dl theory view markdown
Deep learning theory is a complex emerging field  this post contains links displaying some different interesting research directions
theoretical studies
DNNs display many surprising properties
 surprising: more parameters yields better generalization
 surprising: lowering training error should be harder
Many things seem to contribute to the inductive bias of DNNs: SGD, dropout, early stopping, resnets, convolution, more layers…all of these are tangled together and many things correlate with generalization error…what are the important things and how do they contribute?
some more concrete questions:
 what is happening when training err stops going down but val err keeps going down (interpolation regime)?
 what are good statistical markers of an effectively trained DNN?
 how far apart are 2 nets?
 verification  does dnn satisfy a specification (e.g $l_p$ robustness)
 convex barrier  limitation in the tightness of the bounds obtainable by the convex relaxation of the output of a neural network
functional approximation
 dnns are very hard to study in the parameter space (e.g. swapping two parameters changes things like the Hessian), easier to to study in the function space (e.g. the inputoutput relationship)
 nonlinear approximation (e.g. sparse coding)  2 steps
 construct a dictionary function (T)
 learn linear combination of the dictionary elements (g)
 background

$L^2$ function (or function space) is square integrable: $ f ^2 = \int_X f ^2 d\mu$, and $ f $ is its $L_2$norm  Hilbert space  vector space w/ additional structure of inner product which allows length + angle to be measured
 complete  there are enough limits in the space to allow calculus techniques (is a complete metric space)

 composition allows an approximation of a function through level sets (split it up and approximate on these sets)  Zuowei Shen (talk, slides)
 composition operation allows an approximation of a function f through level sets of f –
 one divides up the range of f into equal intervals and approximate the functions on these sets
 nonlinear approximation via compositions (shen 2019)
 how do the weights in each layer help this approximation to be more effective?
 here are some thoughts – If other layers are like the first layer, the weights “whiten” or make the inputs more independent or random projections – that is basically finding PC directions for lowrank inputs.
 are the outputs from later layers more or less low  rank?
 I wonder how this “whitening” helps level set estimation…
 takagi functions
 nonlinear approximation and (deep) relu nets  also comes with slides
inductive bias=implicit regularization: gradient descent finds good minima
DL learns solutions that generalize even though it can find many which don’t due to its inductive bias.
 gd bousquet paper
 in high dims, local minima are usually saddles (ganguli)
 early stopping is very similar to ridge regression
 ridge regression: soln will lie in $pdim$ rowspace of X (span of the rows)
 when $\lambda \to 0$, will give us minnorm soln (basically because we project onto $col(X)$)
 early stopping in least squares
 if we initialize at 0, GD soln will always be in the rowspace of X
 GD will converge to minnorm soln (any soln not in the rowspace will necessarily have larger norm)
 similar to ridge (endpoints are the same) and can related their risks very closely when $\lambda = 1/t$, where $t$ is GD time iterate
 assume gradient flow: take stepsize to 0
 srebro understanding overparameterization
 ex. gunasekar et al 2017: unconstrained matrix completion
 grad descent on U, V yields min nuclear norm solution
 ex. soudry et al 2017
 sgd on logistic reg. gives hard margin svm
 deep linear net gives the same thing  doesn’t actually changed anything
 ex. gunaskar, 2018
 linear convnets give smth better  minimum l1 norm in discrete fourier transform
 ex. savarese 2019
 infinite width relu net 1d input
 weight decay minimization minimizes derivative of TV
 ex. gunasekar et al 2017: unconstrained matrix completion
 implicit bias towards simpler models
 How do infinite width bounded norm networks look in function space? (savarese…srebro 2019)
 minimal norm fit for a sample is given by a linear spline interpolation (2 layer net)
 analytic theory of generalization + transfer (ganguli 19)
 deep linear nets learn important structure of the data first (less noisy eigenvectors)
 similar paper for layer nets
 datasets for measuring causality
 Inferring Hidden Statuses and Actions in Video by Causal Reasoning  about finding causality in the video, not interpretation
 PL condition = PolyakLojawsiewicz condition guarantees global convergence of loca methods

$ \nabla f(x) ^2 \geq \alpha f(x) \geq 0$

semantic biases: what correlations will a net learn?
