We empirically demonstrate that full-batch gradient descent on neural network training objectives typically operates in a regime we call the Edge of Stability. In this regime, the maximum eigenvalue of the training loss Hessian hovers just above the numerical value $2 / \text{(step size)}$, and the training loss behaves non-monotonically over short timescales, yet consistently decreases over long timescales. Since this behavior is inconsistent with several widespread presumptions in the field of optimization, our findings raise questions as to whether these presumptions are relevant to neural network training. We hope that our findings will inspire future efforts aimed at rigorously understanding optimization at the Edge of Stability. Code is available at https://github.com/locuslab/edge-of-stability.

Tensor (multidimensional array) classification problem has become very popular in modern applications such as image recognition and high dimensional spatio-temporal data analysis. Support Tensor Machine (STM) classifier, which is extended from the support vector machine, takes CANDECOMP / Parafac (CP) form of tensor data as input and predicts the data labels. The distribution-free and statistically consistent properties of STM highlight its potential in successfully handling wide varieties of data applications. Training a STM can be computationally expensive with high-dimensional tensors. However, reducing the size of tensor with a random projection technique can reduce the computational time and cost, making it feasible to handle large size tensors on regular machines. We name an STM estimated with randomly projected tensor as Random Projection-based Support Tensor Machine (RPSTM). In this work, we propose a Tensor Ensemble Classifier (TEC), which aggregates multiple RPSTMs for big tensor classification. TEC utilizes the ensemble idea to minimize the excessive classification risk brought by random projection, providing statistically consistent predictions while taking the computational advantage of RPSTM. Since each RPSTM can be estimated independently, TEC can further take advantage of parallel computing techniques and be more computationally efficient. The theoretical and numerical results demonstrate the decent performance of TEC model in high-dimensional tensor classification problems. The model prediction is statistically consistent as its risk is shown to converge to the optimal Bayes risk. Besides, we highlight the trade-off between the computational cost and the prediction risk for TEC model. The method is validated by extensive simulation and a real data example. We prepare a python package for applying TEC, which is available at our GitHub.

The extensive use of smartphones in our everyday lives has created new modes of appropriation and behavior in public spaces. Recognition of these are essential for urban design and planning practices which help us to improve the relationship between humans, technologies, and urban environment. This study aims to research smartphone users in public space by observing their altering visual attention and walking behavior, and, in this way, to reveal the emergent new figures. For this purpose, Korenmarkt square in Ghent, Belgium, was observed for seven days in 10-min time intervals. The gaze and walking behavior of smartphone users were encoded as geo-located and temporal data, analyzed and mapped using statistical and spatial analysis methods. Developing and implementing new methods for identifying the characteristics of smartphone users, this study resulted in a nuanced characterization of novel spatial appropriations. The findings led to a better understanding and knowledge of the different behavior patterns of emergent figures such as post-flaneurs and smartphone zombies while uncovering their altering visual interactions with and movements in the public space. The results evoked questions on how researchers and designers can make use of spatial analysis methods and rethink the public space of the future as a hybrid construct integrating the virtual and the physical.

Access to quality travel time information for roads in a road network has become increasingly important with the rising demand for real-time travel time estimation for paths within road networks. In the context of the Danish road network (DRN) dataset used in this paper, the data coverage is sparse and skewed towards arterial roads, with a coverage of 23.88% across 850,980 road segments, which makes travel time estimation difficult. Existing solutions for graph-based data processing often neglect the size of the graph, which is an apparent problem for road networks with a large amount of connected road segments. To this end, we propose a framework for predicting road segment travel speed histograms for dataless edges, based on a latent representation generated by an adversarially regularized convolutional network. We apply a partitioning algorithm to divide the graph into dense subgraphs, and then train a model for each subgraph to predict speed histograms for the nodes. The framework achieves an accuracy of 71.5% intersection and 78.5% correlation on predicting travel speed histograms using the DRN dataset. Furthermore, experiments show that partitioning the dataset into clusters increases the performance of the framework. Specifically, partitioning the road network dataset into 100 clusters, with approximately 500 road segments in each cluster, achieves a better performance than when using 10 and 20 clusters.

