Non-convex optimization is a critical tool in advancing machine learning, especially for complex models like deep neural networks and support vector machines. Despite challenges such as multiple local minima and saddle points, non-convex techniques offer various pathways to reduce computational costs. These include promoting sparsity through regularization, efficiently escaping saddle points, and employing subsampling and approximation strategies like stochastic gradient descent. Additionally, non-convex methods enable model pruning and compression, which reduce the size of models while maintaining performance. By focusing on good local minima instead of exact global minima, non-convex optimization ensures competitive accuracy with faster convergence and lower computational overhead. This paper examines the key methods and applications of non-convex optimization in machine learning, exploring how it can lower computation costs while enhancing model performance. Furthermore, it outlines future research directions and challenges, including scalability and generalization, that will shape the next phase of non-convex optimization in machine learning.

Decentralized federated learning (DFL) is a collaborative machine learning framework for training a model across participants without a central server or raw data exchange. DFL faces challenges due to statistical heterogeneity, as participants often possess different data distributions reflecting local environments and user behaviors. Recent work has shown that the neural tangent kernel (NTK) approach, when applied to federated learning in a centralized framework, can lead to improved performance. The NTK-based update mechanism is more expressive than typical gradient descent methods, enabling more efficient convergence and better handling of data heterogeneity. We propose an approach leveraging the NTK to train client models in the decentralized setting, while introducing a synergy between NTK-based evolution and model averaging. This synergy exploits inter-model variance and improves both accuracy and convergence in heterogeneous settings. Our model averaging technique significantly enhances performance, boosting accuracy by at least 10% compared to the mean local model accuracy. Empirical results demonstrate that our approach consistently achieves higher accuracy than baselines in highly heterogeneous settings, where other approaches often underperform. Additionally, it reaches target performance in 4.6 times fewer communication rounds. We validate our approach across multiple datasets, network topologies, and heterogeneity settings to ensure robustness and generalizability.

Low-dimensional embeddings are a cornerstone in the modelling and analysis of complex networks. However, most existing approaches for mining network embedding spaces rely on computationally intensive machine learning systems to facilitate downstream tasks. In the field of NLP, word embedding spaces capture semantic relationships \textit{linearly}, allowing for information retrieval using \textit{simple linear operations} on word embedding vectors. Here, we demonstrate that there are structural properties of network data that yields this linearity. We show that the more homophilic the network representation, the more linearly separable the corresponding network embedding space, yielding better downstream analysis results. Hence, we introduce novel graphlet-based methods enabling embedding of networks into more linearly separable spaces, allowing for their better mining. Our fundamental insights into the structure of network data that enable their \textit{\textbf{linear}} mining and exploitation enable the ML community to build upon, towards efficiently and explainably mining of the complex network data.

Kolmogorov-Arnold Networks (KAN) \cite{liu2024kan} were very recently proposed as a potential alternative to the prevalent architectural backbone of many deep learning models, the multi-layer perceptron (MLP). KANs have seen success in various tasks of AI for science, with their empirical efficiency and accuracy demostrated in function regression, PDE solving, and many more scientific problems. In this article, we revisit the comparison of KANs and MLPs, with emphasis on a theoretical perspective. On the one hand, we compare the representation and approximation capabilities of KANs and MLPs. We establish that MLPs can be represented using KANs of a comparable size. This shows that the approximation and representation capabilities of KANs are at least as good as MLPs. Conversely, we show that KANs can be represented using MLPs, but that in this representation the number of parameters increases by a factor of the KAN grid size. This suggests that KANs with a large grid size may be more efficient than MLPs at approximating certain functions. On the other hand, from the perspective of learning and optimization, we study the spectral bias of KANs compared with MLPs. We demonstrate that KANs are less biased toward low frequencies than MLPs. We highlight that the multi-level learning feature specific to KANs, i.e. grid extension of splines, improves the learning process for high-frequency components. Detailed comparisons with different choices of depth, width, and grid sizes of KANs are made, shedding some light on how to choose the hyperparameters in practice.

Predicting phenotypes with complex genetic bases based on a small, interpretable set of variant features remains a challenging task. Conventionally, data-driven approaches are utilized for this task, yet the high dimensional nature of genotype data makes the analysis and prediction difficult. Motivated by the extensive knowledge encoded in pre-trained LLMs and their success in processing complex biomedical concepts, we set to examine the ability of LLMs in feature selection and engineering for tabular genotype data, with a novel knowledge-driven framework. We develop FREEFORM, Free-flow Reasoning and Ensembling for Enhanced Feature Output and Robust Modeling, designed with chain-of-thought and ensembling principles, to select and engineer features with the intrinsic knowledge of LLMs. Evaluated on two distinct genotype-phenotype datasets, genetic ancestry and hereditary hearing loss, we find this framework outperforms several data-driven methods, particularly on low-shot regimes. FREEFORM is available as open-source framework at GitHub: https://github.com/PennShenLab/FREEFORM.

