Training at scale, generalization & characterizing units

Reader-friendly version of the 54 lecture slides: the lecture content in normal flow, with figure descriptions and data tables where the figure carries data. Open the presentation.

Slide 1

CPSY 1291 Computational Methods for Mind, Brain & Behavior

Lecture 10 · Theme 2: Learning representations: from neurons to deep networks

Training at scale, generalization & characterizing units

Thursday, October 15 · Fall 2026

Return to contents

Slide 2

Last lecture: hidden layers and backpropagation

A fully connected network drawn left to right: inputs x1 to xD, hidden layer 1, hidden layer 2 and an output layer y-hat. Every unit connects to every unit of the next layer; the weight matrices W superscript 1, 2 and 3 label the gaps
Open full-size figure

Hidden layers re-represent the inputs, so a readout can separate classes that no single line separates, as in XOR
Backpropagation gives the gradient of the loss for every weight: error at the output, sent backward, error × activity
The training loop: forward pass, loss, backward(), step(), repeated over epochs
Each step of gradient descent used all the training images and a fixed learning rate $\eta$

Return to contents

Slide 3

Today: three questions about a trained network

How do we train it on millions of images? Small batches of examples, a learning rate that shrinks, and Adam
Does what it learned hold for images it never saw? Generalization: how to measure it honestly, and how to improve it
What have its units learned? Measure them the way physiologists measure V1 and IT neurons

Return to contents

Slide 4

Today's three examples

Contour plot of the loss of a linear neuron with two weights, w1 and w2, with a gradient-descent path from a start point to the lowest loss
Open full-size figure

The two-weight neuron
24 simulated examples

Two panels: a degree-3 polynomial that follows the sine curve through 10 noisy points, and a degree-9 polynomial that passes through every point and swings far off between them; under each, its training and test RMSE
Open full-size figure

The curve fit
10 noisy points of a sine curve

A grid of grayscale renderings, one row per category (animal, vehicle, face, vegetable or fruit, house, man-made object), each with two objects seen from three of their 24 viewpoints
Open full-size figure

The IT readout
100 IT sites, six object categories

A neuron, a curve, a readout: the same ideas about training and generalization hold for each, and for a deep network such as AlexNet (end of the lecture)

Return to contents

Slide 5

The IT readout, in full

Recordings: neurons in monkey inferotemporal cortex (IT), the object-recognition area (Bao et al. 2020), the data of the first two assignments
Images: 51 objects × 24 views = 1,224, in six categories (rows of the grid)
Input: each image's average response at 100 IT sites: 100 numbers. Output: its category, one of 6 (chance: 1 in 6)
Every readout today learns this same mapping, from a logistic regression to a small network

Bao, She, McGill & Tsao Nature 2020

Return to contents

Slide 6

How do we train on a million images?

Last lecture, every step used all the training images. That is slow, and they do not fit in memory at once

Return to contents

Slide 7

One example, all of them, or a few at a time?

One step updates every weight once, using the loss averaged over the examples it is computed on
Online, one example per step, like the perceptron and the delta rule: cheap steps, but each gradient is very noisy
Full batch, all $N$ examples per step, like last lecture's loop: an exact gradient, but one step per pass, and a million images do not fit in memory
Minibatch, $B$ examples per step (often 32 to 256): each epoch, shuffle the examples and split them into groups of $B$ , one step per group. Many cheap steps per pass, so far fewer passes

Twelve numbered training examples in order, then the same twelve shuffled, then cut into three batches of four; each batch is one step, and the three steps make one epoch; the next epoch shuffles again
Open full-size figure

Robbins & Monro Ann Math Stat 1951 · PyTorch DataLoader · Recitation 5

Return to contents

Slide 8

Three tips for training your own network

Shrink the learning rate as training goes · Average the gradients: momentum and Adam · Sweep the learning rate first

Return to contents

Slide 9

Tip 1: shrink the learning rate as training goes

Two contour plots of last lecture's two-weight loss, with identical axes, each showing 600 minibatch updates from the same start using the same batches of two examples. Left, constant step size 0.8: the last 300 steps scatter widely around the star that marks the lowest loss, mean distance 0.12. Right, a step that shrinks as 0.8 divided by 1 plus t over 20: the last 300 steps sit in a tight cluster on the star, mean distance 0.01. Faint trails enter from the start, which lies off the plotted area

Open full-size figure

The two-weight neuron, now with minibatches of 2. A constant rate keeps bouncing; a shrinking one settles. Full batches would not need it

PyTorch learning-rate schedulers

Return to contents

Slide 10

Tip 2: average the gradients (momentum)

Contours of the squared-error loss of the two-weight neuron, weight w1 horizontal and w2 vertical on equal scales, with its second input on a scale five times larger, so the loss is a long narrow valley running along w1. From the same start, 40 steps each at the same learning rate: plain gradient descent zigzags up and down across the valley; momentum swings in a few slow, smooth waves along it toward the lowest loss. Up-and-down travel along w2: 4.4 for plain descent, 0.8 for momentum
Open full-size figure

