Practice

Sharpen your skills with hands-on coding, math, and reasoning problems.

Implement the Sigmoid Function

Neural Networks

Spotting Overfitting

Model Evaluation

The Positive Test Result

ReLU, Vectorised

Activation Functions

Softmax Without Overflow

Activation Functions

Cosine Similarity

Intersection over Union

Computer Vision

One Step of Adam

K-Means: The Assignment Step

Simple Moving Average

A Single Q-Learning Update

Reinforcement Learning

Non-Maximum Suppression

Computer Vision

Building the ROC Curve

Metrics & Evaluation

Logistic Regression Training Loop

Pad Sequences

Matrix Transpose

Positional Encoding (sin/cos)

Gradient Descent on a 1D Quadratic

Dropout (Training Mode)

Neural Networks

RMSProp (Single Update Step)

Anchor Box Generation

Computer Vision

Apply a 4×4 Homogeneous Transform

Value Iteration Step

Reinforcement Learning

Expected Value of a Discrete Distribution

Probability and Statistics

Entropy of a Node

Precision and Recall at K

Recommender Systems

Micro-F1

Metrics & Evaluation

Cross-Entropy Loss

Manhattan Distance

Euclidean Distance

GRU Cell Forward Pass

Neural Networks

Causal Masking

Global Average Pooling

Neural Networks

2D Convolution Layer

Neural Networks

Leaky ReLU

Activation Functions

Silhouette Score

Metrics & Evaluation

Min-Max Normalization

Data Processing

Z-Score Standardization

Data Processing

Accuracy, Precision, Recall, F1

Metrics & Evaluation

R² Score

Metrics & Evaluation

Linear Learning Rate Schedule

Batch Normalization

Neural Networks

Hinge Loss

Contrastive Loss

GELU Activation

Activation Functions

Dot Product

Row-wise L2 Normalization

TF-IDF Vectorizer

BM25 Ranking

Eigenvalues of a 2×2 Matrix

Covariance Matrix

Pearson Correlation

Data Processing

Gradient Clipping by Global Norm

AdaGrad (Single Update Step)

Nesterov Momentum

AdamW (Single Update Step)

AdaDelta (Single Update Step)

Nadam (Single Update Step)

Implement Tanh Activation

Activation Functions

Implement Swish Activation

Activation Functions

Implement Huber Loss

Implement Triplet Loss

Implement InfoNCE Loss

Implement Focal Loss

Implement Dice Loss

Implement KL Divergence

Implement Wasserstein Critic Loss

Bernoulli Probability Mass Function & Moments

Probability and Statistics

Binomial Probability Mass Function

Probability and Statistics

Poisson Probability Mass Function & CDF

Probability and Statistics

Geometric Probability Mass Function & Mean

Probability and Statistics

Which Numbers Describe, Which Infer?

Foundations of Statistics

Counting the Continuous Fields

Foundations of Statistics

Two Teams, Two Estimates

Foundations of Statistics

Measuring Only the Athletes

Foundations of Statistics

When the Total Row Lies

Foundations of Statistics

Allocating a Stratified Sample

Sampling & Study Design

Both in the Sample

Sampling & Study Design

Two Stages, One Teacher

Sampling & Study Design

Measuring the Nonresponse Gap

Sampling & Study Design

The App That Reverses Itself

Sampling & Study Design

Mean After a Value Is Deleted

Descriptive Statistics

Median of an Unsorted Log

Descriptive Statistics

Pinning Down x With the Mode

Descriptive Statistics

Pooling Two Group Averages

Descriptive Statistics

When the Mansion Moves In

Descriptive Statistics

Average Speed Over Three Legs

Descriptive Statistics

One Big Innings

Descriptive Statistics

Goals per Match

Descriptive Statistics

Quality Control on the Bottling Line

Descriptive Statistics

Nine Days of Support Tickets

Descriptive Statistics

The Missing Measurement

Descriptive Statistics

Two Ways to Hit the Same Variance

Descriptive Statistics

Reading Shape Off a Frequency Table

Distributions & Shape

Binning the Boundary Values

Distributions & Shape

One Extreme Value, Two Centres

Distributions & Shape

Percentile Rank When Values Repeat

Distributions & Shape

Between Two Percentile Ranks

Distributions & Shape

Pooling Two Error Logs

Highest Number on Two Rolls

Exactly One of Two Genres

Which Direction Is the Condition?

