https://introml.mit.edu

Lecture 5: Features and Neural Networks I

Shen Shen

Oct 5, 2026

2:30pm, Room 10-250

Slides and Lecture Recording

Intro to Machine Learning

Recall:

step

multi-class

\(z = \theta^{\top} x+\theta_0\)

\(z = \theta^{\top} x+\theta_0\)

\(z_k = \theta_k^{\top} x+\theta_{0,k}\)

\(g = \text{step}(z)\)

\(g = \sigma(z)\)

\(g = \text{softmax}(z_1,\ldots,z_K)\)

decision boundary is linear in \(x\)

logistic

[mock-up post generated by Claude Opus 5.5, not real]

[real front-page of NY Times]

Linear classification helped kick off the first wave of AI enthusiasm.

👆

👇

Not linearly separable.

Linear tools cannot solve interesting tasks.

Linear tools cannot, by themselves, solve interesting tasks.

XOR dataset

feature engineering 👉

👈 neural networks

Outline

  • Systematic feature transformations
    • Engineered features
    • Polynomial features
    • Expressive power
  • Neural networks

original features \(x \in \mathbb{R}^d \)

\longrightarrow

new features \(\phi(x) \in \mathbb{R}^{d^{\prime}}\)

nonlinear in \(x\)

linear in \(\phi\)

\longrightarrow

nonlinear

transformation

\(\phi\)

\theta_1\phi_1(x) + \theta_2\phi_2(x) + \dots + \theta_{d'}\phi_{d'}(x)

Linearly separable in \(\phi(x) = x^2\) space

Not linearly separable in \(x\) space

-3
-2
-1
0
1
2
3
4
5
6
7
8
9
\phi(x)

transform via \(\phi(x) = x^2\)

\Downarrow
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
x

Linearly separable in \(\phi(x) = x^2\) space, e.g. predict positive if \(\phi \geq 3\)

Separable by a nonlinear rule in \(x\) space, e.g. predict positive if \(x^2 \geq 3\)

-3
-2
-1
0
1
2
3
4
5
6
7
8
9
\phi(x)
-2
-1
0
1
2
3
4
5
6
7
8
9
x
-3

transform via \(\phi(x) = x^2\)

\Downarrow

data and classifier in \(x\) space

\(x_1^2 + x_2^2 = 10\): nonlinear in \(x\)

\(z = x_1^2 + x_2^2 - 10\), threshold at \(z\!=\!0\)

... in \(\phi\) space, where \(\phi(x) = (x_1^2,\, x_2^2)\)

\(\phi_1 + \phi_2 = 10\): linear in \(\phi\)

\(z = \phi_1 + \phi_2 - 10\), threshold at \(z\!=\!0\)

decision boundary is linear in \(\phi,\) nonlinear in \(x\)

data in \(x\)-space

data in \(\phi\)-space

training data

\(x\) \(y\)
p1 -2 5
p2 1 2
p3 3 10
\Rightarrow

transform via

\(\phi(x)=x^2\)

training data

\(\phi\) \(y\)
p1 4 5
p2 1 2
p3 9 10

nonlinear regression

\(g = \phi + 1\)
\(g = x^2 + 1\)

systematic construction of polynomial features

\(d = 1\),   features: \(x_1\) \(d = 2\),   features: \(x_1, x_2\)
\(k=0\) \(1\) \(1\)
\(k=1\) \(1\) \(x_1\) \(1\) \(x_1,\; x_2\)
\(k=2\) \(1\) \(x_1\) \(x_1^2\) \(1\) \(x_1,\; x_2\) \(x_1^2,\; x_1 x_2,\; x_2^2\)
\(k=3\) \(1\) \(x_1\) \(x_1^2\) \(x_1^3\) \(1\) \(x_1,\; x_2\) \(x_1^2,\; x_1 x_2,\; x_2^2\) \(x_1^3,\; x_1^2 x_2,\; x_1 x_2^2,\; x_2^3\)
\(\vdots\) \(\vdots\) \(\vdots\)
  • The basis holds every monomial of the original features with total degree up to \(k\)
  • Fixing \(d\) and \(k\) fixes the basis.

number of features \(d^{\prime} = \dbinom{d+k}{k}\); \(d = 2,\ k = 3\) gives \(10\)

  • \(n = 9\) data points, each with feature \(x \in \mathbb{R}\) and label \(y \in \mathbb{R}\)
  • data generated from green dashed curve
k=1