The two-weight neuron, $x_2$ on a 5× larger scale: a narrow valley. A rate that does not overshoot across it barely moves along it: plain descent zigzags
Momentum averages recent gradients ( $\mathbf{m}$ ): the up-and-down pushes cancel

Return to contents

Slide 11

Tip 2, continued: Adam, a step size for each weight

The same valley of the two-weight neuron and the same start, w1 horizontal and w2 vertical. Plain gradient descent zigzags faintly in the background; momentum swings in slow waves as on the previous slide; Adam follows one smoother curve down the valley to the lowest loss. Up-and-down travel along w2: plain 4.4, momentum 0.8, Adam 0.6
Open full-size figure

Each weight's step is divided by its typical gradient size: both weights move at a similar pace
Steps to 1% of the starting loss, each at its best learning rate: plain more than 40, momentum 8, Adam 6
Adam tolerates a wider range of learning rates, but $\eta$ is still a setting to choose (Tip 3)

Return to contents

Slide 12

Adam: the update

\mathbf{m}\leftarrow\beta_1\mathbf{m}+(1-\beta_1)\,\nabla\mathcal{L},\qquad \mathbf{v}\leftarrow\beta_2\mathbf{v}+(1-\beta_2)\,(\nabla\mathcal{L})^2,\qquad \boldsymbol{\theta}\leftarrow\boldsymbol{\theta}-\eta\,\frac{\mathbf{m}}{\sqrt{\mathbf{v}}+\epsilon}

θ: all weights and biases · operations are per weight · Adam's first-step correction omitted

$\mathbf{m}$ , the running average of the gradient (its first moment): which way each weight has been pushed lately, like a rolling ball's velocity
$\mathbf{v}$ , the running average of the squared gradient (its second moment): how large each weight's gradients usually are, whatever their sign
Each weight in effect gets its own learning rate, $\eta/\sqrt{v}$ , set by its gradient history. If a weight's recent gradients all point the same way, its step is about $\eta$ long; if they keep flipping sign, its step is short
Defaults $\beta_1=0.9$ , $\beta_2=0.999$ work on most problems; $\epsilon$ prevents division by zero

Kingma & Ba ICLR 2015 · Loshchilov & Hutter ICLR 2019 · Wilson et al. NeurIPS 2017 · PyTorch · Recitation 5

Return to contents

Slide 13

Tip 3: sweep the learning rate first

Training loss on a log scale against epoch for four learning rates of one network reading the six object categories from 100 monkey IT sites. At 10 to the minus 5, dotted grey, the loss barely falls (test accuracy 0.18); at 10 to the minus 3, dashed blue, it falls slowly (test 0.62); at 0.1, solid magenta, it falls to about 0.015 (test 0.91); at 3, dash-dotted black, it jumps off the top of the axis in the first epoch and diverges
Open full-size figure

The IT readout with a small network: 100 IT responses → 64 hidden units (ReLU) → 6 outputs. Training loss: cross-entropy
Plain gradient steps; four runs differing only in $\eta$
Too small: the loss barely moves. Too large: it explodes
$\eta$ is a hyperparameter, set by us. Good values span orders of magnitude: sweep on a log scale

Return to contents

Slide 14

Does what a network learned hold for images it never saw?

Generalization: when does a trained network's fit carry over to new data, and how do we make it?

Return to contents

Slide 15

Recall: train on some images, test on others

Two panels for a readout of animate versus inanimate from 480 monkey IT neurons, trained on 102 images and tested on 306. Left: accuracy bars; with true labels, training 1.00 and test 0.96; with shuffled training labels, training 1.00 and test 0.56 on average, below a dashed majority-class line at 0.63. Right: histogram of the test accuracy over 1,000 label shuffles, centred near 0.55, with its 95th percentile marked near 0.67 and the true-label test accuracy, 0.96, far to the right
Open full-size figure

Recall the baselines lecture: a probe (readout) of animate vs inanimate from 480 IT neurons. Fit it on 102 images, test it on 306 others
With the true labels: 1.00 on the training images, 0.96 on the test images. Then shuffle the training labels and fit the probe again

Bao, She, McGill & Tsao Nature 2020

Return to contents

Slide 16

A perfect training score can hide poor generalization

Shuffled labels were fit perfectly too (1.00), yet scored only 0.56 on new images
Zhang et al. (2017), ImageNet: random labels fit for 95% of 1.2 million photos; new photos 0.1%, chance
Generalization gap: training accuracy minus test accuracy
When does a fit carry over to new data? How do we measure and improve it? (minute paper)

Bao, She, McGill & Tsao Nature 2020 · Zhang, Bengio, Hardt, Recht & Vinyals ICLR 2017

