Survival analysis aims to predict the timing of future events across various fields, from medical outcomes to customer churn. However, the integration of clustering into survival analysis, particularly for precision medicine, remains underexplored. This study introduces SurvMixClust, a novel algorithm for survival analysis that integrates clustering with survival function prediction within a unified framework. SurvMixClust learns latent representations for clustering while also predicting individual survival functions using a mixture of non-parametric experts. Our evaluations on five public datasets show that SurvMixClust creates balanced clusters with distinct survival curves, outperforms clustering baselines, and competes with non-clustering survival models in predictive accuracy, as measured by the time-dependent c-index and log-rank metrics.

This paper presents two models of neural-networks and their training applicable to neural networks of arbitrary width, depth and topology, assuming only finite-energy neural activations; and a novel representor theory for neural networks in terms of a matrix-valued kernel. The first model is exact (un-approximated) and global, casting the neural network as an elements in a reproducing kernel Banach space (RKBS); we use this model to provide tight bounds on Rademacher complexity. The second model is exact and local, casting the change in neural network function resulting from a bounded change in weights and biases (ie. a training step) in reproducing kernel Hilbert space (RKHS) in terms of a local-intrinsic neural kernel (LiNK). This local model provides insight into model adaptation through tight bounds on Rademacher complexity of network adaptation. We also prove that the neural tangent kernel (NTK) is a first-order approximation of the LiNK kernel. Finally, and noting that the LiNK does not provide a representor theory for technical reasons, we present an exact novel representor theory for layer-wise neural network training with unregularized gradient descent in terms of a local-extrinsic neural kernel (LeNK). This representor theory gives insight into the role of higher-order statistics in neural network training and the effect of kernel evolution in neural-network kernel models. Throughout the paper (a) feedforward ReLU networks and (b) residual networks (ResNet) are used as illustrative examples.

State-of-the-art neural network training methods depend on the gradient of the network function. Therefore, they cannot be applied to networks whose activation functions do not have useful derivatives, such as binary and discrete-time spiking neural networks. To overcome this problem, the activation function's derivative is commonly substituted with a surrogate derivative, giving rise to surrogate gradient learning (SGL). This method works well in practice but lacks theoretical foundation. The neural tangent kernel (NTK) has proven successful in the analysis of gradient descent. Here, we provide a generalization of the NTK, which we call the surrogate gradient NTK, that enables the analysis of SGL. First, we study a naive extension of the NTK to activation functions with jumps, demonstrating that gradient descent for such activation functions is also ill-posed in the infinite-width limit. To address this problem, we generalize the NTK to gradient descent with surrogate derivatives, i.e., SGL. We carefully define this generalization and expand the existing key theorems on the NTK with mathematical rigor. Further, we illustrate our findings with numerical experiments. Finally, we numerically compare SGL in networks with sign activation function and finite width to kernel regression with the surrogate gradient NTK; the results confirm that the surrogate gradient NTK provides a good characterization of SGL.

Optimal Transport (OT) based distances are powerful tools for machine learning to compare probability measures and manipulate them using OT maps. In this field, a setting of interest is semi-discrete OT, where the source measure $\mu$ is continuous, while the target $\nu$ is discrete. Recent works have shown that the minimax rate for the OT map is $\mathcal{O}(t^{-1/2})$ when using $t$ i.i.d. subsamples from each measure (two-sample setting). An open question is whether a better convergence rate can be achieved when the full information of the discrete measure $\nu$ is known (one-sample setting). In this work, we answer positively to this question by (i) proving an $\mathcal{O}(t^{-1})$ lower bound rate for the OT map, using the similarity between Laguerre cells estimation and density support estimation, and (ii) proposing a Stochastic Gradient Descent (SGD) algorithm with adaptive entropic regularization and averaging acceleration. To nearly achieve the desired fast rate, characteristic of non-regular parametric problems, we design an entropic regularization scheme decreasing with the number of samples. Another key step in our algorithm consists of using a projection step that permits to leverage the local strong convexity of the regularized OT problem. Our convergence analysis integrates online convex optimization and stochastic gradient techniques, complemented by the specificities of the OT semi-dual. Moreover, while being as computationally and memory efficient as vanilla SGD, our algorithm achieves the unusual fast rates of our theory in numerical experiments.

Determining the form of the Higgs potential is one of the most exciting challenges of modern particle physics. Higgs pair production directly probes the Higgs self-coupling and should be observed in the near future at the High-Luminosity LHC. We explore how to improve the sensitivity to physics beyond the Standard Model through per-event kinematics for di-Higgs events. In particular, we employ machine learning through simulation-based inference to estimate per-event likelihood ratios and gauge potential sensitivity gains from including this kinematic information. In terms of the Standard Model Effective Field Theory, we find that adding a limited number of observables can help to remove degeneracies in Wilson coefficient likelihoods and significantly improve the experimental sensitivity.

