Math for Machine Learning

A dot product multiplies two vectors elementwise and sums the result, producing a single number that grows with agreement between them. That makes it a similarity score, which is why it appears wherever things are compared or combined: a weighted sum of features, a projection onto a direction, or the similarity between two representations. Divide out the magnitudes and you have cosine similarity, which compares direction alone.

Features on wildly different scales distort any method that combines or compares them. Distance-based methods are dominated by the largest-range feature, and gradient descent on an elongated loss surface zigzags and converges slowly. Scaling makes the features comparable, which changes both the geometry the optimizer walks and the meaning of a distance.

A derivative is the rate of change of a function at a point: how much the output moves for a small move in the input. Its sign says which way the function is heading and its magnitude says how steeply. That is exactly the information needed to decide which direction reduces a loss.

A gradient collects the partial derivatives with respect to every parameter into one vector, and it points in the direction of steepest increase. Taking a step in the opposite direction is what reduces the loss. It is a direction with a magnitude, not a value of the function itself.

It can fail because the surface has more than one dip and it settles in a local one, because the learning rate is too large and it overshoots or diverges, or because the rate is too small and it never arrives. Poorly scaled features make the surface elongated so it zigzags, flat regions give it almost no signal, and a badly chosen starting point can put it somewhere unhelpful.

The mean is the centre of a distribution; the variance is how far values typically sit from that centre, in squared units. Two datasets with the same mean can behave completely differently, which is why quoting a mean alone hides the shape. Standard deviation is the square root of the variance and is in the original units.

Conditional probability is the probability of one event given that another is known to have occurred, which restricts attention to the cases where the condition holds. It is where intuition usually fails, because a rare condition changes the answer dramatically: a test with a high detection rate can still produce mostly false positives when the underlying rate is low. Ask what the base rate is before trusting any conditional claim.

Correlation says two quantities move together; causation says changing one changes the other. Correlation can arise from a common cause, from selection in how the data was collected, or from coincidence. To claim causation you need an intervention or a design that rules out the alternative explanations, not a stronger correlation.

Every subject in this stage has review questions. The condition is met when you can answer them out loud, without notes and without looking anything up, in your own words rather than a recited definition. An explanation you can only give with the material open is recognition, not recall.

Complete means each exercise satisfies its stated requirements and its done-when criteria, not that you attempted it. Where an exercise says you should be able to explain or defend something, that explanation is part of completing it. A stage whose source lists no exercises meets this through its owned project milestone instead.

Works means it runs end to end from a clean checkout, on the inputs it was built for, with the failure cases it claims to handle actually handled. It is not a notebook that produced a result once. Someone else following your README should be able to run it without asking you a question.

There is something that would fail if the behaviour regressed, and you have watched it fail. That can be automated tests or a written verification procedure with recorded results, depending on what the stage produces. A test that has only ever passed is evidence of nothing.

The README says what the project is, how to install, run and test it, and why it is built the way it is: the choices you made, the alternatives you rejected, and what you deliberately left out. A list of commands with no reasoning does not meet this condition.

Delete the central piece and write it again from an empty file, without your previous version, a tutorial, or generated code in front of you. It does not have to be identical, it has to work and you have to be able to explain every line. This is the condition that separates having finished from being able.

Record a gate exam and score yourself honestly against the dimensions the roadmap defines: recall, implementation, debugging, testing and design judgment. Elapsed time, hours watched and completed resources count for nothing here. Checking the boxes without a recorded exam does not open the gate.

Ask an AI reviewer to examine the work against this stage's concepts and report foundational misunderstandings, not style. A foundational blocker is a misconception that later stages would build on, such as leakage, a broken abstraction, or an explanation that is confidently wrong. Nothing in that class may remain open when you pass.