Root-finding Approaches for Computing Conformal Prediction Set

Conformal prediction constructs a confidence set for an unobserved response of a feature vector based on previous identically distributed and exchangeable observations of responses and features. It has a coverage guarantee at any nominal level without additional assumptions on their distribution. Its computation deplorably requires a refitting procedure for all replacement candidates of the target response. In regression settings, this corresponds to an infinite number of model fit. Apart from relatively simple estimators that can be written as pieces of linear function of the response, efficiently computing such sets is difficult and is still considered as an open problem. We exploit the fact that, \emph{often}, conformal prediction sets are intervals whose boundaries can be efficiently approximated by classical root-finding algorithm. We investigate how this approach can overcome many limitations of formerly used strategies and we discuss its complexity and drawbacks

Conformalization of Sparse Generalized Linear Models

Given a sequence of observable variables $\{(x_1, y_1), \ldots, (x_n, y_n)\}$, the conformal prediction method estimates a confidence set for $y_{n+1}$ given $x_{n+1}$ that is valid for any finite sample size by merely assuming that the joint distribution of the data is permutation invariant. Although attractive, computing such a set is computationally infeasible in most regression problems. Indeed, in these cases, the unknown variable $y_{n+1}$ can take an infinite number of possible candidate values, and generating conformal sets requires retraining a predictive model for each candidate. In this paper, we focus on a sparse linear model with only a subset of variables for prediction and use numerical continuation techniques to approximate the solution path efficiently. The critical property we exploit is that the set of selected variables is invariant under a small perturbation of the input data. Therefore, it is sufficient to enumerate and refit the model only at the change points of the set of active features and smoothly interpolate the rest of the solution via a Predictor-Corrector mechanism. We show how our path-following algorithm accurately approximates conformal prediction sets and illustrate its performance using synthetic and real data examples.Comment: ICML 202

Discretized conformal prediction for efficient distribution-free inference

In regression problems where there is no known true underlying model, conformal prediction methods enable prediction intervals to be constructed without any assumptions on the distribution of the underlying data, except that the training and test data are assumed to be exchangeable. However, these methods bear a heavy computational cost-and, to be carried out exactly, the regression algorithm would need to be fitted infinitely many times. In practice, the conformal prediction method is run by simply considering only a finite grid of finely spaced values for the response variable. This paper develops discretized conformal prediction algorithms that are guaranteed to cover the target value with the desired probability, and that offer a tradeoff between computational cost and prediction accuracy

New Dimensions for Wound Strings: The Modular Transformation of Geometry to Topology

We show, using a theorem of Milnor and Margulis, that string theory on compact negatively curved spaces grows new effective dimensions as the space shrinks, generalizing and contextualizing the results in hep-th/0510044. Milnor's theorem relates negative sectional curvature on a compact Riemannian manifold to exponential growth of its fundamental group, which translates in string theory to a higher effective central charge arising from winding strings. This exponential density of winding modes is related by modular invariance to the infrared small perturbation spectrum. Using self-consistent approximations valid at large radius, we analyze this correspondence explicitly in a broad set of time-dependent solutions, finding precise agreement between the effective central charge and the corresponding infrared small perturbation spectrum. This indicates a basic relation between geometry, topology, and dimensionality in string theory.Comment: 28 pages, harvmac big. v2: references and KITP preprint number added, minor change

Defect Formation and Critical Dynamics in the Early Universe

We study the nonequilibrium dynamics leading to the formation of topological defects in a symmetry-breaking phase transition of a quantum scalar field with \lambda\Phi^4 self-interaction in a spatially flat, radiation-dominated Friedmann-Robertson-Walker Universe. The quantum field is initially in a finite-temperature symmetry-restored state and the phase transition develops as the Universe expands and cools. We present a first-principles, microscopic approach in which the nonperturbative, nonequilibrium dynamics of the quantum field is derived from the two-loop, two-particle-irreducible closed-time-path effective action. We numerically solve the dynamical equations for the two-point function and we identify signatures of topological defects in the infrared portion of the momentum-space power spectrum. We find that the density of topological defects formed after the phase transition scales as a power law with the expansion rate of the Universe. We calculate the equilibrium critical exponents of the correlation length and relaxation time for this model and show that the power law exponent of the defect density, for both overdamped and underdamped evolution, is in good agreement with the "freeze-out" scenario of Zurek. We introduce an analytic dynamical model, valid near the critical point, that exhibits the same power law scaling of the defect density with the quench rate. By incorporating the realistic quench of the expanding Universe, our approach illuminates the dynamical mechanisms important for topological defect formation. The observed power law scaling of the defect density with the quench rate, observered here in a quantum field theory context, provides evidence for the "freeze-out" scenario in three spatial dimensions.Comment: 31 pages, RevTex, 8 figures in EPS forma