5.3 Stochastic Gradient Descent

Computing the gradient can be very time consuming. However, often it is possible to find a “cheap” approximation of the gradient. Approximating the gradient is still useful as long as it points in roughly the same direction as the true gradient. 

Stochastic gradient descent (often shortened as SGD) is a stochastic approximation of the gradient descent method for minimizing an objective function that is written as a sum of differentiable functions. The word

stochastic here refers to the fact that we acknowledge that we do not know the gradient precisely, but  instead only know a noisy approximation to it. By constraining the probability distribution of the approximate gradients, we can still theoretically guarantee that SGD will converge.

In machine learning, given $n = 1,\ldots,N$ data points, we often consider objective functions that are the sum of the losses $L(\theta)$ incurred by each example $n$. In mathematical notation, we have the form

$L(\theta)=\sum_{n=1}^N L_n(\theta)$ 

where $\theta$ is the vector of parameters of interest.i.e., we want to find $\theta$ that minimizes $L$.An example is the negative log likelihood  function, which is expressed as a sum over log-likelihoods of individual examples so that

$L(\theta)=-\sum_{n=1}^N log(p(y_n|x_n,\theta)$ 

where $x_n \in \mathbb{R}^D$ are the training inputs, $y_n$ are the training targets, and $\theta$ are the parameters of the regression model.

Standard gradient descent, as introduced previously, is a “batch” optimization method, i.e., optimization is performed using the full training set by updating the vector of parameters according to

$\theta_{i+1}=\theta_i-\gamma_i(\bigtriangledown L(\theta_i))^T=\theta_i-\gamma_i\sum_{n=1}^N( \bigtriangledown L_n(\theta_i))^T$

for a suitable step-size parameter $\gamma_i$. Evaluating the sum gradient may require expensive evaluations of the gradients from all individual functions $L_n$. When the training set is enormous and/or no simple formulas exist,evaluating the sums of gradients becomes very expensive.

Consider the term $\sum_{n=1}^{N}\bigtriangledown L_n(\theta_i)$,  we can reduce the amount of computation by taking a sum over a smaller set of  $L_n$. In contrast to batch gradient descent, which uses all $L_n$ for $n = 1,\ldots,N$, we randomly choose a subset of $L_n$ for mini-batch gradient descent. In the extreme case, we randomly select only a single $L_n$  to estimate the gradient. The key insight about why taking a subset of data is sensible is to realize that for gradient descent to converge, we only require that the gradient is an unbiased estimate of the true gradient. In fact the term $\sum_{n=1}^{N} \bigtriangledown L_n(\theta_i)$ is an empirical estimate of the expected value  of the gradient. Therefore, any other unbiased empirical estimate of the expected value, for example using any subsample of the data, would suffice for convergence of gradient descent.

Why should one consider using an approximate gradient? A major reason is practical implementation constraints, such as the size of central processing unit (CPU)/graphics processing unit (GPU) memory or limits on computational time. We can think of the size of the subset used to estimate the gradient in the same way that we thought of the size of a sample when estimating empirical means . Large mini-batch sizes will provide accurate estimates of the gradient, reducing the variance in the parameter update. Furthermore, large mini-batches take advantage of highly optimized matrix operations in vectorized implementations of the cost and gradient. The reduction in variance leads to more stable convergence, but each gradient calculation will be more expensive.

In contrast, small mini-batches are quick to estimate. If we keep the mini-batch size small, the noise in our gradient estimate will allow us to get out of some bad local optima, which we may otherwise get stuck in.In machine learning, optimization methods are used for training by minimizing an objective function on the training data, but the overall goal is to improve generalization performance . Since the goal in machine learning does not necessarily need a precise estimate of the minimum of the objective function, approximate gradients using mini-batch approaches have been widely used. Stochastic gradient descent is very effective in large-scale machine learning problems ,such as training deep neural networks on millions of images,reinforcement learning,or training of large-scale Gaussian process models

Example:

Consider the update equation for stochastic gradient descent. Write down the update when we use a mini-batch size of one.

$\theta_{i+1}=\theta_i-\gamma_i(\bigtriangledown L(\theta_i))^T=\theta_i-\gamma_i\sum_{n=1}^N( \bigtriangledown L_n(\theta_i))^T$

$\theta_{i+1}=\theta_i-\gamma_i(\bigtriangledown L_k(\theta_i))^T$

where $k$ is the index of the example that is randomly chosen.