We study sample quantiles of distributions indexed by estimated parameters, with a on Value-at-Risk related to linear projections of financial returns that whose underlying probability law is heavy-tailed. In this setting, the projection direction and the empirical quantile threshold are estimated from the data, so the standard Bahadur representation under a fixed distribution does not separate the distinct sources of instability. A canonical starting point is Bahadur's representation, which expresses the sample quantile through the empirical distribution function plus a remainder term \cite{bahadur1966}. Empirical-process theory provides a usable scaffolding through the mechanics of half-spaces, symmetric differences, and Glivenko--Cantelli uniform convergence. They yield stability bounds, but absorb changes in projection direction and changes in quantile threshold into a single symmetric-difference measure. Interestingly, a global uniform-convergence requirement is imposed on what is intrinsically a local quantile-stability problem. This paper introduces a Q-Q orthogonality formulation for separating projection-direction and quantile-threshold effects. The object of interest is the difference between the empirical quantile computed using the estimated projection direction and the population quantile computed at the reference projection direction. We decompose this difference into three terms, $\hat q_α(\hat w)-q_α(w_0)=D_1+D_2+D_3$. Here, $D_1$ measures the population quantile movement induced by perturbing the projection direction, $D_2$ measures the empirical quantile fluctuation with the projection direction held fixed, and $D_3$ is the Bahadur-type remainder.

Then each deterministic NN in {πw,b | (w,b) Wπ}is safe if and only if the system of constraints Φ(π,X0,Xu,) is not satisfiable. We prove the equivalent claim that there exists a weight vector (w,b) Wπ for which πw,b is unsafe if and only if Φ(π,X0,Xu,) is satisfiable. First, suppose that there exists a weight vector (w,b) Wπ for which πw,b is unsafe and we want to show that Φ(π,X0,Xu,) is satisfiable. This direction of the proof is straightforward since values of the network's neurons on the unsafe input give rise to a solution of Φ(π,X0,Xu,). Indeed, by assumption there exists a vector of input neuron values x0 X0 for which the corresponding vector of output neuron values xl = πw,b(x0) is unsafe, i.e. xl Xu.

A.1 Dataset Details & Evaluation Metrics As stated earlier, the main application of Extreme Multi-label Text Classification is in e-commerce - product recommendation and dynamic search advertisement - and in document tagging, where the objective of an algorithm is to correctly recommend/advertise among the top-k slots. Thus, for evaluation of the methods, we use precision at k (denoted by P@k), and its propensity scored variant (denoted by PSP@k) [17]. These are standard and widely used metrics by the XMC community [4]. Since P@k treats all the labels equally, it doesn't reveal the performance of the model on tail labels. However, because of the long-tailed distribution in XMC datasets, one of the main challenges is to predict tail labels correctly, which may be more valuable and informative compared to head classes.

By reparameterizing the weights in this way we improve the conditioning of the optimization problem and we speed up convergence of stochastic gradient descent. Our reparameterization is inspired by batch normalization but does not introduce any dependencies between the examples in a minibatch. This means that our method can also be applied successfully to recurrent models such as LSTMs and to noise-sensitive applications such as deep reinforcement learning or generative models, for which batch normalization is less well suited. Although our method is much simpler, it still provides much of the speed-up of full batch normalization. In addition, the computational overhead of our method is lower, permitting more optimization steps to be taken in the same amount of time. We demonstrate the usefulness of our method on applications in supervised image recognition, generative modelling, and deep reinforcement learning.

Deep predictive models of neuronal activity have recently enabled several new discoveries about the selectivity and invariance of neurons in the visual cortex.These models learn a shared set of nonlinear basis functions, which are linearly combined via a learned weight vector to represent a neuron's function.Such weight vectors, which can be thought as embeddings of neuronal function, have been proposed to define functional cell types via unsupervised clustering.However, as deep models are usually highly overparameterized, the learning problem is unlikely to have a unique solution, which raises the question if such embeddings can be used in a meaningful way for downstream analysis.In this paper, we investigate how stable neuronal embeddings are with respect to changes in model architecture and initialization. We find that $L_1$ regularization to be an important ingredient for structured embeddings and develop an adaptive regularization that adjusts the strength of regularization per neuron. This regularization improves both predictive performance and how consistently neuronal embeddings cluster across model fits compared to uniform regularization.To overcome overparametrization, we propose an iterative feature pruning strategy which reduces the dimensionality of performance-optimized models by half without loss of performance and improves the consistency of neuronal embeddings with respect to clustering neurons.Our results suggest that to achieve an objective taxonomy of cell types or a compact representation of the functional landscape, we need novel architectures or learning techniques that improve identifiability.

In this paper we study the problem of learning Rectified Linear Units (ReLUs) which are functions of the form $\vct{x}\mapsto \max(0,\langle \vct{w},\vct{x}\rangle)$ with $\vct{w}\in\R^d$ denoting the weight vector. We study this problem in the high-dimensional regime where the number of observations are fewer than the dimension of the weight vector. We assume that the weight vector belongs to some closed set (convex or nonconvex) which captures known side-information about its structure. We focus on the realizable model where the inputs are chosen i.i.d.~from a Gaussian distribution and the labels are generated according to a planted weight vector. We show that projected gradient descent, when initialized at $\vct{0}$, converges at a linear rate to the planted model with a number of samples that is optimal up to numerical constants. Our results on the dynamics of convergence of these very shallow neural nets may provide some insights towards understanding the dynamics of deeper architectures.

By reparameterizing the weights in this way we improve the conditioning of the optimization problem and we speed up convergence of stochastic gradient descent. Our reparameterization is inspired by batch normalization but does not introduce any dependencies between the examples in a minibatch. This means that our method can also be applied successfully to recurrent models such as LSTMs and to noise-sensitive applications such as deep reinforcement learning or generative models, for which batch normalization is less well suited. Although our method is much simpler, it still provides much of the speed-up of full batch normalization. In addition, the computational overhead of our method is lower, permitting more optimization steps to be taken in the same amount of time. We demonstrate the usefulness of our method on applications in supervised image recognition, generative modelling, and deep reinforcement learning.

Suppose an n x d design matrix in a linear regression problem is given, but the response for each point is hidden unless explicitly requested. The goal is to sample only a small number k << n of the responses, and then produce a weight vector whose sum of squares loss over points is at most 1+epsilon times the minimum. When k is very small (e.g., k=d), jointly sampling diverse subsets of points is crucial. One such method called volume sampling has a unique and desirable property that the weight vector it produces is an unbiased estimate of the optimum. It is therefore natural to ask if this method offers the optimal unbiased estimate in terms of the number of responses k needed to achieve a 1+epsilon loss approximation. Surprisingly we show that volume sampling can have poor behavior when we require a very accurate approximation -- indeed worse than some i.i.d.

We show that it provides significant benefits to RL agents on a suite of gridworld, CARL and MetaWorld tasks.