A major cause of the advances in machine learning are related to big data: Very large datasets.

If you see high variance in the learning curve (cost on training set slowly increases, cost on validation set slowly decreases) as the number of data points is small but increasing, then more data is one of the better ways to improve the system. In that case, optimizing for big data can be worth the effort. If you don't see continued improvements as the number of data points is increased, then larger datasets are, obviously, not worth the effort.

When the number of data sets is large, computing the derivative term for
a standard (aka "Batch") gradient
descent can be expensive because we must sum the slopes for all the data
points to find the direction we want to move. If we instead take steps based
on only one training example at a time, modifying the parameters
~~O~~ for only that one data point, then repeat that process
for each data point, we can get the same effect, but without waiting to update
parameters after scanning ALL the data points. This basically makes the system
converge faster.

For this to work, we must ensure the data is in random order, so an initial step, were we randomize the data, is necessary. Each change in the parameters, after each data point, may not take us in a good direction, but the overall average effect will be the same, as long as the data doesn't "lead us astray" for more than a few points. Randomizing the data order helps to prevent that. Our "walk" toward the minimum cost will probably be more jagged and wandering, but that can actually help avoid local minimums.

To ensure the system is converging on a minimum, we can compute the cost each time we update the parameters, after each data point. Then we can average those costs, and plot them every so many iterations. The result may be noisy, but a general trend should be obvious.

Because individual data points can send us in the wrong direction, it can be difficult to find the end of the "walk" towards the minimum. At or near minimum, the system can still wander around and not really settle on a final best value. In some cases, it may help to slowly decrease your steps size (alpha), or just stop after a certain number of steps.

Instead of updating the parameters for each data point as in Stochastic descent, or only after all data points as with the standard Batch descent, we can use some subset of the data points, hoping that they are representative of the complete data set. Given a set size, b, we compute the average slope of the cost function for this subset of b data points, then update the parameters accordingly, and then repeat using the next b data points.

This method also lends itself to parallel processing. It's cost is higher than Stochastic and lower than Batch.

When the input data continues to flow in, with ever more data points, we can apply each new data point to the learning system as it arrives. For example, if we were using Linear Regression, we can do a single additional change to the parameters with Stochastic gradient descent on that single new data point. In this way, our data set size can be effectively infinite. Another advantage of this system is that it will slowly adapt over time to changes in the environment.

Another very important, but very simple, means of dealing with large datasets
is to split the data over multiple computers (or multiple cores in one computer
if your libraries don't parallalize automatically), calculating the error
slope for each subset seperately on each machine. This brings parallel computing
to bear on the problem. Then we can do the
gradient descent on a "master"
computer re-combine the slopes found by each machine and adjusting our parameters
by that total times alpha / m as usual. e.g. we add

`s = sum(S,2) %sum up the columns, one column from each
machine`

before

theta = theta - alpha .* s

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