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36 matching topics

Empirical risk and loss functions
Distinguish the training objective from predictive performance.
Conditions to keep in mind
A labelled sample and a specified prediction task.
In these courses
- Supervised Learning
Open topic: Empirical risk and loss functions
Linear regression as an optimisation problem
Connect regression coefficients to least squares.
Conditions to keep in mind
A linear predictor in specified features; squared loss.
In these courses
- Supervised Learning
Open topic: Linear regression as an optimisation problem
Normal equations
Derive the least-squares stationarity condition in two ways.
Conditions to keep in mind
Unconstrained least squares with real A and b.
In these courses
- Supervised Learning
- Linear Algebra
Open topic: Normal equations
Orthogonal projection
Characterise the closest vector in a subspace.
Conditions to keep in mind
A finite-dimensional subspace of a Euclidean space.
In these courses
- Supervised Learning
- Linear Algebra
Open topic: Orthogonal projection
Orthogonality and residuals
Express orthogonality using an inner product.
Conditions to keep in mind
The standard Euclidean inner product.
In these courses
- Supervised Learning
- Linear Algebra
Open topic: Orthogonality and residuals
Vectors and linear combinations
Interpret a linear combination geometrically.
Conditions to keep in mind
Real, finite-dimensional vectors of compatible dimensions.
In these courses
- Supervised Learning
- Linear Algebra
Open topic: Vectors and linear combinations
Column space, rank and nullity
Relate independent directions to uniqueness.
Conditions to keep in mind
Finite-dimensional real matrices.
In these courses
- Supervised Learning
- Linear Algebra
Open topic: Column space, rank and nullity
Solvability of linear systems
Decide whether Ax=b is consistent.
Conditions to keep in mind
A is an $m\times n$ real matrix and $b\in\mathbb R^m$ .
In these courses
- Supervised Learning
- Linear Algebra
Open topic: Solvability of linear systems
Derivatives as local approximations
Use a derivative to predict a local change.
Conditions to keep in mind
Differentiability at the point; sufficiently small increments.
In these courses
- Supervised Learning
- Linear Algebra
Open topic: Derivatives as local approximations
Logistic classification and likelihood
Relate a Bernoulli model to cross-entropy.
Conditions to keep in mind
Binary labels; independent observations conditional on inputs for the product likelihood.
In these courses
- Supervised Learning
Open topic: Logistic classification and likelihood
Random variables and independence
Distinguish a random variable, a realisation and independence.
Conditions to keep in mind
A specified probability model and events with defined probabilities.
In these courses
- Supervised Learning
- Statistical Inference
Open topic: Random variables and independence
Gradients and local descent
Justify the negative gradient as a local descent direction.
Conditions to keep in mind
Differentiability, nonzero gradient and a sufficiently small positive step.
In these courses
- Supervised Learning
Open topic: Gradients and local descent
Gradient descent and learning rates
Explain an update rule and its convergence limitations.
Conditions to keep in mind
Ordinary gradient descent; distinguish its assumptions from other optimisers.
In these courses
- Supervised Learning
Open topic: Gradient descent and learning rates
Regularisation and inductive bias
Explain how ridge regularisation changes the objective.
Conditions to keep in mind
Ridge with $\lambda>0$ penalising all coefficients in the stated formula.
In these courses
- Supervised Learning
Open topic: Regularisation and inductive bias
Validation, leakage and generalisation
Design an evaluation that respects how new data arrive.
Conditions to keep in mind
A defined target population and sampling or temporal structure.
In these courses
- Supervised Learning
Open topic: Validation, leakage and generalisation
Supervised-learning examination synthesis
Connect assumptions, objectives, optimisation and evaluation.
Conditions to keep in mind
State the task and data-generating constraints before selecting a method.
In these courses
- Supervised Learning
Open topic: Supervised-learning examination synthesis
Existence and uniqueness
Separate a unique fit from unique coefficients.
Conditions to keep in mind
Finite-dimensional unconstrained least squares.
In these courses
- Linear Algebra
Open topic: Existence and uniqueness
Conditioning and method selection
Distinguish mathematical existence from numerical reliability.
Conditions to keep in mind
Finite precision arithmetic; the condition-number identity assumes full column rank.
In these courses
- Linear Algebra
Open topic: Conditioning and method selection
Least-squares examination synthesis
Combine solvability, geometry and rank in one argument.
Conditions to keep in mind
Real finite-dimensional least squares; distinguish exact and approximate fitting.
In these courses
- Linear Algebra
Open topic: Least-squares examination synthesis
Sampling distributions
Distinguish data variation from estimator variation.
Conditions to keep in mind
A specified repeated-sampling scheme.
In these courses
- Statistical Inference
Open topic: Sampling distributions
Bias, variance and mean squared error
Separate systematic error from sampling variation.
Conditions to keep in mind
Finite second moments for the estimator.
