learning curves

of kernel methods

  • Learn from examples: how many are needed?

  • We consider regression (fitting functions)

  • We study (synthetic) Gaussian random data and real data

  supervised deep learning

  • Performance is evaluated through the generalization error \(\epsilon\)

  • Learning curves decay with number of examples \(n\), often as

  • \(\beta\) depends on the dataset and on the algorithm

Deep networks: \(\beta\sim 0.07\)-\(0.35\) [Hestness et al. 2017]

  learning curves

\(\epsilon\sim n^{-\beta}\)

We lack a theory for \(\beta\) for deep networks!

  • Performance increases with overparametrization

      \(\longrightarrow\) study the infinite-width limit!


[Jacot et al. 2018]

What are the learning curves of kernels like?

  link with kernel learning

(next slides)


[Neyshabur et al. 2017, 2018, Advani and Saxe 2017]

[Spigler et al. 2018, Geiger et al. 2019, Belkin et al. 2019]



  • With a specific scaling, infinite-width limit \(\to\) kernel learning

[Rotskoff and Vanden-Eijnden 2018, Mei et al. 2017, Jacot et al. 2018, Chizat and Bach 2018, ...] 

Neural Tangent Kernel

  • Very brief introduction to kernel methods and real data


  • Gaussian data: Teacher-Student regression

  • Smoothness of Gaussian data

  • Effective dimension and effective smoothness in real data


  • Kernel methods learn non-linear functions
  • Map data to a feature space, where the problem is linear

data \(\underline{x} \longrightarrow \underline{\phi}(\underline{x}) \longrightarrow \) use linear combination of features

only scalar products are needed:                     


  kernel methods

kernel \(K(\underline{x},\underline{x}^\prime)\)


K(\underline{x},\underline{x}^\prime) = \exp\left(-\frac{|\!|\underline{x}-\underline{x}^\prime|\!|^2}{\sigma^2}\right)
K(\underline{x},\underline{x}^\prime) = \exp\left(-\frac{|\!|\underline{x}-\underline{x}^\prime|\!|}{\sigma}\right)



  • Target function  \(\underline{x}_\mu \to Z(\underline{x}_\mu),\ \ \mu=1,\dots,n\)

  • Build an estimator  \(\hat{Z}_K(\underline{x}) = \sum_{\mu=1}^n c_\mu K(\underline{x}_\mu,\underline{x})\)

  • Minimize training MSE \(= \frac1n \sum_{\mu=1}^n \left[ \hat{Z}_K(\underline{x}_\mu) - Z(\underline{x}_\mu) \right]^2\)

  • Estimate the generalization error \(\epsilon = \mathbb{E}_{\underline{x}} \left[ \hat{Z}_K(\underline{x}) - Z(\underline{x}) \right]^2\)

  kernel regression


Regression: performance depends on the target function!


  • With the weakest hypotheses, \(\beta=\frac1d\)


  • With strong smoothness assumptions, \(\beta\geq\frac12\) is independent of \(d\)

Curse of dimensionality!

[Luxburg and Bousquet 2004]

[Smola et al. 1998, Rudi and Rosasco 2017, Bach 2017]

  previous works

\(d\) = dimension of the input space


  real data



2 classes: even/odd

70000 28x28 b/w pictures

2 classes: first 5/last 5

60000 32x32 RGB pictures

Kernel regression on:

dimension \(d = 784\)

dimension \(d = 3072\)

  • Same exponent for Gaussian and Laplace kernel

  • MNIST and CIFAR10 display exponents \(\beta\gg\frac1d\) but \(<\frac12\)

  real data: exponents



  • Controlled setting: Teacher-Student regression

  • Training data are sampled from a Gaussian Process:

          \(Z_T(\underline{x}_1),\dots,Z_T(\underline{x}_n)\ \sim\ \mathcal{N}(0, K_T)\)
          \(\underline{x}_\mu\) are random on a \(d\)-dim hypersphere

  • Regression is done with another kernel \(K_S\)

  kernel teacher-student framework

\(\mathbb{E} Z_T(\underline{x}_\mu) = 0\)

\(\mathbb{E} Z_T(\underline{x}_\mu) Z_T(\underline{x}_\nu) = K_T(|\!|\underline{x}_\mu-\underline{x}_\nu|\!|)\)