Return to contents

Slide 17

Training fits a function from inputs to outputs

A model neuron with two inputs, x1 and x2, with weights w1 and w2, summed with a bias b into z, passed through the nonlinearity g, giving the output a = g(w1 x1 + w2 x2 + b)
Open full-size figure

A neuron, a readout or a network computes an output from an input: $\hat y = f(\mathbf{x};\textcolor{#B5396B}{\boldsymbol\theta})$
Training chooses the parameters $\textcolor{#B5396B}{\boldsymbol\theta}$ so that $\hat y$ matches $y$ on the training pairs $(\mathbf{x}, y)$
Generalization asks whether $f$ is also right for inputs it never saw
With hundreds of inputs we cannot draw $f$ ; with one input we can

Return to contents

Slide 18

Too rigid, or too flexible

Computed: 10 noisy samples of a sine curve, after Bishop (2006), Fig. 1.4

Four panels, each with the same 10 magenta points scattered around a dashed sine-shaped curve, and a blue fitted curve. Degree 0: a flat line, training error 0.63, error on new points 0.76. Degree 1: a falling straight line, 0.48 and 0.56. Degree 3: a curve close to the dashed one, 0.19 and 0.33. Degree 9: a curve through every point that swings far off between and beyond them, training error 0.00, error on new points 0.86. Open green circles: 25 of the new points the error on new points is measured on

Open full-size figure

The curve fit: polynomials of degree $d$ ( $d-1$ bends), 10 training points (magenta), new points (green)
Underfitting: high bias, the error of the average prediction. Overfitting: high variance, the fit changes with the training set
$\text{RMSE}=\sqrt{\tfrac{1}{N}\sum_i (\hat y_i - y_i)^2}$ , in units of $y$ . Training RMSE keeps falling; test RMSE falls, then rises

Bishop Pattern Recognition and Machine Learning, Springer 2006

Return to contents

Slide 19

Same model, more examples

Computed: the same curve and noise, after Bishop (2006), Fig. 1.6

Four panels of the same degree-9 polynomial fitted to more and more noisy points from the dashed sine-shaped curve. N equals 10: the curve swings far off, error on new points 0.86. N equals 15: close to the dashed curve, 0.34. N equals 30: 0.35. N equals 100: the fitted curve follows the dashed one, training error 0.28, error on new points 0.31. Open green circles: 25 of the new points the error on new points is measured on

Open full-size figure

The same degree-9 polynomial, fitted to 10, 15, 30 and 100 points from the same curve
With 10 points it passes through every one; with 100 it cannot, and follows the curve instead
Overfitting depends on flexibility relative to the number of examples

Bishop Pattern Recognition and Machine Learning, Springer 2006

Return to contents

Slide 20

How do we choose a model and measure how well it generalizes?

Every choice made while looking at a score can fit that score's images

Return to contents

Slide 21

Three data splits

A horizontal bar split into three parts. Training, the largest part, in magenta: fit the weights. Validation, in green: choose settings and when to stop. Test, in dark grey: report once, at the end
Open full-size figure

Training set: fit the parameters $\textcolor{#B5396B}{\boldsymbol\theta}$ , every number set from data (weights, biases, a polynomial's coefficients)
Validation set: choose the hyperparameters on it (settings we pick, such as $\eta$ ). Few examples? Cross-validation (next slide)
Test set: used once, at the end. Never choose anything on the test set, or its score is too optimistic
Often broken: ImageNet results are tuned on the set they report; on a fresh test set, 11–14 points lower

Recht, Roelofs, Schmidt & Shankar ICML 2019 · Recitation 6: held-out data

Return to contents

Slide 22

Cross-validation: each example validates once

Five rows, fold 1 to fold 5, each showing the training data cut into five equal parts: one part in green, a different one in each row, the other four in magenta; each row ends in its own score. A separate dark block on the right is the test set, set aside and used once. Caption: train on the magenta parts, score on the green one; the cross-validation score is the mean of the five scores
Open full-size figure

Few examples: a separate validation set wastes data. Cut the training data into $k$ parts; train on $k-1$ , score on the one left out, $k$ times. Leave-one-out: one part per example
Choose the setting with the best mean score, then refit on all the training data. The test set stays out of all of it

scikit-learn: cross-validation

Return to contents

Slide 23

Learning curves: when to stop training

Two panels against epoch, 0 to 400, for a network reading out the six object categories from 100 monkey IT sites, trained on 300 images with a fifth of their labels redrawn at random. Left, loss: training loss in magenta falls to near 0; validation loss in green is lowest near epoch 20, about 0.56, then rises to about 0.97. Right, accuracy: training accuracy rises to 1.0; validation accuracy is highest, 0.82, at epoch 8, marked by a dashed line labelled checkpoint kept, and drifts down to about 0.78. A dotted line marks chance, one in six
Open full-size figure

The IT readout, 300 training images, a fifth of their labels redrawn at random: validation turns as the network fits the wrong labels
Early stopping: stop when the validation score stops improving. Keep the checkpoint (saved parameters) with the best validation score: accuracy here, epoch 8; by loss it would be 20

Return to contents

Slide 24

Why stop before the bottom of the training loss?

