Bilevel optimization (BO) has arisen as a powerful tool for solving many modern machine learning problems. However, due to the nested structure of BO, existing gradient-based methods require second-order derivative approximations via Jacobian- or/and Hessian-vector computations, which can be very costly in practice, especially with large neural network models. In this work, we propose a novel BO algorithm, which adopts Evolution Strategies (ES) based method to approximate the response Jacobian matrix in the hypergradient of BO, and hence fully eliminates all second-order computations. We call our algorithm as ESJ (which stands for the ES-based Jacobian method) and further extend it to the stochastic setting as ESJ-S. Theoretically, we characterize the convergence guarantee and computational complexity for our algorithms. Experimentally, we demonstrate the superiority of our proposed algorithms compared to the state of the art methods on various bilevel problems. Particularly, in our experiment in the few-shot meta-learning problem, we meta-learn the twelve millions parameters of a ResNet-12 network over the miniImageNet dataset, which evidently demonstrates the scalability of our ES-based bilevel approach and its feasibility in the large-scale setting.

Understanding the implicit bias of Stochastic Gradient Descent (SGD) is one of the key challenges in deep learning, especially for overparametrized models, where the local minimizers of the loss function $L$ can form a manifold. Intuitively, with a sufficiently small learning rate $\eta$, SGD tracks Gradient Descent (GD) until it gets close to such manifold, where the gradient noise prevents further convergence. In such a regime, Blanc et al. (2020) proved that SGD with label noise locally decreases a regularizer-like term, the sharpness of loss, $\mathrm{tr}[\nabla^2 L]$. The current paper gives a general framework for such analysis by adapting ideas from Katzenberger (1991). It allows in principle a complete characterization for the regularization effect of SGD around such manifold -- i.e., the "implicit bias" -- using a stochastic differential equation (SDE) describing the limiting dynamics of the parameters, which is determined jointly by the loss function and the noise covariance. This yields some new results: (1) a global analysis of the implicit bias valid for $\eta^{-2}$ steps, in contrast to the local analysis of Blanc et al. (2020) that is only valid for $\eta^{-1.6}$ steps and (2) allowing arbitrary noise covariance. As an application, we show with arbitrary large initialization, label noise SGD can always escape the kernel regime and only requires $O(\kappa\ln d)$ samples for learning an $\kappa$-sparse overparametrized linear model in $\mathbb{R}^d$ (Woodworth et al., 2020), while GD initialized in the kernel regime requires $\Omega(d)$ samples. This upper bound is minimax optimal and improves the previous $\tilde{O}(\kappa^2)$ upper bound (HaoChen et al., 2020).

After deploying a clinical prediction model, subsequently collected data can be used to fine-tune its predictions and adapt to temporal shifts. Because model updating carries risks of over-updating/fitting, we study online methods with performance guarantees. We introduce two procedures for continual recalibration or revision of an underlying prediction model: Bayesian logistic regression (BLR) and a Markov variant that explicitly models distribution shifts (MarBLR). We perform empirical evaluation via simulations and a real-world study predicting COPD risk. We derive "Type I and II" regret bounds, which guarantee the procedures are non-inferior to a static model and competitive with an oracle logistic reviser in terms of the average loss. Both procedures consistently outperformed the static model and other online logistic revision methods. In simulations, the average estimated calibration index (aECI) of the original model was 0.828 (95%CI 0.818-0.938). Online recalibration using BLR and MarBLR improved the aECI, attaining 0.265 (95%CI 0.230-0.300) and 0.241 (95%CI 0.216-0.266), respectively. When performing more extensive logistic model revisions, BLR and MarBLR increased the average AUC (aAUC) from 0.767 (95%CI 0.765-0.769) to 0.800 (95%CI 0.798-0.802) and 0.799 (95%CI 0.797-0.801), respectively, in stationary settings and protected against substantial model decay. In the COPD study, BLR and MarBLR dynamically combined the original model with a continually-refitted gradient boosted tree to achieve aAUCs of 0.924 (95%CI 0.913-0.935) and 0.925 (95%CI 0.914-0.935), compared to the static model's aAUC of 0.904 (95%CI 0.892-0.916). Despite its simplicity, BLR is highly competitive with MarBLR. MarBLR outperforms BLR when its prior better reflects the data. BLR and MarBLR can improve the transportability of clinical prediction models and maintain their performance over time.

Belonging to the family of Bayesian nonparametrics, Gaussian process (GP) based approaches have well-documented merits not only in learning over a rich class of nonlinear functions, but also in quantifying the associated uncertainty. However, most GP methods rely on a single preselected kernel function, which may fall short in characterizing data samples that arrive sequentially in time-critical applications. To enable {\it online} kernel adaptation, the present work advocates an incremental ensemble (IE-) GP framework, where an EGP meta-learner employs an {\it ensemble} of GP learners, each having a unique kernel belonging to a prescribed kernel dictionary. With each GP expert leveraging the random feature-based approximation to perform online prediction and model update with {\it scalability}, the EGP meta-learner capitalizes on data-adaptive weights to synthesize the per-expert predictions. Further, the novel IE-GP is generalized to accommodate time-varying functions by modeling structured dynamics at the EGP meta-learner and within each GP learner. To benchmark the performance of IE-GP and its dynamic variant in the adversarial setting where the modeling assumptions are violated, rigorous performance analysis has been conducted via the notion of regret, as the norm in online convex optimization. Last but not the least, online unsupervised learning for dimensionality reduction is explored under the novel IE-GP framework. Synthetic and real data tests demonstrate the effectiveness of the proposed schemes.

