What is the best machine learning algorithm for large, noisy datasets with interaction between variables?

Probabilistic Random Forest tends to work better then other algorithms on noisy datasets. But the data you are using also plays a major role on whether a algorithm will work or not. Check this paper Probabilistic Random Forest for more details. Happy Learning

Ensemble methods, boosting or bagging, often give predictive accuracies superior to other methods. From my personal experience, I find GBM (ie. Gradient Boosting Regressor over Decision Trees) and LightGBM(faster) often give very accurate predictions.

Check out this diagram on choosing the right estimator.

I would make the following models:

1. a null baseline model
2. a linear regression model with the most highly correlated features
3. create polynomial features and do feature selection to just pick the top 10 or 20 features and try those with a linear regression model.
4. #3 but with ridge regression
5. a LightGBM model with the original features
6. If you think you can still squeeze out some performance and it's worth the time/effort tradeoff, move to neural nets. As long as you have a few layers and a decent number of nodes and a non-linear transformation (e.g. RELU) it should be able to pick up interactions.

If something looks promising, go that direction.