 imagenet models are biased towards texture (and removing texture makes them more robust)
 analyzing semantic robustness
 eval w/ simulations (reviewer argued against this)
 glcm captures superficial statistics
 deeper, unpruned networks are better against noise
 analytic theory of generalization + transfer (ganguli 19)

causality in dnns talk by bottou
 on mnist, color vs shape will learn color
 A mathematical theory of semantic development in deep neural networks
 Emergence of Invariance and Disentanglement in Deep Representations (achille & soatto 2018)
 information in the weights as a measure of complexity of a learned model (information complexity)
 IB Lagrangian between the weights of a network and the training data, as opposed to the traditional one between the activations and the test datum
 explains tradeoff between over/underfitting

On Dropout, Overfitting, and Interaction Effects in Deep Neural Networks (lengerich..caruana, 2020)
 use ANOVA to meaure 1st/2nd/3rd order effects and such
 approximate ANOVA decomp. using boosted trees of depth based on a particular order
 use ANOVA to meaure 1st/2nd/3rd order effects and such
 they find that increasing dropout rate forces nets to emphasize lowerorder effects
 An Investigation of Why Overparameterization Exacerbates Spurious Correlations
 overparameterization can hurt test error on minority groups despite improving average test error when there are spurious correlations in the data
expressiveness: what can a dnn represent?
 complexity of linear regions in deep networks
 Bounding and Counting Linear Regions of Deep Neural Networks
 On the Expressive Power of Deep Neural Networks
 Deep ReLU Networks Have Surprisingly Few Activation Patterns
complexity + generalization: dnns are lowrank / redundant parameters
 measuring complexity
 functional decomposition (molnar 2019)
 decompose function into bias + firstorder effects (ALE) + interactions
 3 things: number of features used, interaction strength, main effect complexity
 functional decomposition (molnar 2019)
 parameters are redundant
 predicting params: weight matrices are lowrank, decompose into UV by picking a U
 pruning neurons
 circulant projection
 rethinking the value of pruning: pruning and training from scratch, upto 30% size
 Lottery ticket: pruning and training from initial random weights, upto 1% size (followup)
 rank of relu activations
 random weights are good
 singular values of conv layers
 TNet: Parametrizing Fully Convolutional Nets with a Single HighOrder Tensor
 generalization
nearest neighbor comparisons
 weighted interpolating nearest neighbors can generalize well (belkin…mitra 2018)
 Statistical Optimality of Interpolated Nearest Neighbor Algorithms
 nearest neighbor comparison
 nearest embedding neighbors
kernels
 To understand deep learning we need to understand kernel learning  overfitted kernel classifiers can still fit the data well
 original kernels (neal 1994) + (lee et al. 2018) + (matthews et al. 2018)
 infinitely wide nets and only top layer is trained
 corresponds to kernel $\text{ker}(x, x’) = \mathbb E_{\theta \sim W}[f(\theta, x) \cdot f(\theta, x’)]$, where $W$ is an intialization distr. over $\theta$
 neural tangent kernel (jacot et al. 2018)
 $\text{ker}(x, x’) = \mathbb E_{\theta \sim W} \left[\left < \frac{f(\theta, x)}{\partial \theta} \cdot \frac{f(\theta, x’)}{\partial \theta} \right> \right]$  evolution of weights over time follows this kernel
 with very large width, this kernel is the NTK at initialization
 stays stable during training (since weights don’t change much)
 at initialization, artificial neural networks (ANNs) are equivalent to Gaussian processes in the infinitewidth limit
 evolution of an ANN during training can also be described by a kernel (kernel gradient descent)
 different types of kernels impose different things on a function (e.g. want more / less low frequencies)
 gradient descent in kernel space can be convex if kernel is PD (even if nonconvex in the parameter space)
 understanding the neural tangent kernel (arora et al. 2019)
 method to compute the kernel quickly on a gpu
 $\text{ker}(x, x’) = \mathbb E_{\theta \sim W} \left[\left < \frac{f(\theta, x)}{\partial \theta} \cdot \frac{f(\theta, x’)}{\partial \theta} \right> \right]$  evolution of weights over time follows this kernel
 Scaling description of generalization with number of parameters in deep learning (geiger et al. 2019)
 number of params = N
