Deep Learning: Difference between revisions

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* [http://www.cs.umd.edu/class/fall2020/cmsc828W/ Course Website]

* [https://www.youtube.com/user/soheilfeiz/videos Lecture Videos]

==Basics==

A refresher of [[Machine Learning]] and Supervised Learning.

A refresher of [[Machine Learning]] and [[Supervised Learning]].

===Empirical risk minimization (ERM)===

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\begin{aligned}

\frac{1}{2}\Vert \nabla f(w) \Vert^2 &= \frac{1}{2}\Vert (F(w)-y)^T \nabla F(w)\Vert^2\\

&=\frac{1}{2}(F(w)=y)^T \nabla F(w) \nabla F(w)^T (F(w)-y)\\

&=\frac{1}{2}(F(w)-y)^T \nabla F(w) \nabla F(w)^T (F(w)-y)\\

&\geq \frac{1}{2} \lambda_{\min}(K(w)) \Vert F(w)-y\Vert ^2\\

&= \lambda_{\min}(K(w)) L(w)\\

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===Linear Regression===

Assume we have a dataset:

Assume we have a dataset:

<math>y_i \in \mathbb{R}</math>

<math>y_i \in \mathbb{R}</math>

<math>x_i \in \mathbb{R}^d</math>

<math>x_i \in \mathbb{R}^d</math>

GD: <math>w(t+1) = w(t) - \eta_{t} \nabla L(w_t)</math> where our gradient is:

GD: <math>w(t+1) = w(t) - \eta_{t} \nabla L(w_t)</math> where our gradient is:

<math>\sum_{i=1}^{n}(y_i - f(w, x_i)) \nabla_{w} f(w_t, x_i) = \sum_{i=1}^{n}(y_i - f(w, x_i)) x_i</math>

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</math>

Is this model linear in w? Yes!

Is this model linear in w? Yes!

Is this model linear in x? No!

Is this model linear in x? No!

Apply GD or convex optimization.

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* <math>\phi</math> is fixed!

* <math>D = O(d^k)</math>

** For ImageNet, d is approx <math>10^5</math> so <math>D=O(10^15)</math>

** For ImageNet, <math display="inline">d</math> is approx <math display="inline">10^5</math> so <math display="inline">D=O(10^{15})</math>

;Kernel Trick:

We may have a closed form solution for <math>\langle \phi(x_i), \phi(x_j) \rangle</math>.

We may have a closed form solution for <math>\langle \phi(x_i), \phi(x_j) \rangle</math>.

This is called the kernel function <math>K(x_i, x_j)</math> or kernel matrix <math>K \in \mathbb{R}^{n \times n}</math>.

This is called the kernel function <math>K(x_i, x_j)</math> or kernel matrix <math>K \in \mathbb{R}^{n \times n}</math>.

K is a PSD matrix.

<math display="inline">K</math> is a PSD matrix.

Idea: In many cases without "explicit" comp of <math>\phi(x_i)</math>, we can compute <math>K(x_i, x_j)</math>.

;Polynomial Kernels

<math>K(x_i, x_j) = (x + x_i^t x_j)^k</math> with <math>\phi(x_i) \in \mathbb{R}^D</math>

<math>K(x_i, x_j) = (x + x_i^t x_j)^k</math> with <math>\phi(x_i) \in \mathbb{R}^D</math>

Here <math>D=O(d^k)</math> but <math>K(x_i, x_j)</math> is <math>O(d)</math>.

Many classical techniques can be ''kernelized'':

SVM to Kernel SVM

* SVM to Kernel SVM

Ridge regression to Kernel ridge regression

* Ridge regression to Kernel ridge regression

PCA to Kernel PCA

* PCA to Kernel PCA

===Neural Networks===

Consider a two-layer neural network.

Consider a two-layer neural network.

We can write the output as:

We can write the output as:

<math>y = f(w, x) = \frac{1}{\sqrt{m}} \sum_{i=1}^{m} b_i \sigma(a_i^t x)</math>

<math>y = f(w, x) = \frac{1}{\sqrt{m}} \sum_{i=1}^{m} b_i \sigma(a_i^t x)</math>

We use quadratic loss: <math>L(w) = \frac{1}{2} \sum_{i=1}^{n} (f(w, x_i) - y_i)^2</math>

We use quadratic loss: <math>L(w) = \frac{1}{2} \sum_{i=1}^{n} (f(w, x_i) - y_i)^2</math>

GD: <math>w(t+1) = w(t) - \eta_{t} \sum_{i=1}^{n} (f(w, x_i) - y_i) \nabla_w f(w_t, x_i)</math>

Init N(0,1)

# Init N(0,1)

Our weights update along a trajectory: w(0), w(1), ...

