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A labelled sample and a specified prediction task.

A linear predictor in specified features; squared loss.

Unconstrained least squares with real A and b.

A finite-dimensional subspace of a Euclidean space.

The standard Euclidean inner product.

Real, finite-dimensional vectors of compatible dimensions.

Finite-dimensional real matrices.

A is an $m\times n$ real matrix and $b\in\mathbb R^m$.

Differentiability at the point; sufficiently small increments.

Binary labels; independent observations conditional on inputs for the product likelihood.

A specified probability model and events with defined probabilities.

Differentiability, nonzero gradient and a sufficiently small positive step.

Ordinary gradient descent; distinguish its assumptions from other optimisers.

Ridge with $\lambda>0$ penalising all coefficients in the stated formula.

A defined target population and sampling or temporal structure.

State the task and data-generating constraints before selecting a method.

Finite-dimensional unconstrained least squares.

Finite precision arithmetic; the condition-number identity assumes full column rank.

Real finite-dimensional least squares; distinguish exact and approximate fitting.

A specified repeated-sampling scheme.

Finite second moments for the estimator.

IID data with finite positive variance for the classical CLT; exact t inference additionally uses normal observations.

A specified parametric model; independence when factorising the likelihood.

The sampling model and interval construction must be specified.

A prespecified test statistic, null distribution and tail convention.

Power is evaluated at a specified alternative parameter value.

A defined estimand, sampling design and model.

Explicit preconditions and a finite input representation.

An invariant at a precisely stated program point and a terminating loop.

An integer-indexed claim with an appropriate starting index.

A specified input-size measure and cost model.

Recursive calls on strictly smaller cases under a well-founded measure.

For the stated merge-sort recurrence, n is a power of two and merging is linear.

A specified optimisation problem; the example is unweighted interval scheduling, with compatibility allowing the next interval to start at the previous finish time.

Positive integer denominations, a nonnegative integer amount and unlimited supply of each coin; states must retain information needed for future choices.

Explicit input restrictions, cost model and optimisation objective.