(artificial, synthetic data)

  teacher-student: simulations

Generalization error

Exponent \(-\beta\)

Can we understand these curves?

  teacher-student: regression

\hat{Z}_S(\underline{x}) = \underline{k}_S(\underline{x}) \cdot \textcolor{darkred}{\mathbb{K}_S^{-1}} \textcolor{gray}{\underline{Z}_T}
(\underline{Z}_T)_\mu = Z_T(\underline{x}_\mu)
(\underline{k}_S(\underline{x}))_\mu = K_S(\underline{x}_\mu, \underline{x})
(\mathbb{K}_S)_{\mu\nu} = K_S(\underline{x}_\mu, \underline{x}_\nu)



Compute the generalization error \(\epsilon\) and how it scales with \(n\)

\epsilon = \textcolor{darkred}{\mathbb{E}_T} \mathbb{E}_{\underline{x}}\, \left[ \hat{Z}_S(\underline{x}) - \textcolor{darkred}{Z_T(\underline{x})} \right]^2 \sim n^{-\beta}
\hat{Z}_S(\underline{x}) = \textcolor{gray}{\underline{k}_S(\underline{x}) \cdot \mathbb{K}_S^{-1} \underline{Z}}
\hat{Z}_S(\underline{x}) = \textcolor{darkred}{\underline{k}_S(\underline{x})} \textcolor{gray}{\cdot \mathbb{K}_S^{-1} \underline{Z}_T}
\hat{Z}_S(\underline{x}) = \underline{k}_S(\underline{x}) \cdot \mathbb{K}_S^{-1} \textcolor{darkred}{\underline{Z}_T}
\hat{Z}_S(\underline{x}) = \underline{k}_S(\underline{x}) \cdot \mathbb{K}_S^{-1} \underline{Z}_T

kernel overlap

Gram matrix

training data

Explicit solution:


\(\hat{Z}_S(\underline{x}) = \sum_{\mu=1}^n c_\mu K_S(\underline{x}_\mu,\underline{x})\)

Minimize \(= \frac1n \sum_{\mu=1}^n \left[ \hat{Z}_S(\underline{x}_\mu) - Z_T(\underline{x}_\mu) \right]^2\)

  teacher-student: theorem (1/2)

To compute the generalization error:

  • We look at the problem in the frequency domain
  • We assume that \(\tilde{K}_S(\underline{w}) \sim |\!|\underline{w}|\!|^{-\alpha_S}\) and \(\tilde{K}_T(\underline{w}) \sim |\!|\underline{w}|\!|^{-\alpha_T}\) as\(|\!|\underline{w}|\!|\to\infty\)

  • SIMPLIFYING ASSUMPTION: We take the \(n\) points \(\underline{x}_\mu\) on a regular \(d\)-dim lattice!
\epsilon \sim n^{-\beta}
\beta=\frac1d \min(\alpha_T - d, 2\alpha_S)

Then we can show that


E.g. Laplace has \(\alpha=d+1\) and Gaussian has \(\alpha=\infty\)

(details: arXiv:1905.10843) 

for \(n\gg1\)

  teacher-student: theorem (2/2)

  • Large \(\alpha \rightarrow\) fast decay at high freq \(\rightarrow\) indifference to local details

  • \(\alpha_T\) is intrinsic to the data (T), \(\alpha_S\) depends on the algorithm (S)

  • If \(\alpha_S\) is large enough, \(\beta\)  takes the largest possible value \(\frac{\alpha_T - d}{d}\)

  • As soon as \(\alpha_S\) is small enough, \(\beta=\frac{2\alpha_S}d\)

(optimal learning)

\beta=\frac1d \min(\alpha_T - d, 2\alpha_S)
  • If Teacher=Student=Laplace

  • If Teacher=Gaussian, Student=Laplace
\beta=\frac1d \min(\alpha_T - d, 2\alpha_S)

What is the prediction for our simulations?

(curse of dimensionality!)

\beta=\frac{\alpha_T-d}d = \frac1d


(\(\alpha_T=\infty, \alpha_S=d+1\))

\beta=\frac{2\alpha_S}d = 2+\frac2d

  teacher-student: comparison (1/2)

Exponent \(-\beta\)

  • Our result matches the numerical simulations
  • There are finite size effects (small \(n\))

(on hypersphere)


Same result with points on regular lattice or random hypersphere?