# Our weights update along a trajectory: w(0), w(1), ...

Each <math>w</math> is a weight matrix.

# Each <math>w</math> is a weight matrix.

Empirical Observation: When the width of the network <math>m</math> is large, the trajectory of the gradient descent is ''almost'' static.

This is called ''lazy'' training.

* Not always the case! Especially for small <math>m</math>.

Since the change in the model weights are not large, we can write the first-order taylor approximation:

Since the change in the model weights are not large, we can write the first-order taylor approximation:

<math>f(w, x) \approx f(w_0, x) + \nabla_{w} f(w_0, x)^t (w - w_x) + ...</math>

<math>f(w, x) \approx f(w_0, x) + \nabla_{w} f(w_0, x)^t (w - w_x) + ...</math>

This model is linear in <math>w</math>.

This model is linear in <math>w</math>.

<math>\phi(x) = \nabla_{w} f(w_0, x)</math>

<math>\phi(x) = \nabla_{w} f(w_0, x)</math>

The kernel <math>K = \langle \phi(x_i), \phi(x_j) \rangle</math> is called the ''Neural Tangent Kernel'' (NTK).

Go back to our 2-layer NN:

These features will not change during the optimization process because we use <math display="inline">w_0</math>

<math>f_m(w, x) = \frac{1}{\sqrt{m}} \sum b_i \sigma(a_i^t x)</math>

<math>\nabla_{a_i} f_m(w, x) = \frac{1}{\sqrt{m}} b_i \sigma'(a_i^t x) x</math>

Go back to our 2-layer NN:

<math>f_m(w, x) = \frac{1}{\sqrt{m}} \sum b_i \sigma(a_i^t x)</math>

<math>\nabla_{a_i} f_m(w, x) = \frac{1}{\sqrt{m}} b_i \sigma'(a_i^t x) x</math>

<math>\nabla_{b_i} f_m(w, x) = \frac{1}{\sqrt{m}} \sigma(a_i^t x)</math>

<math>K_{m}^{(a)}(x, x') = \frac{1}{m} \sum_{i=1}^{m} b_i^2 \sigma'(a_i^tx) \sigma'(a_i^tx) (x x')</math>

<math>K_{m}^{(a)}(x, x') = \frac{1}{m} \sum_{i=1}^{m} b_i^2 \sigma'(a_i^tx) \sigma'(a_i^tx) (x x')</math>

<math>K_{m}^{(b)}(x, x') = \frac{1}{m} \sum_{i=1}^{m} \sigma(a_i^t x) \sigma(a_i^t x')</math>

* <math>a_i</math> and <math>b_i</math> are independent samples at initialization.

Based on law of large numbers, as m goes to infinity,

Based on law of large numbers, as m goes to infinity,

<math>K_{m}^{(a)}(x, x') \to K^{(a)}(x, x') = E \left[ b^2 \sigma'(a^t x) \sigma'(a^t x') (x x') \right]</math>

<math>K_{m}^{(a)}(x, x') \to K^{(a)}(x, x') = E \left[ b^2 \sigma'(a^t x) \sigma'(a^t x') (x x') \right]</math>

<math>K_{m}^{(b)}(x, x') \to K^{(b)}(x, x') = E \left[ \sigma(a^t x) \sigma(a^t x') \right]</math>

<math>K^{(a)}(x, x') = \frac{(x x') E[b^2]}{2 \pi} (\pi - \theta(x, x))</math>

<math>K^{(a)}(x, x') = \frac{(x x') E[b^2]}{2 \pi} (\pi - \theta(x, x))</math>

<math>K^{(b)}(x, x') = \frac{\Vert x \Vert \Vert x' \Vert E[\Vert a \Vert^2]}{2 \pi d} ((\pi - \theta(x, x')) \cos(\theta) + \sin \theta)</math>

;Q: When is this taylor approximation good?

;Q: When is this taylor approximation good?

