Cross-Entropy Loss

The loss applies a negative log-likelihood to the predicted probability of the correct class. As the predicted probability approaches 1, the loss smoothly decays toward 0. As it approaches 0, the loss explodes toward infinity. This asymmetric penalty ensures that confident misclassifications are catastrophically punished, forcing the network to correct high-confidence errors before fine-tuning borderline cases.

Cross-entropy measures the average number of bits needed to encode the true distribution using an optimal code designed for the predicted distribution. Minimizing cross-entropy is equivalent to minimizing the Kullback-Leibler (KL) divergence between the true and predicted distributions. This grounding in information theory makes it the theoretically correct objective for maximum likelihood estimation in classification.

When combined with a softmax activation, the gradient of cross-entropy loss with respect to the logits simplifies to (ŷ - y), the difference between predicted and true probabilities. This elegant linear gradient prevents the vanishing gradient problem that plagues mean squared error in classification. The network receives a strong, clean learning signal proportional to its error, enabling stable and efficient training of deep architectures like ResNets and Transformers.

Two common implementations exist for multi-class modulation recognition:

• Categorical Cross-Entropy: Accepts one-hot encoded labels, requiring the full probability vector. Used when label smoothing or soft targets are needed.
• Sparse Categorical Cross-Entropy: Accepts integer class indices directly, reducing memory overhead for datasets with hundreds of modulation types. Both produce identical gradients; the choice is purely an implementation convenience.

Direct computation of log(softmax(logits)) is numerically unstable due to potential overflow in the exponential. Production implementations use the log-sum-exp trick: subtracting the maximum logit before exponentiation. This transforms the computation into a stable form without altering the mathematical result, preventing NaN gradients that would silently corrupt training of modulation classifiers operating on high-dynamic-range IQ samples.

Standard cross-entropy with hard one-hot targets can drive the model toward overconfident predictions, assigning probability 1.0 to the predicted class. Label smoothing replaces the hard 0/1 targets with softened values (e.g., 0.9 for the correct class, 0.1 distributed among others). This regularization technique prevents the logits from growing unbounded, improves calibration, and increases robustness to noisy channel conditions in modulation recognition tasks.

The softmax function is the critical final layer that transforms raw neural network outputs (logits) into a valid probability distribution over modulation classes. It exponentiates each logit and normalizes by the sum of all exponentiated values, ensuring all outputs are between 0 and 1 and sum to 1. Cross-entropy loss is almost always paired with softmax in multi-class modulation recognition.

• Converts unnormalized logits to probabilities
• Amplifies the largest logit while suppressing smaller ones
• The output vector represents the model's predicted confidence for each modulation scheme

KL divergence measures how one probability distribution diverges from a reference distribution. Minimizing cross-entropy is mathematically equivalent to minimizing the KL divergence between the true label distribution and the model's predicted distribution, plus a constant entropy term. This connection grounds cross-entropy in information theory.

• Asymmetric measure: D_KL(P||Q) ≠ D_KL(Q||P)
• Non-negative; equals zero only when distributions are identical
• Provides the theoretical justification for using cross-entropy as a classification objective

Negative log-likelihood is the core computation within cross-entropy loss. For a given training example, the model's predicted probability for the correct class is extracted, and its negative logarithm is taken. This penalizes confident wrong predictions exponentially more than hesitant ones, driving the model toward high-confidence correct classifications.

• For a correct prediction with probability p, loss = -log(p)
• As p → 1, loss → 0; as p → 0, loss → ∞
• Directly maximizes the likelihood of observing the training data under the model

Categorical cross-entropy is the specific formulation used when each input belongs to exactly one of many modulation classes. The true label is represented as a one-hot encoded vector, and the loss sums the negative log of the predicted probability for the correct class. This is the standard loss function for modulation recognition tasks like distinguishing BPSK, QPSK, 16-QAM, and 64-QAM.

• Assumes mutually exclusive classes
• Pairs with softmax activation
• The most common loss in AMC deep learning pipelines

Binary cross-entropy is used when each modulation label is independent, such as in multi-label scenarios where a signal might simultaneously exhibit multiple properties. It computes the loss separately for each class using a sigmoid activation, rather than softmax. This is relevant for compound signal identification or joint modulation-and-encoding classification.

• Loss = -[y·log(p) + (1-y)·log(1-p)]
• Each class treated as an independent binary decision
• Uses sigmoid activation per output node instead of softmax

Gradient descent is the iterative optimization algorithm that uses the gradient of the cross-entropy loss with respect to every model parameter to update weights. The gradient of cross-entropy with softmax has a particularly clean form: (predicted_probability - true_label), making backpropagation computationally efficient and numerically stable.

• The loss landscape is convex with respect to logits
• Clean gradient signal accelerates convergence
• Often paired with Adam or SGD optimizers in AMC training