Gradients of Vector-Valued and Matrix Functions
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Covers gradients when outputs or parameters are vectors/matrices.
If f: R^n -> R^m, the derivative is the Jacobian.
[
J =
\begin{bmatrix}
\frac{\partial f_1}{\partial x_1} & \dots & \frac{\partial f_1}{\partial x_n} \
\vdots & \ddots & \vdots \
\frac{\partial f_m}{\partial x_1} & \dots & \frac{\partial f_m}{\partial x_n}
\end{bmatrix}
]For scalar f(x):
[
H = \nabla^2 f
]Hessian captures curvature.
December 15, 2025Generative AI
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Generative Artificial Intelligence (GenAI) refers to a class of AI systems that can generate new content such as text, images, audio, video, or code, rather than only making predictions or classifications.
GenAI systems learn patterns and representations from large datasets and use them to produce novel outputs that resemble the data they were trained on.
How Generative AI Differs from Traditional AI
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| Traditional AI | Generative AI |
|---|
| Predicts or classifies | Generates new content |
| Task-specific models | General-purpose models |
| Fixed outputs | Open-ended outputs |
| Often rule-based | Data-driven and probabilistic |
Core Idea of Generative AI
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Instead of learning “what label to assign”, Generative AI learns “how data is structured” and then creates new data following that structure.
Reinforcement Learning (RL)
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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:
- takes actions
- observes outcomes
- receives rewards or penalties
- gradually learns a strategy that maximises long-term reward
Reinforcement Learning teaches an agent how to act, not what to predict.
Useful Gradient Identities
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[
\nabla (a^T x) = a
]
[
\nabla (x^T A x) = (A + A^T)x
]If A symmetric:
[
\nabla (x^T A x) = 2Ax
]These are heavily used in optimisation.
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Deep Reinforcement Learning
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Deep Reinforcement Learning (DRL) studies how an agent learns to make a sequence of decisions by interacting with an environment and receiving feedback through rewards.
Reinforcement learning provides the framework for sequential decision-making. Deep learning extends this framework with powerful function approximators that can handle large or complex state and action spaces.
Deep Reinforcement Learning = Reinforcement Learning + Deep Neural Networks
The learning path begins with classical reinforcement learning foundations and progresses towards value-based deep learning, policy-gradient methods, model-based approaches, and imitation learning.
Inner Products and Dot Product
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An inner product maps two vectors to a single scalar.
It allows us to measure:
- similarity
- vector length
- projections
- orthogonality
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
Definition
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For vectors
\( \mathbf{a}, \mathbf{b} \in \mathbb{R}^n \)
Natural Language Processing
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Natural Language Processing (NLP) studies how computers can analyse, understand, represent, and generate human language.
It combines ideas from linguistics, computer science, machine learning, and deep learning to work with text and language-based information.
Natural Language Processing = Linguistics + Computation + Machine Learning
The learning path begins with language understanding and vector representations, progresses through language modelling, tagging, and parsing, and then moves towards transformers, knowledge graphs, Retrieval-Augmented Generation, and modern NLP applications.
Artificial and Computational Intelligence
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Modular Structure
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1. Introduction
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- Artificial Intelligence foundations
- Overview of modern AI
- AI application domains
2. Introduction to Intelligent Agents
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- Notion of agents and environments
- Rationality
- Nature of environments
- Structure of agents
3. Problem-Solving Agents Using Search
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- Problem formulation
- Uninformed and informed search algorithms
- Heuristics
- Greedy Best-First Search
- A* Search and optimality of A*
- Heuristic accuracy and algorithm performance
- Admissible heuristics from relaxed problems
- Pattern databases and experience
- Learning heuristics
- Local search and optimisation
- Hill Climbing
- Local Beam Search
- Genetic Algorithms
- Ant Colony Optimisation
- Neural Architecture Search
- Neuroevolution
4. Game Playing
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- Minimax Algorithm
- Alpha-Beta Pruning
- Monte Carlo Tree Search
- Stochastic Games
5. Knowledge Representation Using Logic
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- Propositional and Predicate Logic
- TT-Entail and theorem proving
- Logic representation of intelligent agents
- Proof by resolution
- DPLL Algorithm
- Agents based on Propositional Logic
- Unification
- Forward Chaining
- Backward Chaining
- Resolution
6. Multi-Agent Decision Making
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- Properties of multi-agent environments
- Multi-agent planning
- Non-Cooperative Game Theory
- Cooperative Game Theory
- Collective decision making
7. Probabilistic Representation and Reasoning
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- Representing knowledge in uncertain domains
- Semantics of Bayesian Networks
- Exact inference in Bayesian Networks
- Approximate inference in Bayesian Networks
8. Probabilistic Reasoning Over Time
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- Time and uncertainty
- Inference in temporal models
- Hidden Markov Models
- Learning HMMs
- Dynamic Bayesian Networks
9. Ethics in AI
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- Explainable AI
- Logically Explained Networks
- Explainable Bayesian Networks
Experiments
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| # | Experiment |
|---|
| 1 | Implement Uninformed Search Algorithms such as BFS and DFS |
| 2 | Implement the A* Algorithm for Informed Search |
| 3 | Implement Local Search Techniques using a Genetic Algorithm |
| 4 | Implement the Minimax Algorithm for Adversarial Search in game playing |
| 5 | Represent knowledge using logic and perform reasoning using Prolog |
| 6 | Experiment with Bayesian Networks and exact inference |
| 7 | Experiment with the application of a Hidden Markov Model in Natural Language Processing |
- Programming languages: Python, Prolog
- Tools and libraries: Jupyter, NumPy, SciPy, Pandas, pgmpy, NLTK
- Environments: Google Colab, SWI-Prolog Online
Book References
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Primary Textbook
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- Stuart Russell and Peter Norvig, Artificial Intelligence: A Modern Approach, 4th Edition, Pearson Education, 2020.
Reference Books
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- Ryszard S. Michalski, Jaime G. Carbonell and Tom M. Mitchell, Machine Learning: An Artificial Intelligence Approach, Elsevier, 2014.
- Dan W. Patterson, Introduction to AI and Expert Systems, Prentice Hall of India, New Delhi, 2010.
- Elaine Rich and Kevin Knight, Artificial Intelligence, 2nd Edition, Tata McGraw Hill Publishing Company, New Delhi, 2003.
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