Pardon my matrix calculus, been some time since I last used it.
From my answer to Is Gradient Descent possible for kernelized SVMs (if so, why do people use Quadratic Programming)?, we can write the primal SVM (Hinge-loss with squared-$\ell_2$ regularization) objective as:
$$J(\mathbf{u}, b) = C {\displaystyle \sum\limits_{i=1}^{m} max\left(0, 1 - y_i (\mathbf{u}^t \cdot \mathbf{K}_i + b)\right)} + \dfrac{1}{2} \mathbf{u}^t \cdot \mathbf{K} \cdot \mathbf{u}$$
We want to minimize this quantity. Deriving it with regards to $\mathbf{u}$ results in:
$$\frac{\partial J(\mathbf{u}, b)}{\partial \mathbf u} = C {\displaystyle \sum\limits_{i=1}^{m}
\left [y_i (\mathbf{u}^t \cdot \mathbf{K}_i + b) \lt 1 \right ] \cdot (- y_i \cdot\mathbf{K}_i}) + \mathbf{K}\cdot\mathbf{u}$$
The derivative regarding $b$ results in:
$$\frac{\partial J(\mathbf{u}, b)}{\partial b} = C {\displaystyle \sum\limits_{i=1}^{m} \left [y_i (\mathbf{u}^t \cdot \mathbf{K}_i + b) \lt 1 \right ] \cdot\left(- y_i \cdot b\right)}$$
Where $\left[\cdot\right]$ is the Iverson bracket. Notice that the derivative is undefined when $y_i (\mathbf{u}^t \cdot \mathbf{K}_i + b) \equiv 1$, and in keeping with SGD tradition a reasonable value can be assigned to the step, or the sample can be skipped.
From these equations, you can derive SGD gradients.
The applicability of batch or stochastic gradient descent really depends on the error manifold expected.
Batch gradient descent computes the gradient using the whole dataset. This is great for convex, or relatively smooth error manifolds. In this case, we move somewhat directly towards an optimum solution, either local or global. Additionally, batch gradient descent, given an annealed learning rate, will eventually find the minimum located in it's basin of attraction.
Stochastic gradient descent (SGD) computes the gradient using a single sample. Most applications of SGD actually use a minibatch of several samples, for reasons that will be explained a bit later. SGD works well (Not well, I suppose, but better than batch gradient descent) for error manifolds that have lots of local maxima/minima. In this case, the somewhat noisier gradient calculated using the reduced number of samples tends to jerk the model out of local minima into a region that hopefully is more optimal. Single samples are really noisy, while minibatches tend to average a little of the noise out. Thus, the amount of jerk is reduced when using minibatches. A good balance is struck when the minibatch size is small enough to avoid some of the poor local minima, but large enough that it doesn't avoid the global minima or better-performing local minima. (Incidently, this assumes that the best minima have a larger and deeper basin of attraction, and are therefore easier to fall into.)
One benefit of SGD is that it's computationally a whole lot faster. Large datasets often can't be held in RAM, which makes vectorization much less efficient. Rather, each sample or batch of samples must be loaded, worked with, the results stored, and so on. Minibatch SGD, on the other hand, is usually intentionally made small enough to be computationally tractable.
Usually, this computational advantage is leveraged by performing many more iterations of SGD, making many more steps than conventional batch gradient descent. This usually results in a model that is very close to that which would be found via batch gradient descent, or better.
The way I like to think of how SGD works is to imagine that I have one point that represents my input distribution. My model is attempting to learn that input distribution. Surrounding the input distribution is a shaded area that represents the input distributions of all of the possible minibatches I could sample. It's usually a fair assumption that the minibatch input distributions are close in proximity to the true input distribution. Batch gradient descent, at all steps, takes the steepest route to reach the true input distribution. SGD, on the other hand, chooses a random point within the shaded area, and takes the steepest route towards this point. At each iteration, though, it chooses a new point. The average of all of these steps will approximate the true input distribution, usually quite well.
Best Answer
Ridge Regression python package has several solver options, and is not employing the same method as you. Your implementation is the very basic of gradient descent method that employs constant learning coefficient I presume, i.e. you don't have any strategy for adaptively setting your learning coefficient. And in sensitive cases as yours (i.e. large numbers), this can easily lead to different results. Library methods, in general, are products of highly experienced researchers and developers and highly stable in cases of numerical challenges.