal{S}||\mathcal{A}|}{(1-\gamma)^4\varepsilon^2}$ (up to some log factor) for any $0<\varepsilon <1$, and (2) developing a matching lower bound to confirm the sharpness of our result. Our findings unveil both the effectiveness and limitation of Q-learning: its sample complexity matches that of speedy Q-learning without requiring extra computation and storage, albeit still being considerably higher than the minimax lower bound.