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Just Interpolate: Kernel "Ridgeless" Regression Can Generalize
Tengyuan Liang, Alexander Rakhlin
TL;DR
The paper addresses why kernel regression can generalize when its minimum-norm solution interpolates the training data. It develops a random-matrix-based, data-dependent risk analysis and identifies implicit regularization from kernel curvature, high dimensionality, and favorable spectral geometry. The results explain good interpolation performance under suitable conditions and provide MNIST evidence, while also identifying settings where interpolation performs poorly.
Problem
Existing theory largely centered on explicitly regularized kernel regression and did not explain when or why minimum-norm interpolation works.
Method
The paper combines random matrix theory with spectral analysis of empirical kernel and covariance matrices to derive data-dependent upper bounds for interpolating kernel estimators.
Results
The analysis identifies implicit regularization from kernel curvature, high-dimensionality, and favorable data geometry, with MNIST experiments suggesting good out-of-sample performance at λ = 0.
Takeaways & Limitations
Interpolation can be statistically sound when spectral decay and related geometric conditions provide favorable implicit control of variance and bias.
Takeaways & Limitations
Interpolation can perform poorly for data such as high-dimensional spherical Gaussian samples, where eigenvalues do not decay and the variance is large.
Abstract
from arXiv · showhide
In the absence of explicit regularization, Kernel "Ridgeless" Regression with nonlinear kernels has the potential to fit the training data perfectly. It has been observed empirically, however, that such interpolated solutions can still generalize well on test data. We isolate a phenomenon of implicit regularization for minimum-norm interpolated solutions which is due to a combination of high dimensionality of the input data, curvature of the kernel function, and favorable geometric properties of the data such as an eigenvalue decay of the empirical covariance and kernel matrices. In addition to deriving a data-dependent upper bound on the out-of-sample error, we present experimental evidence suggesting that the phenomenon occurs in the MNIST dataset.
1 Introduction
The paper asks why minimum-norm kernel interpolants can generalize despite fitting training data exactly, and attributes this behavior to implicit regularization in high-dimensional, favorably structured settings.
- Conventional kernel regression adds explicit regularization to limit overfitting and balance bias against variance.
- Empirically, minimum-norm interpolation often achieves the best out-of-sample performance at λ = 0, although its mechanism has been unclear.
- For virtually all tested MNIST digit pairs, test mean squared error decreases monotonically as λ decreases.
- The paper identifies implicit regularization arising from kernel curvature, high-dimensional inputs, and favorable data geometry.
- Its analysis uses random matrix theory to explain why interpolation can perform well, contrasting with prior work focused mainly on explicitly regularized estimators.
2 Preliminaries
The paper formulates minimum-norm kernel interpolation in an RKHS, represents it through kernel and integral operators, and derives data-dependent risk bounds for full-rank kernel matrices.
- Problem Formulation: The setup observes n i.i.d. covariate-response pairs and assumes the conditional expectation function lies in an RKHS with positive definite kernel K.
- Problem Formulation: With an invertible kernel matrix K(X, X), the interpolating estimator has a closed-form representation based on kernel evaluations against the training data.
- Problem Formulation: The paper provides high-probability, data-dependent upper bounds on integrated squared risk, which also imply excess square-loss bounds.
- Notation and Background on RKHS: The kernel is analyzed through an integral operator whose eigenfunctions and nonnegative eigenvalues provide a spectral representation K(x, z) = e(x)∗Te(z).
- Notation and Background on RKHS: Sample kernel operators and their eigenvalues connect the population spectral description to the empirical interpolation estimator.
3 Main Result
The paper gives high-probability, data-dependent guarantees for minimum-norm interpolation under high-dimensional and favorable spectral conditions. The bound attributes implicit control of variance and bias to kernel curvature, data geometry, and kernel-matrix structure rather than a tunable ridge parameter.
- Assumptions: The analysis assumes a high-dimensional regime with d/n bounded above and below, along with moment, noise, and smooth nonlinear-kernel conditions.The main theorem is stated for inner-product kernels and can be extended to RBF kernels under suitable modifications.
- Main theorem: Under assumptions (A.1)-(A.4), the interpolation estimator satisfies a high-probability upper bound with respect to the design matrix X.The guarantee holds for d large enough and probability at least 1 − 2δ − d−2.
- Main theorem: The bound’s remainder term is ϵ(n, d) = O(d−m/(8+m) log^4.1 d) + O(n−1/2 log^0.5(n/δ)).This term quantifies finite-dimensional and finite-sample contributions to the theorem’s guarantee.