\( h = \textcolor{#4a6fa5}{\theta_0} + \textcolor{#4a6fa5}{\theta_1} \textcolor{#888}{x} \)

Learn 2 parameters — degree-1

k=2

\( h = \textcolor{#4a6fa5}{\theta_0} + \textcolor{#4a6fa5}{\theta_1} \textcolor{#888}{x} + \textcolor{#4a6fa5}{\theta_2} \textcolor{#888}{x^2} \)

Learn 3 parameters — degree-2

Underfitting

Appropriate model

Overfitting

k=1
k=2
k=10

high error on train set

high error on test set

low error on train set

low error on test set

low error on train set

high error on test set

  • \(k:\) a hyperparameter that determines the capacity (expressiveness) of the hypothesis class.
  • Models with many rich features and free parameters tend to have high capacity but also greater risk of overfitting.
  • How to choose \(k?\) Validation/cross-validation.

Previously:

\downarrow
x
g
\downarrow
\rightarrow
\boxed{h}
\rightarrow

🧠⚙️

hypothesis class
loss function
hyperparameters

\(\left\{\left(x^{(i)}, y^{(i)}\right)\right\}_{i=1}^{n}\)

Linear

Learning Algorithm

\downarrow
x
g
\downarrow
\rightarrow
\boxed{h}
\rightarrow

\(\left\{\left(\phi(x^{(i)}), y^{(i)}\right)\right\}_{i=1}^{n}\)

Linear

Learning Algorithm

🧠⚙️

hypothesis class
loss function
hyperparameters

\downarrow
\phi(x)

today, so far:

\xrightarrow{\hspace{1.5cm}}

🧠⚙️
feature

transformation \(\phi(x)\)

can we automate 👆? 

i.e. fold it into the learning algorithm? 

\(\left\{\left(x^{(i)}, y^{(i)}\right)\right\}_{i=1}^{n}\)

\downarrow
x
g
\downarrow
\rightarrow
\boxed{h}
\rightarrow

Nonlinear

Learning Algorithm

🧠⚙️

hypothesis class
loss function
hyperparameters

neural networks:

\(\left\{\left(x^{(i)}, y^{(i)}\right)\right\}_{i=1}^{n}\)

the nonlinearity \(\phi\) is learned, not hand-designed

Something to think about:

What fixed transformation \(\phi\) can make this decision boundary in \(x\) space?

Outline

  • Systematic feature transformations
  • Neural networks
    • Terminology
    • Design choices

👋 heads-up:

all neural network diagrams focus on a single data point

Outlined the fundamental concepts of neural networks:

expressiveness

efficient learning

  • ​Nonlinear feature transformation
  • Composing simple nonlinearities amplifies this effect
  • Backpropagation 
\left\{ \begin{array}{l} \\ \\ \end{array} \right.
\left\} \begin{array}{l} \\ \\ \end{array} \right.

layered structure

We abstract this into a simple mathematical unit.

importantly, linear in \(\phi\), nonlinear in \(x\)

  • ​Nonlinear feature transformation
\phi\left(\left[x_1; x_2\right]\right)=\left[|x_1-x_2|\right]
\phi\left(\left[x_1; x_2\right]\right)=\left[|x_1-x_2|\right] = 2.5

transform via

  • "Composing" simple transformations
\sigma_1 = \sigma(5 x_1 -5 x_2 + 1)
\sigma_2 = \sigma(-5 x_1 + 5 x_2 + 1)

some appropriately weighted sum

  • "Composing" simple transformations
=

A neuron:

\(w\): what the algorithm learns

  • \(x\): \(d\)-dimensional input
a = f(z)
\Sigma

A neuron:

  • \(a\): post-activation output
  • \(f\): activation function
  • \(w\): weights (i.e. parameters)
  • \(z\): pre-activation output

\(f\): what we engineers choose

\dots
x_1
x_2
x_d
x = \left[ \begin{array}{c} \\ \\ \\ \\ \\ \\ \\ \end{array} \right]
w_1
w_d
\dots
w_2
= w^Tx
z
f(\cdot)
= f(w^Tx)