Influence functions serve as crucial tools for assessing sample influence in model interpretation, subset training set selection, noisy label detection, and more. By employing the first-order Taylor extension, influence functions can estimate sample influence without the need for expensive model retraining. However, applying influence functions directly to deep models presents challenges, primarily due to the non-convex nature of the loss function and the large size of model parameters. This difficulty not only makes computing the inverse of the Hessian matrix costly but also renders it non-existent in some cases. Various approaches, including matrix decomposition, have been explored to expedite and approximate the inversion of the Hessian matrix, with the aim of making influence functions applicable to deep models. In this paper, we revisit a specific, albeit naive, yet effective approximation method known as TracIn. This method substitutes the inverse of the Hessian matrix with an identity matrix. We provide deeper insights into why this simple approximation method performs well. Furthermore, we extend its applications beyond measuring model utility to include considerations of fairness and robustness. Finally, we enhance TracIn through an ensemble strategy. To validate its effectiveness, we conduct experiments on synthetic data and extensive evaluations on noisy label detection, sample selection for large language model fine-tuning, and defense against adversarial attacks.

Neural networks can identify low-dimensional relevant structures within high-dimensional noisy data, yet our mathematical understanding of how they do so remains scarce. Here, we investigate the training dynamics of two-layer shallow neural networks trained with gradient-based algorithms, and discuss how they learn pertinent features in multi-index models, that is target functions with low-dimensional relevant directions. In the high-dimensional regime, where the input dimension $d$ diverges, we show that a simple modification of the idealized single-pass gradient descent training scenario, where data can now be repeated or iterated upon twice, drastically improves its computational efficiency. In particular, it surpasses the limitations previously believed to be dictated by the Information and Leap exponents associated with the target function to be learned. Our results highlight the ability of networks to learn relevant structures from data alone without any pre-processing. More precisely, we show that (almost) all directions are learned with at most $O(d \log d)$ steps. Among the exceptions is a set of hard functions that includes sparse parities. In the presence of coupling between directions, however, these can be learned sequentially through a hierarchical mechanism that generalizes the notion of staircase functions. Our results are proven by a rigorous study of the evolution of the relevant statistics for high-dimensional dynamics.

Machine learning models for continuous outcomes often yield systematically biased predictions, particularly for values that largely deviate from the mean. Specifically, predictions for large-valued outcomes tend to be negatively biased, while those for small-valued outcomes are positively biased. We refer to this linear central tendency warped bias as the "systematic bias of machine learning regression". In this paper, we first demonstrate that this issue persists across various machine learning models, and then delve into its theoretical underpinnings. We propose a general constrained optimization approach designed to correct this bias and develop a computationally efficient algorithm to implement our method. Our simulation results indicate that our correction method effectively eliminates the bias from the predicted outcomes. We apply the proposed approach to the prediction of brain age using neuroimaging data. In comparison to competing machine learning models, our method effectively addresses the longstanding issue of "systematic bias of machine learning regression" in neuroimaging-based brain age calculation, yielding unbiased predictions of brain age.

Gradient descent and its variants are de facto standard algorithms for training machine learning models. As gradient descent is sensitive to its hyperparameters, we need to tune the hyperparameters carefully using a grid search, but it is time-consuming, especially when multiple hyperparameters exist. Recently, parameter-free methods that adjust the hyperparameters on the fly have been studied. However, the existing work only studied parameter-free methods for the stepsize, and parameter-free methods for other hyperparameters have not been explored. For instance, the gradient clipping threshold is also a crucial hyperparameter in addition to the stepsize to prevent gradient explosion issues, but none of the existing studies investigated the parameter-free methods for clipped gradient descent. In this work, we study the parameter-free methods for clipped gradient descent. Specifically, we propose Inexact Polyak Stepsize, which converges to the optimal solution without any hyperparameters tuning, and its convergence rate is asymptotically independent of L under L-smooth and $(L_0, L_1)$-smooth assumptions of the loss function as that of clipped gradient descent with well-tuned hyperparameters. We numerically validated our convergence results using a synthetic function and demonstrated the effectiveness of our proposed methods using LSTM, Nano-GPT, and T5.

Graph Neural Networks have achieved remarkable accuracy in semi-supervised node classification tasks. However, these results lack reliable uncertainty estimates. Conformal prediction methods provide a theoretical guarantee for node classification tasks, ensuring that the conformal prediction set contains the ground-truth label with a desired probability (e.g., 95%). In this paper, we empirically show that for each node, aggregating the non-conformity scores of nodes with the same label can improve the efficiency of conformal prediction sets. This observation motivates us to propose a novel algorithm named Similarity-Navigated Adaptive Prediction Sets (SNAPS), which aggregates the non-conformity scores based on feature similarity and structural neighborhood. The key idea behind SNAPS is that nodes with high feature similarity or direct connections tend to have the same label. By incorporating adaptive similar nodes information, SNAPS can generate compact prediction sets and increase the singleton hit ratio (correct prediction sets of size one). Moreover, we theoretically provide a finite-sample coverage guarantee of SNAPS. Extensive experiments demonstrate the superiority of SNAPS, improving the efficiency of prediction sets and singleton hit ratio while maintaining valid coverage.