As Jan says in a comment, AlexNet uses cross entropy as the loss function.
It's important to note, though, that a Convolutional Neural Network describes the architecture of the network, not the goal of the network. It is the goal of a network that determines the loss function.
CNN architectures can be used for many tasks with different loss functions:
- multi-class classification as in AlexNet
- Typically cross entropy loss
- Typically Squared Error loss
- image segmentation
- Can use cross entropy loss as well, but can also use several other kinds of loss functions
- reinforcement learning
- In Deep Q-Networks, the "Expected discounted accumulated future reward" can be used
- generative adversarial networks (generating images)
- The Jensen–Shannon divergence was used in the original implementation