Benjamin Recht ICLR 2017 - Invited Talk - what can deep learning learn from linear regression?.

**Concept Graph & Resume using Claude 3 Opus | Chat GPT4 | Gemini Adv | Llama 3:**

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classDef theory fill:#d4f9d4, font-weight:bold, font-size:14px;
classDef optimization fill:#d4d4f9, font-weight:bold, font-size:14px;
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A[Benjamin Recht

ICLR 2017] --> B[Deep learning: remarkable

breakthroughs, capabilities. 1] A --> C[Trustable, predictable, safe

models needed. 2] A --> D[No consensus on optimization

difficulty causes. 3] A --> I[Population risk = training error

+ generalization gap. 8] A --> K[Bias-variance tradeoff: complexity reduces

bias, increases variance. 9] A --> S[Kernel trick allows exact

minimum norm computation. 16] D --> E[Good local minima have

mostly near-zero eigenvalues. 4] D --> F[Saddle points difficult to

converge to randomly. 5] D --> G[Zero training error easily

achieved without regularization. 6] G --> H[Test error divergence indicates overfitting. 7] I --> J[Zero training error doesn't

necessarily mean overfitting. 8] K --> L[Large unregularized nets can

fit arbitrary labels. 10] L --> M[Capacity is very high. 10] K --> N[Regularization, early stopping help

generalization, not necessary. 11] K --> O[Overfitting issue even for

simple overparameterized models. 12] O --> P[Infinitely many global minima

with zero eigenvalues. 13] S --> T[Kernel regression performs well

on MNIST, CIFAR-10. 17, 18] S --> Q[SGD converges to minimum

norm solution. 14] Q --> R[Minimum norm leverages useful

parameter structure/regularity. 15] K --> U[Shallow models 19] K --> V[Margin is inverse of

parameter norm. 20] V --> W[Minimum norm solutions change

little with perturbations. 21] K --> X[Theory bounds test error

in terms of margin. 22] K --> Y[Analyzing optimization without regularization

provides clarity. 23] K --> Z[Saddle points may not

be major problem. 24] K --> AA[Interpolation doesn't necessarily lead

to poor generalization. 25] C --> AB[Large margin classification promising

for deep learning generalization. 26] C --> AC[Algorithmic stability leads to

model stability, generalization. 27] C --> AD[Statistical learning theory provides

insight, demystifies observations. 28] C --> AE[Theoretical understanding enables safe,

reliable deployment. 29] C --> AF[Predictable, robust, trustworthy models

critical as impact grows. 30] class A,B deeplearning; class C,I,J,K,L,M,N,O,P,U,V,W,X,Y,Z,AA,AB,AC,AD,AE,AF theory; class D,E,F,G,H,Q,R optimization; class S,T kernels;

ICLR 2017] --> B[Deep learning: remarkable

breakthroughs, capabilities. 1] A --> C[Trustable, predictable, safe

models needed. 2] A --> D[No consensus on optimization

difficulty causes. 3] A --> I[Population risk = training error

+ generalization gap. 8] A --> K[Bias-variance tradeoff: complexity reduces

bias, increases variance. 9] A --> S[Kernel trick allows exact

minimum norm computation. 16] D --> E[Good local minima have

mostly near-zero eigenvalues. 4] D --> F[Saddle points difficult to

converge to randomly. 5] D --> G[Zero training error easily

achieved without regularization. 6] G --> H[Test error divergence indicates overfitting. 7] I --> J[Zero training error doesn't

necessarily mean overfitting. 8] K --> L[Large unregularized nets can

fit arbitrary labels. 10] L --> M[Capacity is very high. 10] K --> N[Regularization, early stopping help

generalization, not necessary. 11] K --> O[Overfitting issue even for

simple overparameterized models. 12] O --> P[Infinitely many global minima

with zero eigenvalues. 13] S --> T[Kernel regression performs well

on MNIST, CIFAR-10. 17, 18] S --> Q[SGD converges to minimum

norm solution. 14] Q --> R[Minimum norm leverages useful

parameter structure/regularity. 15] K --> U[Shallow models 19] K --> V[Margin is inverse of

