Loss Landscape and

Performance in Deep Learning

M. Geiger, A. Jacot, S. d’Ascoli,  M. Baity-Jesi,

L. Sagun, G. Biroli, C. Hongler, M. Wyart

Stefano Spigler

arXivs:  1901.01608;  1810.09665;  1809.09349

(Supervised) Deep Learning

\(<-1\)

\(>+1\)

  • Learning from examples: train set
     
  • Is able to predict: test set
     
  • Not understood why it works so well!

\(f(\mathbf{x})\)

  • How many data are needed to learn?
     
  • What network size?

Set-up: Architecture

  • Deep net \(f(\mathbf{x};\mathbf{W})\) with \(\textcolor{red}{N}\sim h^2L\) parameters

depth \(L\)

width \(\color{red}h\)

\(f(\mathbf{x};\mathbf{W})\)

  • Alternating linear and nonlinear operations!

\(W_\mu\)

Set-up: Dataset

  • \(\color{red}P\) training data:
                                                                          \(\mathbf{x}_1, \dots, \mathbf{x}_P\)

     
  • Binary classification:
                                                                          \(\mathbf{x}_i \to \mathrm{label}\ y_i = \pm1\)

     
  • Independent test set to evaluate performance




    Example - MNIST (parity):

70k pictures, digits \(0,\dots,9\);

use parity as label

\(\pm1=\) cats/dogs, yes/no, even/odd...

Outline

Vary network size \(\color{red}N\) (\(\sim\color{red}h^2\)):


 

  1. Can networks fit all the \(\color{red}P\) training data?


     
  2. Can networks overfit? Can \(\color{red}N\) be too large?



    \(\to\)  Long term goal: how to choose \(\color{red}N\)?

\(h\)

Learning

  • Find parameters  \(\mathbf{W}\)  such that  \(\mathrm{sign} f(\mathbf{x}_i; \mathbf{W}) = y_i\)  for \(i\in\) train set




     

  • Minimize some loss!






     

  • \(\mathcal{L}(\mathbf{W}) = 0\) if and only if \(y_i f(\mathbf{x}_i;\mathbf{W}) > 1\) for all patterns

\displaystyle \mathcal{L}(\mathbf{W}) = \sum_{i=1}^P {\color{red}\ell\left( {\color{black}y_i f(\mathbf{x}_i;\mathbf{W})} \right)}

(classified correctly with some margin)

Binary classification: \(y_i = \pm1\)

Hinge loss:

Learning dynamics = descent in loss landscape

  • Minimize loss    \(\longleftrightarrow\)    gradient descent
     

  • Start with random initial conditions!
     


  •  

Random, high dimensional, not convex landscape!

  • Why not stuck in bad local minima?
     
  • What is the landscape geometry?


     
  • Many flat directions are found!

bad local minimum?

Soudry, Hoffer '17; Sagun et al. '17; Cooper '18; Baity-Jesy et al. '18 - arXiv:1803.06969

in practical settings:

Analogy with granular matter: Jamming

Upon increasing density  \(\to\)  transition

sharp transition with finite-range interactions

  • random initial conditions
     
  • minimize energy \(\mathcal{L}\)
     
  • either find \(\mathcal{L}=0\) or \(\mathcal{L}>0\)

Random packing:

this is why we use the hinge loss!

Shallow networks \(\longleftrightarrow\) packings of spheres: Franz and Parisi, '16

 

Deep nets \(\longleftrightarrow\) packings of ellipsoids!

(if signals propagate through the net)


\(\color{red}N^\star < c_0 P\)

typically \(c_0=\mathcal{O}(1)\)

Theoretical results: Phase diagram

  • When \(N\) is large, \(\mathcal{L}=0\)
     
  • Transition at \(N^\star\)

\(\color{red}N^\star < c_0 P\)

\(\color{red}N^\star\)

network size

dataset size

Empirical tests: MNIST (parity)

Geiger et al. '18 - arXiv:1809.09349;

Spigler et al. '18 - arXiv:1810.09665

  • Above \(\color{red}N^*\) we have \(\mathcal{L}=0\)


     
  • Solid line is the bound \(\color{red}N^* < c_0 P\)

No local minima are found when overparametrized!

\(P\)

\(N\)

dataset size

network size

\(\color{red}N^\star < c_0 P\)

Landscape curvature

Spectrum of the Hessian (eigenvalues)

We don't find local minima when overparametrized...                                                                                                                    ...shape of the landscape?

