Gatsby Computational Neuroscience Unit, University College London

Based on
“Demystifying MMD GANs”
[ICLR-18]

and
“On gradient regularizers for MMD GANs”
[arXiv:1805.11565],
with:

Sheffield Machine Learning seminar, 30 May 2018

(Swipe or arrow keys to move in slides; for a menu to jump; to show more.)

Given samples from distribution over

Want model that can produce new samples from

Don't necessarily care about likelihoods, interpretability, …

Deep network (params ) mapping from noise to images

DCGAN generator [Radford+ ICLR-16]

is e.g. uniform on

Choose by minimizing…something

- Can't evaluate likelihood of samples under model
- Likelihood maybe not the best choice anyway [Theis+ ICLR-16]
- Doesn't hurt likelihoods much to take 99% white noise

- Instead, we'll minimize some

Max-likelihood objective vs WGAN objective [Danihelka+ 2017]

( is RKHS with kernel )

Can do optimization in closed form:

0.6 | 0.5 | 0.2 | 0.4 | 0.2 | ||

0.6 | 0.7 | 0.4 | 0.1 | 0.1 | ||

0.5 | 0.7 | 0.3 | 0.1 | 0.2 | ||

0.2 | 0.4 | 0.3 | 0.7 | 0.8 | ||

0.4 | 0.1 | 0.1 | 0.7 | 0.6 | ||

0.2 | 0.1 | 0.2 | 0.8 | 0.6 |

Estimate generator based on SGD
with minibatches

, ,
loss

Hard to pick a good kernel for images

- Use a class of
*deep kernels*: - Choose most-discriminative out of those kernels:
- Initialize random generator and representation
- Repeat:
SGD step in to minimize
- times:
- Take SGD step in to maximize

- Take SGD step in to minimize

- times:

- Repeat:
SGD step in to minimize

- WGANs [Arjovsky+ ICML-17],
WGAN-GPs [Gulrajani+ NIPS-17]:
- Train a neural network for critic
- Enforce Lipschitz constraint on (more on this later)
- Run SGD on minibatches , with that critic

- Train a neural network for critic

- Consider linear-kernel MMD GAN, :
- WGAN has:
- Linear-kernel MMD GAN and WGAN almost the same
- MMD GAN “offloads” some of the critic's work to closed-form optimization in the RKHS

- Bellemare+ [2017] say that:
- WGANs have biased gradients
- which can lead SGD to wrong minimum even in expectation
- but Cramér GANs have unbiased gradients
- Cramér GAN MMD GAN with particular kernel

- We show:
- Gradients of
*fixed*critic, , are unbiased - Gradients of optimized critic, , are biased
*Exact same situation*for WGAN and MMD GAN

- Gradients of

- For almost all feedforward architectures ,
- Works for ReLU, max-pooling, …

- For any distributions , with ,
- For most kernels used in practice,
- Includes linear kernel, RBF, RQ, distance kernel, …

- For Lebesgue-almost all parameters :
- , so:
- , unbiased (WGANs)
- unbiased

- Can't use standard argument: needs differentiable in a neighborhood of for almost all inputs
- But take
- Fixed : differentiable everywhere but

- might have probability

- Instead: Show gradient exchanges when differentiable in for almost all inputs, using Lipschitz properties of the network
- Then show the set of non-differentiable has 0 measure, with a bit of geometry
- Example: for , is a critical point:

- By Fubini theorem, only need to show:
- For fixed input , differentiable for almost all

- Recall: , with piecewise smooth :

- case 1: inside domain of analyticity

- case 2: on boundary

- case 3: crosses the boundary:
- in a union of manifolds of 0 measure

- Recall
- Estimator splits data:
- Pick on train set, estimate on test set
- GAN test set: current minibatch

- Estimator splits data:
- Showed: unbiased for any fixed
- Now: is biased
- Estimators have non-constant bias iff gradients are biased
- Will show is biased

- Eval on test set is unbiased:
- But training introduces bias:
- If , estimator must be biased down

- Probably not a big deal in practice
- No (direct) bias due to minibatch size
- Can decrease bias by training critic longer

- But informs theory
- Convergence based on SGD of made difficult

- Beautiful argument of Bickel & Lehman [Ann. Math. Stat 1969]:
- Let ,
- Suppose ; then
- But isn't a polynomial
- So no unbiased can exist, though could

