What is the relationship between hard-sigmoid function and vanishing gradient descent problem?
The vanishing gradient problem appears with activation functions that squash their input in a very small output range, like sigmoids and tanh. The input range is large and therefore the probability for it to fall in the squashing part of the sigmoid is very high. Chain several sigmoids and there you have vanishing gradients,

Typical examples of such a type of activations are sigmoid and hyperbolic tangent. Hard sigmoids are no exception. Their main difference with normal sigmoids is that they are computationally cheaper.

ReLU's are different. Half of their range is linear. That's why they don't suffer from vanishing gradients. However, they have their own problem: dying ReLU. This happens when the input falls far in the left side and cannot escape.