Geiger et al. '18 - arXiv:1809.09349

Local curvature:

second order approximation

Information captured by Hessian matrix:        \(\mathcal{H}_{\mu\nu} = \frac{\partial^2}{\partial_{\mathbf{W}_\mu}\partial_{\mathbf{W}_\nu}} \mathcal{L}(\mathbf{W})\)

w.r.t parameters \(W\)

Flat directions

Spectrum

Over-parametrized

Jamming

Under-parametrized

\(\sim\sqrt{\mathcal{L}}\)

\(\leftrightarrow\)

eigenvalues

eigenvalues

eigenvalues

\(\mathcal{L} = 0\)

Flat

Geiger et al. '18 - arXiv:1809.09349

Almost flat

\(N>N^\star\)

\(N\approx N^\star\)

\(N<N^\star\)

Spectrum

Spectrum

From numerical simulations:

(at the transition)

Dirac \(\delta\)'s

\left.\phantom{\int}\right\}

depth

Outline

Vary network size \(\color{red}N\) (\(\sim\color{red}h^2\)):


 

  1. Can networks fit all the \(\color{red}P\) training data?


     
  2. Can networks overfit? Can \(\color{red}N\) be too large?



    \(\to\)  Long term goal: how to choose \(\color{red}N\)?

\(h\)

Yes, deep networks fit all data if \(N>N^*\ \longrightarrow\)   jamming transition

Generalization

Spigler et al. '18 - arXiv:1810.09665

Ok, so just crank up \(N\) and fit everything?

 

Generalization?  \(\to\)  Compute test error \(\epsilon\)

But wait... what about overfitting?

overfitting

\(N\)

\(N^*\)

Test error \(\epsilon\)

Train error

example: polynomial fitting

\(N \sim \mathrm{polynomial\ degree}\)

Overfitting?

Spigler et al. '18 - arXiv:1810.09665

  • Test error decreases monotonically with \(N\)!

     
  • Cusp at the jamming transition

Advani and Saxe '17;

Spigler et al. '18 - arXiv:1810.09665;

Geiger et al. '19 - arXiv:1901.01608

"Double descent"

test error

\(N\)

\(N/N^*\)

(after the peak)

\(P\)

\(N\)

dataset size

network size

We know why: Fluctuations!

Ensemble average

  • Random initialization  \(\to\)  output function \(f_{\color{red}N}\) is stochastic

     
  • Fluctuations: quantified by  average  and  variance

ensemble average over \(n\) instances:

\bar f^n_N(\mathbf{x}) \equiv \frac1n \sum_{\alpha=1}^n f_N(\mathbf{x}; \mathbf{W}_\alpha)

\(\phantom{x}\)

\(f_N(\mathbf{W}_1)\)

\(f_N(\mathbf{W}_2)\)

\(f_N(\mathbf{W}_3)\)

\(-1\)

\(-1\)

\(+1\)

\overbrace{\phantom{wwwwwwwwwwwwwwwwi}}

\(\bar f_N\)

\(-1!\)

\(\frac{{\color{red}-1-1}{\color{blue}+1}}{3}\cdots\)

|\!|f_N - \bar f^n_N|\!|^2 \sim N^{-\frac12}

Explained in a few slides

Define some norm over the output functions:

ensemble variance (fixed \(n\)):

\(\phantom{x}\)

Fluctuations increase error

\( \{f(\mathbf{x};\mathbf{W}_\alpha)\} \to \left\langle\epsilon_N\right\rangle\)

Remark:

Geiger et al. '19 - arXiv:1901.01608

  • Test error increases with fluctuations


     
  • Ensemble test error is nearly flat after \(N^*\)!

\(\bar f^n_N(\mathbf{x}) \to \bar\epsilon_N\)

test error of ensemble average

average test error

\(\neq\)

normal average

ensemble average

test error \(\epsilon\)

test error \(\epsilon\)

(CIFAR-10 \(\to\) regrouped in 2 classes)

(MNIST parity)

Scaling argument!

