Understanding Domain Shift
When the World Changes
Face ID: A Success Story
Apple's Face ID. Trained on over a billion images.
Works in the dark.
Works with glasses.
Works with hats.
Ship it!
✓ Unlocked
Then 2020 Happened
✗ Locked
COVID-19 → everyone wears masks
Your phone keeps asking for your passcode.
Again. And again. And again.
Millions of frustrated users worldwide.
Face ID Meets Masks
Training
→
Deployment
Face ID didn't get worse. The input to the model changed.
Apple had to ship iOS 15.4 with "Face ID with a Mask" to fix this.
This is Domain Shift
Also called: distribution shift, dataset shift
Wait... Is This Just Overfitting?
Let's look at a trained neural network on MNIST digits
Training samples
Accuracy: 97%
Test samples
Accuracy: 81%
Looks like overfitting... or is it?
What the MNIST Network Learned
Activation maps at each layer — what each filter responds to for a digit "4":
Layer 1 (32 filters): strokes, edges — digit still recognizable
Layer 2 (64 filters): stroke fragments, spatial patterns
Layer 3 (128 filters): abstract, sparse codes
The network builds hierarchical features — not memorizing individual digits.
Training ≠ Deployment
Training: upright, centered
Test: rotated and shifted
The distribution changed, not the model complexity
This is domain shift, not overfitting
Domain Shift
| Overfitting | Domain Shift | |
|---|---|---|
| Symptom | Training ✓, testing ✗ | |
| Analogy | Studies only past exams, tested on same course | Masters C01, tested on C011 exam |
| Root cause | Model memorizes noise | \(P_{\text{source}}(x,y) \neq P_{\text{target}}(x,y)\) |
| Fixes | Regularization, dropout, simpler model | Different — that's today & next lecture |
Source domain = training | Target domain = deployment
What Does \(P_{\text{source}} \neq P_{\text{target}}\) Look Like?
Face ID: \(x\) = face scan, \(y\) = unlock / reject
Randomly sample one unlock attempt...
| \(x\) | \(y\) | \(P_{\text{source}}\) (pre-2020) | \(P_{\text{target}}\) (mid-2020) |
|---|---|---|---|
| full face | unlock | 47% | 8% |
| masked face | unlock | 0.1% | 42% |
An \((x, y)\) pair that was 0.1% of training is now 42% of deployment.1
1 Contrived numbers for illustration.
Domain Shift is Everywhere
Robot platform and distribution shift settings: same task, different object and station configurations
Tedrake et al. (2025)
The Sim-to-Real Gap
Top: real world | Middle & bottom: simulation counterparts
Maddukuri et al. (2025)
Satellite Imagery: Crop Classification
Large, regular fields → high accuracy
Tiny, irregular fields → much worse
Wang, Waldner & Lobell (2022)
Pattern 1: Covariate Shift
The inputs change, but the rules don't
Covariate Shift: Wildlife Monitoring
Training: one location, one season
Deployment: new locations, new times
Same Deer, Different Pixels
A deer in a different place, at a different time, is still a deer.
Same ears. Same legs. Same antlers.
Didn't change
The relationship between "deer features" and "deer"
Changed
The background — location, lighting, vegetation
The rules are the same — only the inputs look different.
Covariate Shift: Definition
\(P(x)\) changes — inputs look different
\(P(y|x)\) stays the same — rules unchanged
The deer detector's knowledge is still valid — a deer is still a deer.
It just needs adjusting for different-looking inputs.
The most common type of domain shift.
Why the Joint Changes: \(P(x)\) Shifts
\(P(x, y) = P(y|x) \cdot P(x)\)
| \(P(\text{deer} | \text{infrared image})\) | \(P(\text{infrared image})\) | \(P(\text{infrared, deer})\) | |
|---|---|---|---|
| Training | 20% | 5% | 1% |
| Deployment | 20% | 90% | 18% |
Same conditional × different input frequency = different joint → \(P_{\text{source}} \neq P_{\text{target}}\)
Covariate Shift: You've Seen It
| Example | \(x\) | \(P(x)\) shifted | \(P(y|x)\) same |
|---|---|---|---|
| Face ID + masks | face scan | masked faces everywhere | your identity didn't change |
| MNIST rotation | digit image | rotated pixels | a 3 is still a 3 |
| Sim-to-real robotics | camera image | sim vs real textures | same grasping task |
| Wildlife monitoring | trail camera | day vs infrared night | a deer is still a deer |
Same rules, different-looking inputs → covariate shift.
Fixing Covariate Shift
Weight each training example by: \(w(x) = \frac{P_{\text{target}}(x)}{P_{\text{source}}(x)}\)
Common in target → boost | Rare in target → reduce
Other Fixes (Preview)
Reweighting isn't the only approach:
🔧
Finetuning
Continue training on target data
Deer detector + a few night images
🎲
Data Augmentation
Simulate target conditions during training
Synthetically darken daytime photos
⚡
Test-Time Adaptation
Adapt on-the-fly at deployment
No target labels needed
We'll cover these in detail next lecture.
Pattern 2: Label Shift
Semester
📚
40%
40%
😴
30%
30%
🏃
15%
15%
🥳
15%
15%
→
P(y) changed
Summer
📚
10%
10%
😴
25%
25%
🏃
30%
30%
🥳
35%
35%
👩💻 — studying or fun? Same image, different odds.