 above 0 training err, larger number of params reduces variance but doesn’t actually help
 ensembling with smaller N fixes problem
 the improvement of generalization performance with N in this classification task originates from reduced variance of fN when N gets large, as recently observed for meansquare regression
 On the Inductive Bias of Neural Tangent Kernels (bietti & mairal 2019)
 Kernel and Deep Regimes in Overparametrized Models (Woodworth…Srebro 2019)
 transition between kernel and deep regimes
 The HSIC Bottleneck: Deep Learning without BackPropagation (Ma et al. 2019)
 directly optimize information bottleneck (approximated by HSIC) yields pretty good results
random projections
 sgd for polynomials
 deep linear better than just linear
 relation to bousquet  fitting random polynomials
 hierarchical sparse coding for images (can’t just repeat sparse coding, need to include input again)
implicit dl + optimization
 implicit deep learning (el ghaoui et al. 2019)
 $\hat y (u) = Cx + D u $, where $x = \phi(Ax + Bu)$
 here, $u$ is a new input
 $x$ is a hidden state which represents some hidden features (which depends on $u$)
 wellposedness  want x to be unique for a given u
 $A, B$ are matrices which let us compute $x$ given $u$
 $C, D$ help us do the final prediction (like the final linear layer)
 ex. feedforward nets
 consider net with $L$ layers
 $x_0 = u$
 $x_{l + 1} = \phi_l (W_l x_l)$
 $\hat y (u) = W_L x_L$
 rewriting in implicit form
 $x = (x_L, …, x_1)$  concatenate all the activations into one big vector
 ex. $Ax + Bu= \begin{bmatrix} W_{L1}x_{L1} \ W_{L2} x_{L2} \ \vdots \ W_1x_1 \ \mathbf 0\end{bmatrix} + \begin{bmatrix} 0 \ 0 \ \vdots \ 0 \ W_0 u \end{bmatrix}$
 consider net with $L$ layers
 $\hat y (u) = Cx + D u $, where $x = \phi(Ax + Bu)$
 lifted neural networks (askari et al. 2018)
 can solve dnn $\hat y = \phi(W_2 \phi (W_1X_0))$ by rewriting using constraints:
 $X_1 = \phi(W_1 X_0)$
 $X_2 = \phi(W_2 X_1)$
 $\begin{align} &\min (y  \hat y)^2\s.t. X_1 &= \phi(WX_0)\X_2 &= \phi(WX_1)\end{align}$
 can be written using Lagrangian multipliers: $\min (y  \hat y)^2 + \lambda_1( X_1  \phi(WX_0)) + \lambda_2(X_2  \phi(WX_1))$
 can solve dnn $\hat y = \phi(W_2 \phi (W_1X_0))$ by rewriting using constraints:
 Fenchel Lifted Networks: A Lagrange Relaxation of Neural Network Training (gu et al. 2018)
 in the lifted setting above, can replace Lagrangian with simpler expression using Fenchel conjugates
 robust optimization basics
 immunize optimization problems against uncertainty in the data
 do so by having worstcase constraints (e.g. $a < 5, \forall a$)
 local robustness  maximize radius surrounding parameter subject to all constraints (no objective to maximize)
 global robustness  maximize objective subject to robustness constraint (trades off robustness with objective value)
 nonprobabilistic robust optimization models (e.g. Wald’s maximin model: $\underset{x}{\max} \underset{u}{\min} f(x, u)$
 also are probabilistically robust optimization
 distributional robustness  using moments in the dl work
 ch 4 and ch10 of robust optimization book (bental, el ghaoui, & nemirovski 2009)
 Certifying Some Distributional Robustness with Principled Adversarial Training (sinha, namkoong, & duchi 2018)
 want to guarantee performance under adversarial input perturbations
 considering a Lagrangian penalty formulation of perturbing the underlying data distribution in a Wasserstein ball
 during training, augments model parameter updates with worstcase perturbations of training data
 little extra cost and achieves guarantees for smooth losses
 On Distributionally Robust ChanceConstrained Linear Programs (calafiore & el ghaoui 2006)
 linear programs where data (in the constraints) is random
 want to enforce the constraints up to a given prob. level
 can convert the prob. constraints into convex 2ndorder cone constraints
 under distrs. for the random data, can guarantee constraints
 Differentiable Convex Optimization Layers (agrawal et al. 2019)
statistical physics
 Statistical Mechanics of Deep Learning (bahri et al. 2019)
 what is the advantage of depth  connect to dynamical phase transitions
 there are several function which require only polynomial nodes in each layer for deep nets, but exponential for shallow nets
 what is the shape of the loss landscape  connect to random Gaussian processes, sping glasses, and jamming
 how to pick a good parameter initialization?