If the Hessian has bounded eigenvalues. (Hessian Control)

;Analyze GD:

<math>\eta \to 0</math> Gradient-flow

<math>\eta \to 0</math> Gradient-flow

<math>w(t+1) = w(t) - \eta \nabla_{w} L(w(t)) \implies \frac{w(t+1) - w(t)}{\eta} = - \nabla_{w} L(w(t))</math>

<math>w(t+1) = w(t) - \eta \nabla_{w} L(w(t)) \implies \frac{w(t+1) - w(t)}{\eta} = - \nabla_{w} L(w(t))</math>

<math>\to \frac{dw(t)}{dt} = -\nabla_{w} L(w(t))</math>

<math>\frac{dw(t)}{dt} = -\nabla_{w} \hat{y}(w) (\hat{y}(w) - y)</math>

<math>\frac{dw(t)}{dt} = -\nabla_{w} \hat{y}(w) (\hat{y}(w) - y)</math>

<math>

\begin{aligned}

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</math>

If we let <math>u = \hat{y} - y</math>, then <math>\frac{du}{dt} \approx -K(w_i) u</math>.

If we let <math>u = \hat{y} - y</math>, then <math>\frac{du}{dt} \approx -K(w_i) u</math>.

This ODE implies <math>u(t) = u(0)\exp(-K(w_i)t)</math>.

This ODE implies <math>u(t) = u(0)\exp(-K(w_i)t)</math>.

In the over-parameterized case, <math>K(w_0) > 0 </math> (positive definite).

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We need pairs of similar images and dissimilar images.

SimCLR [Chen ''et al.'' 2020]

;SimCLR [Chen ''et al.'' 2020]

# Create two correlated views of an image <math>x</math>: <math>\tilde{x}_i</math> and <math>\tilde{x}_j</math>.

#* Random cropping + resize

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Training is <math>\min_{f,g} L = \frac{1}{N} \sum_{k=1}^{N} \frac{l(2k-1,2k) + l(2k, 2k-1)}{2}</math>.

==~~Misc~~==

Practical observations:

[[~~Visible~~ to::~~users~~]]

* Composition of data augmentation is important.

* Larger batch sizes and longer training helps self-supervised learning!

* Optimization over the MLP projection head <math>g</math> helps!

** f is able to keep some information useful for the classification (e.g. color or orientation)

Empirical results:

* For image net, top-1 accuracy increases as number of parameters increases.

* After learning embeddings <math>f</math>, you don't need much labeled data for supervised-training.

===Theory of self-supervised learning===

;[Arora ''et al.'' 2019]

Modeling of similar pairs:

* <math>x \sim D_{C1}(x)</math>

* <math>x^+ \sim D_{C1}(x)</math> is semantically similar

* <math>x^- \sim D_{C2}(x)</math> negative samples

Downstream task:

* pairwise classification

** nature picks two classes C1, C2

** generate samples from C1 & C2 and evaluate the classification loss

* Assume <math>m \to \infty</math> so just look at population loss

Notations:

* <math>L_{sup}(f)</math> is the supervised learning loss with optimal last layer

* <math>l=E_{(x, x^+) \sim D_{sim}, x^- \sim D_{net}}[\log(1+\exp(f(x)^T f(x^-) - f(x)^Tf(x^+)]</math> is a logistic loss

Result 1:

* <math>L_{sup}^{mean}(f) \leq \frac{1}{1-\tau}(L_{un}(f)-\tau)</math> for all <math>f \in F</math>

** <math>\tau</math> is the collision probability for pair of random classes.

Idea result:

* <math>L_{sup}(f^*) \leq \alpha L_{sup}(f)</math> forall <math>f</math>.

* In general it is impossible to get this.

;[Tosh et al. 2020]

This work connects self-supervised learning with multi-view representation theory.

Start with (x, z, y) where x,z are two views of the data and y is the label.

* x and z should share redundant information w.r.t. y. I.e. predicting y from x or z individually should be equivalent to predicting y from both.

* <math>e_x = E(E[Y|X] - E[Y|X,Z])^2</math> and <math>e_z = E(E[Y|Z] - E[Y|X,Z])^2</math> should be small

* Formulate contrastive learning as an artificial ''classification'' problem:

** Classify between <math>(x, z, 1)</math> and <math>(x, \tilde{z}, 0)</math>.

** <math>g^*(x,z) = \operatorname{argmin}(\text{classification loss}) = \frac{P_{X,Z}(x,z)}{P_X(x) P_Z(z)}</math>.

* x,z have redundant info from Y so if we first predict z from x, then use the result to predict Y, we should be good

<math>

\begin{aligned}

\mu(x) &= E[E[Y|Z]|X=x] \\

&= \int E[Y|Z=z] P_{Z|X}(z|x) dz\\

&= \int E[Y|Z=z] g^*(x,z) P_Z(z) dz

\end{aligned}

</math>

Lemma: <math>E[(\mu(x) - E[Y | x,z])^2] \leq e_x + e_z + 2\sqrt{e_x e_z}</math>

Landmark embedding: <math>g^*</math> is computed via contrastive learning.

