Calculus, Gradients and Convex Optimization · AI Engineering

Derivatives, autodiff, GD, momentum, Adam, Newton and convexity tests. From-scratch Python and PyTorch.

Forward and backward through one neuron

Nodes and connections

Node	Role	Kind
x	input
z1 = w*x	multiply
z2 = w*x + b	add bias
a = sigmoid(z2)	activation
dL/dw	dL/dz2 * x
dL/db	dL/dz2 * 1
dL/dz2	dL/da * a(1-a)
L	cross-entropy loss

From	To	Connection
x	z1 = w*x	*w
z1 = w*x	z2 = w*x + b	+b
z2 = w*x + b	a = sigmoid(z2)
a = sigmoid(z2)	L
L	dL/dz2	backward
dL/dz2	dL/dw	x
dL/dz2	dL/db	1

Sources

phases/01-math-foundations/04-calculus-for-ml/docs/en.md

Reverse-mode autodiff in Value.backward

Record each operation makes a new Value with its children and a _backward closure.
Sort build_topo puts every node after the nodes it depends on.
Seed set self.grad = 1.0 on the output. This is dy/dy = 1.
Walk back call v._backward() for each node in reversed topological order.
Accumulate each closure adds to .grad with +=. A value used twice gets the sum.
Reset set p.grad = 0.0 for every parameter before the next loss.backward().
Update p.data -= lr * p.grad for each parameter.

Sources

Tasks in code

Task	Code
Central difference	`numerical_derivative(f, x, h=1e-7)`
Gradient of many inputs	`numerical_gradient(f, point, h=1e-7)`
Hessian of f(x, y)	`hessian_2d(f, x, y, h=1e-5)`
Check an autodiff gradient	`gradient_check(build_expr, x_val, h=1e-7)`
XOR network, 4 hidden	`model = MLP([2, 4, 1])`
Run an optimizer	`optimize(optimizer, func, grad_func, start)`
Test convexity by sampling	`check_convexity(f, dim)`
Newton steps in 2D	`newtons_method(grad_f, hessian_f, x0)`

Sources

Optimizers from scratch

GradientDescent(lr=0.001)w = w - lr * gradient

SGDMomentum(lr=0.001, momentum=0.9)v = beta * v + gradient

Adam(lr=0.001, beta1=0.9, beta2=0.999, epsilon=1e-8)two moments per weight

m_hat = [m / (1 - self.beta1 ** self.t) for m in self.m]bias correction

p - self.lr * mh / (vh ** 0.5 + self.epsilon)per-weight step size

adam_history = optimize(Adam(lr=0.01), rosenbrock, rosenbrock_gradient, start)

Sources

Autograd and optimizers in PyTorch

x = torch.tensor(2.0, requires_grad=True)

y = x ** 2 + 3 * x + 1

y.backward()reverse mode through the recorded graph

print(x.grad)7.0 = 2*x + 3

sgd = torch.optim.SGD(model.parameters(), lr=0.01, momentum=0.9)

adam = torch.optim.Adam(model.parameters(), lr=0.001)

adamw = torch.optim.AdamW(model.parameters(), lr=0.001, weight_decay=0.01)

scheduler = torch.optim.lr_scheduler.CosineAnnealingLR(adam, T_max=100)

Sources

Key formulas

Central difference: f'(x) \approx \frac{f(x + h) - f(x - h)}{2h}.
Chain rule: \frac{dy}{dx} = f'(g(x)) \cdot g'(x). Sum the paths where the graph branches.
Gradient descent: w = w - \text{lr} \cdot \text{gradient}.
Momentum: v = \beta \cdot v + \text{gradient}, then w = w - \text{lr} \cdot v.
Adam: w = w - \text{lr} \cdot \hat{m} / (\sqrt{\hat{v}} + \epsilon) with bias-corrected moments.
Newton: x_{\text{new}} = x - H^{-1} \cdot \text{gradient}.
Convex: f(tx + (1-t)y) \le t f(x) + (1-t) f(y) for t \in [0, 1].
Lagrangian: L(x, \lambda) = f(x) + \lambda g(x). Set its gradient to zero.

