The hip high-dimensional version of classic offline optimisation.

Something I am learning about belatedly; be warned. These are not reference-grade notes.

Batch and stochastic methods for minimising loss when you have a lot of data, or a lot of parameters, and using it all at once is silly, or when you want to iteratively improve your solution as data comes in, and you have access to a gradient for your loss, ideally automatically calculated <{filename}autodiff.rst>_. It’s not clear at all that it should work, except by collating all your data and optimising offline, except that much of modern machine learning shows that it does.

Sometimes this apparently stupid trick it might even be fast for small-dimensional cases, so you may as well try.

Technically, “online” optimisation in bandit/RL problems might imply that you have to “minimise regret online”, which has a slightly different meaning and, e.g. involves seeing each training only as it arrives along some notional arrow of time, yet wishing to make the “best” decision at the next time, and possibly choosing your next prediction in order to trade-off exploration versus exploitation etc.

In, e.g., SGD you can see your data as often as you want and in whatever order, but you only look at a little bit at a time. Usually the data is given and predictions make no difference to what information is available to you.

Some of the same technology pops up in each of these, but I am really thinking about SGD here.

There are many more permutations and variations used in practice.

## Gradient Descent

The workhorse of many large-ish scale machine-learning techniques and especially deep neural networks. See Sebastian Ruder’s explanation.

### Conditional Gradient

a.k.a. Frank-Wolfe algorithm: Don’t know much about this.

### Stochastic

### Variance-reduced

Zeyuan Allen-Zhu : Faster Than SGD 1: Variance Reduction:

SGD is well-known for large-scale optimization. In my mind, there are two (and only two) fundamental improvements since the original introduction of SGD: (1) variance reduction, and (2) acceleration. In this post I’d love to conduct a survey regarding (1),

### Second order (Quasi-newton at scale)

LiSSA attempts to make 2nd order gradient descent methods practical (AgBH16):

linear time stochastic second order algorithm that achieves linear convergence for typical problems in machine learning while still maintaining run-times theoretically comparable to state-of-the-art first order algorithms. This relies heavily on the special structure of the optimization problem that allows our unbiased hessian estimator to be implemented efficiently, using only vector-vector products.

David McAllester observes:

Since \(H^{t+1}y^t\) can be computed efficiently whenever we can run backpropagation, the conditions under which the LiSSA algorithm can be run are actually much more general than the paper suggests. Backpropagation can be run on essentially any natural loss function.

## Momentum, Nesterov Accelerated

Magical high speed convergence for smooth functions: Improved Nesterov accelaration and a lovely explanation of what the hell it is.

Normally considered for Gradient Descent, but also works for coordinate descent (Nest12a) and second order (Nest07).

Do momentum methods in the SGD setting fit here?

## …With sparsity

Hmmm.

## Sundry Hacks

### Weight Normalization

Pragmatically, controlling for variability in your data can be very hard in, e.g. deep learning so you might normalise it by the batch variance. Salimans and Kingma (SaKi16) have a more satisfying approach to this.

We present weight normalization: a reparameterization of the weight vectors in a neural network that decouples the length of those weight vectors from their direction. By reparameterizing the weights in this way we improve the conditioning of the optimization problem and we speed up convergence of stochastic gradient descent. Our reparameterization is inspired by batch normalization but does not introduce any dependencies between the examples in a minibatch. This means that our method can also be applied successfully to recurrent models such as LSTMs and to noise-sensitive applications such as deep reinforcement learning or generative models, for which batch normalization is less well suited. Although our method is much simpler, it still provides much of the speed-up of full batch normalization. In addition, the computational overhead of our method is lower, permitting more optimization steps to be taken in the same amount of time.

They provide an open implemention for keras, Tensorflow and lasagne.

## Parallel

Classic, basic SGD takes walks through the data set example-wise or feature-wise - but this doesn’t work in parallel, so you tend to go for mini-batch gradient descent so that you can at least vectorize. Apparently you can make SGD work in “true” parallel across communication-constrained cores, but I don’t yet understand how.

## Implementations

Specialized optimisation software.

See also statistical software.

- vowpal wabbit solve a bunch of optimisations well in an online setting.
- keras.optimizers support varied SGD-type online/minibatch optimizers for python + Tensorflow/Theano.
- tensorflow.train is a very similar list of online/SGD optimisers

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