How to embed x? <math>\phi(x) = (g^*(x, z_1),...,g^*(x,z_m))</math>

z's are randomly generated from <math>P_Z</math>.

Each <math>g^*</math> output is a single number.

* <math>\phi()</math> is good embedding if a linear classifier of <math>\phi</math> is approx <math>\mu(x)</math>.

** I.e. <math>\exists w : w^t \phi(x) \approx \mu(x)</math>

** <math>w^t \phi(x) = \frac{1}{m} \sum_{i=1}^{m} E[Y|Z_i] g^*(x, z_i) \stackrel{m \to \infty}{\rightarrow} \int E[Y|Z=z] g^*(x,z)P_{Z}*z dz = \mu(x)</math>

Direct embedding

Instead of learning a bivariate <math>g(x,z)</math>, learn <math>\eta(x)^T \psi(z)</math>.

Lemma: For every <math>\eta: x \to \mathbb{R}^m</math> and <math>\psi:z \to \mathbb{R}^m</math>, there exists <math>w \in \mathbb{R}^m</math> such that <math>E[(w^t \eta(x) - \mu(x))^2] \leq E[Y^s \epsilon_{direct}(\eta, \psi)</math> where <math>\epsilon_{direct} = [(\eta(x)^t \psi(x) - g^*(x,z))^2</math>.

Can we write <math>g^*(x,z)</math> as <math>\eta(x)^T \psi(z)</math>?

* If there is a hidden variable H s.t. X and Z are conditionally independent given H.

# Case 1: H is a discrete variable then <math>g^*(x, z) = \eta^*(x)^t \psi(z^*)</math>

# Case 2: There exists <math>\eta,\psi</math> such that <math>E(\eta(x)^t \psi(z) - g^*(x,z))^2 \leq o(\frac{1}{m})</math>.

==Meta Learning==

In many applications, we don't have a large training dataset. However, humans can adapt and learn ''on the fly''.

The key is to use prior knowledge in performing new tasks.

How can we train AI/ML models similarly?

Goal of meta learning: Train a model on different learning tasks such that it can solve new tasks using only a small number of training samples.

Few shot classification problem:

Inputs: <math>D_{metatrain} = \{(D_i^{train}, D_i^{test})\}_{i=1}^{n}</math>.

<math>

\begin{cases}

D_{i}^{train} = \{(x_j^i, y_j^i) \}_{j=1}^{k}\\

D_{i}^{test} = \{(x_j^i, y_j^i) \}_{j=1}^{k'}

\end{cases}

</math>

* <math>i</math> is the index of the task

* <math>j</math> is the index of the sample

K-shot classification means k training samples per label. Some papers use k as the size of the whole training set.

Q: How to use <math>D_{metatrain}</math> to solve meta-test tasks more effectively?

* Use <math>D_{metatrain}</math> to learn meta parameters <math>\theta</math> such that:

* Base learner <math>A</math> outputs task-specific model parameters <math>\phi_i = A(D_{i}^{train}, \theta)</math> good for <math>D_{i}^{test}</math>.

* Training procedure:

** Loss: <math>\min_{\theta} \sum_{i=1}^{n} loss(D_{i}^{test}, \phi_i)</math> where <math>\phi_i = A(D_{i}^{train}, \phi)</math>.

* Test time: given a new task <math>(D^{train}, D^{test})</math>. Apply <math>A(D^{train}, \theta^*)</math> to get <math>\phi</math>.

[Fin et al. 2017]

* Idea: train a model that can be easy to fine-tune on new tasks

* one-step fine tuning: <math>\theta - \alpha \nabla L(\theta, D_{i}^{train})</math> gives us <math>\phi_{i}</math>

* Evaluate <math>\phi_i</math> on <math>D_i^{test}</math>

* <math>\min_{\theta} \sum_{i=1}^{n} L(\phi_i, D_i^{test}) = L(\theta - \alpha \nabla L(\theta, D_i^{train}), D_i^{test})</math>

1. Model-agnostic meta learning (MAML)

* Use GD to optimize over <math>\theta</math>

** <math>\nabla_\theta = \sum_{i=1}^{n}(\nabla_{\theta} \phi_i) \nabla_{\phi} L(\phi_i, D_i^{test})</math>

** <math>(\nabla_{\theta} \phi_i)</math> involves second-order derivatives which are expensive.

* First-order MAML: just ignore <math>\nabla_{\theta} \phi_i</math> term. Replace it with the identity matrix.