Sources

Derivatives to know

Command	What it does
x^2	2x. MSE loss
wx + b	x for w, 1 for b, w for x. Linear layer
e^x	e^x. Softmax, attention
ln(x)	1/x. Cross-entropy loss
sigmoid	f(x)(1-f(x)). Sigmoid activation
tanh	1 - tanh^2. Classic activation
x^n	n x^(n-1). Used by `__pow__`
sin(x^2)	cos(x^2) * 2x. Chain rule example

Sources

Pick a method

Command	What it does
Gradient descent	First derivatives. O(n) memory. Baseline
SGD with momentum	Best final accuracy with more tuning
Adam	Diagonal curvature from gradient moments. Deep learning default
AdamW	Adam with decoupled weight decay. Transformers
Newton's method	Full Hessian. O(n^3) per step. Small convex problems
L-BFGS	Last m gradient differences. Medium convex problems
K-FAC	Kronecker-factored Fisher. Large-batch training

Sources

Numbers and defaults

Setting	Default	Note
Adam `lr`	0.001	Start here for most problems
Adam `beta2`	0.999	`beta1` is 0.9. Moment decay rates
Adam `epsilon`	1e-8	Stops division by zero
SGD momentum `lr`	0.01	With momentum 0.9
AdamW `weight_decay`	0.01	Decoupled from the gradient step
Mini-batch size	32-256	Good gradient estimate, moderate noise
Finite difference `h`	1e-7	In `numerical_derivative`
Gradient check difference	< 1e-5	Autodiff against numerical

Sources

Troubleshooting

Symptom	Cause	Fix
Loss grows or becomes nan	Learning rate too large	Reduce the learning rate
Training is too slow	Learning rate too small	Increase it. Add a schedule
Zigzag in a narrow valley	High condition number of the Hessian	Add momentum, or use Newton
Gradient near 0, no progress	Saddle point	Momentum and mini-batch noise
Gradients grow each step	`.grad` adds with `+=`	Set `p.grad = 0.0` before `backward`
Autodiff disagrees with numeric	Bug in a `_backward` rule	Run `gradient_check`
Noisy second derivative	Step h too small	Use `h=1e-4` for both calls
Newton too costly for a big model	O(n^2) memory, O(n^3) inverse	Use L-BFGS or Adam

Sources

Questions

Why reverse mode for networks?: Many inputs and one loss. One backward pass gives all gradients.
How do I test convexity?: f'' >= 0 in 1D. Hessian positive semidefinite in many dimensions.
Why does SGD work on non-convex losses?: Most critical points are saddle points. Most minima are good.
Why does L1 give sparse weights?: The diamond has corners on the axes.
What is complementary slackness?: \lambda_i \, g_i(x) = 0. Support vectors have \lambda > 0.

Sources

Tips

Use a learning rate schedule for runs longer than a few epochs.
Cosine annealing and warmup suit transformers and large models.
Newton's method solves a quadratic in one step.
Gradient checks cost 2 forward passes per parameter. Do not run them in production training.
A Hessian with mixed eigenvalue signs at a critical point marks a saddle.
Flat minima generalize better than sharp minima.

Sources

Original source

---
id: calculus-optimization
title: Calculus, Gradients and Convex Optimization
code: CG
subtitle: Derivatives, autodiff, GD, momentum, Adam, Newton and convexity tests. From-scratch Python and PyTorch.
hue: 30
sources: phases/01-math-foundations/04-calculus-for-ml, phases/01-math-foundations/05-chain-rule-and-autodiff, phases/01-math-foundations/08-optimization, phases/01-math-foundations/18-convex-optimization
source_hash: e8999348285411d1
---

## Forward and backward through one neuron
type: architecture
from: phases/01-math-foundations/04-calculus-for-ml/docs/en.md

```arch
node | x | 1,1 | x | input
node | z1 | 2,1 | z1 = w*x | multiply
node | z2 | 3,1 | z2 = w*x + b | add bias
node | a | 4,1 | a = sigmoid(z2) | activation
node | dw | 1,2 | dL/dw | dL/dz2 * x
node | db | 2,2 | dL/db | dL/dz2 * 1
node | dz2 | 3,2 | dL/dz2 | dL/da * a(1-a)
node | loss | 4,2 | L | cross-entropy loss
edge | x | z1 | *w
edge | z1 | z2 | +b
edge | z2 | a |
edge | a | loss |
edge | loss | dz2 | backward
edge | dz2 | dw | x
edge | dz2 | db | 1
```

## Reverse-mode autodiff in Value.backward
type: flow
from: phases/01-math-foundations/05-chain-rule-and-autodiff/docs/en.md, phases/01-math-foundations/05-chain-rule-and-autodiff/code/autodiff.py

1. **Record**: each operation makes a new `Value` with its children and a `_backward` closure.
2. **Sort**: `build_topo` puts every node after the nodes it depends on.
3. **Seed**: set `self.grad = 1.0` on the output. This is dy/dy = 1.
4. **Walk back**: call `v._backward()` for each node in reversed topological order.
5. **Accumulate**: each closure adds to `.grad` with `+=`. A value used twice gets the sum.
6. **Reset**: set `p.grad = 0.0` for every parameter before the next `loss.backward()`.
7. **Update**: `p.data -= lr * p.grad` for each parameter.

