A Library of Mirrors: Deep Neural Nets in Low Dimensions are Convex Lasso Models with Reflection Features

We prove that training neural networks on 1-D data is equivalent to solving a convex Lasso problem with a fixed, explicitly defined dictionary matrix of features. The specific dictionary depends on the activation and depth. We consider 2-layer networks with piecewise linear activations, deep narrow ReLU networks with up to 4 layers, and rectangular and tree networks with sign activation and arbitrary depth. Interestingly in ReLU networks, a fourth layer creates features that represent reflections of training data about themselves. The Lasso representation sheds insight to globally optimal networks and the solution landscape.

Bellman Conformal Inference: Calibrating Prediction Intervals For Time Series

We introduce Bellman Conformal Inference (BCI), a framework that wraps around any time series forecasting models and provides approximately calibrated prediction intervals. Unlike existing methods, BCI is able to leverage multi-step ahead forecasts and explicitly optimize the average interval lengths by solving a one-dimensional stochastic control problem (SCP) at each time step. In particular, we use the dynamic programming algorithm to find the optimal policy for the SCP. We prove that BCI achieves long-term coverage under arbitrary distribution shifts and temporal dependence, even with poor multi-step ahead forecasts. We find empirically that BCI avoids uninformative intervals that have infinite lengths and generates substantially shorter prediction intervals in multiple applications when compared with existing methods.

Conformal Inference for Online Prediction with Arbitrary Distribution Shifts

Conformal inference is a flexible methodology for transforming the predictions made by any black-box model (e.g. neural nets, random forests) into valid prediction sets. The only necessary assumption is that the training and test data be exchangeable (e.g. i.i.d.). Unfortunately, this assumption is usually unrealistic in online environments in which the processing generating the data may vary in time and consecutive data-points are often temporally correlated. In this article, we develop an online algorithm for producing prediction intervals that are robust to these deviations. Our methods build upon conformal inference and thus can be combined with any black-box predictor. We show that the coverage error of our algorithm is controlled by the size of the underlying change in the environment and thus directly connect the size of the distribution shift with the difficulty of the prediction problem. Finally, we apply our procedure in two real-world settings and find that our method produces robust prediction intervals under real-world dynamics.

Testing for Outliers with Conformal p-values

This paper studies the construction of p-values for nonparametric outlier detection, taking a multiple-testing perspective. The goal is to test whether new independent samples belong to the same distribution as a reference data set or are outliers. We propose a solution based on conformal inference, a broadly applicable framework which yields p-values that are marginally valid but mutually dependent for different test points. We prove these p-values are positively dependent and enable exact false discovery rate control, although in a relatively weak marginal sense. We then introduce a new method to compute p-values that are both valid conditionally on the training data and independent of each other for different test points; this paves the way to stronger type-I error guarantees. Our results depart from classical conformal inference as we leverage concentration inequalities rather than combinatorial arguments to establish our finite-sample guarantees. Furthermore, our techniques also yield a uniform confidence bound for the false positive rate of any outlier detection algorithm, as a function of the threshold applied to its raw statistics. Finally, the relevance of our results is demonstrated by numerical experiments on real and simulated data.