At Least One Check Fails

How Small Can the Overlap Be?

Even First Roll, Sum of Seven

Counting Outcomes That Map to the Same Value

Random Variables

First Head on an Even Flip

Random Variables

The Peak of a Biased PMF

Random Variables

Endpoints of a Discrete CDF

Random Variables

Normalising a Two-Block Density

Random Variables

Rounding a Continuous Wait

Random Variables

Bernoulli Standard Deviation

Probability Distributions

Half of the Empirical Rule

Probability Distributions

At Least Two Defectives

Probability Distributions

Poisson on a Shorter Interval

Probability Distributions

Waiting, Already Waited

Probability Distributions

The Peak of the Posterior

Probability Distributions

Recovering n and p

Probability Distributions

Two Tails, Two Unknowns

Probability Distributions

Standard Error of a Bottling Line

Statistical Inference

From Population Variance to Standard Error

Statistical Inference

Reading σ Back Out of a Confidence Interval

Statistical Inference

Two Surveys, Two Different Answers

Statistical Inference

Rainfall, Skew, and the CLT

Statistical Inference

The Scale That Always Adds 5 Grams

Statistical Inference

Distance From the Null Model

Hypothesis Testing

Reading Power Off a Simulation

Hypothesis Testing

Two Tails, One Decision

Hypothesis Testing

When Sigma Is Unknown

Hypothesis Testing

From Sums of Squares to F

Hypothesis Testing

Counting the Significant Findings

Hypothesis Testing

Switching Tails After the Fact

Hypothesis Testing

Which Way Did the Line Miss?

Correlation & Regression

Pearson's r From Scratch

Correlation & Regression

Fit the Line, Then Predict

Correlation & Regression

A Perfect Pattern With Nothing to Show

Correlation & Regression

One Step Downhill

Correlation & Regression

The Residual Nobody Wrote Down

Correlation & Regression

Two Scalers, One Value

Statistics for ML

The Constant That Minimises Squared Error

Statistics for ML

The Feature That Correlation Cannot See

Statistics for ML

Picking a Model From Its Decomposition

Statistics for ML

What the Leaked Scaler Gives Away

Statistics for ML

Which Machine Made the Defect?

Bayesian Inference

Where the Posterior Settles

Bayesian Inference

Two Analysts, One Posterior

Bayesian Inference

Reverse-Engineering the Prior

Bayesian Inference

The Word That Cancels the Case

Bayesian Inference

Entropy of a Loaded Spinner

Information Theory & Stochastic Processes

Buying Eight Times the Precision

Information Theory & Stochastic Processes

Cross-Entropy Over a Mini-Batch

Information Theory & Stochastic Processes

Reasoning Backwards Through a Markov Chain

Information Theory & Stochastic Processes

Both Directions of a KL Divergence

Information Theory & Stochastic Processes

The Probability of the Other Class

Statistics for Deep Learning

A Loss Hidden Behind a Shift

Statistics for Deep Learning

How Much Dropout Wobbles a Sum

Statistics for Deep Learning

Where the Moving-Average Window Sits

The Seasonal Index Nobody Reported

Forecasting Back Through a Difference

Ten Tastings and a Strict Alpha

Mixed: Distributions & Hypothesis Testing

Which Loom Wove This Cloth?

Mixed: Distributions & Bayesian Inference

Shrinking an Interval by a Factor of 2.8

Mixed: Sampling Distributions & Confidence Intervals

Slope and Leftover Variation from Summary Statistics

Mixed: Regression & Dispersion

How Far Is a Coin-Flip Count from Uniform?

Mixed: Distributions & Information Theory

Ask the Registrar, Ask a Student

Mixed: Sampling & Expected Value

Two Features, Two Scales, One Comparison

Mixed: Normal Distribution & Standardization

Spam in a Half-Hour

Mixed: Probability & Distributions

The Unbiased Model That Loses

Mixed: ML Statistics & Estimation

A Walk That Never Settles

Mixed: Time Series & Random Variables

One Interval, Two Null Hypotheses

Mixed: Confidence Intervals & Hypothesis Testing

Tall Fathers, Slightly Shorter Sons

Mixed: Regression & Descriptive Statistics

How Low Could the Renewal Rate Really Be?