## Tasks in code
type: task-table
from: phases/01-math-foundations/04-calculus-for-ml/code/derivatives.py, phases/01-math-foundations/05-chain-rule-and-autodiff/code/autodiff.py, phases/01-math-foundations/05-chain-rule-and-autodiff/docs/en.md, phases/01-math-foundations/08-optimization/code/optimizers.py, phases/01-math-foundations/18-convex-optimization/code/convex.py

| Task | Code |
|---|---|
| Central difference | `numerical_derivative(f, x, h=1e-7)` |
| Gradient of many inputs | `numerical_gradient(f, point, h=1e-7)` |
| Hessian of f(x, y) | `hessian_2d(f, x, y, h=1e-5)` |
| Check an autodiff gradient | `gradient_check(build_expr, x_val, h=1e-7)` |
| XOR network, 4 hidden | `model = MLP([2, 4, 1])` |
| Run an optimizer | `optimize(optimizer, func, grad_func, start)` |
| Test convexity by sampling | `check_convexity(f, dim)` |
| Newton steps in 2D | `newtons_method(grad_f, hessian_f, x0)` |

## Optimizers from scratch
type: annotated-code
lang: python
from: phases/01-math-foundations/08-optimization/docs/en.md, phases/01-math-foundations/08-optimization/code/optimizers.py

```python
GradientDescent(lr=0.001)  # w = w - lr * gradient
SGDMomentum(lr=0.001, momentum=0.9)  # v = beta * v + gradient
Adam(lr=0.001, beta1=0.9, beta2=0.999, epsilon=1e-8)  # two moments per weight
m_hat = [m / (1 - self.beta1 ** self.t) for m in self.m]  # bias correction
p - self.lr * mh / (vh ** 0.5 + self.epsilon)  # per-weight step size
adam_history = optimize(Adam(lr=0.01), rosenbrock, rosenbrock_gradient, start)
```

## Autograd and optimizers in PyTorch
type: annotated-code
lang: python
from: phases/01-math-foundations/05-chain-rule-and-autodiff/docs/en.md, phases/01-math-foundations/08-optimization/docs/en.md

```python
x = torch.tensor(2.0, requires_grad=True)
y = x ** 2 + 3 * x + 1
y.backward()  # reverse mode through the recorded graph
print(x.grad)  # 7.0 = 2*x + 3
sgd = torch.optim.SGD(model.parameters(), lr=0.01, momentum=0.9)
adam = torch.optim.Adam(model.parameters(), lr=0.001)
adamw = torch.optim.AdamW(model.parameters(), lr=0.001, weight_decay=0.01)
scheduler = torch.optim.lr_scheduler.CosineAnnealingLR(adam, T_max=100)
```

## Key formulas
type: concepts
from: phases/01-math-foundations/04-calculus-for-ml/docs/en.md, phases/01-math-foundations/05-chain-rule-and-autodiff/docs/en.md, phases/01-math-foundations/08-optimization/docs/en.md, phases/01-math-foundations/18-convex-optimization/docs/en.md

- **Central difference**: $f'(x) \approx \frac{f(x + h) - f(x - h)}{2h}$.
- **Chain rule**: $\frac{dy}{dx} = f'(g(x)) \cdot g'(x)$. Sum the paths where the graph branches.
- **Gradient descent**: $w = w - \text{lr} \cdot \text{gradient}$.
- **Momentum**: $v = \beta \cdot v + \text{gradient}$, then $w = w - \text{lr} \cdot v$.
- **Adam**: $w = w - \text{lr} \cdot \hat{m} / (\sqrt{\hat{v}} + \epsilon)$ with bias-corrected moments.
- **Newton**: $x_{\text{new}} = x - H^{-1} \cdot \text{gradient}$.
- **Convex**: $f(tx + (1-t)y) \le t f(x) + (1-t) f(y)$ for $t \in [0, 1]$.
- **Lagrangian**: $L(x, \lambda) = f(x) + \lambda g(x)$. Set its gradient to zero.