Label Shift: Cold Diagnosis
Same cold detection model, different patient populations:
Children's Hospital
Kids get 6–8 colds per year
MGH (adult general hospital)
Adults get 2–3 colds per year
\(P(x|y)\) same — a cold looks the same in kids and adults
\(P(y)\) changed — prevalence is very different
Also called: prior probability shift, target shift
Why the Joint Changes: \(P(y)\) Shifts
\(P(x, y) = P(x|y) \cdot P(y)\)
| \(P(\text{runny nose} | \text{cold})\) | \(P(\text{cold})\) | \(P(\text{runny nose, cold})\) | |
|---|---|---|---|
| Children's | 80% | 40% | 32% |
| MGH | 80% | 10% | 8% |
Same conditional × different prior = different joint → \(P_{\text{source}} \neq P_{\text{target}}\)
Label Shift as Bayes' Rule
3Blue1Brown, The medical test paradox
Sensitivity = 90%, specificity = 91%. You test positive.
| \(P(\text{disease})\) | \(P(\text{disease} | +)\) | |
|---|---|---|
| 10% prevalence | 10% | ~50% |
| 1% prevalence | 1% | ~9% |
10% case: \(\frac{0.9 \cdot 0.10}{0.9 \cdot 0.10 + 0.09 \cdot 0.90} = \frac{0.09}{0.171} \approx 0.53 \approx 50\%\)
1% case: \(\frac{0.9 \cdot 0.01}{0.9 \cdot 0.01 + 0.09 \cdot 0.99} \approx 0.09\)
Same test, same \(P(x|y)\) — the prior \(P(y)\) changes everything.
That's label shift.
\[ P(D|+) = \frac{P(+|D)P(D)}{P(+|D)P(D) + P(+|\neg D)P(\neg D)} \]
Fixing Label Shift
Adjust predictions: \(P_{\text{target}}(y|x) \propto P_{\text{source}}(y|x) \cdot \frac{P_{\text{target}}(y)}{P_{\text{source}}(y)}\)
Common in target → boost | Rare in target → reduce
MNIST example demo
Two Patterns, One Framework
Covariate Shift
\(P(x)\) changes
\(P(y|x)\) stays same
World looks different, works the same
e.g. deer detector, sim-to-real, Face ID
Fix: reweight, finetune, augment
Label Shift
\(P(y)\) changes
\(P(x|y)\) stays same
World works the same, mix is different
e.g. cold diagnosis, MNIST class bias
Fix: adjust priors, rebalance
Identifying which pattern → determines the fix.
Which Type of Shift?
Spam filter: trained on Gmail, deployed on corporate email
→ Covariate — different writing style, same spam vs not-spam
Sentiment analysis: trained on restaurant reviews, deployed on electronics reviews
→ Covariate — different vocabulary, same positive/negative meaning
Disease screening: trained in flu season, deployed in summer
→ Label shift — same symptoms, way fewer sick patients
Satellite imagery: trained in France, deployed in India
→ Both! — different terrain (covariate) and different crop mix (label)
Your Turn: Loan Approval Model
Credit scoring model trained at a major bank in New York, deployed at a regional bank in rural Midwest.
What might be different?
- Applicant profiles (income levels, employment types)
- Default rates (different economic conditions)
Both types! — Different applicant features AND different default rates
The Diagnostic Framework
When we encounter a new deployment scenario, ask:
- Do the inputs look different? → Check for covariate shift
- Can a domain classifier separate source vs target? → measurable input shift
Probe: train \(d(x)\) to predict source vs target.
Near chance (~50%) → little detectable input shift; high accuracy → covariate shift signal.
- Are the class frequencies different? → Check for label shift
- Compare class distributions source vs target → class imbalance signals label shift
Probe: compare \(\hat{P}(y)\) across source and target.
Similar frequencies → no label shift; large gaps → label shift signal.
- Did the underlying rules change? → Might need to retrain
Case Study: Waymo Phoenix → SF
Flat, wide, sunny → hills, fog, narrow streets
Diagnosing Waymo's Shift
| Question | Answer | Diagnosis |
|---|---|---|
| Inputs look different? | Yes — fog, hills, narrow streets | Covariate shift |
| Class frequencies different? | Yes — more cyclists, pedestrians | Label shift |
| Rules changed? | No — a stop sign is still a stop sign | No concept shift |
Domain-classifier check: source-vs-target frame classifier performs well → measurable input shift.
Both types — real-world shifts are often mixed.
How Waymo Adapted
- Collected SF data — thousands of miles with safety drivers
- Reweighted — upweight rare SF scenarios (fog, hills) ← covariate shift fix
- Rebalanced — adjust for higher cyclist/pedestrian frequency ← label shift fix
- Finetuned — Phoenix model as starting point + SF data
- Augmented — synthetic fog, hills, unusual traffic
After years of adaptation, Waymo now operates commercially in SF.
Summary
Domain shift is not overfitting — the world changed, not your model's complexity.
Covariate shift (\(P(x)\) changes, \(P(y|x)\) same) is the most common pattern — fix with reweighting, finetuning, or augmentation.
Label shift (\(P(y)\) changes, \(P(x|y)\) same) requires adjusting class priors, not model architecture.
Diagnosis before treatment — identify what changed, then pick the matching fix.
Models are frozen snapshots of one distribution; the world keeps moving, so anticipate and adapt.
Reference: Shift Types
| Shift Type | What Changes? | What Stays Same? | Fix Strategy | Today's Examples |
|---|---|---|---|---|
| Covariate | \(P(x)\) | \(P(y|x)\) | Reweight, finetune, augment | Face ID, deer detector, sim-to-real |
| Label | \(P(y)\) | \(P(x|y)\) | Adjust priors, rebalance | Cold diagnosis, MNIST class bias |
| Concept (preview) | \(P(y|x)\) | Nothing guaranteed | Retrain, continual learning | Next lecture! |
Next Time: Making Models Robust
Now that we can diagnose domain shift... how do we fix it?
Data Augmentation
Ensembles
Monitoring
Uncertainty
Test-Time Adaptation
AI Educators Pilot - Lecture - Domain Shift
By Shen Shen