Left: the curve fit's 10 training points and the dashed sine curve, with a degree-9 polynomial trained by gradient descent shown at three moments: after 10 steps, a smooth curve that is still too flat; after 1,467 steps, a curve close to the sine; after 10 to the 10 steps, the curve that passes through every point and swings far off between them. Right: training RMSE and test RMSE against gradient steps on a log axis; training RMSE falls to zero, test RMSE is lowest, 0.33, at step 1,467, marked stop, and rises to 0.86

Open full-size figure

The curve fit, degree 9, trained step by step: early steps fit the broad shape, late steps the noise
Stopped at its best step, degree 9 does as well as degree 3 (test RMSE 0.33 for both)

Return to contents

Slide 25

A test accuracy from few images is imprecise

Test accuracy of the running readout, six categories from 100 monkey IT sites, against the number of test images, 25 to 800, on a log axis. Each point has a vertical bootstrap interval. With 25 test images the interval runs from 0.76 to 1.00; with 800, from 0.83 to 0.88. A dashed line marks the accuracy on all 924 other images, 0.86
Open full-size figure

The IT readout as a logistic regression, trained on 300 images; the dashed line is its accuracy on all 924 other images (0.86)
The bootstrap (baselines lecture): resample the test images with replacement; the middle 95% of the accuracies is a 95% confidence interval
With 25 test images, the accuracy could be anywhere from 0.76 to 1.00; with 800, from 0.83 to 0.88
Compare models on enough test images that their intervals do not overlap

Return to contents

Slide 26

Shortcut learning: texture over shape

Reproduced from Geirhos et al. 2019, Fig. 1: three images with ResNet-50's top three answers. (a) A texture image of elephant skin: Indian elephant 81.4 percent, indri 10.3, black swan 8.2. (b) A photograph of a tabby cat: tabby cat 71.1 percent, grey fox 17.3, Siamese cat 3.3. (c) The cat's shape with elephant-skin texture, a texture-shape cue conflict: Indian elephant 63.9 percent, indri 26.4, black swan 9.6
Open full-size figure

Even a precise test accuracy can mislead. Out of distribution (OOD): test images from a different source than the training images
Shortcut learning: a cue that predicts the label in training but not in the world (texture, background)
Cue-conflict images like (c): people chose shape in 95.9%, ResNet-50 (an ImageNet network) in 22.1%

Geirhos et al. ICLR 2019 · Geirhos et al. Nat Mach Intell 2020

Return to contents

Slide 27

How do we make a network generalize better?

A penalty on the weights, more data, and switching units off

Return to contents

Slide 28

A penalty on the weights

\mathcal{L}(\textcolor{#B5396B}{\boldsymbol\theta}) \;=\; \underbrace{\mathcal{L}_{\text{train}}(\textcolor{#B5396B}{\boldsymbol\theta})}_{\text{fit}} \;+\;\; \underbrace{\lambda\vphantom{\lVert\textcolor{#B5396B}{\boldsymbol\theta}\rVert^2}}_{\text{strength}}\;\;\underbrace{\lVert\textcolor{#B5396B}{\boldsymbol\theta}\rVert^2}_{\text{size of the parameters}}

Computed: the curve fit's 10 points, after Bishop (2006), Fig. 1.7

Four panels of the degree-9 polynomial fitted to the same 10 points, with an increasing penalty. No penalty: through every point, swinging far off, error on new points 0.86. Lambda 10 to the minus 7: a smooth curve close to the dashed one, 0.34. Lambda 10 to the minus 3: smooth, 0.34. Lambda 1: an almost flat line, training error 0.49, error on new points 0.61. Open green circles: 25 of the new points the error on new points is measured on

Open full-size figure

The degree-9 polynomial, its coefficients penalized (weight decay): test RMSE falls from 0.86 to 0.34. Too large a $\lambda$ underfits

Bishop Pattern Recognition and Machine Learning, Springer 2006

Return to contents

Slide 29

Without a penalty, a readout's weights keep growing

Two panels for the running readout, six categories from 100 IT sites, as a logistic regression trained on 300 images. Left: the size of the weights against gradient steps on a log axis; without a penalty it keeps rising, to about 10.7; with weight decay it levels off near 3.5. Right: the training cross-entropy on a log axis; without a penalty it falls toward zero, to 0.0015; with weight decay it levels off near 0.11
Open full-size figure

The IT readout (logistic): all 300 training images right; a longer $\textcolor{#B5396B}{\mathbf{w}}$ only adds confidence
A penalty stops it: same accuracy, less confidence. (If the training classes overlapped, $\textcolor{#B5396B}{\mathbf{w}}$ would stop)