## Derivatives to know
type: command-table
from: phases/01-math-foundations/04-calculus-for-ml/docs/en.md, phases/01-math-foundations/05-chain-rule-and-autodiff/docs/en.md

| Command | What it does |
|---|---|
| x^2 | 2x. MSE loss |
| wx + b | x for w, 1 for b, w for x. Linear layer |
| e^x | e^x. Softmax, attention |
| ln(x) | 1/x. Cross-entropy loss |
| sigmoid | f(x)(1-f(x)). Sigmoid activation |
| tanh | 1 - tanh^2. Classic activation |
| x^n | n x^(n-1). Used by `__pow__` |
| sin(x^2) | cos(x^2) * 2x. Chain rule example |

## Pick a method
type: command-table
from: phases/01-math-foundations/04-calculus-for-ml/docs/en.md, phases/01-math-foundations/08-optimization/docs/en.md, phases/01-math-foundations/18-convex-optimization/docs/en.md

| Command | What it does |
|---|---|
| Gradient descent | First derivatives. O(n) memory. Baseline |
| SGD with momentum | Best final accuracy with more tuning |
| Adam | Diagonal curvature from gradient moments. Deep learning default |
| AdamW | Adam with decoupled weight decay. Transformers |
| Newton's method | Full Hessian. O(n^3) per step. Small convex problems |
| L-BFGS | Last m gradient differences. Medium convex problems |
| K-FAC | Kronecker-factored Fisher. Large-batch training |

## Numbers and defaults
type: defaults
from: phases/01-math-foundations/04-calculus-for-ml/code/derivatives.py, phases/01-math-foundations/05-chain-rule-and-autodiff/docs/en.md, phases/01-math-foundations/08-optimization/docs/en.md, phases/01-math-foundations/08-optimization/code/optimizers.py

| Setting | Default | Note |
|---|---|---|
| Adam `lr` | 0.001 | Start here for most problems |
| Adam `beta2` | 0.999 | `beta1` is 0.9. Moment decay rates |
| Adam `epsilon` | 1e-8 | Stops division by zero |
| SGD momentum `lr` | 0.01 | With momentum 0.9 |
| AdamW `weight_decay` | 0.01 | Decoupled from the gradient step |
| Mini-batch size | 32-256 | Good gradient estimate, moderate noise |
| Finite difference `h` | 1e-7 | In `numerical_derivative` |
| Gradient check difference | < 1e-5 | Autodiff against numerical |

## Troubleshooting
type: troubleshooting
from: phases/01-math-foundations/04-calculus-for-ml/docs/en.md, phases/01-math-foundations/05-chain-rule-and-autodiff/docs/en.md, phases/01-math-foundations/08-optimization/docs/en.md, phases/01-math-foundations/18-convex-optimization/docs/en.md

| Symptom | Cause | Fix |
|---|---|---|
| Loss grows or becomes nan | Learning rate too large | Reduce the learning rate |
| Training is too slow | Learning rate too small | Increase it. Add a schedule |
| Zigzag in a narrow valley | High condition number of the Hessian | Add momentum, or use Newton |
| Gradient near 0, no progress | Saddle point | Momentum and mini-batch noise |
| Gradients grow each step | `.grad` adds with `+=` | Set `p.grad = 0.0` before `backward` |
| Autodiff disagrees with numeric | Bug in a `_backward` rule | Run `gradient_check` |
| Noisy second derivative | Step h too small | Use `h=1e-4` for both calls |
| Newton too costly for a big model | O(n^2) memory, O(n^3) inverse | Use L-BFGS or Adam |

## Questions
type: qa
from: phases/01-math-foundations/05-chain-rule-and-autodiff/docs/en.md, phases/01-math-foundations/18-convex-optimization/docs/en.md

- **Why reverse mode for networks?** Many inputs and one loss. One backward pass gives all gradients.
- **How do I test convexity?** f'' >= 0 in 1D. Hessian positive semidefinite in many dimensions.
- **Why does SGD work on non-convex losses?** Most critical points are saddle points. Most minima are good.
- **Why does L1 give sparse weights?** The diamond has corners on the axes.
- **What is complementary slackness?** $\lambda_i \, g_i(x) = 0$. Support vectors have $\lambda > 0$.

## Tips
type: tips
from: phases/01-math-foundations/04-calculus-for-ml/docs/en.md, phases/01-math-foundations/05-chain-rule-and-autodiff/docs/en.md, phases/01-math-foundations/08-optimization/docs/en.md, phases/01-math-foundations/18-convex-optimization/docs/en.md

- Use a learning rate schedule for runs longer than a few epochs.
- Cosine annealing and warmup suit transformers and large models.
- Newton's method solves a quadratic in one step.
- Gradient checks cost 2 forward passes per parameter. Do not run them in production training.
- A Hessian with mixed eigenvalue signs at a critical point marks a saddle.
- Flat minima generalize better than sharp minima.