What matters is how nearest-neighbor distance \(\delta\) scales with \(n\)

  nearest-neighbor distance

In both cases  \(\delta\sim n^{\frac1d}\)

Finite size effects: asymptotic scaling only when \(n\) is large enough



  • For Gaussian data, \(\alpha_T-d \equiv 2s\) is a measure of smoothness \(\sim\) # of continuous derivatives

  • Can we say something about real data?
\beta \approx \frac{\text{smoothness}\ \ \textcolor{darkred}{\alpha_T-d = 2s}}{\text{dimension}\ \ \textcolor{darkred}{d}}
\textrm{(optimal)}\ \ \beta=\frac{\alpha_T - d}d

  what about real data?

1. Effective dimension is much smaller:

\beta=\frac1{d_\mathrm{eff}} \min(\alpha_T - d_\mathrm{eff}, 2\alpha_S) \quad \Longrightarrow \quad \beta=\frac{\alpha_T - d_\mathrm{eff}}{d_\mathrm{eff}}

\(\delta\sim n^{\frac1{d_\mathrm{eff}}}\)

2. We find the same exponent regardless of the student:

d_\mathrm{eff}^\mathrm{MNIST} \approx 15, \quad d_\mathrm{eff}^\mathrm{CIFAR10} \approx 35

Assuming this formula holds

  kernel pca

  • \(\mathbb{K}_S\) is the Gram matrix,    \(\lambda_1\geq\lambda_2\geq\dots\) are its eigenvalues,
    \((\underline{\phi}_\rho)_{\rho\geq1}\) are its eigenvectors

  • Given a Teacher Gaussian process \(\underline{Z}_T\), we can project it on this basis to compute

  • \(q_\rho\) is a Gaussian variable, with
q_\rho \equiv \underline{Z}_T\cdot\underline{\phi}_\rho
\mathbb{E} q_\rho = 0, \quad \mathbb{E} q_\rho^2 \sim \rho^{-\frac{\alpha_T}d}

Guess: measure \(\alpha_T\) in real data from this projetion!

\(\frac{\alpha_T}d = 1 + \frac{2s}d\)                    

\(= 1+\frac1d\,\) for Laplace

\(=1\) for Gaussian     

  projection of real data

q_\rho \equiv \textcolor{red}{\underline{y}}\cdot\underline{\phi}_\rho \quad \longrightarrow \quad \textrm{plot} \ q_\rho^2 \sim \rho^{-\textcolor{red}{c}}

Measure effective smoothness in real data

Fit \(c=\frac{\alpha_T}d\) from the projection


  exponent of real data (1/3)

  • We can then try and predict the exponent \(\beta\) and the smoothness!
  • Smoothness \(2s = \alpha_T-d_\mathrm{eff} = d_\mathrm{eff}(c-1)\)
  • Exponent \(\beta=\frac{\alpha_T-d_\mathrm{eff}}{d_\mathrm{eff}} = c-1\)

\(\beta\approx0.36 \ \ \ \ 2s\approx5.4\)

\(\beta\approx0.07 \ \ \ \ 2s\approx2.45\)

  exponent of real data (2/3)

  • Bordelon, et al 2020 derived an approximate formula for the test error

  • For large \(n\), kernel regression learns only the largest \(n\) modes. Error comes from the remaining modes:

\epsilon \approx \sum_{\rho\geq n} q_\rho^2 \sim \sum_{\rho\geq n} \rho^{-c} \sim n^{-\textcolor{red}{(c-1)}}

  exponent of real data (3/3)

\epsilon \approx \sum_{\rho\geq n} q_\rho^2

eigenmodes are extracted from a Gram matrix with a larger training set of size \(\tilde{n}\)


  • Learning curves of real data decay as power laws with exponents

  • We justify how different kernels can lead to the same exponent \(\beta\)

  • We link \(\beta\) to the smoothness and dimension of the Gaussian data

  • Real data live in manifolds of small effective dimension and we can define an effective smoothness that correlates with \(\beta\)

  • Open question: what fixes the smoothness in real data?
\frac1d \ll \beta < \frac12

Learning curves of kernel methods

By Stefano Spigler

Learning curves of kernel methods

Group meeting, July 2020

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