Foundations of Statistics

Equal Interviews, Unequal Zones

Sampling & Study Design

Two Regions, Two Sampling Rates

Sampling & Study Design

How Big Is the Most Popular Choice?

Descriptive Statistics

Rebuilding a Mean From Deviations

Descriptive Statistics

The Gap Between Mean and Median

Descriptive Statistics

The Score the Final Has to Carry

Descriptive Statistics

A Week of Readings, Two Scales

Descriptive Statistics

How Late Is the 8:15 Bus?

Descriptive Statistics

Two Measures of the Same Spread

Descriptive Statistics

Merging Two Sections

Descriptive Statistics

Relative Frequency of One Bin

Distributions & Shape

How Far the Tail Drags the Mean

Distributions & Shape

Percentile Rank After the Log Is Cleaned

Distributions & Shape

Resolved Within Four Hours

A Topic Shows Up Twice

Python but Not SQL

Which Line Made the Defect?

Three Jobs, One Tag

A PMF Proportional to the Value

Random Variables

An Interval from a Shifted Quadratic CDF

Random Variables

A Window of Rolls Until the First Six

Random Variables

Splitting a Triangular Density in Half

Random Variables

Backing Out the Late-Delivery Rate

Probability Distributions

How Much Do Clicks Wobble?

Probability Distributions

Reading the Rate Off the Quiet Windows

Probability Distributions

The Fifty-Fifty Crash Time

Probability Distributions

Four Bolts, All In Spec

Probability Distributions

How Sharp Is the Posterior?

Probability Distributions

A Point Estimate for a City's Water Use

Statistical Inference

How Many More Parcels Must Be Weighed?

Statistical Inference

The Cutoff Only 5% of Shifts Exceed

Statistical Inference

Upper Limit of a t-Interval for Concrete Strength

Statistical Inference

Where the Rejection Region Starts

Hypothesis Testing

Same Operators, Two Layouts

Hypothesis Testing

F From Means and Variances

Hypothesis Testing

How Often Would It Catch the Effect?

Hypothesis Testing

How Far Off Is the Delivery Model, Typically?

Correlation & Regression

Correlation From a Summary Table

Correlation & Regression

Fit the Line, Then Measure the Miss

Correlation & Regression

The Loss After One Step Downhill

Correlation & Regression

Standardising a Held-Out Row

Statistics for ML

Borrowing the Irreducible Error

Statistics for ML

Which Supplier Sent the Unlabelled Crate?

Bayesian Inference

The Peak of the Posterior After Two Batches

Bayesian Inference

Smoothing the Counts Before You Classify

Bayesian Inference

Extra Bits Per Token

Information Theory & Stochastic Processes

Bits Per Reading in the Long Run

Information Theory & Stochastic Processes

The Same Activation, Two Different Outputs

Statistics for Deep Learning

How Much Does Yesterday Explain Today?

The Arrow That Moved

The Component That Went Missing

Only the Sum and the Difference Survived

Twenty Units the Other Way

Centering a Customer

Twice at Five Units Out

Scale, Add, Then Dot

Vector Operations & Similarity

Recover the Missing Coordinate

Vector Operations & Similarity

Unit Vector of a Combination

Vector Operations & Similarity

The Angle Between Two Taste Profiles

Vector Operations & Similarity

Which Candidate Is Retrieved First?

Vector Operations & Similarity

Completing the Analogy

Vector Operations & Similarity

Building Row 3 From a Rule

The Mean of a Feature

Blending Two Weight Matrices

Making a Matrix Its Own Transpose

Centering a Batch

Two Scalars, Two Transposes

Shape of a Product and the Entry at Row 2, Column 3

Matrix Multiplication

Matrix Times Vector, Checked Two Ways

Matrix Multiplication

Cubing a Matrix Without Cubing Its Entries

Matrix Multiplication

Two Groupings, Two Very Different Costs

Matrix Multiplication

One Hidden Layer, a Batch of Two

Matrix Multiplication

What the Binomial Formula Leaves Behind

Matrix Multiplication

Rebuilding a Vector From Two Landings

Linear Transformations

Two Known Outputs, One Unknown

Linear Transformations

A Thirty-Degree Turn

Linear Transformations

Three Machines in a Row

Linear Transformations

Does the Layer Pull Them Apart?