\(z\): scalar

\(a\): scalar

Choose activation \(f(z)=z\)

learnable parameters (weights)

e.g. linear regressor represented as a computation graph

= z
= w^Tx
w_1
w_d
\dots
x_1
x_2
x_d
\dots
x = \left[ \begin{array}{c} \\ \\ \\ \\ \\ \\ \\ \end{array} \right]
w_2
\Sigma
f(\cdot)
g
z

Choose activation \(f(z)=\sigma(z)\)

learnable parameters (weights)

e.g. linear logistic classifier represented as a computation graph

= \sigma(z)
= w^Tx
w_1
w_d
\dots
x_1
x_2
x_d
\dots
w_2
\Sigma
f(\cdot)
g
z
x = \left[ \begin{array}{c} \\ \\ \\ \\ \\ \\ \\ \end{array} \right]
\dots

A layer:

learnable weights

A layer:

a_1
z_1
\Sigma
f(\cdot)
z_2
\Sigma
f(\cdot)
a_2
z_m
a_m
\Sigma
f(\cdot)
\dots
x_1
x_2
x_d
  • (# of neurons) = layer's output dimension.
  • activations apply element-wise, so \(z_1\) doesn't affect \(a_2\) (softmax in the output layer is the common exception).
  • all neurons in a layer typically share the same \(f\) (mixed \(f\) is possible but complicates the math).
  • layers are typically fully connected: each input \(x_i\) influences every \(a_j\).
\dots

layer

linear combo

activations

A (fully-connected, feed-forward) neural network: 

\dots
\dots

layer

\dots
x_1
x_2
x_d

input

\Sigma
f(\cdot)
\Sigma
f(\cdot)
\Sigma
f(\cdot)
\Sigma
f(\cdot)
\Sigma
f(\cdot)
\Sigma
f(\cdot)
\Sigma
f(\cdot)

neuron

learnable weights

Engineers choose:

  • activation \(f\) in each layer
  • # of layers
  • # of neurons in each layer

hidden

output

\sigma_1 = \sigma(5 x_1 -5 x_2 + 1)
\sigma_2 = \sigma(-5 x_1 + 5 x_2 + 1)

some appropriately weighted sum

=
recall this example
x_1
x_2
1
\Sigma
f(\cdot)
\Sigma
f(\cdot)
\Sigma
f(\cdot)

\(f =\sigma(\cdot)\)

\(f(\cdot) \) identity function

W^1
W^2
\sigma_1 = \sigma(5 x_1 -5 x_2 + 1)
\sigma_2 = \sigma(-5 x_1 + 5 x_2 + 1)
5
-5
1
-5
5
1
-3
-3

\(-3(\sigma_1 +\sigma_2)\)

can be represented as:

  • Activation \(f\) is chosen as the identity function
  • Evaluate the loss \(\mathcal{L} = (g^{(i)}-y^{(i)})^2\)
  • Repeat for each data point, average the \(n\) individual losses

e.g. forward pass of a linear regressor

y^{(i)}
f(\cdot)
\underbrace{\quad \quad \quad }
\dots
= w^Tx
w_1
w_d
\dots
w_2
\Sigma
z
= z
g
\dots
x^{(i)}_1
x^{(i)}_2
x^{(i)}_d
x^{(i)} \quad = \left[ \begin{array}{c} \\ \\ \\ \\ \\ \\ \\ \end{array} \right]
\mathcal{L}(g, y)
\mathcal{L}(g, y)
\mathcal{L}(g, y)
\dots
\dots
\dots
n
  • Activation \(f\) is chosen as the sigmoid function
  • Evaluate the loss \(\mathcal{L} = - [y^{(i)} \log g^{(i)}+\left(1-y^{(i)}\right) \log \left(1-g^{(i)}\right)]\)
  • Repeat for each data point, average the \(n\) individual losses

e.g. forward pass of a linear logistic classifier

y^{(i)}
f(\cdot)
\underbrace{\quad \quad \quad }
\dots
= w^Tx
w_1
w_d
\dots
w_2
\Sigma
z
=\sigma(z)
g
\dots
x^{(i)}_1
x^{(i)}_2
x^{(i)}_d
x^{(i)} \quad = \left[ \begin{array}{c} \\ \\ \\ \\ \\ \\ \\ \end{array} \right]
\mathcal{L}(g, y)
\mathcal{L}(g, y)
\mathcal{L}(g, y)
\dots
\dots
\dots
n
x^{(1)}
y^{(1)}
f^1
\begin{aligned} & W^1 \\ \end{aligned}
\begin{aligned} & W^2 \\ \end{aligned}
\begin{aligned} & W^L \\ \end{aligned}
f^2
f^L
g^{(1)}

