Spectral structure learning for clinical time series

We develop and evaluate a structure learning algorithm for clinical time series. Clinical time series are multivariate time series observed in multiple patients and irregularly sampled, challenging existing structure learning algorithms. We assume that our times series are realizations of StructGP, a k-dimensional multi-output or multi-task stationary Gaussian process (GP), with independent patients sharing the same covariance function. StructGP encodes ordered conditional relations between time series, represented in a directed acyclic graph. We implement an adapted NOTEARS algorithm, which based on a differentiable definition of acyclicity, recovers the graph by solving a series of continuous optimization problems. Simulation results show that up to mean degree 3 and 20 tasks, we reach a median recall of 0.93% [IQR, 0.86, 0.97] while keeping a median precision of 0.71% [0.57-0.84], for recovering directed edges. We further show that the regularization path is key to identifying the graph. With StructGP, we proposed a model of time series dependencies, that flexibly adapt to different time series regularity, while enabling us to learn these dependencies from observations.

An Uncertainty Principle for Linear Recurrent Neural Networks

We consider linear recurrent neural networks, which have become a key building block of sequence modeling due to their ability for stable and effective long-range modeling. In this paper, we aim at characterizing this ability on a simple but core copy task, whose goal is to build a linear filter of order $S$ that approximates the filter that looks $K$ time steps in the past (which we refer to as the shift-$K$ filter), where $K$ is larger than $S$. Using classical signal models and quadratic cost, we fully characterize the problem by providing lower bounds of approximation, as well as explicit filters that achieve this lower bound up to constants. The optimal performance highlights an uncertainty principle: the optimal filter has to average values around the $K$-th time step in the past with a range~(width) that is proportional to $K/S$.

Statistical Collusion by Collectives on Learning Platforms

As platforms increasingly rely on learning algorithms, collectives may form and seek ways to influence these platforms to align with their own interests. This can be achieved by coordinated submission of altered data. To evaluate the potential impact of such behavior, it is essential to understand the computations that collectives must perform to impact platforms in this way. In particular, collectives need to make a priori assessments of the effect of the collective before taking action, as they may face potential risks when modifying their data. Moreover they need to develop implementable coordination algorithms based on quantities that can be inferred from observed data. We develop a framework that provides a theoretical and algorithmic treatment of these issues and present experimental results in a product evaluation domain.

Building Bridges between Regression, Clustering, and Classification

Regression, the task of predicting a continuous scalar target y based on some features x is one of the most fundamental tasks in machine learning and statistics. It has been observed and theoretically analyzed that the classical approach, meansquared error minimization, can lead to suboptimal results when training neural networks. In this work, we propose a new method to improve the training of these models on regression tasks, with continuous scalar targets. Our method is based on casting this task in a different fashion, using a target encoder, and a prediction decoder, inspired by approaches in classification and clustering. We showcase the performance of our method on a wide range of real-world datasets.

The Surprising Agreement Between Convex Optimization Theory and Learning-Rate Scheduling for Large Model Training

We show that learning-rate schedules for large model training behave surprisingly similar to a performance bound from non-smooth convex optimization theory. We provide a bound for the constant schedule with linear cooldown; in particular, the practical benefit of cooldown is reflected in the bound due to the absence of logarithmic terms. Further, we show that this surprisingly close match between optimization theory and practice can be exploited for learning-rate tuning: we achieve noticeable improvements for training 124M and 210M Llama-type models by (i) extending the schedule for continued training with optimal learning-rate, and (ii) transferring the optimal learning-rate across schedules.

