# Empirical Risk Minimization and Stochastic Gradient Descent for Relational Data

arXiv: https://arxiv.org/abs/1806.10701 · HTML: https://mlsanity.com/paper/1806.10701/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 Jun 2018.

## Built on by (most cited first, 4 of 4)

- [Asymptotics of Network Embeddings Learned via Subsampling](https://mlsanity.com/paper/2107.02363/) (2021)
- [Using Embeddings for Causal Estimation of Peer Influence in Social Networks](https://mlsanity.com/paper/2205.08033/) (2022)
- [Asymptotics of ℓ2 Regularized Network Embeddings](https://mlsanity.com/paper/2201.01689/) (2022)
- [Community Detection Guarantees Using Embeddings Learned by Node2Vec](https://mlsanity.com/paper/2310.17712/) (2023)

## Abstract (arXiv)

Empirical risk minimization is the main tool for prediction problems, but its extension to relational data remains unsolved. We solve this problem using recent ideas from graph sampling theory to (i) define an empirical risk for relational data and (ii) obtain stochastic gradients for this empirical risk that are automatically unbiased. This is achieved by considering the method by which data is sampled from a graph as an explicit component of model design. By integrating fast implementations of graph sampling schemes with standard automatic differentiation tools, we provide an efficient turnkey solver for the risk minimization problem. We establish basic theoretical properties of the procedure. Finally, we demonstrate relational ERM with application to two non-standard problems: one-stage training for semi-supervised node classification, and learning embedding vectors for vertex attributes. Experiments confirm that the turnkey inference procedure is effective in practice, and that the sampling scheme used for model specification has a strong effect on model performance. Code is available at https://github.com/wooden-spoon/relational-ERM.