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Machine learning2 topics

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Machine learning

Gradient descent

Course1 built1 learned

Chapter 01What gradient descent doeswhat-gradient-descent-does

OBJECTIVES

•Read each optimizer step as a downhill move on the loss surface.
•Explain why following the negative gradient lowers the loss.
•Spot when small steps converge versus overshoot the minimum.

TLDR

The gradient is the steepest-uphill vector; we step along its negative, scaled by η.

INTUITION

The slope the hiker feels is the gradient ∇L - the vector of partial derivatives, one per parameter. It points in the direction of steepest increase, so −∇L is the steepest decrease. The update rule writes that down: θ ← θ − η∇L.

In practice ∇L is averaged over the batch, so the step size doesn't silently scale with how many samples you fed it.

for epoch in range(epochs):

    grad = compute_grad(X, y, theta)   # averaged over the batch

    theta = theta - lr * grad

WORKED_EXAMPLEfrom Ask: session 2

Start at 1e-3 and watch the curve: smoothly down → try larger; sawtooth or NaN → halve it. This is exactly the symptom in your latest Ask - picked up again formally in Ch.3.

CONNECTIONS

Builds on §1.1 (Beginner) - the blindfolded-hiker analogy. Here “downhill” stops being a feeling and becomes a vector.

Chapter 02The update rule & learning ratethe-update-rule-learning-rate

OBJECTIVES

•Write the update rule θ ← θ − η∇L and name every term.
•Predict how the learning rate η reshapes the trajectory.
•Diagnose divergence from the shape of the loss curve.

TLDR

Average the gradient so η is batch-independent; scale features so one η fits every dimension.

INTUITION

Summing the gradient over a batch makes the effective step grow with batch size, so 1e-3 behaves differently at batch 16 versus 256. Averaging decouples η from n. Feature scaling does the spatial version of the same fix: if one feature ranges 0-1 and another 0-1000, the loss surface is a stretched ravine and no single η suits both axes.

•The learning rate is the single most important knob.
•Standardize features so the surface is round, not a ravine - then one η works everywhere.

WORKED_EXAMPLE

With a sum, doubling the batch doubles each step - same η, twice the overshoot. Dividing by len(y) keeps the step the same size regardless of batch, which is why frameworks default to the mean.

DEFINITION

When the inner loop is the bottleneck - a custom kernel, an embedded target, no NumPy - the identical update drops to a systems language. The math is unchanged; you just own the memory and the vectorization. Ask in C++ or Rust and the course writes the snippet in that language instead.

// theta -= lr * (Xt * (X*theta - y)) / n

void gd_step(const MatXf& X, const VecXf& y,

             VecXf& theta, float lr) {

  VecXf resid = X * theta - y;        // batch error

  VecXf grad  = X.transpose() * resid;

  grad /= static_cast<float>(y.size());   // average

  theta.noalias() -= lr * grad;       // in-place

}

CONNECTIONS

Builds on §2.1 (Beginner) - the Goldilocks step. Here it becomes η acting on a batch-averaged gradient, plus why scaling features matters.

Add your own →

the rest of this chapter, and 1 more chapter in this course

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Built on how people actually learn

Built on how
people actually learn

Active recallSpaced repetition - FSRSThe Feynman techniqueLearning from errorno telemetry - no training on you

One question in

A whole livingcourse out.

Ask in one box, then rate it.
One box, a real answer, then the self-rate — the hand-off to Theory only appears once you commit your guess.
learn / gradient-descent / ask
Ask: one box
youMy loss goes to NaN after about 5 steps of gradient descent. I think my gradient is wrong.
coachThat pattern — drop, then growing overshoot — is the signature of too large a step, not a wrong gradient. What do you know about how the safe step relates to the curvature of the loss?
coachRate your guess before moving on. Committing first is what turns reading into recall.
youI got it, confidence 3 of 5
usage clip coming soon
One course, three depths.
One course flips across depth tabs — the hiker's intuition → the update rule → the proof it is optimal.
learn / gradient-descent / theory
Theory: practitioner
From “downhill” to a vector
∇L is the vector of partial derivatives: it points at the steepest increase, so −∇L is the steepest decrease. θ ← θ − η∇L writes that down.
BeginnerThe blindfolded hiker
PractitionerFrom “downhill” to a vector
PhD−∇L is the steepest-descent direction
usage clip coming soon
Drills from your own bug.
A fresh multiple-choice drill built from your bug; pick the option, get the why.
learn / gradient-descent / practice
Drill: pick difficulty
Your loss explodes to NaN within a few steps. Most likely cause?
A. learning rate too small
B. learning rate too large✓
C. too few features
D. batch size of 1
why — a step above the stability threshold makes each update overshoot, amplifying the error every iteration until it diverges.
usage clip coming soon
Catch the planted flaw.
A confident 'fix' seeded with two real bugs; catch the flipped sign and the runaway learning rate.
learn / gradient-descent / interrogate
Wrong on purpose: 2 flaws planted
gradient_descent.py, "this is the fix"
def gradient_descent(grad, x0, lr=0.1, steps=100):
x = x0
for _ in range(steps):
g = grad(x)
x = x + lr * g # ← flaw 1
return x
# caller
best = gradient_descent(grad, x0, lr=10.0) # ← flaw 2
print(best)
flaw 1: sign flipped, this climbs; flaw 2: lr far above the 2/β stability bound
usage clip coming soon
Explain it back, get graded.
You type a plain-English explanation; the rubric ticks direction and step size, flags the missing stability bound.
learn / gradient-descent / feynman
In your own words
Gradient descent walks downhill on the loss. The gradient points uphill, so you step the other way. The learning rate is how big the step is — too big and you overshoot, too small and it's slow.
✓Negative gradient is the descent direction
✓Learning rate scales the step
✗Stability is bounded by curvature (η < 2/L)
coachYou have the role of the step size. You did not say what BOUNDS it — the curvature L, with η < 2/L. That gap is what gets scheduled back.
usage clip coming soon

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Play the forgetting curve.

Reviews: 2

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Hover for memory level — click to review

FSRSThe modern spaced-repetition scheduler, running under the hood

The engine

Two ideas do the teaching

Everything else - the four graded surfaces, the schedule - is just machinery around these two.

Learning from error

Spotting a flaw forces you to hold the correct model in your head and compare. So one surface hands you an answer with a deliberate, real mistake and asks you to find it - the moment you catch it, you have understood it.

The Feynman technique

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Four graded retrieval surfaces - your Ask guess, Practice, Interrogate the flaw, and Feynman - feed one living course and one schedule. Theory and Answer are where you read; the graded surfaces are where you prove it.

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Start from wrong.Leave understanding.

Chapter 01What gradient descent doeswhat-gradient-descent-does

Ask in one box, then rate it.

One course, three depths.

Drills from your own bug.

Catch the planted flaw.

Explain it back, get graded.

Any phasesany time.