While momentum-based methods, in conjunction with stochastic gradient descent (SGD), are widely used when training machine learning models, there is little theoretical understanding on the generalization error of such methods. In this work, we first show that there exists a convex loss function for which algorithmic stability fails to establish generalization guarantees when SGD with standard heavy-ball momentum (SGDM) is run for multiple epochs. Then, for smooth Lipschitz loss functions, we analyze a modified momentum-based update rule, i.e., SGD with early momentum (SGDEM), and show that it admits an upper-bound on the generalization error. Thus, our results show that machine learning models can be trained for multiple epochs of SGDEM with a guarantee for generalization. Finally, for the special case of strongly convex loss functions, we find a range of momentum such that multiple epochs of standard SGDM, as a special form of SGDEM, also generalizes. Extending our results on generalization, we also develop an upper-bound on the expected true risk, in terms of the number of training steps, the size of the training set, and the momentum parameter. Experimental evaluations verify the consistency between the numerical results and our theoretical bounds and the effectiveness of SGDEM for smooth Lipschitz loss functions.

Additive noise models are a class of causal models in which each variable is defined as a function of its causes plus independent noise. In such models, the ordering of variables by marginal variances may be indicative of the causal order. We introduce varsortability as a measure of agreement between the ordering by marginal variance and the causal order. We show how varsortability dominates the performance of continuous structure learning algorithms on synthetic data. On real-world data, varsortability is an implausible and untestable assumption and we find no indication of high varsortability. We aim to raise awareness that varsortability easily occurs in simulated additive noise models. We provide a baseline method that explicitly exploits varsortability and advocate reporting varsortability in benchmarking data.

We study the generation of prediction intervals in regression for uncertainty quantification. This task can be formalized as an empirical constrained optimization problem that minimizes the average interval width while maintaining the coverage accuracy across data. We strengthen the existing literature by studying two aspects of this empirical optimization. First is a general learning theory to characterize the optimality-feasibility tradeoff that encompasses Lipschitz continuity and VC-subgraph classes, which are exemplified in regression trees and neural networks. Second is a calibration machinery and the corresponding statistical theory to optimally select the regularization parameter that manages this tradeoff, which bypasses the overfitting issues in previous approaches in coverage attainment. We empirically demonstrate the strengths of our interval generation and calibration algorithms in terms of testing performances compared to existing benchmarks.

Variable importance measures are the main tools to analyze the black-box mechanism of random forests. Although the Mean Decrease Accuracy (MDA) is widely accepted as the most efficient variable importance measure for random forests, little is known about its theoretical properties. In fact, the exact MDA definition varies across the main random forest software. In this article, our objective is to rigorously analyze the behavior of the main MDA implementations. Consequently, we mathematically formalize the various implemented MDA algorithms, and then establish their limits when the sample size increases. In particular, we break down these limits in three components: the first two are related to Sobol indices, which are well-defined measures of a variable contribution to the output variance, widely used in the sensitivity analysis field, as opposed to the third term, whose value increases with dependence within input variables. Thus, we theoretically demonstrate that the MDA does not target the right quantity when inputs are dependent, a fact that has already been noticed experimentally. To address this issue, we define a new importance measure for random forests, the Sobol-MDA, which fixes the flaws of the original MDA. We prove the consistency of the Sobol-MDA and show its good empirical performance through experiments on both simulated and real data. An open source implementation in R and C++ is available online.

As predictive models are increasingly being deployed in high-stakes decision making (e.g., loan approvals), there has been growing interest in post hoc techniques which provide recourse to affected individuals. These techniques generate recourses under the assumption that the underlying predictive model does not change. However, in practice, models are often regularly updated for a variety of reasons (e.g., dataset shifts), thereby rendering previously prescribed recourses ineffective. To address this problem, we propose a novel framework, RObust Algorithmic Recourse (ROAR), that leverages adversarial training for finding recourses that are robust to model shifts. To the best of our knowledge, this work proposes the first solution to this critical problem. We also carry out detailed theoretical analysis which underscores the importance of constructing recourses that are robust to model shifts: 1) we derive a lower bound on the probability of invalidation of recourses generated by existing approaches which are not robust to model shifts. 2) we prove that the additional cost incurred due to the robust recourses output by our framework is bounded. Experimental evaluation on multiple synthetic and real-world datasets demonstrates the efficacy of the proposed framework and supports our theoretical findings.

The Zoetrope Genetic Programming (ZGP) algorithm is based on an original representation for mathematical expressions, targeting evolutionary symbolic regression.The zoetropic representation uses repeated fusion operations between partial expressions, starting from the terminal set. Repeated fusions within an individual gradually generate more complex expressions, ending up in what can be viewed as new features. These features are then linearly combined to best fit the training data. ZGP individuals then undergo specific crossover and mutation operators, and selection takes place between parents and offspring. ZGP is validated using a large number of public domain regression datasets, and compared to other symbolic regression algorithms, as well as to traditional machine learning algorithms. ZGP reaches state-of-the-art performance with respect to both types of algorithms, and demonstrates a low computational time compared to other symbolic regression approaches.