Reptile (Michal et al 2018)

2. <math>A</math> is a simple linear/non-parametric learning on data embeddings computed via <math>f_{\theta}</math>

[Lee et al. 2019]

* <math>f_{\theta}</math> is used to compute embeddings

* <math>A</math> is a linear classifier (e.g. SVM)

* Use dual form of SVM so # of optimization variables = # training samples * # classes

3. <math>A</math> is a non-parametric learning

* Embedding: <math>\tilde{x} = f_{\theta}(x)</math>

[Snell et al. 2017)

* Define prototypes (cluster centers)

** <math>c_k = \frac{1}{|D_i^{tr}|} \sum_{(x,y) \in S_k} f_{\theta}(x)</math>

** <math>P_{\theta}(y=k|x) = \frac{\exp(-d(f_{\theta}(x), c_k))}{\sum_{k'} \exp(-d(f_{\theta}(x), c_k))}</math>

4: <math>A</math> is a black box (e.g. LSTM).

==LSTMs and transformers==

Lecture 23 (November 19, 2020)

===Recurrent Neural Networks (RNNs)===

Hidden state: <math>h_t = \tanh(W_{hh} h_{t-1} + W_{xh} x_t)</math>

Prediction at t: <math>y_t = W_{hy} h_{t}</math>

;Backpropagation through time

If <math>W</math> has largest singular value < 1, then gradient vanishes.

If <math>W</math> has largest singular value > 1, then gradient explodes.

Typically, gradient vanishes because of initialization of <math>W</math>.

===Long Short Term Memory (LSTMs)===

Goal is to solve the vanishing and exploding gradient problem.

LSTM has several gates:

* Input gate: <math>i_t = \sigma(W_{xi}x_t + W_{hi}h_{t-1} + b_i)</math>

* Forget gate: <math>f_t = \sigma(W_{xf}x_t + W_{hf}h_{t-1} + b_f)</math>

* Output Gate: <math>o_t = \sigma(W_{xo}x_t + W_{ho}h_{t-1} + b_o)</math>

* Cell state: <math>c_t = f_t \odot c_{t-1} + i_t \odot \tilde{c}_t</math>

* Hidden state: <math>h_t = o_t \odot \tanh(c_t)</math>

;Bidirectional RNNs

First LSTM takes input in correct order.

Second LSTM takes input in reverse order.

Concatenate outputs from both LSTMs.

===Attention===

Goal: To help memorize long source sentences in machine translation.

;Encoder-decoder attention

;Self-Attention

===Transformer===

;Positional encoding

;Self-Attention

Have queries, keys, and values.

Multiply queries with keys, pass through softmax. Then times values.

Yields attention of every work with respect to another.

Initially, transformers had n=8 heads giving 8 queries, keys, and values.

;Architecture

Stack encoders.

==Interpretability==

;Interpretability Methods

* Built-in model interpretability

* Feature level interpretability

* Instance based explanations

We will focus on feature level interpretability.

===Feature Level Interpretability===

These are given through saliency maps.

* Perturbation-based: Perturb the input to get another output and compute the difference.

* Gradient-based

===Gradient-based Methods===

Take the derivative of the output with respect to the input.

;Limitations

* Too local and sensitive to slight perturbations

* Saturated outputs lead to unintuitive gradients

* Discontinuous gradients are problematic

;SmoothGrad

* Add gaussian noise to input and average the gradient.

;Integrated Gradients

* Average the gradients along path from baseline to input.

;DeepLift

* We don't care about gradient but the slope relative to the ''reference'' state

;Limitations

* Models must be able to compute the gradient of the output with respect to the input

* Interpretation of neural networks is fragile

** Saliency maps are uninterpretable for adversarial examples on clean models and adversarially trained models.

* Needs white-box gradient access to the model.

===Evaluation of interpretability methods===

* Human evaluation

** Can humans evaluate saliency?

* Accuracy drop after removing ''salient'' features

* Sanity checks

** Model parameter randomization test - compare output of saliency method on trained vs untrained method to make sure saliency depends on model parameters.

