[Laplacian eigenvectors](https://paperswithcode.com/paper/laplacian-eigenmaps-and-spectral-techniques) represent a natural generalization of the [Transformer](https://paperswithcode.com/method/transformer) positional encodings (PE) for graphs as the eigenvectors of a discrete line (NLP graph) are the cosine and sinusoidal functions. They help encode distance-aware information (i.e., nearby nodes have similar positional features and farther nodes have dissimilar positional features).

Hence, Laplacian Positional Encoding (PE) is a general method to encode node positions in a graph. For each node, its Laplacian PE is the k smallest non-trivial eigenvectors.

The full architecture of CGNN is presented at [CGNN's official site](https://tony-y.github.io/cgnn/architectures/).

CGNN

Crystal Graph Neural Networks for Data Mining in Materials Science

Laplacian PE

Benchmarking Graph Neural Networks

**Decentralized Distributed Proximal Policy Optimization (DD-PPO)** is a method for distributed reinforcement learning in resource-intensive simulated environments. DD-PPO is distributed (uses multiple machines), decentralized (lacks a centralized server), and synchronous (no computation is ever `stale'), making it conceptually simple and easy to implement. 

Proximal Policy Optimization, or [PPO](https://paperswithcode.com/method/ppo), is a policy gradient method for reinforcement learning. The motivation was to have an algorithm with the data efficiency and reliable performance of [TRPO](https://paperswithcode.com/method/trpo), while using only first-order optimization. 

Let $r\_{t}\left(\theta\right)$ denote the probability ratio $r\_{t}\left(\theta\right) = \frac{\pi\_{\theta}\left(a\_{t}\mid{s\_{t}}\right)}{\pi\_{\theta\_{old}}\left(a\_{t}\mid{s\_{t}}\right)}$, so $r\left(\theta\_{old}\right) = 1$. TRPO maximizes a “surrogate” objective:

$$ L^{v}\left({\theta}\right) = \hat{\mathbb{E}}\_{t}\left[\frac{\pi\_{\theta}\left(a\_{t}\mid{s\_{t}}\right)}{\pi\_{\theta\_{old}}\left(a\_{t}\mid{s\_{t}}\right)})\hat{A}\_{t}\right] = \hat{\mathbb{E}}\_{t}\left[r\_{t}\left(\theta\right)\hat{A}\_{t}\right] $$

As a general abstraction, DD-PPO implements the following:
at step $k$, worker $n$ has a copy of the parameters, $\theta^k_n$, calculates the gradient, $\delta \theta^k_n$, and updates $\theta$ via 

$$ \theta^{k+1}\_n =  \text{ParamUpdate}\Big(\theta^{k}\_n, \text{AllReduce}\big(\delta \theta^k\_1, \ldots, \delta \theta^k\_N\big)\Big) = \text{ParamUpdate}\Big(\theta^{k}\_n, \frac{1}{N}  \sum_{i=1}^{N} { \delta \theta^k_i}   \Big) $$

where $\text{ParamUpdate}$ is any first-order optimization technique (e.g. gradient descent) and $\text{AllReduce}$ performs a reduction (e.g. mean) over all copies of a variable and returns the result to all workers.
Distributed DataParallel scales very well (near-linear scaling up to 32,000 GPUs), and is reasonably simple to implement (all workers synchronously running identical code).

Source	Benchmarking Graph Neural Networks
Year	2000
Data Source	CC BY-SA - https://paperswithcode.com