Many time series can be modeled as a sequence of segments representing high-level discrete states, such as running and walking in a human activity application. Flexible models should describe the system state and observations in stationary ``pure-state'' periods as well as transition periods between adjacent segments, such as a gradual slowdown between running and walking. However, most prior work assumes instantaneous transitions between pure discrete states. We propose a dynamical Wasserstein barycentric (DWB) model that estimates the system state over time as well as the data-generating distributions of pure states in an unsupervised manner. Our model assumes each pure state generates data from a multivariate normal distribution, and characterizes transitions between states via displacement-interpolation specified by the Wasserstein barycenter. The system state is represented by a barycentric weight vector which evolves over time via a random walk on the simplex. Parameter learning leverages the natural Riemannian geometry of Gaussian distributions under the Wasserstein distance, which leads to improved convergence speeds. Experiments on several human activity datasets show that our proposed DWB model accurately learns the generating distribution of pure states while improving state estimation for transition periods compared to the commonly used linear interpolation mixture models.

Node embeddings are a powerful tool in the analysis of networks; yet, their full potential for the important task of node clustering has not been fully exploited. In particular, most state-of-the-art methods generating node embeddings of signed networks focus on link sign prediction, and those that pertain to node clustering are usually not graph neural network (GNN) methods. Here, we introduce a novel probabilistic balanced normalized cut loss for training nodes in a GNN framework for semi-supervised signed network clustering, called SSSNET. The method is end-to-end in combining embedding generation and clustering without an intermediate step; it has node clustering as main focus, with an emphasis on polarization effects arising in networks. The main novelty of our approach is a new take on the role of social balance theory for signed network embeddings. The standard heuristic for justifying the criteria for the embeddings hinges on the assumption that "an enemy's enemy is a friend". Here, instead, a neutral stance is assumed on whether or not the enemy of an enemy is a friend. Experimental results on various data sets, including a synthetic signed stochastic block model, a polarized version of it, and real-world data at different scales, demonstrate that SSSNET can achieve comparable or better results than state-of-the-art spectral clustering methods, for a wide range of noise and sparsity levels. SSSNET complements existing methods through the possibility of including exogenous information, in the form of node-level features or labels.

Survival analysis is a critical tool for the modelling of time-to-event data, such as life expectancy after a cancer diagnosis or optimal maintenance scheduling for complex machinery. However, current neural network models provide an imperfect solution for survival analysis as they either restrict the shape of the target probability distribution or restrict the estimation to pre-determined times. As a consequence, current survival neural networks lack the ability to estimate a generic function without prior knowledge of its structure. In this article, we present the metaparametric neural network framework that encompasses existing survival analysis methods and enables their extension to solve the aforementioned issues. This framework allows survival neural networks to satisfy the same independence of generic function estimation from the underlying data structure that characterizes their regression and classification counterparts. Further, we demonstrate the application of the metaparametric framework using both simulated and large real-world datasets and show that it outperforms the current state-of-the-art methods in (i) capturing nonlinearities, and (ii) identifying temporal patterns, leading to more accurate overall estimations whilst placing no restrictions on the underlying function structure.

Modern pattern recognition tasks use complex algorithms that take advantage of large datasets to make more accurate predictions than traditional algorithms such as decision trees or k-nearest-neighbor better suited to describe simple structures. While increased accuracy is often crucial, less complexity also has value. This paper proposes a training data selection algorithm that identifies multiple subsets with simple structures. A learning algorithm trained on such a subset can classify an instance belonging to the subset with better accuracy than the traditional learning algorithms. In other words, while existing pattern recognition algorithms attempt to learn a global mapping function to represent the entire dataset, we argue that an ensemble of simple local patterns may better describe the data. Hence the sub-setting algorithm identifies multiple subsets with simple local patterns by identifying similar instances in the neighborhood of an instance. This motivation has similarities to that of gradient boosted trees but focuses on the explainability of the model that is missing for boosted trees. The proposed algorithm thus balances accuracy and explainable machine learning by identifying a limited number of subsets with simple structures. We applied the proposed algorithm to the international stroke dataset to predict the probability of survival. Our bottom-up sub-setting algorithm performed on an average 15% better than the top-down decision tree learned on the entire dataset. The different decision trees learned on the identified subsets use some of the previously unused features by the whole dataset decision tree, and each subset represents a distinct population of data.

We propose a new class of random feature methods for linearizing softmax and Gaussian kernels called hybrid random features (HRFs) that automatically adapt the quality of kernel estimation to provide most accurate approximation in the defined regions of interest. Special instantiations of HRFs lead to well-known methods such as trigonometric (Rahimi and Recht, 2007) or (recently introduced in the context of linear-attention Transformers) positive random features (Choromanski et al., 2021). By generalizing Bochner's Theorem for softmax/Gaussian kernels and leveraging random features for compositional kernels, the HRF-mechanism provides strong theoretical guarantees - unbiased approximation and strictly smaller worst-case relative errors than its counterparts. We conduct exhaustive empirical evaluation of HRF ranging from pointwise kernel estimation experiments, through tests on data admitting clustering structure to benchmarking implicit-attention Transformers (also for downstream Robotics applications), demonstrating its quality in a wide spectrum of machine learning problems.

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