* Synthetic Data

* Data randomization test

** Train on random labels and see if saliency depends on relationship between input & output.

Temporal saliency Rescaling

* If you remove this feature, how is the gradient going to change.

==Reinforcement Learning==

Lecture 27 (December 1, 2020)

;Goal: Sequential decision making problems

* Current decisions can affect future decisions

Examples: games, robotics, self-driving, finance

Agent takes an action <math>a_t</math> in the environment which leads to a new state <math>\delta_{t+1}</math>.

;Definitions

* <math>S</math> is the state space. This can be discrete or continuous.

* <math>O</math> are the observations.

* <math>A</math> is the set of actions which can be discrete or continuous.

* <math>P</math> are the transition probabilities which model the environment.

Example:

* <math>|S|</math> = 2

* <math>|A|</math> = 2

In general, <math>S_t \sim P(S_t=s | s_0, a_0, s_1, a_1, ..., a_{t-1})</math>.

The given variables are the trajectory <math>T_{t-1}</math>.

Modeling this can be very complex for large <math>T</math>.

Thus, we apply a markov assumption:

* <math>S_t \sim P(S_t=s | a_{t-1}, s_{t-1})</math>

Objective: To maximize the rewards

* <math>R_t \stackrel{\triangle}{=} R(s_t, a_t)</math>

Two regimes for rewards

* Finite horizon; T is finite

* Infinite horizon; T goes to infinity

* Discount factor

===Classical RL===

Finite Markov Decision Process (MDP)

* S: Finite number of states

* A: Finite number of actions

* <math>R_t = R(a_t, S_t)</math>

* r discount factor

Goal: Choose actions to maximize the total rewards.

* Policy function <math>\pi</math> determines how actions should be taken.

* The policy function could yield a dist on actions or can be deterministic. We will focus on deterministic actions.

* Assume this is time invariant.

* Value function determines the expected reward if we start at state s and follow policy <math>\pi</math>.

* Q function (action-value function)

** <math>Q_\pi (s, a) = E[R_0 + \gamma R_1+... | A_0=a, S_0=s, \pi]</math>

* If we have <math>\pi</math>, can we evaluate <math>V_{\pi}(s)</math>?

** <math>V_{\pi}(s) = E[R_0 + \gamma R_1 + ... | S_0 = s, \pi]</math>

* Use Bellman's equation

** <math>V_{\pi}(s) = R_0 + \gamma \sum_{s'} P_{\pi}(s' | s) V_{\pi}(s')</math>

** <math>V_{\pi} = R + \gamma P_{\pi} V_{\pi}</math>

** <math>V_{\pi} = (I - \gamma P_{\pi})^{-1}R</math>

** <math>P_{\pi}</math> is a stochastic matrix. It is not symmetric so eigenvalues can be complex.

All eigen values are <math>\Vert \lambda_i \Vert \leq 1</math> with one eigenvalue norm exactly 1.

This implies <math>I - \gamma P_{\pi}</math> is always invertible.

However, computing the inverse is computationally expensive since it scales with <math>|S|^3</math>.

===Value-Iteration (Bellman Recursion)===

<math>V_{\pi} = R + \gamma P_{\pi} V_{\pi}</math>

Define an operator <math>L_{\pi}v = R + \gamma P_{\pi}v</math>

<math>v_pi = L_{\pi} v_{\pi}</math> so <math>v_{\pi}</math> is a fixed point to <math>L_{\pi}</math>.

;Claim: <math>L_{\pi}</math> is a contraction.

Proof: Take <math>v_1, v_2 \in \mathbb{R}^d</math>.

<math>

\begin{aligned}

L_{\pi}v_1 - L_{\pi}v_2 &= (R + \gamma P_{\pi}v_1) - (R + \gamma P_{\pi}v_2) \\

&= \gamma P_{\pi}(v_1 - v_2)

\end{aligned}

</math>

<math>

\begin{aligned}

\Vert L_{\pi}v_1 - L_{\pi}v_2 \Vert &= \Vert \gamma P_{\pi}(v_1 - v_2) \Vert\\

& \leq \gamma \Vert P_{\pi} \Vert \Vert v_1 - v_2 \Vert \\

& \leq \Vert v_1 - v_2 \Vert

\end{aligned}

</math>

By Banach's Fixed Point Theorem, we can converge to the fixed point iteratively by repeatedly applying <math>L_{\pi}</math>.