A fully Bayesian treatment of complicated predictive models (such as deep neural networks) would enable rigorous uncertainty quantification and the automation of higher-level tasks including model selection. However, the intractability of sampling Bayesian posteriors over many parameters inhibits the use of Bayesian methods where they are most needed. Thermodynamic computing has emerged as a paradigm for accelerating operations used in machine learning, such as matrix inversion, and is based on the mapping of Langevin equations to the dynamics of noisy physical systems. Hence, it is natural to consider the implementation of Langevin sampling algorithms on thermodynamic devices. In this work we propose electronic analog devices that sample from Bayesian posteriors by realizing Langevin dynamics physically. Circuit designs are given for sampling the posterior of a Gaussian-Gaussian model and for Bayesian logistic regression, and are validated by simulations. It is shown, under reasonable assumptions, that the Bayesian posteriors for these models can be sampled in time scaling with $\ln(d)$, where $d$ is dimension. For the Gaussian-Gaussian model, the energy cost is shown to scale with $ d \ln(d)$. These results highlight the potential for fast, energy-efficient Bayesian inference using thermodynamic computing.

Graph embeddings play a critical role in graph representation learning, allowing machine learning models to explore and interpret graph-structured data. However, existing methods often rely on opaque, high-dimensional embeddings, limiting interpretability and practical visualization. In this work, we introduce Topological Evolution Rate (TopER), a novel, low-dimensional embedding approach grounded in topological data analysis. TopER simplifies a key topological approach, Persistent Homology, by calculating the evolution rate of graph substructures, resulting in intuitive and interpretable visualizations of graph data. This approach not only enhances the exploration of graph datasets but also delivers competitive performance in graph clustering and classification tasks. Our TopER-based models achieve or surpass state-of-the-art results across molecular, biological, and social network datasets in tasks such as classification, clustering, and visualization.

Breast cancer's complexity and variability pose significant challenges in understanding its progression and guiding effective treatment. This study aims to integrate protein sequence data with expression levels to improve the molecular characterization of breast cancer subtypes and predict clinical outcomes. Using ProtGPT2, a language model designed for protein sequences, we generated embeddings that capture the functional and structural properties of proteins sequence. These embeddings were integrated with protein expression level to form enriched biological representations, which were analyzed using machine learning methods like ensemble K-means for clustering and XGBoost for classification. Our approach enabled successful clustering of patients into biologically distinct groups and accurately predicted clinical outcomes such as survival and biomarkers status, achieving high performance metrics, notably an F1 score of 0.88 for survival and 0.87 for biomarkers status prediction. Analysis of feature importance highlighted key proteins like KMT2C, GCN1, and CLASP2, linked to hormone receptor and Human Epidermal Growth Factor Receptor 2 (HER2) expression, which play a role in tumor progression and patient outcomes, respectively. Furthermore, protein-protein interaction networks and correlation analyses revealed the interdependence of proteins that may influence breast cancer subtype behaviors. These findings suggest that integrating protein sequence and expression data provides valuable insights into tumor biology and has significant potential to enhance personalized treatment strategies in breast cancer care.

Recent retrieval-augmented models enhance basic methods by building a hierarchical structure over retrieved text chunks through recursive embedding, clustering, and summarization. The most relevant information is then retrieved from both the original text and generated summaries. However, such approaches face limitations with dynamic datasets, where adding or removing documents over time complicates the updating of hierarchical representations formed through clustering. We propose a new algorithm to efficiently maintain the recursive-abstractive tree structure in dynamic datasets, without compromising performance. Additionally, we introduce a novel post-retrieval method that applies query-focused recursive abstractive processing to substantially improve context quality. Our method overcomes the limitations of other approaches by functioning as a black-box post-retrieval layer compatible with any retrieval algorithm. Both algorithms are validated through extensive experiments on real-world datasets, demonstrating their effectiveness in handling dynamic data and improving retrieval performance.

Contextual Self-Modulation (CSM) is a potent regularization mechanism for the Neural Context Flow (NCF) framework which demonstrates powerful metalearning of physical systems. However, CSM has limitations in its applicability across different modalities and in high-data regimes. In this work, we introduce two extensions: iCSM, which expands CSM to infinite-dimensional tasks, and StochasticNCF, which improves scalability. These extensions are demonstrated through comprehensive experimentation on a range of tasks, including dynamical systems with parameter variations, computer vision challenges, and curve fitting problems. StochasticNCF enables the application of both CSM and iCSM to high-data scenarios by providing an unbiased approximation of meta-gradient updates through a sampled set of nearest environments. Additionally, we incorporate higher-order Taylor expansions via Taylor-Mode automatic differentiation, revealing that higher-order approximations do not necessarily enhance generalization. Finally, we demonstrate how CSM can be integrated into other meta-learning frameworks with FlashCAVIA, a computationally efficient extension of the CAVIA meta-learning framework (Zintgraf et al. 2019). FlashCAVIA outperforms its predecessor across various benchmarks and reinforces the utility of bi-level optimization techniques. Together, these contributions establish a robust framework for tackling an expanded spectrum of meta-learning tasks, offering practical insights for out-of-distribution generalization. Our opensourced library, designed for flexible integration of self-modulation into contextual meta-learning workflows, is available at github.com/ddrous/self-mod. Meta-learning has emerged as a powerful paradigm in machine learning, addressing the limitations of conventional approaches that train a single algorithm for a specific task. This innovative technique aims to develop models capable of rapid adaptation to novel but related tasks with minimal data, a process often referred to as "learning to learn" (Wang et al., 2021). By leveraging common information across multiple training environments (or meta-knowledge), meta-learning algorithms can efficiently adapt to new scenarios without starting from scratch (Hospedales et al., 2021). The success of meta-learning has been demonstrated in various domains, including dynamical system reconstruction (Norcliffe et al., 2021), program induction (Devlin et al., 2017), out-of-distribution (OoD) generalization (Yao et al., 2021), and continual learning (Hurtado et al., 2021).