Interpolation Can Be Harmless

That warning quietly fuses two different claims: first, that the model fits the noise in its training data exactly; second, that the fitted noise must corrupt its predictions elsewhere. Modern practice pulled the two apart — large neural networks routinely interpolate their training data and still generalize — and the question became why.

Overparameterization is what separates the claims. When there are more feature directions than sample constraints, an interpolating model is obliged to fit the noise, but it retains freedom in where to put it. Generalization then depends on the geometry of that choice, not on the mere fact of interpolation. Our work made that geometry exactly calculable in deliberately tractable linear and Fourier models — first for regression, then for classification, then for many classes, adversarial perturbations, and learning from a weaker teacher's noisy labels.

This question was being attacked from several directions at once. Belkin, Hsu, Ma, and Mandal mapped double descent; Bartlett, Long, Lugosi, and Tsigler characterized benign overfitting through covariance spectra; and Hastie, Montanari, Rosset, and Tibshirani developed precise asymptotics for ridgeless interpolation. Belkin and Hsu later joined us on the classification paper below. What distinguishes the line of work in this story is its instrument: a Fourier sampling model that exposes the mechanism through exact calculation, paired with phase maps that reduce competing effects to comparable exponents.

Our decisive conceptual move, in the ISIT 2019 paper and its expanded JSAIT version, was to turn a high-dimensional puzzle into an elementary problem from signal processing. Take $n$ regularly spaced samples of a one-dimensional signal, and fit them using more Fourier frequencies than samples. Overparameterization is now literally undersampling.

Sampling theory says exactly what goes wrong — and here, what goes right. Frequencies separated by multiples of the sampling rate, $f$ and $f + kn$, are exact aliases: on the training grid they are indistinguishable, taking identical values at every sample. Under the uniform test distribution used in the toy, however, the corresponding Fourier features are exactly orthogonal. The sampled data can be decomposed into discrete Fourier modes, and minimum-$\ell_2$-norm interpolation responds to the degeneracy in the most even-handed way possible: it divides the coefficient required for each sampled mode among the frequencies in that mode's alias class.

Parseval's identity then makes the bookkeeping exact. Noise that has been spread over an expanding set of orthogonal directions contributes almost nothing to mean-squared test error — even though every noisy training sample was fit perfectly. The fitted noise is not gone; it has been dissipated into directions that are individually tiny and mutually orthogonal under the test distribution.

The same dissipation that disposes of noise can also erase what one wanted to learn. True signal bleeds into aliases just as noise does: the energy of the low-frequency component the data actually came from is likewise split among its high-frequency impostors.

This led us to a pair of complementary bookkeeping quantities. Survival measures how much of the true signal direction remains in the learned predictor. Contamination measures how much the many false directions — the aliases carrying their shares of signal and noise — perturb a prediction at a test point. Harmless interpolation requires both sides of a bargain: enough extra directions to absorb the noise, together with enough inductive preference for the true directions — for instance, feature weights that favor low frequencies — to keep the signal alive while the noise dissipates.

This is why the story cannot be compressed to "noise gets spread across many dimensions." Which directions are aliased with which, and how the interpolator allocates energy among them, is precisely the sampling geometry — and it is that geometry, not high dimensionality alone, that decides whether interpolation is benign or catastrophic.

This Fourier construction was part of the work from the beginning, not an illustration we attached afterward. The ISIT 2019 paper already pointed toward the Fourier setting; the full arXiv/JSAIT treatment developed the regular-grid Ultra Toy and its exact alias calculations. Its value is that nothing important is hidden behind a large random matrix: one can watch minimum-norm interpolation allocate every sampled Fourier mode among features that are identical on the training grid and orthogonal under the test-distribution inner product.

The regression side is not confined to one carefully chosen waveform. In the lab below, every smooth target in the menu is supported entirely on the default $s=7$ favored low-frequency directions. Widen the favored band to see precisely which additional Fourier directions receive the same preference; change the target and the same aliasing mechanism persists. Then switch to the clean $-1$-to-$+1$ step: that deliberately misspecified target is the natural bridge to the classification paper.

Exact aliasing and exact orthogonality collapse the mechanism to a calculation simple enough to serve as a conjecture engine. The Fourier "ultra-toy" suggested — and we then established for Gaussian features in the JMLR 2021 paper written with Mikhail Belkin and Daniel Hsu (along with students Vidya Muthukumar, Adhyyan Narang, and Vignesh Subramanian) — that regression and classification obey different thresholds.

