The vanishing gradient problem is a difficulty encountered in artificial neural networks during training through backpropagation. When layers have activation functions such as the sigmoid function or hyperbolic tangent, their gradients can be near zero. This results in a very small change in weights and biases, especially in earlier layers of the network. As a result, these layers learn very slowly, or stop learning altogether, leading to suboptimal performance. This issue is more pronounced in deep networks, with many layers. Solutions to the vanishing gradient problem include the use of ReLU activation functions, batch normalization, and careful weight initialization.

Continuing on, the vanishing gradient problem has a significant impact on the speed and effectiveness of training deep neural networks. It can cause the network to get stuck in the initial stages of learning, preventing it from effectively recognizing patterns in the input data.

ReLU (Rectified Linear Unit) activation functions are one solution that have gained popularity. They are linear and non-saturating, meaning they do not suffer from the vanishing gradient problem to the same extent as sigmoid or hyperbolic tangent functions.

Batch normalization is another technique used to combat the vanishing gradient problem. It normalizes the input layer by adjusting and scaling the activations. This makes the network more stable and allows higher learning rates, accelerating the learning process.

Careful weight initialization is also crucial. Techniques like Xavier and He initialization can be used to ensure that weights are neither too small (which can lead to vanishing gradients) nor too large (which can lead to exploding gradients).

Despite these solutions, the vanishing gradient problem remains a challenge in the field of deep learning and is an active area of research.