Rethinking Early Stopping: Refine, Then Calibrate

Machine learning classifiers often produce probabilistic predictions that are critical for accurate and interpretable decision-making in various domains. The quality of these predictions is generally evaluated with proper losses like cross-entropy, which decompose into two components: calibration error assesses general under/overconfidence, while refinement error measures the ability to distinguish different classes. In this paper, we provide theoretical and empirical evidence that these two errors are not minimized simultaneously during training. Selecting the best training epoch based on validation loss thus leads to a compromise point that is suboptimal for both calibration error and, most importantly, refinement error. To address this, we introduce a new metric for early stopping and hyperparameter tuning that makes it possible to minimize refinement error during training. The calibration error is minimized after training, using standard techniques. Our method integrates seamlessly with any architecture and consistently improves performance across diverse classification tasks.

Variational Inference on the Boolean Hypercube with the Quantum Entropy

In this paper, we derive variational inference upper-bounds on the log-partition function of pairwise Markov random fields on the Boolean hypercube, based on quantum relaxations of the Kullback-Leibler divergence. We then propose an efficient algorithm to compute these bounds based on primal-dual optimization. An improvement of these bounds through the use of ''hierarchies,'' similar to sum-of-squares (SoS) hierarchies is proposed, and we present a greedy algorithm to select among these relaxations. We carry extensive numerical experiments and compare with state-of-the-art methods for this inference problem.

Efficient Optimization Algorithms for Linear Adversarial Training

Adversarial training can be used to learn models that are robust against perturbations. For linear models, it can be formulated as a convex optimization problem. Compared to methods proposed in the context of deep learning, leveraging the optimization structure allows significantly faster convergence rates. Still, the use of generic convex solvers can be inefficient for large-scale problems. Here, we propose tailored optimization algorithms for the adversarial training of linear models, which render large-scale regression and classification problems more tractable. For regression problems, we propose a family of solvers based on iterative ridge regression and, for classification, a family of solvers based on projected gradient descent. The methods are based on extended variable reformulations of the original problem. We illustrate their efficiency in numerical examples.

Optimizing Estimators of Squared Calibration Errors in Classification

In this work, we propose a mean-squared error-based risk that enables the comparison and optimization of estimators of squared calibration errors in practical settings. Improving the calibration of classifiers is crucial for enhancing the trustworthiness and interpretability of machine learning models, especially in sensitive decision-making scenarios. Although various calibration (error) estimators exist in the current literature, there is a lack of guidance on selecting the appropriate estimator and tuning its hyperparameters. By leveraging the bilinear structure of squared calibration errors, we reformulate calibration estimation as a regression problem with independent and identically distributed (i.i.d.) input pairs. This reformulation allows us to quantify the performance of different estimators even for the most challenging calibration criterion, known as canonical calibration. Our approach advocates for a training-validation-testing pipeline when estimating a calibration error on an evaluation dataset. We demonstrate the effectiveness of our pipeline by optimizing existing calibration estimators and comparing them with novel kernel ridge regression-based estimators on standard image classification tasks.

Statistical and Geometrical properties of regularized Kernel Kullback-Leibler divergence

In this paper, we study the statistical and geometrical properties of the Kullback-Leibler divergence with kernel covariance operators (KKL) introduced by Bach [2022]. Unlike the classical Kullback-Leibler (KL) divergence that involves density ratios, the KKL compares probability distributions through covariance operators (embeddings) in a reproducible kernel Hilbert space (RKHS), and compute the Kullback-Leibler quantum divergence. This novel divergence hence shares parallel but different aspects with both the standard Kullback-Leibler between probability distributions and kernel embeddings metrics such as the maximum mean discrepancy. A limitation faced with the original KKL divergence is its inability to be defined for distributions with disjoint supports. To solve this problem, we propose in this paper a regularised variant that guarantees that the divergence is well defined for all distributions. We derive bounds that quantify the deviation of the regularised KKL to the original one, as well as finite-sample bounds. In addition, we provide a closed-form expression for the regularised KKL, specifically applicable when the distributions consist of finite sets of points, which makes it implementable. Furthermore, we derive a Wasserstein gradient descent scheme of the KKL divergence in the case of discrete distributions, and study empirically its properties to transport a set of points to a target distribution.