In a neural network, each neuron has a set of parameters associated with it, which typically include a weight for each input connection and a bias term.
The weight parameter determines the strength of the connection between a neuron's input and its output, and is typically learned through a process called backpropagation during training. The bias term represents the neuron's inherent "activation level" and is also learned through training.
Together, the weight and bias parameters determine how the neuron responds to its inputs and how it contributes to the overall behavior of the neural network.
Why do we use linear equations in neural networks?
In the context of neural networks, the reason we use a linear equation for computing the weighted sum of the inputs is that this allows us to model linear relationships between the inputs and outputs. For example, in a simple regression problem where we are trying to predict a continuous output variable based on a set of input features, a linear model can often provide a good approximation of the underlying relationship between the inputs and outputs.
However, for more complex problems where the relationships between the inputs and outputs are nonlinear, we need to use more sophisticated models that can capture these nonlinearities. This is where more advanced neural network architectures, such as those with multiple layers or with nonlinear activation functions, come into play. These models allow us to capture more complex and nuanced relationships between the inputs and outputs, and can often achieve better performance than linear models.
most neural networks used in practice are not purely linear
most neural networks used in practice are not purely linear but rather employ some form of nonlinearity in their computations. This is because many real-world problems that we want to solve with neural networks have nonlinear relationships between the inputs and outputs.
For example, in a classification task where we want to predict the class label of an input data point, the relationship between the input features and the output class labels is often nonlinear. In order to capture these nonlinearities, we use nonlinear activation functions in the neurons of the neural network.
Common activation functions used in neural networks include the sigmoid function, the hyperbolic tangent (tanh) function, and the rectified linear unit (ReLU) function, among others. These activation functions introduce nonlinearity into the neural network computations and allow us to model complex and nonlinear relationships between the inputs and outputs.
Why do we use activation functions in NN?
Nonlinearity: Activation functions introduce nonlinearity into the computations of neural networks, allowing them to model complex nonlinear relationships between the inputs and outputs. Without activation functions, neural networks would be limited to linear transformations of the inputs, which would be unable to model many real-world problems.
Mapping to a range: Activation functions are often designed to map the output of a neuron to a specific range or set of values. For example, the sigmoid function maps its inputs to a range between 0 and 1, which is useful in binary classification problems where we want to predict the probability of a data point belonging to a particular class.
Smoothness: Activation functions can be designed to be smooth and differentiable, which is important for efficient training of neural networks using techniques like gradient descent. The derivatives of the activation functions are used in the backpropagation algorithm to compute the gradients of the loss function with respect to the weights and biases of the network.
Sparsity: Certain activation functions, such as the ReLU function, can induce sparsity in the activations of the neurons, which can improve the efficiency and interpretability of the neural network.
Overall, activation functions play a critical role in the computations of neural networks and are essential for enabling them to model complex nonlinear relationships between the inputs and outputs.
Nonlinear activation functions are typically used in neural networks to introduce nonlinearity into the computations of the network, even when the underlying functions being modeled are linear. This is because linear functions alone are often insufficient to model complex relationships between the inputs and outputs.
For example, consider a neural network that is being used to model a simple linear regression problem, where we want to predict a continuous output variable based on a set of input features. In this case, the underlying relationship between the inputs and the output is linear, and we could in principle use a linear activation function (i.e., no activation function at all) for the output neuron. However, in practice, we often use a nonlinear activation function such as the ReLU or sigmoid function for the output neuron, even though they are not strictly necessary for this problem. This is because using a nonlinear activation function can improve the network's ability to generalize to new data and can prevent overfitting.
Similarly, in the hidden layers of a neural network, nonlinear activation functions are used to introduce nonlinearity into the computations of the network and enable it to model complex relationships between the inputs and outputs, even when the underlying functions being modeled are linear. Without these nonlinear activation functions, the neural network would be limited to linear transformations of the inputs, which would be unable to capture many real-world phenomena.
why we are not commonly using a non-linear function as a base for a neural network?
we want the overall function computed by the network to be differentiable so that we can use gradient-based optimization techniques like backpropagation to train the network. Nonlinear functions can be highly non-smooth, which makes it difficult to use them for training neural networks.
Instead, we typically use a linear function as the base for a neural network and introduce nonlinearity through the use of activation functions in the neurons. The linear function allows us to compute a weighted sum of the inputs, which is then transformed by the activation function to introduce nonlinearity into the computation.
The use of a linear function as the base of a neural network also has the advantage of making the overall function computationally efficient and easy to optimize. Linear functions are simple and easy to compute, which makes training a neural network with a linear base more efficient than using a highly complex non-linear function as the base.
Overall, the combination of a linear function as the base and non-linear activation functions in the neurons allows us to create a powerful and flexible function approximator that can model a wide range of complex relationships between the inputs and outputs.
Several steps we can take to increase the chances of getting good performance from a neural network:
Use a large and diverse dataset for training: The more data you have, the better your neural network will be able to generalize to new examples. It's important to use a dataset that is representative of the problem you're trying to solve, and that includes a wide range of examples to cover the variability of real-world scenarios.
Choose an appropriate neural network architecture: Different neural network architectures are better suited for different types of problems. For example, convolutional neural networks are often used for image recognition tasks, while recurrent neural networks are often used for sequence modeling tasks. It's important to choose an architecture that is appropriate for the problem you're trying to solve.
Tune the hyperparameters: Hyperparameters are the settings that control the behavior of the neural network, such as the learning rate, batch size, and regularization strength. It's important to tune these hyperparameters to ensure that the neural network is performing optimally.
Regularize the model: Regularization techniques such as L1 and L2 regularization can be used to prevent overfitting, which occurs when the neural network becomes too complex and starts to memorize the training data instead of learning to generalize to new examples.
Monitor the performance during training: It's important to monitor the performance of the neural network during training to detect any issues early on. This can be done by measuring the loss on a validation set, or by monitoring other metrics such as accuracy or F1 score.
Test the neural network on a held-out test set: Once the neural network has been trained, it's important to test it on a held-out test set to evaluate its performance on new examples that were not seen during training. This will give you an estimate of how well the neural network is likely to perform in the real world.
By following these steps, you can increase the chances of getting good performance from a neural network, but it's important to keep in mind that there is always a trade-off between model complexity and generalization, and there may be limits to how well a neural network can perform on a given task.
Bias in Neural Network
In a neural network, bias is a term that represents the ability of a neuron to activate even when there is no input.
Mathematically, bias is a constant term that is added to the weighted sum of inputs of a neuron before applying the activation function. The bias term allows the neuron to have some activation even when all the input values are zero.
The bias term is an important component of a neural network, as it allows the network to learn more complex and nuanced patterns in the data. Without the bias term, the network would only be able to model linear relationships between the input and output.
The bias term is a learnable parameter, which means that its value is updated during training along with the weights of the network. The network learns the optimal value of the bias term which allows it to make accurate predictions on the training data.
In summary, bias is a term in a neural network that allows neurons to activate even when there is no input, and it is a learnable parameter that is updated during training.
Typically in the training process of a neural network, the bias values are initialized randomly, just like the weights. The neural network then learns the optimal values of the bias terms through the process of backpropagation, where the gradient of the loss function with respect to the bias terms is calculated and used to update their values.
The choice of how to initialize the bias terms can have an impact on the performance of the neural network, so it is important to choose an appropriate initialization method based on the specific problem being solved.