- Toy problem in [Mescheder+ ICML-18]:
- Point mass target , model
- Representation ,
- Gaussian kernel

- Toy problem in [Mescheder+ ICML-18]:
- Point mass target , model
- Representation ,
- Gaussian kernel

- Taking gives
- But if we restrain ourselves to where optimal critic is bounded Lipschitz, implies , and is continuous and a.e. differentiable

- First attempt: only optimize over that are -Lipschitz
- Hard to specify Lipschitz constant of deep nets
- WGAN [Arjovsky+ ICML-17] tried with simple box constraint
- Also original MMD GAN paper [Li+ NIPS-17]

- WGAN-GP [Gulrajani+ NIPS-17]:
penalize non-Lipschitzness
(with drawn in between the and minibatches)
- Looser constraint than Lipschitz
- Tends to work better in practice
- But barely changes anything on the toy problem!

- Remember
- New Gradient-Constrained MMD:
- Variance constraint makes it like a Sobolev norm
- Doesn't quite constrain the Lipschitz constant

- Say we have samples
- Let
- Then with has , has , has
- Dropping kernel matrix gives
- Solving this linear system takes time!

- It's a reasonable distance to optimize:
- …but this took days to run
- even with a low-rank incomplete-Cholesky approximation

- Using a bit of RKHS theory, can write
- Define lower bound on of

- When ,

MMD-GP

SMMD

- Optimization failure we sometimes see on SMMD, GC-MMD:
- Generator doing reasonably well
- Critic filters become low-rank
- Generator corrects it by breaking everything else
- Generator gets stuck

- ; learn and freely
- Encourages diversity without limiting representation

- , so that ,
- Works well for original GANs [Miyato+ ICLR-18]
- …but doesn't work at all as only constraint in a WGAN
- Limits representation too much
- In the toy problem, constrains to

- No likelihoods, so…how to compare models?
- Main approach:

look at a bunch of pictures and see if they're pretty or not- Easy to find (really) bad samples
- Hard to see if modes are missing / have wrong probabilities
- Hard to compare models beyond certain threshold

- Need better, quantitative methods

- Current standard quantitative method
- Based on ImageNet classifier label predictions
- Classifier should be confident on individual images
- Predicted labels should be diverse across sample

- No notion of target distribution
- Scores completely meaningless on LSUN, Celeb-A, SVHN, …
- Not great on CIFAR-10 either

- Fit normals to Inception hidden layer activations of and
- Compute Fréchet (Wasserstein-2) distance between fits
- Meaningful on not-ImageNet datasets
- Estimator extremely biased, tiny variance
- ,

- between Inception hidden layer activations
- Use default polynomial kernel:
- Unbiased estimator: more able to compare estimates
- Reasonable estimates with fewer samples

- Models need appropriate learning rate schedule to work well
- Automate with three-sample MMD test [Bounliphone+ ICLR-16]:

CIFAR-10 | Small | Big |
---|---|---|

WGAN-GP | .116 ± .002 | .026 ± .001 |

MMD GAN-GP | .032 ± .001 | .027 ± .001 |

LSUN Bedrooms | Small | Big |
---|---|---|

WGAN-GP | .370 ± .003 | .039 ± .002 |

MMD GAN-GP | .091 ± .002 | .028 ± .002 |

SN-SMMD GAN

KID: 0.006

KID: 0.006

WGAN-GP

KID: 0.022

KID: 0.022

SN-SMMDGAN

KID: 0.035

KID: 0.035

SN-GAN

KID: 0.044

KID: 0.044

BGAN

KID: 0.047

KID: 0.047

**Demystifying MMD GANs**
[ICLR-18]

Mikołaj Bińkowski^{*},
__Dougal J. Sutherland__^{*},
Michael Arbel,
Arthur Gretton

github.com/mbinkowski/MMD-GAN

- MMD GANs do some optimization in closed form
- Can handle smaller critic networks

- Bias situation is the same between WGAN and MMD GAN
- Evaluation and learning rate adaptation with KID

**On gradient regularizers for MMD GANs**
[arXiv:1805.11565]

Michael Arbel,
__Dougal J. Sutherland__,
Mikołaj Bińkowski,
Arthur Gretton

github.com/MichaelArbel/Scaled-MMD-GAN

- Gradient control is important
- Scaled MMD does it in closed form, seems to help a lot

- Spectral normalization plays nice with SMMD