- Interpretation: Unlike explicit regularization, the estimator has no tunable λ; the nonlinear kernel itself implicitly controls the bias-variance terms through curvature and data properties.The upper bound is data-dependent and can serve as a certificate when bounds on σ^2 and ∥f*∥_H^2 are available.
- Interpretation: The variance term is small when the data covariance eigenvalues decay, while the bias term is small when the kernel matrix eigenvalues decay rapidly or the matrix is effectively low rank.The curvature quantity γ provides implicit regularization for the variance term; α and β are constants, while γ scales with (Tr(Σ_d)/d)^2.
- Limitations: The analysis leaves the upper bound data-dependent because non-asymptotic spectral-density characterization remains unavailable for general spectral decay.The discussion also notes that relaxing the i.i.d. coordinate assumption required for the favorable high-dimensional effect is left for future work.
4 Behavior of the Data-dependent Bound
The data-dependent bounds exhibit a bias-variance trade-off governed by spectral decay, dimensionality, and implicit regularization. Favorable decay can make both terms small in either n > d or d > n regimes.
- Regimes: The analysis studies data-dependent variance and bias bounds separately in the n > d and d > n regimes.The bounds are examined numerically and theoretically, with residual terms treated as negligible when n and d are sufficiently large.
- n > d: As spectral decay becomes slower, implicit regularization increases, decreasing variance while increasing bias.This trade-off is summarized for the n > d setting through the parameter κ controlling approximate low-rankness.
- Numerical behavior: Extremely fast spectral decay can cause large variance, whereas very slow decay can cause large bias and hurt generalization.An intermediate spectral-decay regime provides the best trade-off and generalization error.
- n > d: For low-rank covariance structure, both variance and bias bounds vanish as ϵ → 0 when n > d.The low-rank example sets Σd to have ϵd nonzero eigenvalues and obtains r = ϵ^2.
- n > d: For approximate low rank, both terms vanish when ϵ ≍(d/n)1/2 and n ≫d.For nonparametric decay λj(Σd) = j^-α, balancing the bounds yields an upper bound n^-α/(2α+1).
- d > n: In the d ≫n regime, choosing r = (n/d)2/3 makes both variance and bias bounds vanish.This choice implies Tr(Σd)/d = O((n/d)1/3).
5 Proofs
The proofs decompose interpolation error into bias and variance, then derive data-dependent bounds for each component. The resulting analysis relates variance to covariance structure and bias to kernel spectral properties.
- Error decomposition: The mean squared error is decomposed into bias and variance, and both terms must be small for convergence.The paper provides a data-dependent bound for each component in separate proof sections.
- 5.2 Variance: Theorem 2 gives a high-probability upper bound for the variance under assumptions (A.1)–(A.4).The stated probability is at least 1 − δ − d^-2, with d sufficiently large under the accompanying condition.
- 5.2 Variance: The variance analysis explains implicit regularization through kernel curvature, eigenvalue decay, and high-dimensional geometry.For centered data, the first term simplifies, and the resulting formula captures an effective rank of the training data.
- 5.2 Variance: The effective-rank measure used here differs from the classical effective rank for explicitly regularized kernel regression.The classical quantity uses population integral-operator eigenvalues and a ridge-dependent function gr(t) = t/(r + t).
- 5.3 Bias: Theorem 3 bounds the bias under bounded diagonal kernel values and i.i.d. inputs.Its proof uses spectral decomposition, projections onto empirical kernel eigenspaces, empirical-process bounds, and Rademacher complexity.
- 5.3 Bias: In the noiseless regression setting, the variance contribution is zero, and tuning the rank parameter k balances the remaining bound.Pk is the best-rank-k projection based on X, while P⊥k is its orthogonal projection.
6 Experiments
Experiments on MNIST and synthetic data examine when minimum-norm interpolation generalizes well. They find strong MNIST performance and a spectral-decay-dependent trade-off in synthetic settings.
- MNIST: Among all 45 MNIST digit-pair experiments, the no-regularization interpolation estimator performs best.Each experiment uses roughly n ≈10000 samples.
- MNIST: In all but one MNIST digit pair, interpolation performs best across the finer grid of λ values.The grid includes λ ∈{0, 0.01, 0.02, 0.04, 0.08, 0.16, 0.32, 0.64, 1.28}.
- MNIST: The five highlighted MNIST experiments show fast decay in both empirical kernel-matrix and sample-covariance eigenvalues.This supports the theoretical finding that interpolation can perform well when eigenvalues decay rapidly.