Linear Transformations

Six Passes Through the Same Machine

Linear Transformations

Splitting a Bill Into Two Unknowns

Systems of Linear Equations

Reading Ax = b

Systems of Linear Equations

Three Equations, One Point

Systems of Linear Equations

The Contradictory Pair

Systems of Linear Equations

Three Points, One Curve

Systems of Linear Equations

Turning the Dial on k

Systems of Linear Equations

The Triangle After the Transform

Determinants & Inverses

Expanding a 3x3 by Cofactors

Determinants & Inverses

Two Values That Kill the Inverse

Determinants & Inverses

Reverse-Engineering k From the Inverse

Determinants & Inverses

How Far Is That Outlier, Really?

Determinants & Inverses

Squeezed by an Inverse

Determinants & Inverses

The Only Subspace in the Room

Vector Spaces, Span & Basis

Rebuilding a Vector From Two Others

Vector Spaces, Span & Basis

Four Vectors, Two Hidden Relations

Vector Spaces, Span & Basis

How Much Freedom Do Three Equations Leave?

Vector Spaces, Span & Basis

The Dummy Variable Trap

Vector Spaces, Span & Basis

Two Subspaces That Are Secretly One

Vector Spaces, Span & Basis

The Direction the Matrix Won't Turn

Eigenvalues & Eigenvectors

Reading the Stretch Factor Off

Eigenvalues & Eigenvectors

Tuning k Until the Roots Collide

Eigenvalues & Eigenvectors

Back-Substituting an Eigenvector

Eigenvalues & Eigenvectors

How Much Variance Does PC1 Keep?

Eigenvalues & Eigenvectors

Three Eigenvalues, Two Questions

Eigenvalues & Eigenvectors

The Value That Makes Them Perpendicular

Orthogonality & Projections

Casting One Vector Onto Another

Orthogonality & Projections

Coordinates in an Orthogonal Basis

Orthogonality & Projections

Completing a Rotation Matrix

Orthogonality & Projections

The Error the Model Cannot Explain

Orthogonality & Projections

How Far Is the Point From the Plane?

Orthogonality & Projections

The Last Pivot of an LU Factorization

Matrix Decompositions

Factor Once, Solve Fifty Times

Matrix Decompositions

Three Steps Through the Eigenbasis

Matrix Decompositions

Least Squares Through Gram-Schmidt

Matrix Decompositions

How Many Singular Values to Keep

Matrix Decompositions

Conditioning of a Triangular Weight Matrix

Matrix Decompositions

Scaling a Weight Vector

Norms & Distance

L1 Minus L-Infinity

Norms & Distance

Shrinking a Point Into Every Unit Ball

Norms & Distance

A Point on the Way, Measured Two Ways

Norms & Distance

Same Data, Two Penalties, Two Winners

Norms & Distance

Three Metrics, Three Different Neighbours

Norms & Distance

One Column of Ones, Three Columns Total

Linear Algebra for ML

Scoring a Guessed Weight Vector

Linear Algebra for ML

What the Residual Is Perpendicular To

Linear Algebra for ML

The Same Temperature, Recorded Twice

Linear Algebra for ML

Centred Features, Clean Fit

Linear Algebra for ML

Rescuing a Singular Fit with Ridge

Linear Algebra for ML

Flattening a Batch of Images

Linear Algebra for Deep Learning

Two Layers, One Matrix

Linear Algebra for Deep Learning

Where ReLU Breaks Additivity

Linear Algebra for Deep Learning

Predicting a Step with the Jacobian

Linear Algebra for Deep Learning

Two Transposes Backward

Linear Algebra for Deep Learning

The Jacobian of a ReLU Network

Linear Algebra for Deep Learning

Almost Everything Is an Edge Case

Dimensionality Reduction & Embeddings

The First Principal Direction

Dimensionality Reduction & Embeddings

The Eigenvalue That Went Missing

Dimensionality Reduction & Embeddings

Nearest Neighbour in Embedding Space

Dimensionality Reduction & Embeddings

One Million Samples, How Many Features?

