RL is learning by trial and error.
Reinforcement Learning (RL) is a type of machine learning where an autonomous agent learns to make decisions by interacting with an environment.
Instead of being told the correct answer, the agent:
Reinforcement Learning teaches an agent how to act, not what to predict.
If A symmetric:
[ \nabla (x^T A x) = 2Ax ]These are heavily used in optimisation.
An inner product maps two vectors to a single scalar.
It allows us to measure:
flowchart TD
T["Inner<br/>products<br/>(types)"] --> DOT["Euclidean<br/>Dot product"]
T --> WIP["Weighted<br/>inner product"]
T --> FN["Function-space<br/>(integral)"]
T --> HERM["Complex<br/>Hermitian"]
T --> MAT["Matrix<br/>inner product<br/>(Frobenius)"]
DOT --> Rn["Vectors in<br/>
<span>
\( \mathbb{R}^n \)
</span>
"]
WIP --> SPD["SPD matrix<br/>W"]
FN --> L2["L2 space<br/>functions"]
HERM --> Cn["Vectors in<br/>C^n"]
MAT --> Mnm["Matrices<br/>R^{m×n}"]
style T fill:#90CAF9,stroke:#1E88E5,color:#000
style DOT fill:#C8E6C9,stroke:#2E7D32,color:#000
style WIP fill:#C8E6C9,stroke:#2E7D32,color:#000
style FN fill:#C8E6C9,stroke:#2E7D32,color:#000
style HERM fill:#C8E6C9,stroke:#2E7D32,color:#000
style MAT fill:#C8E6C9,stroke:#2E7D32,color:#000
style Rn fill:#CE93D8,stroke:#8E24AA,color:#000
style SPD fill:#CE93D8,stroke:#8E24AA,color:#000
style L2 fill:#CE93D8,stroke:#8E24AA,color:#000
style Cn fill:#CE93D8,stroke:#8E24AA,color:#000
style Mnm fill:#CE93D8,stroke:#8E24AA,color:#000
For vectors
\( \mathbf{a}, \mathbf{b} \in \mathbb{R}^n \)
Backpropagation applies the chain rule:
Chain rule:
[ \frac{dL}{dx} = \frac{dL}{dy} \cdot \frac{dy}{dx} ]
flowchart LR
x --> y
y --> L
Automatic differentiation computes exact derivatives efficiently using computational graphs.
Once we define an inner product, we can define the angle between two vectors.
Angles allow us to measure how aligned or different two vectors are in space.
Key Idea: Angle measures similarity between vectors. Orthogonality means complete independence (no similarity).
For vectors in n-dimensional space:
January 26, 2026A selection of notes that didn’t fit elsewhere or are being worked on!.
July 4, 2024Artificial Intelligence is often described in three stages, based on capability and scope:
examples