[closed] Setting optimization parameters

You might want to look at this question and this question; although most of the answers are focused on hyperparameters for bayesian models some methods are useful for your sort of problem.

Generally, what you suggest seems good, although you should worry about the relative size of the smaller set. The smaller the training set is (compared to the number of parameters in your model), the more you need regularization to perform well, so this might bias you towards excessive regularization (which is less harmful than too little regularization, but this means you're throwing performance away). Since the regularization shouldn't depend much on the learning rate, you could maybe test many values for the learning rate on the small set and use the full training set to set the regularization coefficient (using a few good values of the learning rate), probably following the strategy described by Bengio in this answer. A good thing to keep in mind is that you don't make your life worse by changing the regularization coefficient of an already trained model (of course, you would need to retrain it to find another optimum). Since the performance versus regularization coefficient curve seems to be well behaved, you can do a grid search or some sort of line minimization algorithm to find the best value. I suggest, however, that you measure performance of a separate validation set to do this, instead of measuring the actual loss on your training set, to avoid overfitting.

So, tl;dr: use a validation set to measure performance, instead of looking at your surrogate loss directly; optimize the learning rate on the smaller version of the training set if it saves you time; and search for the best value of the regularization coefficient using as much training data as you can, without necessarily resetting the model before trying another value. Always keep count of the performance on the validation set, as well.