What a neural layer actually computes

A simple neuron computes a weighted sum, adds a bias, and applies an activation: h = activation(w dot x + b). The weights determine which input directions matter; the bias shifts the point at which the activation changes behavior. Training adjusts those parameters to reduce a chosen loss. Nothing in this formula requires the neuron to correspond to a human-readable concept.

Take x = [2, -1], w = [0.5, 1], and b = 0.2. The affine value is 0.5 times 2 plus 1 times -1 plus 0.2, or 0.2. ReLU returns 0.2 because the input is positive. A second neuron with different weights can react to another direction of the same input. Stacking their outputs produces a learned feature vector.

Two affine layers with no nonlinear activation between them can be collapsed into one affine layer. If h = W1x + b1 and y = W2h + b2, substitution gives y = W2W1x + W2b1 + b2. More parameters and more computation do not, by themselves, create a nonlinear decision boundary. The activation between layers is what prevents this collapse.

A small construction demonstrates the difference. For binary inputs a and b, set h1 = ReLU(a+b) and h2 = ReLU(a+b-1). Then output h1 - 2h2 is zero for [0,0] and [1,1], but one for [0,1] and [1,0]. This implements XOR on those four points. The network does not discover the weights in the example; they are deliberately chosen to expose the mechanism.

If each input has d features and a layer has h units, its weight matrix has h rows and d columns in the convention used here. Its bias has h entries. For a batch of n examples, frameworks often arrange data as n by d and multiply by the transpose of that weight matrix. Other conventions are valid, but changing conventions halfway through an implementation creates hard-to-diagnose bugs.

Count parameters before discussing model size. A dense layer from 3 inputs to 4 outputs has 12 weights and 4 biases, for 16 parameters. A second layer from 4 to 2 adds 8 weights and 2 biases. Batch size changes how many examples are processed together, not how many learned parameters the layer contains.

ReLU outputs zero for negative inputs and passes positive inputs through. Its derivative is zero on the negative side and one on the positive side, with a convention needed at zero. A unit that remains negative for every training example may receive no useful gradient through its activation. Smooth bounded activations such as tanh avoid that exact dead region but can saturate, producing small derivatives at large magnitudes.

Initialization, normalization, learning rate, and architecture interact with these effects. Starting every hidden unit with identical weights can preserve symmetry, so the units learn redundant features. Extremely large initial weights can push bounded activations into saturation. These are mechanisms to investigate when training stalls, rather than reasons to declare one activation universally superior.

The last layer should match the output you intend to model. A scalar real-valued regression target needs a different interpretation from a vector of class logits. A softmax turns class logits into a normalized categorical distribution; independent sigmoid outputs instead permit multiple labels to be active at once. Choosing the wrong output interpretation can impose an unintended constraint before training even begins.

The runnable example verifies the XOR construction and counts the parameters of a separate two-layer architecture. It gives you a forward-pass reference with no library hiding the arithmetic. The exercise changes the target so you must reason about the features the hidden layer created. Later, backpropagation will automate how parameters change, but it will not decide whether the network's output contract matches the product's objective.

The two ReLU features implement XOR using hand-selected weights; there is no training here. A separate parameter-count calculation shows that a 3-to-4-to-2 dense network has 26 parameters including biases. Change one hidden bias and inspect all four binary inputs to understand its role.