Return to contents

Slide 30

More data, or altered copies of it

Four versions of one photograph of a pink orchid: the original; flipped left to right; cropped and enlarged; and with its colour faded and raised
Open full-size figure

More training examples always help generalization: the same model overfits less (the polynomial with 10 vs 100 points)
No more data? Make some. Data augmentation: train on altered copies that keep the label (flipped, cropped, recoloured). AlexNet trained on 2,048 versions of each photo
An augmentation states what should not change the label: a flipped orchid is still an orchid

Krizhevsky, Sutskever & Hinton NeurIPS 2012

Return to contents

Slide 31

Regularization: penalties and dropout

Three copies of the same small network: in the first two, a different random half of the hidden units is switched off for one training step each, with their connections removed; in the third, at test time, all units are on
Open full-size figure

With all the data used, regularization: limit how freely the network fits the training data, like the L2 weight penalty
Dropout: at each training step, switch off a random half of the hidden units; all are on at test time
In practice a mix: L2, dropout, sometimes L1. Each cuts the network's freedom to fit noise

Srivastava et al. JMLR 2014

Return to contents

Slide 32

Try it: overfit, then regularize

Last lecture's network, 30 training points, 15% of labels flipped. Train: held-out loss falls, then rises. Then add λ, dropout, or more points

Regularization playground: last lecture's network on two moons, 30 training points (filled) with 15% of labels flipped and 200 held-out points (open circles) scored on their true labels. Controls for the number of training points, hidden units, learning rate, flipped labels, a weight penalty and dropout. The plot shows the training loss and the held-out loss after each epoch — open interactive version

Return to contents

Slide 33

A model of a neuron: regression with a penalty

Two panels for one monkey IT recording site and 512 features of the trained network's late layer. Left: predicted against recorded responses for a ridge fit; training images faint grey, held-out images red, both scattered along the diagonal, R-squared 0.71 on training and 0.69 on held-out images. Right: R-squared against the penalty lambda on a log axis; with almost no penalty, training R-squared is 0.85 and held-out R-squared 0.28; as the penalty grows, training R-squared falls while held-out R-squared rises to meet it near 0.69 at the penalty chosen on the training images, then both fall
Open full-size figure

Computed: one monkey IT site (Bao et al. 2020), 1,224 object images, 512 late-layer units of ResNet-18; a random fifth held out

Predict one IT site from 512 network units: ridge regression, a weighted sum plus the weight penalty. $\lambda$ by cross-validation (RidgeCV); $R^2$ on held-out images

Return to contents

Slide 34

Does a trained network see like the visual cortex?

Nobody told AlexNet about V1. Measure its units the way physiologists measure neurons, and compare

Return to contents

Slide 35

What a trained network's first layer learns

First-layer weights of AlexNet shown as small color images: oriented edges, stripe patterns and color blobs
Open full-size figure

AlexNet, trained to label 1.2 million photos, was never shown a receptive field
Its first-layer filters, each drawn as a small image: oriented edges, stripes, colour blobs
Do these look like the receptive fields of V1 cells? And how would we measure "look like"?

Krizhevsky, Sutskever & Hinton NeurIPS 2012

Return to contents

Slide 36

Real V1 receptive fields look like Gabors

Reproduced from Ringach 2002, Fig. 2: six macaque V1 simple cells in three blocks. Each row shows the measured receptive field (Data), the best-fitting two-dimensional Gabor (Fit) and what is left over (Residual). The fields range from round blobs to elongated stripes of alternating sign; each fit reproduces its field's shape, and the residuals are faint, unstructured speckle. A colour bar runs from minus 1, blue, through 0, green, to plus 1, red
Open full-size figure

First the brain. A cell's receptive field: the input pattern that drives it. Data: six macaque V1 simple cells
Daugman (1985): each is a Gabor, a grating seen through a smooth window. Fit: the best Gabor; Residual: what is left. The residual is mostly noise

Ringach J Neurophysiol 2002 · Daugman J Opt Soc Am A 1985 · Jones & Palmer J Neurophysiol 1987

Return to contents

Slide 37

The Gabor function, fitted, and two numbers from it

G(x,y)=A\,\exp\!\Big(-\frac{x'^2}{2\sigma_x^2}-\frac{y'^2}{2\sigma_y^2}\Big)\cos\!\big(2\pi f x'+\phi\big)

Left, a toy receptive field with pixel noise, the Gabor fitted to it, and the residual, which is only noise; R-squared 0.87. Right, three noise-free toy Gabors: A with nx 0.20 and ny 0.20, a small spot; B with nx 0.35 and ny 0.50, a few short stripes; C with nx 0.90 and ny 1.20, many long stripes

Open full-size figure

Fit $A,\sigma_x,\sigma_y,f,\phi$ , orientation and centre by least squares; $R^2$ scores the fit
Keep $n_x=\sigma_x f$ , $n_y=\sigma_y f$ : the window's width and length in cycles

Return to contents

Slide 38

Where do V1 simple cells sit?