 bounding generalization error
 often, generalization bounds take the form $\epsilon_{test} \leq \epsilon_{train} + \frac {\mathcal C (\mathcal F)} p$, where $\mathcal C (\mathcal F)$ is the complexity of a function class and $p$ is the number of examples
 ex. VCdimension, Rademacher complexity
 alternative framework: algorithmic stability  will generalize if map is stable wrt perturbations of the data $\mathcal D$
 altenative: PAC bounds suggest if distr. of weights doesn’t change much during training, generalization will be succesful
 often, generalization bounds take the form $\epsilon_{test} \leq \epsilon_{train} + \frac {\mathcal C (\mathcal F)} p$, where $\mathcal C (\mathcal F)$ is the complexity of a function class and $p$ is the number of examples
 deep linear networks
 student learns biggest singular values of the inputoutput correlation matrix $\Sigma = \sum_i y_i x_i^T$, so it learns the important stuff first and the noise last
 infinitewidth limit
 if parameters are random, indcues a prior distribution $P(\mathcal F)$ over the space of functions
 in th limit of infinite width, this prior is Gaussian, with a specific correlation kernel

learning is similar to learning the Bayesian posterior $P(f data)$, but connecting this to sgd is still not clear
 what is the advantage of depth  connect to dynamical phase transitions
 Neural Mechanics: Symmetry and Broken Conservation Laws in Deep Learning Dynamics (kunin et al. 2020)
 no assumptions about DNN architecture or gradientflow
 instead, assumptions on symmetries embedded in a network’s architecture constrain training dynamics
 similar to Noether’s thm in physics
 can much better analytically describe learning dynamics
 weights have a differentiable symmetry in the loss if the loss doesn’t change under a certain differentiable transformation of the weights
 ex. translation symmetry  for layer before softmax, we get symmetries across weights that shift all outputs
 ex. scale symmetry  inputs to batch normalization are invariant to scaling
 ex. rescale symmetry  scaling up/down 2 things which multiply or add
 modeling discretization
empirical studies
interesting empirical papers
 modularity (“lottery ticket hypothesis”)
 contemporary experience is that it is difficult to train small architectures from scratch, which would similarly improve training performance  lottery ticket hypothesis: large networks that train successfully contain subnetworks that–when trained in isolation–converge in a comparable number of iterations to comparable accuracy
 ablation studies
 deep learning is robust to massive label noise
 are all layers created equal?
 Truth or backpropaganda? An empirical investigation of deep learning theory
adversarial + robustness
 robustness may be at odds with accuracy (madry 2019)
 adversarial training helps w/ little data but hurts with lots of data
 adversarially trained models have more meaningful gradients (and their adversarial examples actually look like other classes)
 Generalizability vs. Robustness: Adversarial Examples for Medical Imaging
 Towards Robust Interpretability with SelfExplaining Neural Networks
 Adversarial Attacks and Defenses in Images, Graphs and Text: A Review (xu et al. 2019)
 Adversarial examples are inputs to machine learning models that an attacker intentionally designed to cause the model to make mistakes
 threat models
 poisoning attack (insert fake samples into training data) vs. evasion attack (just evade at test time)
 targeted attack (want specific class) vs. nontargeted attack (just change the prediction)
 adversary’s knowledge
 whitebox  adversary knows everything
 blackbox  can only feed inputs and get outputs
 graybox  might have white box for limited amount of time
 security evaluation
 robustness  minimum norm perturbation to change class
 adversarial loss  biggest change in loss within some epsilon ball
 AugMix: A Simple Data Processing Method to Improve Robustness and Uncertainty (hendrycks et al. 2020)
 do a bunch of transformations and average images to create each training image
 certifying robustness
 bound gradients of f around x  solve SDP for 2layer net (raghunathan, steinhardt, & liang 2018, iclr)
 relax SDP and then applies to multilayer nets (raghunathan et al. 2018)
 improved optimizer for the SDP (dathathri et al. 2020)
 interval bound propagation  instead of passing point, pass an interval where it could be
 works well for word substitions (jia et al. 2019)
 RobEn  cluster words that are confusable + give them the same encoding when you train (jones et al. 2020)
 then you are robust to confusing the words
