Detecting drift in image data

What is being tested#

Let $P_{\text{ref}}$ and $P_{\text{cur}}$ be distributions over inputs $x$, with labels $y$. The decomposition that matters:

• Covariate shift: $P(x)$ changes, $P(y \mid x)$ fixed. Lighting, optics, sensor. Detectable from inputs alone.
• Prior shift: $P(y)$ changes, $P(x \mid y)$ fixed. Seasonal mix, new site.
• Concept drift: $P(y \mid x)$ changes. The definition of the label moved — a new defect category, a changed inspection standard.

Only the first two are detectable without labels. Concept drift is invisible to every method below, which is the fundamental limitation of unsupervised monitoring and the reason a labelled sample is not optional.

Maximum mean discrepancy#

For a characteristic kernel $k$ with feature map $\phi$,

$$ \mathrm{MMD}^2(P, Q) = \left\lVert \mathbb{E}{x \sim P}[\phi(x)] - \mathbb{E}{y \sim Q}[\phi(y)] \right\rVert_{\mathcal{H}}^2 $$

with the unbiased estimator

$$ \widehat{\mathrm{MMD}}^2 = \frac{1}{m(m-1)}\sum_{i \ne j} k(x_i, x_j) + \frac{1}{n(n-1)}\sum_{i \ne j} k(y_i, y_j) - \frac{2}{mn}\sum_{i,j} k(x_i, y_j) $$

Gretton et al. (2012) established this as a consistent two-sample test: for a characteristic kernel, $\mathrm{MMD}(P,Q) = 0$ if and only if $P = Q$. The null distribution has no closed form, so significance comes from a permutation test, as in the code above. The RBF bandwidth is conventionally set by the median heuristic — $\gamma = 1/\text{median}(\lVert x_i - x_j\rVert^2)$ — which is a heuristic, not a principled choice, and Sutherland et al. (2017) show that learning the kernel gives substantially more power.

Cost is $O((m+n)^2 d)$ per permutation, which is why the example subsamples to 200 points per side.

The classifier two-sample test#

Lopez-Paz and Oquab (2017) formalised the intuition: train a classifier to separate the two samples and test whether its held-out accuracy exceeds chance. Under $H_0: P = Q$, accuracy is binomially distributed around $1/2$, giving an exact test.

Its two practical advantages are decisive for production monitoring. It yields an interpretable effect size — an AUC of 0.57 and one of 0.97 mean different things, where two p-values below 1e-5 do not. And it is inspectable: the classifier's coefficients or feature importances point at what changed, which is the first question anyone asks after an alert.

Its weakness is finite-sample power: with a weak classifier or few samples it misses shifts a kernel test would catch. Use both.

The multiple-testing and power problem#

Running $d$ marginal tests at level $\alpha$ gives a family-wise error rate of $1 - (1-\alpha)^d$ under independence — about 0.80 for $d = 32$, $\alpha = 0.05$. Bonferroni ($\alpha/d$) controls it conservatively; Benjamini-Hochberg controls the false discovery rate with more power and is the better default for monitoring.

The deeper issue is the one the sample-size experiment demonstrates. For a fixed true difference $\delta > 0$, the power of any consistent test tends to 1 as $n \to \infty$. Since $\delta$ is never exactly zero in production, every drift test fires eventually, given enough traffic. The measured progression — 0, then 2, then 16 dimensions flagged for a shift that moves AUC from 0.543 to 0.567 — is this asymptotic behaviour in miniature.

The correct construction is therefore an equivalence test, not a difference test: define a tolerance region and alert when the estimated effect size, with its confidence interval, leaves it. Effect size first, significance as a gate.

Detecting novelty, which is a different problem#

The runnable table shows aggregate tests performing worst on the new-class scenario. That is structural: batch tests measure a difference between distributions, and a novel class contaminating a fraction $\epsilon$ of the stream moves aggregate statistics by roughly $O(\epsilon)$.

Per-image out-of-distribution scoring is the right tool:

• Maximum softmax probability (Hendrycks and Gimpel, 2017) — the baseline; weak, because networks are overconfident on unfamiliar inputs.
• ODIN (Liang et al., 2018) — temperature scaling plus input perturbation.
• Mahalanobis distance (Lee et al., 2018) — class-conditional Gaussian fit to penultimate features; strong and cheap.
• Energy score (Liu et al., 2020) — $-T\log\sum_c e^{f_c(x)/T}$; better calibrated than the softmax maximum.
• kNN distance in feature space (Sun et al., 2022) — non-parametric, competitive, and needs a stored reference set.

Yang et al. (2022), OpenOOD, benchmark these consistently and are worth reading before choosing. Run one of them per image, and aggregate the fraction of images exceeding a threshold as your novelty signal — a rate, not a test.

Practical architecture#

Reference window. A fixed sample from a period of verified good performance, stored as embeddings rather than images. Version it with the model, and refresh it when you retrain.

Two cadences. Cheap per-image statistics logged continuously; expensive batch tests on a schedule with a fixed window size, so the sample-size effect is held constant. A varying window size makes your alarm threshold meaningless.

Feature choice. Penultimate-layer embeddings are the standard, and they inherit the model's blind spots — a model that ignores a change will show no drift in its features even when the change matters. Pairing them with raw-pixel statistics covers different failures.

Labels. Sample, label, measure, monthly. Everything above is an early-warning system for deciding when to look; only labels close the loop. See monitoring and drift for the surrounding practice.

Tooling#

alibi-detect implements MMD, the classifier test, learned-kernel MMD and several OOD detectors with a consistent interface. evidently and deepchecks target tabular data primarily and are weaker on images. torchdrift is PyTorch-native and lighter. For most teams the forty lines above, run on stored embeddings, is sufficient and easier to reason about than a framework.

Papers#

• Gretton et al., A Kernel Two-Sample Test, JMLR 2012
• Lopez-Paz and Oquab, Revisiting Classifier Two-Sample Tests, ICLR 2017 — arxiv.org/abs/1610.06545
• Rabanser et al., Failing Loudly: An Empirical Study of Methods for Detecting Dataset Shift, NeurIPS 2019 — arxiv.org/abs/1810.11953
• Lee et al., A Simple Unified Framework for Detecting Out-of-Distribution Samples, NeurIPS 2018 — arxiv.org/abs/1807.03888
• Liu et al., Energy-based Out-of-distribution Detection, NeurIPS 2020 — arxiv.org/abs/2010.03759
• Yang et al., OpenOOD: Benchmarking Generalized Out-of-Distribution Detection, NeurIPS 2022 — arxiv.org/abs/2210.07242
• Sutherland et al., Generative Models and Model Criticism via Optimized MMD, ICLR 2017 — arxiv.org/abs/1611.04488

What to learn next#

• Monitoring and drift — the surrounding production practice.
• Testing robustness to corruptions — measuring in advance what the drift you detected will cost.
• Train and serve preprocessing mismatch — a shift that looks like drift and is entirely your own doing.