Neural Network Foundations, Explained: Updating Weights with Gradient Descent & Backpropagation

In neural networks, connection weights are adjusted in order to help reconcile the differences between the actual and predicted outcomes for subsequent forward passes. But how, exactly, do these weights get adjusted?
By Matthew Mayo, KDnuggets.

Recall that in order for a neural networks to learn, weights associated with neuron connections must be updated after forward passes of
Figure 1. Very simple neural network, with its 16 neuron weights.

If we want precision to 3 decimal places, we have a possible 100016 -- or 1048 -- weight combinations. Brute forcing all of these possibilities would take... a while. Clearly a better approach is required.

Imagine that a cost function is used to determine our error (the difference between actual and predicted values), based on a given weight. Consider the cost function illustrated in Figure 2.


Figure 2. Cost function (Source).

Now, let's take as true the assertion that the lowest point on that cost function is the optimal value (minima), representing where the rate of change of the function is exactly zero. Our objective is then to determine the value which produces this rate of change of zero. How is this determined? Well, let's start somewhere on that function, with some value, and then use some method for determining where on the curve we are relative to the minima, which will then provide us with some clue as to what our next move should be, in order to make an attempt at reaching the bottom, where the rate of change is zero (which is optimal).

Conceptually, using the slope of the angle of our cost function at our current location can tell us if we are headed in the right direction. As per basic algebra, a negative slope tells us we are headed downward (good!), while a positive slope says that our previous step has overshot our goal (moved beyond the optimal and back up the other side of the function).

OK, great. But how do we determine these slopes? As it turns out, gradient is actually a synonym for derivative, while derivative is the rate of change of a function. Well, that sounds suspiciously like exactly what we want. Descent indicates that we are spelunking our way to the bottom of a cost function using these changing gradients. And how do we get derivatives? By using the process of differentiation.

How far should we move in a direction, meaning how should we determine our learning rate (or step size)? That's a different story. But step size will have an effect on the how long it takes to reach the optimal value, how many steps it takes to get there, and how direct or indirect our journey is.

So, what about stochastic gradient descent (SGD)?

The process of gradient descent is very formulaic, in that it takes the entirety of a dataset's forward pass and cost calculations into account in total, after which a wholesale propagation of errors backward through the network to neurons is made. This process would result in the same errors and subsequent propagated errors each and every time it is undertaken. Plain vanilla gradient descent is deterministic.

However, stochastic means randomly determined. Instead of a rote processing of data, SGD uses a random sampling of the
Figure 3. Function with multiple local minima and maxima (Source).

Gradient descent is susceptible to local minima since every data instance from the dataset is used for determining each weight adjustment in our neural network. The entire batch of data is used for each step in this process (hence its synonymous name, batch gradient descent). Gradient descent does not allow for the more free exploration of the function surface required in order to move beyond local minima. By considering the data en masse, our gradient decent is less susceptible to extremes and outliers, which is not desirable when on the hunt for the global minima.

SGD gets around this by making weight adjustments after every data instance. A single data instance makes a forward pass through the neural network, and the weights are updated immediately, after which a forward pass is made with the next data instance, etc. This makes our gradient decent process more volatile, with greater fluctuations, but which can escape local minima and help ensure that a global cost function minima is found. Global minima is not guaranteed, but SGD has a better chance of locating it.

Mini-batch gradient descent, as you may have guessed by this point, is a happy medium; it is generally faster than SGD, yet allows for more fluctuations and volatility than does batch gradient descent.

Updating Weights - Key Points
  • Weights associated with neuron connections
  • The error represents the difference between actual and predicted values
  • This error is required at neurons to make weight adjustments, and are propagated backward through the network after calculation -- backpropagation of error
  • Gradient descent is used to more efficiently determine optimal weights by acting as a guide when searching for a cost function's optimal value
  • Stochastic gradient descent is a randomization of data sampling on which a single selection is used for error backpropagation (and weight updates)
     
  • Gradient descent - assists in determining error while searching for optimal value to plug into cost function
  • Backpropagation - distributes these errors backward through the network to be used by neurons for adjusting individual weights


Sources:
  [1] Vector Calculus: Understanding the Gradient
  [2] Gradient Descent (and Beyond)
  [3] Find Limits of Functions in Calculus

 
Related:

  • Neural Network Foundations, Explained: Activation Function
  • Deep Learning and Neural Networks Primer: Basic Concepts for Beginners
  • An Intuitive Guide to Deep Network Architectures