A.1 Proof of Proposition 1 Proof of Proposition 1. Recall that h denotes the vanilla activations of the network, those we obtain with no noise injection. Let us not inject noise in the final, predictive, layer of our network such that the noise on this layer is accumulated from the noising of previous layers. Let us first consider the Taylor series expansion of the loss function with the accumulated noise defined in Proposition 1. Denoting =[ This can be deduced from the slightly opaque Fa ` a di Bruno's formula, which states that for multivariate derivatives of a composition of functions f: R The final equality comes from the moments of a mean 0 Gaussian, where j takes the values of the multi-index. Though these equalities can already offer insight into the regularising mechanisms of GNIs, they are not easy to work with and will often be computationally intractable. We will include these terms in our remainder term C .

(see Fig 2). See Fig 3 for a demonstration of this. We will also make sure to include the justification for this in the Appendix. Fourier domain is less when injecting noise only on data. Y ou make an interesting point about Fig 1: All models were trained with a relatively low learning rate (lr) of 0.001, In light of this we have run the baseline with lr=0.1 and found that Exploring this connection further would be a very interesting stream of research.

Quantum Computing offers a new paradigm for efficient computing and many AI applications could benefit from its potential boost in performance. However, the main limitation is the constraint to linear operations that hampers the representation of complex relationships in data. In this work, we propose an efficient implementation of quantum splines for non-linear approximation. In particular, we first discuss possible parametrisations, and select the most convenient for exploiting the HHL algorithm to obtain the estimates of spline coefficients. Then, we investigate QSpline performance as an evaluation routine for some of the most popular activation functions adopted in ML. Finally, a detailed comparison with classical alternatives to the HHL is also presented.

While modern day economics, and broadly social science research, is often faced with high dimensional estimation problems in which the number of potential explanatory variables is large, often larger than the number of sample observations, the extant literature for high dimensional methods has focused developments mainly on for conditional mean models. Moving beyond the conditional mean, by estimating quantile regression on the other hand, allows to gauge potentially heterogeneous effects of variables directly across the conditional response distribution. While highly influential in the risk-management and finance literature in calculating risk measures such as VaR (i.e., the loss a portfolio's value incurs at a specific probability level), quantile regression has experienced a recent surge in popularity within the macroeconomic literature to quantify risks and vulnerabilities of output growth in response to summary measures of financial health, aptly named growth-at-risk (GaR) (Adrian et al., 2019; Figueres and Jarociński, 2020; Adams et al., 2020). As an important distinction to literature that focuses on forecasting crisis periods directly such as through Markov-switching models (Hubrich and Tetlow, 2015; Guérin and Marcellino, 2013) or probit models (McCracken et al., 2021), GaR instead gives information about the accumulation of risks facing an economy. Since sources of risk can be numerous, high dimensional quantile problems are becoming ever more pertinent to policy makers and practitioners alike which has spurned methods that deal with variable selection and shrinkage for the quantile regression problem (Chernozhukov et al., 2010; Kohns and Szendrei, 2020; Hasenzagl et al., 2020).

Integration over non-negative integrands is a central problem in machine learning (e.g. for model averaging, (hyper-)parameter marginalisation, and computing posterior predictive distributions). Bayesian Quadrature is a probabilistic numerical integration technique that performs promisingly when compared to traditional Markov Chain Monte Carlo methods. However, in contrast to easily-parallelised MCMC methods, Bayesian Quadrature methods have, thus far, been essentially serial in nature, selecting a single point to sample at each step of the algorithm. We deliver methods to select batches of points at each step, based upon those recently presented in the Batch Bayesian Optimisation literature. Such parallelisation significantly reduces computation time, especially when the integrand is expensive to sample.

Model selection is a crucial issue in machine-learning and a wide variety of penalisation methods (with possibly data dependent complexity penalties) have recently been introduced for this purpose. However their empirical performance is generally not well documented in the literature. It is the goal of this paper to investigate to which extent such recent techniques can be successfully used for the tuning of both the regularisation and kernel parameters in support vector regression (SVR) and the complexity measure in regression trees (CART). This task is traditionally solved via V-fold cross-validation (VFCV), which gives efficient results for a reasonable computational cost. A disadvantage however of VFCV is that the procedure is known to provide an asymptotically suboptimal risk estimate as the number of examples tends to infinity. Recently, a penalisation procedure called V-fold penalisation has been proposed to improve on VFCV, supported by theoretical arguments. Here we report on an extensive set of experiments comparing V-fold penalisation and VFCV for SVR/CART calibration on several benchmark datasets. We highlight cases in which VFCV and V-fold penalisation provide poor estimates of the risk respectively and introduce a modified penalisation technique to reduce the estimation error.