This paper investigates the asymptotic distribution of the maximum-likelihood estimate (MLE) in multinomial logistic models in the high-dimensional regime where dimension and sample size are of the same order.

However, simply pooling everyone's data and sharing with each other can lead to free-riding [

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Input-Convex Neural Networks (ICNNs) are networks that guarantee convexity in their input-output mapping. These networks have been successfully applied for energy-based modelling, optimal transport problems and learning invariances. The convexity of ICNNs is achieved by using non-decreasing convex activation functions and non-negative weights. Because of these peculiarities, previous initialisation strategies, which implicitly assume centred weights, are not effective for ICNNs. By studying signal propagation through layers with non-negative weights, we are able to derive a principled weight initialisation for ICNNs. Concretely, we generalise signal propagation theory by removing the assumption that weights are sampled from a centred distribution. In a set of experiments, we demonstrate that our principled initialisation effectively accelerates learning in ICNNs and leads to better generalisation. Moreover, we find that, in contrast to common belief, ICNNs can be trained without skip-connections when initialised correctly. Finally, we apply ICNNs to a real-world drug discovery task and show that they allow for more effective molecular latent space exploration.

We first provide the detailed algorithm for binomial estimation. Before doing so we recall a few definitions. Our algorithms will crucially rely on the use of Gray codes, which we define next. We next describe the detailed protocol below. The server: One of the 3 following cases must hold.

These jump from their maximum possible value to zero, at well denullned signal-to-noise thresholds whose asymptotic values we determine exactly.

However, simply pooling everyone's data and sharing with each other can lead to free-riding [