In these courses
- Statistical Inference
Open topic: Bias, variance and mean squared error
Standard errors and normal approximation
Identify the assumptions behind a mean interval.
Conditions to keep in mind
IID data with finite positive variance for the classical CLT; exact t inference additionally uses normal observations.
In these courses
- Statistical Inference
Open topic: Standard errors and normal approximation
Maximum-likelihood estimation
Construct a likelihood and check its maximiser.
Conditions to keep in mind
A specified parametric model; independence when factorising the likelihood.
In these courses
- Statistical Inference
Open topic: Maximum-likelihood estimation
Confidence intervals and coverage
Interpret a confidence procedure under repeated sampling.
Conditions to keep in mind
The sampling model and interval construction must be specified.
In these courses
- Statistical Inference
Open topic: Confidence intervals and coverage
Hypothesis tests and p-values
Interpret a p-value relative to a specified null model.
Conditions to keep in mind
A prespecified test statistic, null distribution and tail convention.
In these courses
- Statistical Inference
Open topic: Hypothesis tests and p-values
Type I error, power and multiplicity
Distinguish false rejection from sensitivity to an alternative.
Conditions to keep in mind
Power is evaluated at a specified alternative parameter value.
In these courses
- Statistical Inference
Open topic: Type I error, power and multiplicity
Statistical-inference examination synthesis
Connect an inferential claim to its estimand and assumptions.
Conditions to keep in mind
A defined estimand, sampling design and model.
In these courses
- Statistical Inference
Open topic: Statistical-inference examination synthesis
Problem specifications and correctness
State valid inputs and the required output before choosing an algorithm.
Conditions to keep in mind
Explicit preconditions and a finite input representation.
In these courses
- Algorithm Design
Open topic: Problem specifications and correctness
Loop invariants
Prove a loop correct using an invariant.
Conditions to keep in mind
An invariant at a precisely stated program point and a terminating loop.
In these courses
- Algorithm Design
Open topic: Loop invariants
Induction and proof structure
State a base case and an inductive implication.
Conditions to keep in mind
An integer-indexed claim with an appropriate starting index.
In these courses
- Algorithm Design
Open topic: Induction and proof structure
Asymptotic complexity
State and justify an asymptotic resource bound.
Conditions to keep in mind
A specified input-size measure and cost model.
In these courses
- Algorithm Design
Open topic: Asymptotic complexity
Recursive correctness and termination
Separate recursive meaning from termination.
Conditions to keep in mind
Recursive calls on strictly smaller cases under a well-founded measure.
In these courses
- Algorithm Design
Open topic: Recursive correctness and termination
Divide and conquer
Derive a recurrence from an algorithm structure.
Conditions to keep in mind
For the stated merge-sort recurrence, n is a power of two and merging is linear.
In these courses
- Algorithm Design
Open topic: Divide and conquer
Greedy strategies and exchange arguments
Justify a greedy choice or find a counterexample.
Conditions to keep in mind
A specified optimisation problem; the example is unweighted interval scheduling, with compatibility allowing the next interval to start at the previous finish time.
In these courses
- Algorithm Design
Open topic: Greedy strategies and exchange arguments
Dynamic programming and state design
Define a sufficient state, recurrence and evaluation order.
Conditions to keep in mind
Positive integer denominations, a nonnegative integer amount and unlimited supply of each coin; states must retain information needed for future choices.
In these courses
- Algorithm Design
Open topic: Dynamic programming and state design
Algorithm examination synthesis
Present a complete algorithmic argument.
Conditions to keep in mind
Explicit input restrictions, cost model and optimisation objective.
In these courses
- Algorithm Design
Open topic: Algorithm examination synthesis

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Find a topic

Empirical risk and loss functions

Linear regression as an optimisation problem

Normal equations

Orthogonal projection

Orthogonality and residuals

Vectors and linear combinations

Column space, rank and nullity

Solvability of linear systems

Derivatives as local approximations

Logistic classification and likelihood

Random variables and independence

Gradients and local descent

Gradient descent and learning rates

Regularisation and inductive bias

Validation, leakage and generalisation

Supervised-learning examination synthesis

Existence and uniqueness

Conditioning and method selection

Least-squares examination synthesis

Sampling distributions

Bias, variance and mean squared error

Standard errors and normal approximation

Maximum-likelihood estimation

Confidence intervals and coverage

Hypothesis tests and p-values

Type I error, power and multiplicity

Statistical-inference examination synthesis

Problem specifications and correctness

Loop invariants

Induction and proof structure

Asymptotic complexity

Recursive correctness and termination

Divide and conquer

Greedy strategies and exchange arguments

Dynamic programming and state design

Algorithm examination synthesis