Author: Sharath Overview This paper discusses how the BNNs behave as we scale to the large models. This paper discusses the "soap-bubble" issue in case of high dimensional probability spaces and how MFVI suffers from this. As a way to tackle this issue, the authors propose a new variational posterior approximation in hyperspherical coordinate system and show that this overcomes the soap-bubble issue when we sample from this posterior. The geometry of high dimensional spaces One of the properties of high dimensional spaces is that there is much more volume outside any given neighbourhood than inside of it. Betacount et al explained this behaviour visually with two intuitive examples. For first example let us consider partitioning our parameter space in equal rectangular intervals as shown below. We can see that as we increase the dimensions the distribution of volume around the center decreases. This becomes almost negligible as compared to the its neighbourhood in high dimensional cases where $D$ is very large. We can observe a similar behaviour if we consider spherical view of parameter space, where the exterior volume grows even larger than the interior in high dimensional spaces as shown in the figure. . How this intuition of volumes in high dimensions explain soap bubble phenomenon?
7/4/2021Author: Sharath What is a convex function? Let's try to define a convex function formally and geometrically. Formally, a function $f$ is said to be a convex function if the domain of $f$ is a convex set and if it satisfies the following $\forall x \ \text{and} \ y \in \text{dom} f$; \begin{equation} f(\theta x + (1 - \theta)y) \leq \theta f(x) + (1 - \theta)f(y) \end{equation} Geometrically it means that the value of a function at the convex combination of two points of the function always lies below the convex combination of the values at the corresponding points. It means that if we draw a line at any two points $(x, y) \in \text{dom} f$, then this line/chord always lies above the function $f$.
7/4/2021Author: Sharath What is a dynamical system? It is any system that evolves and changes through time governed by a set of rules. Using dynamical systems we can study the long term behavior of an evolving system. Formally, it is a triplet $(X, T, \phi)$ where $X$ denotes the state space, $T$ denotes the time space and $\phi: X \times T \rightarrow X$ is the flow (this is the rule that governs the evolution). There are few properties of flow: $\phi(X, 0) = X$ Principle of compositionality: $\phi(\phi(x, t), s) = \phi(x, t+s)$
7/4/2021Author: Sharath Chandra Paper Link tags: simulation, interaction-networks, robotics In this paper the authors proposed a hybrid dynamics model, Simulation-Augemented Interaction Networks, where they incorporated Interaction Networks into a physics engine for solving real world complex robotics control tasks. Brief Outline: Most of the physics based simulators serves as a good platform for carrying out robot planning and control tasks. But no simulator is a perfect because it has it's own modelling errors. So, most of the physics engines (mujoco, bullet, gazebo etc.,) demonstrate some descrepencies between their predictions and actual real world predictions. To decrease these errors, many methods have been propsed in the literature. Some of the methods include randomizing the simulation environments, famously known as Domain Randomization. In this paper, model errors are tackled by learning a residual model between the real world and simulator. In other words, instead of adding pertubations to the environment parameters, here we utilize some real world data to correct the simulator. Even though this method uses some real world data, this method is shown to be sample efficient and have better generalization capabilities. Interaction Networks
7/4/2021or
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