===Optimal Policy===

Elementwise max: <math>\max_{\pi} v_{\pi}(s)</math> leads to the optimal policy <math>\pi^*(s)</math>.

Bellman (1957) showed that for an MDP, there exists an optimal policy that is deterministic and such that <math>v_{\pi^*}(s) \geq v_{\pi}(s)</math> for all <math>s, \pi</math>.

The policy may not be unique but if two policies are equal, they have the same value function.

Intermediate questions:

* Given <math>v^*</math>, can we compute <math>Q^*</math>?

<math>

\begin{aligned}

Q^*(s, a) &= \max_{\pi} Q(s,a)\\

&= \max_{\pi} R(s) + \gamma \sum_{s'} p(s' | s,a) v_{\pi}(s')\\

&= R(s) + \gamma \sum_{s'} P(s' | s, a) \max_{\pi}(v_{\pi}(s'))\\

\end{aligned}

</math>

* Given <math>Q^*</math>, can we compute <math>v^*</math>.

Bellman's optimality states that <math>v^*(s) = \max_{a} Q^*(s, a)</math>.

<math>\pi^*(s) = \operatorname*{argmax}_{a} Q^*(s, a)</math>

;How to compare optimal policies?

First approach: Value Iteration

<math>V^*(s) = \max_{a} Q^*(s, a) = \max_{a} [R(s) + \gamma \sum_{s'} p(s'|s,a) v^*(s')]</math>

Define an operator: <math>L(v) = \max_{\pi}[R + \gamma P_{\pi} v]</math>

<math>v^*</math> is a fixed point of <math>L(v)</math>.

Claim: <math>L</math> is also a contraction.

Thus <math>v^*</math> can be computed by repeated application of <math>L</math>.

===Value Iteration===

* Start with a random <math>v^{(0)} \in \mathbb{R}^d</math>

* <math>v^{(r+1)}(s) = \max_{a} [R(s) + \gamma \sum_{s'} p(s'|s,a) v^{r}(s')]</math>

* Repeat until <math>\Vert v^{(r+1)} - v^{(r)} \Vert \leq \epsilon</math>

===Policy Iteration===

Directly optimize policy.

* Start with a random policy <math>\pi^{(0)}</math>

* Evaluate the policy <math>v_{\pi}</math> using closed form or value iteration

* Evaluate the Q-Function <math>Q_{\pi}(s, a) = R(s) + \gamma \sum_{s'} P(s'|s,a) V_{\pi}(s')</math>

* Find the best action to be taken at state <math>s</math>:

** <math>a^*(s) = \operatorname{argmax}_{a} Q_{\pi}(s, a)</math>

* Update <math>\pi</math> using <math>\pi(s) = a^*(s)</math>

* Repeat until convergence.

Policy iteration is guaranteed to converge to an optimal policy.

Oftentimes converges faster than value iteration.

===Deep Reinforcement Learning===

;Relaxing some unrealistic assumptions

# Evaluate <math>v_{\pi}(s)</math>

#* <math>Q_{\pi}(s, a) = R(s, a) + \gamma E_{s' \sim P(s'|s,a)} v_{\pi}(s')</math>

# Improve the policy

#* <math>\operatorname{argmax}_{a_t} Q_{\pi}(s_t, a_t)</math>

#* Assumption <math>|S|=d</math>

# How to represent <math>V(s)</math>?

#* Can we use a neural network to represent <math>V(s)</math>? Yes

;How to train <math>v_{\phi}</math>?

* Start with an old <math>v_{\phi}</math>, compute <math>Q_{\pi}(s,a)</math>.

** <math>Q_{\pi}(s,a) = R(s,a) + \gamma E[v_{\phi}^{old}(s')]</math>

* Fit <math>v_{\phi}</math> to <math>\max_{a}Q_{\pi}(s,a)</math> using a quadratic loss:

** <math>L(\phi) = \frac{1}{2} \Vert V_{\phi}(s) - \max_{a} Q_{\pi}(s,a) \Vert^2</math>

* Iterate

;Similarly we can parameterize the Q function

Compare target <math>y_i \leftarrow R(s_i, a_i) + \gamma E[v_{\phi}(s_i')]</math>

We need to know transition probabilities <math>P(s'|s,a)</math> to compute the expectation (model-based RL).

