This study addresses the challenge of detecting anomalies in multivariate time series data. Considering a bag (e.g., multi-sensor data) consisting of two-dimensional spaces of time points and multivariate instances (e.g., individual sensors), we aim to detect anomalies at both the bag and instance level with a unified model. To circumvent the practical difficulties of labeling at the instance level in such spaces, we adopt a multiple instance learning (MIL)-based approach, which enables learning at both the bag- and instance- levels using only the bag-level labels. In this study, we introduce time-aware and instance-learnable MIL (simply, TAIL-MIL). We propose two specialized attention mechanisms designed to effectively capture the relationships between different types of instances. We innovatively integrate these attention mechanisms with conjunctive pooling applied to the two-dimensional structure at different levels (i.e., bag- and instance-level), enabling TAIL-MIL to effectively pinpoint both the timing and causative multivariate factors of anomalies. We provide theoretical evidence demonstrating TAIL-MIL's efficacy in detecting instances with two-dimensional structures. Furthermore, we empirically validate the superior performance of TAIL-MIL over the state-of-the-art MIL methods and multivariate time-series anomaly detection methods.

Multiple Instance Learning (MIL) has been increasingly adopted to mitigate the high costs and complexity associated with labeling individual instances, learning instead from bags of instances labeled at the bag level and enabling instance-level labeling. While existing research has primarily focused on the learnability of MIL at the bag level, there is an absence of theoretical exploration to check if a given MIL algorithm is learnable at the instance level. This paper proposes a theoretical framework based on probably approximately correct (PAC) learning theory to assess the instance-level learnability of deep multiple instance learning (Deep MIL) algorithms. Our analysis exposes significant gaps between current Deep MIL algorithms, highlighting the theoretical conditions that must be satisfied by MIL algorithms to ensure instance-level learnability. With these conditions, we interpret the learnability of the representative Deep MIL algorithms and validate them through empirical studies.