Scatter of Gabor width nx against length ny, both in cycles, for 93 macaque V1 simple cells: most cluster near the origin, below about 0.6 on both axes, with a few longer fields. Three blue stars mark the toy units A, B and C of the Gabor-fit slide: A and B sit among the cells, C lies beyond most of them
Open full-size figure

Ringach (2002) fit Gabors to 93 simple cells of macaque V1 (cells that respond to a grating only at some phases) and released the fits
Most cells sit below 0.6 on both axes (medians 0.32 and 0.41): a window of one or two stripes
Toy unit C, with many long stripes, lies beyond most real cells
The same two numbers can place any receptive field on this plot, a network unit's included

Ringach J Neurophysiol 2002

Return to contents

Slide 39

Quantified: V1 cells and AlexNet's first layer

The same scatter and axes as the previous slide, with the toy units removed: macaque V1 simple cells as grey squares near the origin, and AlexNet's first-layer filters that a Gabor fits well as open green circles; the filters overlap the cells but sit systematically larger
Open full-size figure

Filled squares: macaque V1 simple cells. Open circles: AlexNet's first-layer filters, each fit by a Gabor
The same two numbers place a cell and a network unit on one plot
Do the network's filters look like V1 cells, and where do they differ?

Ringach J Neurophysiol 2002 · Krizhevsky, Sutskever & Hinton NeurIPS 2012

Return to contents

Slide 40

What does a unit respond to? Its tuning curve

Top: the weights of four hand-built units, three Gabors at different orientations and window lengths, labelled with orientation selectivity indices 1.00, 1.00 and 0.90, and a centre-surround unit with index 0.00. Bottom: their tuning curves to gratings at 24 orientations, each scaled to its peak; the three Gabors peak at different orientations, the long one narrowly, the short one broadly, and the centre-surround unit is flat
Open full-size figure

Hubel and Wiesel's method: show gratings (stripes) at many orientations and record the response. Here, four units built by hand, so we can check the method
Top: each unit's weights, its receptive field: three Gabors and a round centre-surround unit
Orientation selectivity index (OSI), (max − min) / (max + min) of the responses: near 1 when the weakest response is near zero. Bandwidth: the half-width of the tuning curve at half its height
A and C both have indices near 1, yet C is far more broadly tuned: the index does not measure tuning width

Return to contents

Slide 41

Why rectify before averaging over phase?

Bar chart for one Gabor unit and a grating at its preferred orientation shown at four phases: the weighted sums are plus one, plus one, minus one, minus one and average to zero; after rectifying, the responses are one, one, zero, zero and average to one half. Thumbnails of the four gratings and of the unit's weights sit under the bars
Open full-size figure

To measure a unit's orientation tuning alone, average its response over where the stripes sit (phase). A raw weighted sum flips sign after half a cycle, and the average cancels
A neuron's firing rate is never negative: model it as rectified, like a ReLU. A rectified unit keeps its preference, as a real cell does. The Gabor fit itself uses the weights and needs no rectification

Return to contents

Slide 42

Deeper in: how selective is one IT site for objects?

Responses of two IT sites to 51 objects, 24 views each: dots for single views, bars for object means. Site A, object selectivity index 0.60, has object means that differ widely, highest for a swan, shown as a thumbnail; site B, index 0.22, has object means that barely differ compared with the spread of the views

Open full-size figure

Past the first layer, weights no longer look like images: measure selectivity from responses alone. Show each of 51 objects from 24 views (the images of the first assignments); split one IT site's variance into between objects and within an object
Object selectivity index = between / total: near 1 when the response depends on which object, near 0 when the object means barely differ compared with the spread of the views
Site A: the bars (one per object, its mean over 24 views) differ much more than the views of one object do: 0.60. Site B: the bars are nearly level and the views vary as much: 0.22
This ranks units by how they respond. Whether a unit is needed is another measurement: next slide

Bao, She, McGill & Tsao Nature 2020

Return to contents

Slide 43

Is a selective unit needed? Remove it and see

Selectivity: how a unit responds. Necessity: what a readout loses without it. Different measurements
Ablation: set a unit's activity to zero and measure the drop in a readout's accuracy
Control: remove the same number of randomly chosen units. Selective units matter more only if removing them costs more than removing random ones
Keep the readout fixed, and you ask whether it relied on those units. Refit it, and you ask whether the remaining units can still separate the classes

Return to contents

Slide 44

Recap

Training at scale: minibatches give cheap, noisy gradients; a schedule shrinks the step; Adam adapts it per weight. Sweep the learning rate first, on a log scale
Generalization: performance on data the model never saw. Overfitting depends on flexibility relative to the number of examples
Splits: train to fit, validate to choose and to stop, test once, with a bootstrap interval
Beyond the test set: shortcuts fail on new kinds of images. Large networks can memorize random labels and still generalize on real labels
Regularization: weight decay, data augmentation, dropout and early stopping can reduce overfitting. Choose their settings on validation data
Characterizing a unit: tuning curves (rectified before averaging over phase), Gabor fits placed beside V1 cells, object selectivity from responses. Ablation with a random control asks whether a unit is needed

Return to contents

Slide 45

Minute paper

Submit on Canvas → Minute papers → Minute Paper 11 (access code read out in class).