Dimensionality Reduction & Embeddings

What PCA Throws Away

Dimensionality Reduction & Embeddings

One Parameter, Three Fates

Mixed: Determinants & Systems of Equations

Three Numbers, Three Eigenvalues

Mixed: Eigenvalues & Determinants

Two Hidden Dependencies

Mixed: Rank & Linear Independence

The Line That Misses Every Point

Mixed: Least Squares & Orthogonal Projection

From Singular Values to Explained Variance

Mixed: SVD & PCA

Which Ball Touches the Line First?

Mixed: Norms & Regularization

Reflect, Shear, Rotate

Mixed: Matrix Multiplication & Linear Transformations

Four Steps Without Multiplying Four Times

Mixed: Eigendecomposition & Change of Basis

A Jacobian Through a Dead ReLU

Mixed: Jacobians & Layer Composition

A Column That Was Already There

Mixed: Design Matrix & Normal Equations

West, and Somewhere North

The Leg the Log Lost

How Unusual Is This Viewer?

The Length Nobody Measured

Vector Operations & Similarity

Scoring a Candidate Against a Session

Vector Operations & Similarity

Adding Two Directions

Vector Operations & Similarity

Five Across, Four Down

Mixing Until One Weight Disappears

Two Shapes That Only Match After a Flip

Grams First, Rupees Second

Matrix Multiplication

Two Orders, Two Landing Points

Matrix Multiplication

A Batch Through Two Layers

Matrix Multiplication

Two Landings, One Distance

Linear Transformations

The Augmentation Pipeline

Linear Transformations

The Better of Two Weight Matrices

Linear Transformations

The Row That Wasn't in the Matrix

Systems of Linear Equations

When One Equation Says Nothing New

Systems of Linear Equations

Four Records, One Missing Reading

Systems of Linear Equations

Scaled, Transposed, Swapped

Determinants & Inverses

Only One Row of the Inverse

Determinants & Inverses

The Coefficient Ridge Pulls Back

Determinants & Inverses

Two Concentrates, Three Targets

Vector Spaces, Span & Basis

The Calibration Value That Collapses the Array

Vector Spaces, Span & Basis

Four Weights, Two Real Directions

Vector Spaces, Span & Basis

The Factor After Three Ticks

Eigenvalues & Eigenvectors

Calibrate k, Then Find the Fastest Mode

Eigenvalues & Eigenvectors

The Score Along the First Principal Direction

Eigenvalues & Eigenvectors

How Far the Drone Sits Off the Wire

Orthogonality & Projections

What Two Patterns Cannot Explain

Orthogonality & Projections

Three Columns, Two Directions

Orthogonality & Projections

The Last Pivot and Its Multiplier

Matrix Decompositions

Five Steps Through a Two-Unit Layer

Matrix Decompositions

What the Rank-One Copy Remembers

Matrix Decompositions

Clip the Gradient, Then Find Its Biggest Step

Norms & Distance

Ranked by One Metric, Scored by Another

Norms & Distance

Two Penalties, One Gap

Norms & Distance

Two Features, One Matrix Inverse

Linear Algebra for ML

Three Drivers for Every Van

Linear Algebra for ML

How Many Parameters After the Flatten?

Linear Algebra for Deep Learning

Three Layers, One Matrix, One Zero

Linear Algebra for Deep Learning

One Step on the Input, Not the Weights

Linear Algebra for Deep Learning

The Eigenvalue That Got Smudged

Dimensionality Reduction & Embeddings

Which Way Does the Cloud Point?

Dimensionality Reduction & Embeddings

Compress, Decompress, Measure the Damage

Dimensionality Reduction & Embeddings

Rewinding a Volume-Doubling Transform

Mixed: Determinants & Systems of Equations

Ridge Through the Spectrum

Mixed: Eigenvalues & Determinants

How Far Is y From the Column Space?

Mixed: Orthogonal Projection & Least Squares

The Missing Singular Value

Mixed: SVD & PCA

Two Dead Units and a Nudge

Mixed: Jacobians & Layer Composition

Six Columns In, How Many Numbers Out?

Linear Algebra for ML