 bound gradients of f around x  solve SDP for 2layer net (raghunathan, steinhardt, & liang 2018, iclr)
 unlabeled data + selftraining helps
 training robust models has higher sample complexity than training standard models
 tradeoff between robustness and accuracy (raghunathann et al. 2020)
 adding valid data can hurt even in wellspecified, convex setting
 robust selftraining eliminates this tradeoff in linear regression
 sample complexity can be reduced with unlabeled examples (carmon et al. 2019)
 distributional robust optimization
 instead of minimizing training err, minimize maximum training err over different perturbations
 hard to pick the perturbation set  can easily be too pessimistic
 these things possibly magnify disparities
 larger models
 selective classification
 feature noise
 removing spurious features
 group DRO (sagawa et al. 2020)  maximize error for worst group
 need to add regularization to keep errors from going to 0
 training overparameterized models makes this problem worse
 abstain from classifying based on confidence (jones et al. 2020)
 makes error rates worse for worst group = selective classification can magnify disparities
 adding feature noise to each group can magnify disparities
 domain adaptation
 standard domain adaptation: labeled (x, y) in source and unlabeled x in target
 gradual domain adaptation  things change slowly over time, can use gradual selftraining (kumar et al. 2020)
 InNOut (xie et al. 2020)  if we have many features, rather than using them all as features, can use some as features and some as targets when we shift, to learn the domain shift
tools for analyzing
 dim reduction: svcca, diffusion maps
 viz tools: bei wang
 visualizing loss landscape
 1d: plot loss by extrapolating between 2 points (start/end, 2 ends)
 goodfellow et al. 2015, im et al. 2016
 exploring landscape with this technique
 2d: plot loss on a grid in 2 directions
 important to think about scale invariance (dinh et al. 2017)
 want to scale direction vector to have same norm in each direction as filter
 use PCA to find important directions (ex. sample w at each step, pca to find most important directions of variance)
misc theoretical areas
 deep vs. shallow rvw
 probabilistic framework
 information bottleneck: tishby paper + david cox followup
 Emergence of Invariance and Disentanglement in Deep Representations
 manifold learning
comparing representations
simple papers
adam vs sgd
 svd parameterization rnn paper: inderjit paper
 original adam paper: kingma 15
 regularization via SGD (layers are balanced: du 18)
 marginal value of adaptive methods: recht 17
 comparing representations: svcca
 montanari 18 pde mean field view
 normalized margin bounds
memorization background
 memorization on single training example
 memorization in dnns
 “memorization” as the behavior exhibited by DNNs trained on noise, and conduct a series of experiments that contrast the learning dynamics of DNNs on real vs. noise data
 look at critical samples  adversarial exists nearby
 networks that generalize well have deep layers that are approximately linear with respect to batches of similar inputs
 networks that memorize their training data are highly nonlinear with respect to similar inputs, even in deep layers
 expect that with respect to a single class, deep layers are approximately linear
 example forgetting paper
 secret sharing
 memorization in overparameterized autoencoders
 autoencoders don’t lean identity, but learn projection onto span of training examples = memorization of training examples
 sometimes output individual training images, not just project onto space of training images
 https://arxiv.org/pdf/1810.10333.pdf
 https://arxiv.org/pdf/1909.12362.pdf
 https://pdfs.semanticscholar.org/a624/6278cb5e2d0ab79fe20fe20a41c586732a11.pdf
 Autoencoders: reconstruction versus compression
probabilistic inference
 multilayer idea
 equilibrium propagation
architecture search background
 ideas: nested search (retrain each time), joint search (make arch search differentiable), oneshot search (train big net then search for subnet)
bagging and boosting
 analyzing bagging (buhlmann and yu 2002)
 boosting with the L2 loss (buhlmann & yu 2003)
 boosting algorithms as gradient descent (mason et al. 2000)
basics
 good set of class notes
 demos to gain intuition
 colah

overview / reviews
 some people involved: nathan srebro, sanjeev arora, jascha sohldickstein, tomaso poggio, stefano soatto, ben recht, olivier bousquet, jason lee, simon shaolei du
 interpretability: cynthia rudin, rich caruana, been kim, nicholas papernot, finale doshivelez
 neuro: eero simoncelli, haim sompolinsky