\(\dots\)

f^2\left(\hspace{2cm}; \mathbf{W}^2\right)
f^1(\mathbf{x}^{(i)}; \mathbf{W}^1)
f^L\left(\dots \hspace{3.5cm}; \dots \mathbf{W}^L\right)

Forward pass: evaluate, given the current parameters

  • the model outputs \(g^{(i)}\) =  
  • the loss incurred on the current data \(\mathcal{L}(g^{(i)}, y^{(i)})\)
  • the training error \(J = \frac{1}{n} \sum_{i=1}^{n}\mathcal{L}(g^{(i)}, y^{(i)})\)
\mathcal{L}(g^{(1)}, y^{(1)})
\mathcal{L}(g, y)
\mathcal{L}(g^{(n)}, y^{(n)})
\underbrace{\quad \quad \quad \quad \quad }
\dots
\dots
\dots
n

linear combination

loss function

(nonlinear) activation

\dots

Outline

  • Systematic feature transformations
  • Neural networks
    • Terminology
    • Design choices

Choosing the hidden-layer activation \(f\)

\(\sigma\) used to be the most popular

  firing rate of a neuron

  elegant gradient \(\sigma^{\prime}(z)=\sigma(z) \cdot(1-\sigma(z))\)

\operatorname{ReLU}(z)=\left\{\begin{array}{ll} 0 & \text { if } z<0 \\ z & \text { otherwise } \end{array}\\ \\ \right.
=\max (0, z)

very simple form, and so is its gradient.

nowadays, the default choice:

With learnable \(\textcolor{#38761d}{w}\) and \(\textcolor{#38761d}{w_0}\), which of these can \(\mathrm{ReLU}(\textcolor{#38761d}{w}\,x + \textcolor{#38761d}{w_0})\) draw?

✓

✓

Plotted against \(z,\) \(\mathrm{ReLU}(z)\) looks like this:

compositions of ReLU(s) can be quite expressive, e.g.,

=

can create bump

with enough of these bumps appropriately scaled and shifted:

asymptotically, can approximate any continuous function arbitrarily well (for regression), therefore can also approximate arbitrary decision boundaries (for classification)

=

asymptotically: as the number of neurons grows. With enough neurons, even one hidden layer gets within any error of any continuous function on a bounded domain.

just 4 ReLU neurons already
carve out a non-trivial surface

the same holds for higher dimensions

=
  • Width: # of neurons in layers 
  • Depth: # of layers
  • Typically, increasing either the width or depth (with nonlinear activation) makes the model more expressive, but it also increases the risk of overfitting.

For neural networks, though, how size relates to overfitting is still open research; see the double-descent curve and scaling laws.

(The demo doesn't show in the PDF; the link below works.)

output layer design choices

  • # neurons, activation, and loss depend on the high-level goal. 
  • Typically straightforward.
  • Multi-class setup: to predict exactly one of \(K\) classes, the output layer has \(K\) neurons, softmax activation, and cross-entropy loss

 

 

 

 

  • For other multi-class settings, see hw/lab.

e.g., say \(K=5\) classes

input \(x\)

hidden layer(s)

\dots

output layer

\begin{bmatrix} -1.0 \\ -0.5 \\ 0.0 \\ 0.5 \\ 1.0 \end{bmatrix}
\begin{bmatrix} 0.058 \\ 0.096 \\ 0.158 \\ 0.260 \\ 0.429 \end{bmatrix}
\operatorname{softmax}\left(z_j\right)\\=\frac{\exp \left(z_j\right)}{\sum_{i=1}^K \exp \left(z_i\right)}

Summary

  • Linear models are convenient but lack expressiveness for most real-world tasks.

  • A fixed nonlinear feature transformation lets us use linear methods on complex problems, but designing features by hand gets tedious.

  • Neural networks automate feature learning: layers alternate parameterized linear maps with nonlinear activations (typically ReLU).

  • For classification outputs, we apply sigmoid (binary) or softmax (multi-class).

Next time: How do we train neural networks? (gradient descent + backpropagation)

feature transformations feature transformationpolynomial featuresfeature engineering

neural networks neuronactivation functionlayerhidden layerforward pass

design choices ReLUwidthdepth