Write three brief points in your own words:

In plain words: how can a readout score 100% on its training images and still do poorly on new ones?
Something you do not yet understand, or a question still open
Another idea you found interesting, and why it matters for brains, behavior or AI

Credit for a thoughtful attempt, not for being correct.

Return to contents

Slide 47

Appendix

Slides cut from the lecture for time: schedules, bias and variance, many dimensions, five-photo results, the PyTorch checkpoint loop

Return to contents

Slide 48

Four ways to shrink the learning rate

Two stacked panels. Top: the learning rate against step for four schedules: constant, step decay that halves it at fixed intervals, 1 over 1 plus t over tau, and cosine decay. Bottom: the training loss for each; the shrinking schedules settle lower and more smoothly than the constant one
Open full-size figure

One readout, trained four times with the same minibatches; only the schedule of $\eta$ differs. Top: the step size at each step. Bottom: the training loss
Constant never changes. Step decay halves $\eta$ every 200 steps. Cosine lowers it smoothly to near 0. $1/(1+t/\tau)$ shrinks it a little at every step
Inset, last 200 steps: every shrinking schedule ends steadier than the constant step
A common recipe for image networks: start at 0.1 and divide by 10 every 30 epochs, or follow a cosine

Return to contents

Slide 49

Same model, different training sets

Computed: the polynomial example; 20 fits shown, numbers from 2,000 training sets

Three panels, each with 20 thin blue curves fitted to 20 training sets of 10 noisy points, their average in red, and the true curve dashed. Degree 1, too simple: the lines agree with each other but their average misses the curve; the gap between the average and the true curve is shaded and labelled bias. Degree 3, about right: the fits cluster tightly around the true curve. Degree 9, too flexible: the average lies on the true curve but the individual fits swing wildly
Open full-size figure

Draw many training sets from one source (same 10 inputs, new noise each time) and fit each. Thin blue lines: one fit per training set; red: their average; dashed: the true curve
Degree 1: the fits agree, but their average misses the curve: bias, the error of the average prediction, the shaded gap $f(x)-\mathbb{E}[\hat y]$ . Degree 9: right on average, but the fitted curve itself changes from one training set to the next: variance

Return to contents

Slide 50

Bias and variance

\mathbb{E}\big[(y-\hat y)^2\big] = \underbrace{\textcolor{#B5396B}{\big(f(x)-\mathbb{E}[\hat y]\big)^2}}_{\textcolor{#B5396B}{\text{bias}^2}} + \underbrace{\textcolor{#3F7A6B}{\mathbb{E}\big[(\hat y-\mathbb{E}[\hat y])^2\big]}}_{\textcolor{#3F7A6B}{\text{variance}}} + \sigma^2_{\text{noise}}, \qquad y = f(x) + \text{noise}

Squared error against the degree of the polynomial, 0 to 9. Bias squared in magenta falls from 0.50 at degree 0 to near 0 from degree 3 on. Variance in green rises slowly, then steeply to 0.55 at degree 9. Their sum plus the noise, in dark ink, is U-shaped, lowest at degree 3, 0.13. A dotted line marks the noise, 0.09
Open full-size figure

Computed: 2,000 training sets

$f$ : the true curve; $\hat y$ : the fitted prediction at $x$ ; $\mathbb{E}$ averages over training sets. $\sigma^2_{\text{noise}}$ : the noise variance, $0.30^2 = 0.09$ (squared errors here, not RMSE)
Bias falls and variance rises with the degree: the best model balances the two, here at degree 3

Geman, Bienenstock & Doursat Neural Comput 1992

Return to contents

Slide 51

Many dimensions, few examples

Accuracy against the number of input dimensions on a log axis, from 1 to 1024, with 100 training points. Training accuracy, in magenta, stays near 1.0 throughout. Accuracy on new points, in green, starts at 1.0 and falls to about 0.58 at 1024 dimensions; the shaded gap between the curves widens. A dashed line marks 100 dimensions, as many as training points
Open full-size figure

Simulated points: the label is the sign of the first input; every other input is random noise. A linear readout, 100 training points
Training accuracy stays near 1.0; accuracy on new points falls toward chance
The curse of dimensionality (baselines lecture): the examples needed grow quickly with the number of dimensions. With 1,024 weights and 100 examples, the readout fits the training labels partly through the noise dimensions
The fix: a representation, or a limit on the readout, that ignores the irrelevant dimensions

