Take a look at the image below from Off Convex. In a convex function (leftmost image), there is only one local minimum, which is also the global minimum. But in a non-convex function (rightmost image), there may be multiple local minima and often joining two local minima is a saddle point. If you are approaching from a higher point, the gradient is comparatively flatter, and you risk getting stuck there, especially if you are moving only in one direction.
Now the thing is, whether you are optimizing using mini-batch or stochastic gradient descent, the underlying non-convex function is the same, and the gradient is a property of the this function. When doing mini-batch, you consider many samples at a time and take the gradient step averaged over all of them. This reduces variance. But if the average gradient direction is still pointing in the same direction as the saddle point, then you still risk getting stuck there. The analogy is, if you're taking 2 steps forward and 1 step back, averaging over those, you ultimately end up taking 1 step forward.
If you perform SGD instead, you take all the steps one after the other, but if you're still moving in a single direction, you can reach the saddle point and find that the gradient on all sides is fairly flat and the step size is too small to go over this flat part. This doesn't have anything to do with whether you considered a bunch of points at once, or one by one in random order.
Take a look at the visualization here. Even with SGD, if the fluctuations occur only along one dimension, with the steps getting smaller and smaller, it would converge at the saddle point. In this case, the mini-batch method would just reduce the amount of fluctuation but would not be able to change the gradient's direction.
SGD can sometimes break out of simple saddle points, if the fluctuations are along other directions, and if the step size is large enough for it to go over the flatness. But sometimes the saddle regions can be fairly complex, such as in the image below.
The way methods like momentum, ADAGRAD, Adam etc are able to break out of this, is by considering the past gradients. Consider momentum,
$$
v_t = \gamma v_{t-1} + \eta \nabla_{theta} J(\theta)
$$
which adds a portion of the last gradient, $v_{t-1}$. In case you've just been going back and forth in one direction, essentially changing signs, it ends up dampening your progress. While if there has consistently been positive progress in one direction, it ends up building up and going down that way.
Better optimization does not necessarily mean a better model. In the end what we care about is how well the model generalizes, and not necessarily how good the performance on the training set is. Fancier optimization techniques usually perform better and converge faster on the training set, but do not always generalize as well as basic algorithms. For example this paper shows that SGD can generalize better than ADAM optimizer. This can also be the case with some second order optimization algorithms.
[Edit] Removed the first point as it does not apply here. Thanks to bayerj for pointing this out.
Best Answer
There's not really a more intuitive way to think about this. Suppose that you have the eigendecomposition of the Hessian for $P$ an orthonormal matrix of eigenvectors and $D$ diagonal matrix of eigenvalues. $$ \begin{align} \nabla^2 f(x) &= PDP^\top \\ \left[\nabla^2 f(x)\right]^{-1} &= PD^{-1}P^\top \end{align} $$ This is relevant to Newton's method because the update is given by $$ \begin{align} x^{(t+1)} &= x^{(t)}-\left[\nabla^2 f(x^{(t)})\right]^{-1}\nabla f(x^{(t)}) \\ &= x^{(t)}-PD^{-1}P^\top\nabla f(x^{(t)}) \end{align} $$
Saddle points have gradient 0, and Newton's method seeks points with gradient 0. If the problem is non-convex, then depending on the starting point, you may find yourself in the "basin of attraction" of the saddle point.
I also think that this post is of interest:
Gradient descent on non-convex functions
Why is Newton's method not widely used in machine learning?