Regression needs the predicted value to be right: survival must stay near one and contamination near zero. Classification only needs the surviving signed signal to dominate the contamination — a weakened signal still classifies correctly as long as it outvotes the noise at the test point. The consequence is a genuine intermediate regime in which the very same minimum-norm predictor fails as a regressor while succeeding as a classifier.

There is a second, stranger consequence. With sufficient effective overparameterization, every training point becomes a support vector, and the hard-margin SVM — trained on a completely different loss — coincides exactly with the minimum-norm least-squares interpolator. In these regimes, the choice of training loss can matter far less than the geometry both losses inherit.

Our second major tool was a deliberately physics-style use of asymptotics. Instead of sending sample size and dimension to infinity along one fixed aspect ratio, we write each structural quantity as a power of the number of training points $n$: the number of features scales as $n^p$, the number of favored directions as $n^r$, the strength of the favoring enters through an exponent $q$, and — in the later work — the number of classes grows as $n^t$.

Every competing mechanism now owns an exponent, and comparing exponents turns complicated non-asymptotic bounds into linear boundaries between regions: a phase map. One region where regression and classification both succeed; one where classification alone succeeds; one where nothing does. The maps did not merely summarize finished theorems — they organized the search for theorems, separating effects that tangle together at any single problem size and identifying which result should be true in each region. We then progressively filled in these maps across the multiclass papers: NeurIPS 2022 established achievable regions, ISIT 2023 supplied a matching converse, and the NeurIPS 2023 paper sharpened the lower bounds into strong converses that pin the boundaries exactly.

A smaller observation we made in the JSAIT regression paper deserves its own moment, because it makes the regularizing role of extra features concrete. The familiar "pseudo-observation" view of ridge regression makes the data matrix taller: append fake training examples with zero targets, and the ridge solution appears.

The alternative view makes it wider: append one sample-specific feature per training point — a scaled identity block — and solve an ordinary minimum-norm interpolation problem. Those private features can absorb each training point's residual, but they contribute nothing at a test point, where they are never active. The two constructions impose exactly the same spectral shift, so ridge regression is minimum-norm interpolation with extra feature directions. Additional dimensions, used this way, do not enlarge a model so much as regularize it — the same moral the aliasing story tells, in matrix form.

Once survival, contamination, and the phase-map method exist, they travel.

Adversarial examples. The aliased component is not spread uniformly in space but concentrated in sharp, Gibbs-like spikes near the training samples. Contamination can be small on average — so ordinary classification succeeds — while being locally enormous, so that tiny input perturbations near training data flip predictions. Low test error and proliferating adversarial failures are not in tension; in this model they are the same geometry read at two different scales.

Many classes. Multiclass classification asks the surviving signal to win more comparisons with less margin. We extended the phase maps to a growing number of classes, located the new boundaries, and — in the NeurIPS 2023 paper — proved matching lower bounds and strong converses, so the map's frontiers are exact rather than conjectured.

Weak-to-strong generalization. A Fourier counterpart makes the connection vivid. If a weak alias is present in the teacher model, then giving the student more pseudo-labeled samples puts that alias into the samples themselves. The student could learn it. But the inductive bias does not preserve every teacher-side direction equally: the weak alias does not survive strongly enough to be learned, while the true direction does. In the formal Gaussian-feature model, the same competition is measured through survival and contamination. The student can fit the imperfect supervision exactly without inheriting every error that produced it, and can therefore generalize beyond the teacher.

The through-line of the whole story is a method as much as a result: build the smallest world in which a mechanism operates exactly, calculate it there, and let the exponents say where each phenomenon begins and ends. The Fourier toy and the phase maps remain live instruments for us, and this arc is still being extended.

Our ISIT 2019 paper introduced the harmless-interpolation problem and already pointed toward the Fourier setting. Its expanded arXiv/JSAIT version developed the Fourier Ultra Toy, exact alias calculations, survival and contamination, and the wide-feature view of ridge. The JMLR 2021 paper used that Ultra Toy to conjecture the classification/regression separation and support-vector proliferation, then established the corresponding results in the Gaussian model. The 2021 signal-processing paper made the mechanism spatially visible, including its adversarial consequences. The NeurIPS 2022 and ISIT 2023 papers opened up the multiclass phase maps, the NeurIPS 2023 paper closed them with precise asymptotics and strong converses, and the ICLR 2025 paper carried benign overfitting to weak-to-strong generalization.