Return to contents

Slide 52

Learning a new category from five photos

Reproduced from Sorscher, Ganguli and Sompolinsky 2022, Fig. 1A and B. A: four photographs each of two animals the network was never trained on, a coati and a numbat, and a new photograph marked with a question mark. B: the primate ventral stream from the retina to IT drawn above a deep network trained on ImageNet; each image becomes a point in the network's feature space, the labelled examples of each animal are averaged into a prototype, and a straight boundary halfway between the two prototypes labels the new image
Open full-size figure

ResNet-50 trained on ImageNet, frozen; 1,000 new categories (ImageNet-21k) it never saw
5 photos each; an animal's average is its prototype, and a new photo takes the nearer one's label: 98.6% correct (raw pixels: 66%)

Sorscher, Ganguli & Sompolinsky PNAS 2022

Return to contents

Slide 53

Five photos are enough, in a good representation

Reproduced from Sorscher, Ganguli and Sompolinsky 2022: accuracy of the prototype readout on new categories against the number of examples per category, for the trained network, an untrained network and raw pixels, and the five-photo accuracy read from monkey IT and V4 recordings compared with network layers
Open full-size figure

Pairs of the 1,000 new categories: 98.6% from 5 photos (untrained network 56%, pixels 66%)
Monkey IT (168 sites, 64 objects): 84% (AlexNet 80%, ResNet-50 93%). Category geometry predicts the errors

Sorscher, Ganguli & Sompolinsky PNAS 2022 · Majaj, Hong, Solomon & DiCarlo J Neurosci 2015

Return to contents

Slide 54

In PyTorch: keep the best checkpoint

best_acc = -float("inf")                    # best validation accuracy so far
for epoch in range(num_epochs):             # one epoch = one pass over the training set
    model.train()                           # training mode (dropout on)
    for X, y in train_loader:               # one minibatch at a time
        optimizer.zero_grad()               # clear last step's gradients
        loss_fn(model(X), y).backward()     # loss on this batch, then backprop
        optimizer.step()                    # update every weight
    model.eval()                            # evaluation mode (dropout off)
    with torch.no_grad():                   # no gradients needed to score
        correct = sum((model(X).argmax(1) == y).sum().item()
                      for X, y in val_loader)   # count right answers
    val_acc = correct / len(val_loader.dataset)
    if val_acc > best_acc:                  # better than every earlier epoch?
        best_acc = val_acc
        torch.save(model.state_dict(), "best.pt")   # save the weights: a checkpoint
model.load_state_dict(torch.load("best.pt"))        # end: reload the best weights

Return to contents

Training at scale, generalization & characterizing units

Training at scale, generalization & characterizing units

Last lecture: hidden layers and backpropagation

Today: three questions about a trained network

Today's three examples

The IT readout, in full

How do we train on a million images?

Last lecture, every step used all the training images. That is slow, and they do not fit in memory at once

One example, all of them, or a few at a time?

Three tips for training your own network

Shrink the learning rate as training goes · Average the gradients: momentum and Adam · Sweep the learning rate first

Tip 1: shrink the learning rate as training goes

Tip 2: average the gradients (momentum)

Tip 2, continued: Adam, a step size for each weight

Adam: the update

Tip 3: sweep the learning rate first

Does what a network learned hold for images it never saw?

Generalization: when does a trained network's fit carry over to new data, and how do we make it?

Recall: train on some images, test on others

A perfect training score can hide poor generalization

Training fits a function from inputs to outputs

Too rigid, or too flexible

Same model, more examples

How do we choose a model and measure how well it generalizes?

Every choice made while looking at a score can fit that score's images

Three data splits

Cross-validation: each example validates once

Learning curves: when to stop training

Why stop before the bottom of the training loss?

A test accuracy from few images is imprecise

Shortcut learning: texture over shape

How do we make a network generalize better?

A penalty on the weights, more data, and switching units off

A penalty on the weights

Without a penalty, a readout's weights keep growing

More data, or altered copies of it

Regularization: penalties and dropout

Try it: overfit, then regularize

A model of a neuron: regression with a penalty

Does a trained network see like the visual cortex?

Nobody told AlexNet about V1. Measure its units the way physiologists measure neurons, and compare

What a trained network's first layer learns

Real V1 receptive fields look like Gabors

The Gabor function, fitted, and two numbers from it

Where do V1 simple cells sit?

Quantified: V1 cells and AlexNet's first layer

What does a unit respond to? Its tuning curve

Why rectify before averaging over phase?

Deeper in: how selective is one IT site for objects?

Is a selective unit needed? Remove it and see

Recap

Further reading for the research trail

Minute paper

Appendix

Slides cut from the lecture for time: schedules, bias and variance, many dimensions, five-photo results, the PyTorch checkpoint loop

Four ways to shrink the learning rate

Same model, different training sets

Bias and variance

Many dimensions, few examples

Learning a new category from five photos

Five photos are enough, in a good representation

In PyTorch: keep the best checkpoint