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Abstract Stochastic bilevel optimization finds widespread applications in machine learning, including meta-learning, hyperparameter optimization, and neural architecture search. To extend stochastic bilevel optimization to distributed data, several decentralized stochastic bilevel optimization algorithms have been developed. However, existing methods often suffer from slow convergence rates and high communication costs in heterogeneous settings, limiting their applicability to real-world tasks. To address these issues, we propose two novel decentralized stochastic bilevel gradient descent algorithms based on simultaneous and alternating update strategies. Our algorithms can achieve faster convergence rates and lower communication costs than existing methods. Importantly, our convergence analyses do not rely on strong assumptions regarding heterogeneity. More importantly, our theoretical analyses clearly disclose how the computation and communication regarding the Hessian-inverse-vector product under the heterogeneous setting affects the convergence rate. To the best of our knowledge, this is the first time such favorable theoretical results have been achieved with mild assumptions in the heterogeneous setting. Furthermore, we demonstrate how to establish the convergence rate for the alternating update strategy when combined with the variance-reduced gradient. Finally, experimental results confirm the efficacy of our algorithms. 

Surface-enhanced Raman spectroscopy (SERS) has evolved from a fundamental optical phenomenon to a powerful, molecule-specific analytical technique capable of detecting ultra-trace-level species across biomedicine, catalysis, environmental monitoring, and national security applications. In this review, we summarize recent advances in SERS probe design and fabrication along three major directions: (i) engineering plasmonic hotspots with enhanced field confinement to achieve stronger and more uniform signals; (ii) analyte-directed strategies that precisely position and retain target molecules via tailored surface chemistries, nanoscale confinement, and on-surface reactions for single hotspot SERS; and (iii) hybrid architectures integrating plasmonic metals with functional materials, including high entropy materials, semiconductors, and graphene and other 2D materials, to synergistically couple electromagnetic and chemical enhancement mechanisms. Despite significant progress, key challenges remain for practical applications outside laboratories, including substrate reproducibility and stability, diverse analyte compatibility, unknown molecule identification and standardized quantitative performance in complex environments. We highlight emerging solutions, such as large-area nanomanufacturing for controlled nanoscale gaps, high-resolution Raman mapping for spatial–temporal characterization, density-functional-theory-guided molecular interpretation, and machine-learning-enabled spectral analysis. Advances in foundational AI models and data-driven discovery are positioning SERS to become an increasingly versatile platform, from decoding unknown molecular structures to analyzing complicated multi-component systems for environmental, biomedical, and national security applications with high sensitivity and selectivity. 

Although the tropical intraseaonal variability (TISV), as the most important predictability sources for subseasonal-to-seasonal (S2S) prediction, is dominated by Madden-Julian oscillation (MJO), its significant fraction does not always share the canonical MJO features, especially when the convective activity arrives at Maritime Continent. In this study, using principal oscillation pattern (POP) analysis on the combined fields of daily equatorial convection and zonal wind, two distinct leading TISV modes with relatively slower e-folding decay rates are identified. One is an oscillatory mode with the period of 51 days and e-folding time of 19 days, capturing the eastward propagating (EP) feature of the canonical MJO. The other is a non-oscillatory damping mode with e-folding time of 13.6 days, capturing a standing dipole (SD) with convection anomalies centered over the Maritime Continent and tropical central Pacific, respectively. Compared to the EP mode, the leading moisture anomalies at low level to the east of convection center are diminish for the SD mode, and instead, the strong negative anomalies of moisture and subsidence motion emerge in the tropical central Pacific area, which may be responsible for the distinct propagation features. Without filtering methods used, timeseries of the two POPs could be applied to the real-time monitoring of EP and SD events in the phase-space diagram. The two modes can serve as the simple and objective approach for a better characterization for diverse natures of TISV beyond the canonical MJO description, which may further shed light on dynamics of the TISV and its predictability. 

In the past decade, topological data analysis has emerged as a powerful algebraic topology approach in data science. Although knot theory and related subjects are a focus of study in mathematics, their success in practical applications is quite limited due to the lack of localization and quantization. We address these challenges by introducing knot data analysis (KDA), a paradigm that incorporates curve segmentation and multiscale analysis into the Gauss link integral. The resulting multiscale Gauss link integral (mGLI) recovers the global topological properties of knots and links at an appropriate scale and offers a multiscale geometric topology approach to capture the local structures and connectivities in data. By integration with machine learning or deep learning, the proposed mGLI significantly outperforms other state-of-the-art methods across various benchmark problems in 13 intricately complex biological datasets, including protein flexibility analysis, protein–ligand interactions, human Ether-à-go-go-Related Gene potassium channel blockade screening, and quantitative toxicity assessment. Our KDA opens a research area—knot deep learning—in data science.