Geiger et al. '19 - arXiv:1901.01608

decision boundaries:

Smoothness of test error as function of decision boundary  +  symmetry:

\left\langle\epsilon_N\right\rangle - \bar\epsilon_N \sim |\!|f_N - \bar f_N|\!|^2 \sim N^{-\frac12}

normal average

ensemble average

Infinitely-wide networks: Initialization

Neal '96; Williams '98; Lee et al '18; Schoenholz et al. '16

  • Weights:  each initialized as \(W_\mu\sim {\color{red}h}^{-\frac12}\mathcal{N}(0,1)\)

     
  • Neurons sum \(\color{red}h\) signals of order \({\color{red}h}^{-\frac12}\)  \(\longrightarrow\)  Central Limit Theorem

     
  • Output function becomes a Gaussian Random Field

width \(\color{red}h\)

\(f(\mathbf{x};\mathbf{W})\)

input dim \(\color{red}d\)

\(W^{(1)}\sim d^{-\frac12}\mathcal{N}(0,1)\)

as \(h\to\infty\)

\(W_\mu\)

Infinitely-wide networks: Learning

 Jacot et al. '18

  • For small width \(h\): \(\nabla_{\mathbf{W}}f\) evolves during training
     
  • For large width hh\(h\): \(\nabla_{\mathbf{W}}f\) is constant during training

For an input x\mathbf{x} the function f(x;W)f(\mathbf{x};\mathbf{W}) lives on a curved manifold

The manifold becomes linear!

      Lazy learning:

  • weights don't change much:
     
  • enough to change the output \(f\) by \(\sim \mathcal{O}(1)\)!\partial_{\mathbf{W}}
|\!|\mathbf{W}^t - \mathbf{W}^{t=0}|\!|^2 \sim \frac1h

Neural Tangent Kernel

  • Gradient descent implies:
\displaystyle \frac{\mathrm{d}}{\mathrm{d}t} f(\mathbf{x}{\color{gray};\mathbf{W}^t}) = \sum_{i=1}^P \ \Theta^t(\mathbf{x},\mathbf{x}_i) \ \ {\color{gray} y_i \ell^\prime(y_i f(\mathbf{x}_i;\mathbf{W}^t))}
\Theta^t(\mathbf{x},\mathbf{x}^\prime) = \nabla_{\mathbf{W}} f(\mathbf{x}{\color{gray};\mathbf{W}^t}) \cdot \nabla_{\mathbf{W}} f(\mathbf{x}^\prime{\color{gray};\mathbf{W}^t})

The formula for the kernel \(\Theta^t\) is useless, unless...

Theorem. (informal)

\lim_{\mathrm{width}\ h\to\infty} \Theta^t(\mathbf{x},\mathbf{x}^\prime) \equiv \Theta_\infty(\mathbf{x},\mathbf{x}^\prime)

Deep learning  \(=\)  learning with a kernel as \(h\to\infty\)

 Jacot et al. '18

\(\phantom{x}\)

\overbrace{\phantom{wwwwwwwwwwwwwwwwwwi}}

convolution with a kernel

\(\phantom{wwwwwwww}\)

Finite \(N\) asymptotics?

Geiger et al. '19 - arXiv:1901.01608;

Hanin and Nica '19;

Dyer and Gur-Ari '19

  • Evolution in time is small:


     
  • Fluctuations are much larger:

Then: 

|\!|f_N-\bar f_N|\!|^2 \sim \left(\Delta\Theta^{t=0}\right)^2 \sim N^{-\frac12}

The output function fluctuates similarly to the kernel

\(\Delta\Theta^{t=0} \sim 1/\sqrt{h} \sim N^{-\frac14}\)

at \(t=0\)

\(|\!|\Theta^t - \Theta^{t=0}|\!|_F \sim 1/h \sim N^{-\frac12}\)

\displaystyle f(\mathbf{x}{\color{gray};\mathbf{W}^t}) = \int \mathrm{d}t \sum_{i=1}^P \ \Theta^t(\mathbf{x},\mathbf{x}_i) \ \ {\color{gray} y_i \ell^\prime(y_i f(\mathbf{x}_i;\mathbf{W}^t))}

Conclusion

1. Can networks fit all the \(\color{red}P\) training data?

  • Yes, deep networks fit all data if \(N>N^*\ \longrightarrow\)   jamming transition



     
  • Initialization induces fluctuations in output that increase test error
     
  • No overfitting: error keeps decreasing past \(N^*\) because fluctuations diminish

check Geiger et al. '19 - arXiv:1906.08034 for more!

check Spigler et al. '19 - arXiv:1905.10843 !

2. Can networks overfit? Can \(\color{red}N\) be too large?

3. How does the test error scale with \(\color{red}P\)?

\(\to\)  Long term goal: how to choose \(\color{red}N\)?

(tentative)   Right after jamming, and do ensemble averaging!

Loss Landscape and Performance in Deep Learning

By Stefano Spigler

Loss Landscape and Performance in Deep Learning

Talk given in ICTS, Bangalore, January 2020

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