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 at Emory, June 2020

  • 231
Loading comments...

More from Stefano Spigler