parameter norm. 20] V --> W[Minimum norm solutions change

little with perturbations. 21] K --> X[Theory bounds test error

in terms of margin. 22] K --> Y[Analyzing optimization without regularization

provides clarity. 23] K --> Z[Saddle points may not

be major problem. 24] K --> AA[Interpolation doesn't necessarily lead

to poor generalization. 25] C --> AB[Large margin classification promising

for deep learning generalization. 26] C --> AC[Algorithmic stability leads to

model stability, generalization. 27] C --> AD[Statistical learning theory provides

insight, demystifies observations. 28] C --> AE[Theoretical understanding enables safe,

reliable deployment. 29] C --> AF[Predictable, robust, trustworthy models

critical as impact grows. 30] class A,B deeplearning; class C,I,J,K,L,M,N,O,P,U,V,W,X,Y,Z,AA,AB,AC,AD,AE,AF theory; class D,E,F,G,H,Q,R optimization; class S,T kernels;

**Resume: **

**1.-**Deep learning has made remarkable breakthroughs in the last 5 years, enabling capabilities far beyond what was envisioned in 2011.

**2.-**To push deep learning into products affecting lives, we need a theoretical understanding to ensure models are trustable, predictable and behave safely.

**3.-**There's no consensus on what makes optimizing deep models hard. Hypotheses include difficulty finding global minima as network size increases.

**4.-**Another hypothesis is that good local minima have mostly near-zero eigenvalues in the Hessian, with very few positive or negative ones.

**5.-**Conventional wisdom is that saddle points make convergence difficult, but they are actually very difficult to converge to if initialized randomly.

**6.-**Optimizing deep models isn't hard - zero training error can be easily achieved on MNIST and CIFAR-10 without regularization.

**7.-**Test error diverging from near-zero train error indicates overfitting. But classification error can plateau at a reasonable level without diverging.

**8.-**Fundamental theory of ML: Population risk = Training error + Generalization gap. Zero training error doesn't necessarily mean overfitting.

**9.-**Bias-variance tradeoff: Increased model complexity reduces bias but increases variance. Deep learning operates in the "high variance" regime.

**10.-**Deep models with vastly more parameters than data points can fit arbitrary label patterns, even random noise. Capacity is very high.

**11.-**Regularization, early stopping, data augmentation etc. help generalization but aren't necessary conditions. Large unregularized nets outperform regularized shallower nets.

**12.-**Overfitting is an issue even for simple models like linear regression when number of parameters exceeds number of data points.

**13.-**With more parameters than data points, there are infinitely many global minima, all with the same Hessian with many zero eigenvalues.

**14.-**SGD converges to the minimum norm solution out of all global minima for linear regression problems.

**15.-**Minimum norm is reasonable for generalization because it picks a solution that leverages useful structure/regularity in parameters.

**16.-**Kernel trick allows exact minimum norm solution to be computed for kernel regression on datasets like MNIST in a few minutes.

**17.-**Kernel regression with no regularization or preprocessing gets 1.2% test error on MNIST, 0.6% error with a wavelet transform.

**18.-**On CIFAR-10, kernel regression on top of random convolutional features gets 16% test error, or 14% with some regularization.

**19.-**Shallow models like kernel machines can do surprisingly well just by interpolating the training data, questioning the need for depth.

**20.-**For linear models, margin is the inverse of parameter norm. Maximum margin means being as far as possible from the data.

**21.-**Minimum norm (max margin) solutions change very little with small perturbations, corresponding to "flat" optima. Sharp optima change a lot.

**22.-**Some theory bounds test error in terms of margin. Challenge is getting practical margin bounds for deep nets.

**23.-**Regularization makes the connection between optimization and generalization unclear. Analyzing optimization without regularization can provide clarity.

**24.-**Saddle points may not actually be a major problem for optimization, despite being an active research focus recently.

**25.-**Interpolating the training data doesn't necessarily lead to overfitting and poor generalization.

**26.-**Large margin classification is a promising framework for thinking about generalization in deep learning.

**27.-**Algorithmic stability likely leads to model stability and generalization. Stable training algorithms tend to result in models that generalize.

**28.-**Well-established ideas in statistical learning theory can provide insight into deep learning and help demystify recent empirical observations.

**29.-**The theoretical community hopes that a better formal understanding of deep learning will enable it to be deployed safely and reliably.

**30.-**As deep learning increasingly impacts all of society, it is critical that the models be predictable, robust and trustworthy.

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