- Synthetic example: Interpolation can perform poorly when eigenvalues do not decay, including the spherical-Gaussian example.In that high-dimensional setting, the variance term itself is large.
- Synthetic example: For synthetic data, varying spectral decay κ reveals a sweet spot where the generalization-error trade-off is optimized.The experiments vary κ for high-dimensionality ratios and include both n > d and d > n cases.
7 Further Discussion
The discussion frames ridgeless regression as a form of implicit regularization shaped by kernel choice and data geometry. It also identifies open questions about optimal rates and when interpolation dominates explicit regularization.
- Optimization connection: The minimum-norm interpolating solution can be obtained by kernel-matrix inversion or gradient descent initialized at zero.The analysis is discussed alongside work on implicit regularization in optimization.
- Implicit regularization: The paper characterizes a novel implicit regularization effect in interpolating minimum-norm solutions.The effect is linked to spectral decay and the geometry of the data.
- Implicit regularization: Parametrizing spectral decay recovers a familiar bias-variance trade-off controlled by implicit regularization.Changing kernel bandwidth may likewise expose this trade-off even though the estimator always interpolates.
- Open questions: The paper leaves open whether suitable bandwidth choices achieve optimal estimation rates under appropriate assumptions.It also leaves open when the interpolating minimum-norm solution dominates the regularized solution.
A Propositions
This appendix develops concentration and random-matrix propositions supporting the paper’s non-asymptotic analysis. The results rely on stated moment, boundedness, dimensional, and probability conditions.
- Technical propositions: The appendix begins with a concentration result for quadratic forms under a mild moment condition.The result is used as a technical ingredient in the subsequent analysis.
- Technical propositions: Propositions A.1 and A.2 provide high-probability bounds under bounded-entry and related assumptions for sufficiently large d.Several bounds use probability at least 1 − d−2, with additional dependence on δ and logarithmic factors.
- Scope of the propositions: The appendix emphasizes that the adapted error rate can be very slow, motivating a non-asymptotic rather than only convergence-in-probability formulation.This clarification is presented as the appendix’s contribution relative to the cited result.
- Proof strategy: The proof adapts El Karoui’s random-kernel-matrix analysis by decomposing approximation error into diagonal, second-order off-diagonal, and third-order terms.The appendix notes that convergence for the second-order term is obtained only in probability.
- Proof strategy: A Chebyshev-bound argument controls the second-order contribution under the appendix’s stated conditions.The displayed bound includes δ-dependent terms and yields a probability control of δ.
B Lemmas and Corollaries
The lemmas and corollaries establish high-probability controls used to approximate nonlinear kernel quantities and derive variance and bias bounds. Their assumptions include Gaussian or weak-moment data and sufficiently large dimension.
- Kernel approximation: Lemma B.1 gives a Gaussian-case high-probability result under the stated covariance and moment assumptions.The guarantee is with probability at least 1 − d−2 over the draw of X.
- Kernel approximation: Lemma B.2 extends the corresponding result to a weak-moment setting under assumptions (A.1), (A.2), and (A.4).Its proof starts from an entry-wise Taylor expansion for the smooth kernel.
- Concentration steps: Bernstein and related concentration arguments control kernel-vector deviations conditionally on the observed sample.The resulting bounds hold with exponential tail probabilities and, after further steps, with probability at least 1 − d−2 over X.
- Kernel approximation: The lemmas define Z as the norm difference between the nonlinear kernel vector and its linear approximation and derive tail and high-probability bounds for Z.The bounds are obtained through conditional probability estimates and changes of variables.
- Corollaries: The appendix uses matrix-ordering, random-matrix, chi-squared, and concentration arguments in proofs of the variance and bias corollaries.These steps support the corresponding upper bounds rather than introducing a tunable regularization parameter.
- Auxiliary concentration: Lemma B.4 supplies a bounded-function concentration result involving Rademacher variables, proved using McDiarmid’s inequality and symmetrization.The statement gives probability at least 1 − 2δ under its boundedness condition.
C MNIST Result
The MNIST digit-pair results report errors in percentage, with error varying across digit pairs and Lambda values.
- Errors are reported as percentages; for example, 2.921 corresponds to 2.921%.
- For digit pair [0, 2], error increases from 2.921 at Lambda=0 to 4.689 at Lambda=0.1 and 7.584 at Lambda=1.
- The table lists errors for multiple MNIST digit pairs under Lambda=0, Lambda=0.1, and Lambda=1.