**Bayesian Layer**

We describe Bayesian Layers, a module designed for fast experimentation with neural network uncertainty. It extends neural network libraries with layers capturing uncertainty over weights (Bayesian neural nets), pre-activation units (dropout), activations (‘stochastic output layers’), and the function itself (Gaussian processes). With reversible layers, one can also propagate uncertainty from input to output such as for flow-based distributions and constant-memory backpropagation. Bayesian Layers are a drop-in replacement for other layers, maintaining core features that one typically desires for experimentation. As demonstration, we fit a 10-billion parameter ‘Bayesian Transformer’ on 512 TPUv2 cores, which replaces attention layers with their Bayesian counterpart. … **Linear Quadratic Regulator**

The theory of optimal control is concerned with operating a dynamic system at minimum cost. The case where the system dynamics are described by a set of linear differential equations and the cost is described by a quadratic function is called the LQ problem. One of the main results in the theory is that the solution is provided by the linear-quadratic regulator (LQR), a feedback controller whose equations are given below. The LQR is an important part of the solution to the LQG (linear-quadratic-Gaussian) problem. Like the LQR problem itself, the LQG problem is one of the most fundamental problems in control theory. … **Unrooted Tree**

In a context where trees are supposed to have a root, a tree without any designated root is called a free tree / unrooted tree. … **Speculative Query Planning (Spec-QP)**

Organisations store huge amounts of data from multiple heterogeneous sources in the form of Knowledge Graphs (KGs). One of the ways to query these KGs is to use SPARQL queries over a database engine. Since SPARQL follows exact match semantics, the queries may return too few or no results. Recent works have proposed query relaxation where the query engine judiciously replaces a query predicate with similar predicates using weighted relaxation rules mined from the KG. The space of possible relaxations is potentially too large to fully explore and users are typically interested in only top-k results, so such query engines use top-k algorithms for query processing. However, they may still process all the relaxations, many of whose answers do not contribute towards top-k answers. This leads to computation overheads and delayed response times. We propose Spec-QP, a query planning framework that speculatively determines which relaxations will have their results in the top-k answers. Only these relaxations are processed using the top-k operators. We, therefore, reduce the computation overheads and achieve faster response times without adversely affecting the quality of results. We tested Spec-QP over two datasets – XKG and Twitter, to demonstrate the efficiency of our planning framework at reducing runtimes with reasonable accuracy for query engines supporting relaxations. …

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**03**
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Jan 2020

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