With model-free RL:

We can approximate as <math>E[v(s_i')] \approx v(s_i') = \max_a Q(s_i', a')</math>

This is called ''Q-Learning''.

;What if we have continuous actions:

* Approach 1: Use a function class such that <math>\max_{a}Q(s,a)</math> is easy to solve

** [Gu ''et al.'' 2016] use quadratic functions.

* Approach 2: Learn another network to approximate the maximizer: <math>\max_{a} Q(s,a')</math>

===Policy Gradient Method===

Lecture 29 (Dec 8, 2020)

Probability of observing a trajectory:

<math>P_{\theta}(\tau) = P(s_1) \prod_{t=1}^{\tau} \pi_{\theta}(a_t | s_t) P(s_{t+1} | s_t, a_t)</math>

Reward for a trajectory:

The average reward is:

<math>J(\theta) = E[R(\tau)] = \sum E[R(s_t, a_t)]</math>

Our goal is to maximize the average reward: <math>\max_{\theta} J(\theta)</math>.

Gradient of the average reward:

<math>

\begin{aligned}

\nabla_{\theta} J(\theta) &= \nabla_{\theta} E[R(\tau)] \\

&= \nabla_{\theta} \int P_{\theta}(\tau) R(\tau) d\tau \\

&= \int \nabla_{\theta} P_{\theta}(\tau) R(\tau) d\tau \\

&= \int P_{\theta}(\tau) \nabla_{\theta} \log P_\theta(\tau) R(\tau) d\tau\\

&= E[\log P_\theta(\tau) R(\tau)]

\end{aligned}

</math>

<math>

\nabla_{\theta} \log P_{\theta}(\tau) = \sum_{t=1}^{\tau} \nabla_{\theta} \log \pi_{\theta}(a_t | s_t)

</math>

Implies,

<math>

\begin{aligned}

\nabla_{\theta} J(\theta) &= E[...]\\

&\approx \frac{1}{N} \sum_{i=1}^{N}(\sum \nabla_{\theta} \log \pi(a_t^{(i)} | s_t^{(i)}) ....

\end{aligned}

</math>

;Summary

* Sample trajectories

* Approximate <math>\nabla_{\theta} J(\theta)</math>

* <math>\theta \leftarrow \theta + \alpha \nabla J(\theta)</math>

;Intuition

<math>E[\nabla_{\theta} \log P_{\theta}(\tau) R(\tau)]</math>

Formalizing ''trial & error''.

[Finn & Levin, ICML]

===Issues with Policy Gradient===

;High variance of gradient estimation

;Solutions

* Subtract a baseline

* Reward-to-go

<math>

\begin{aligned}

\nabla_{\theta} J(\theta) &\approx \frac{1}{N} \sum_{i=1}^{N} \nabla_{\theta} \log P_{\theta}(\tau) R(\tau)\\

&= \frac{1}{N} \sum_{i=1}^{N} \left(\sum_{t=1}^{T} \nabla_{\theta} \pi_{\theta}(a_t^{(i)}| s_t^{(i)}) \right) \left(\sum_{t'=1}^T R(s_{t'}^{(i)}, a_{t'}^{(i)})\right)\\

&= \frac{1}{N} \sum_{i=1}^{N} \sum_{t=1}^{T} \nabla_{\theta} \pi_{\theta}(a_t^{(i)}| s_t^{(i)}) \left(\sum_{t'=1}^T R(s_{t'}^{(i)}, a_{t'}^{(i)})\right)\\

&\approx \frac{1}{N} \sum_{i=1}^{N} \sum_{t=1}^{T} \nabla_{\theta} \pi_{\theta}(a_t^{(i)}| s_t^{(i)}) \left(\sum_{t'=t}^T R(s_{t'}^{(i)}, a_{t'}^{(i)})\right)\\

\end{aligned}

</math>

;Some parameters can change <math>\pi_{\theta}</math> more than others so it's hard to choose a fixed learning rate.

Use natural policy gradient: <math>\theta' \leftarrow \theta - \eta F^{-1}\nabla L(\theta)</math>

===Actor-critic algorithms===

Have an actor <math>\pi_{\theta}</math>.

Have a critic <math>V_{\phi}/Q</math>

<math>\nabla_{\theta} J(\theta) \approx \frac{1}{N} \sum_{i=1}^{N} \sum_{t=1}^{T} \nabla_{\theta} \log \pi_{\theta}(a_t^{(i)} | s_t^{(i)}) \left(Q(s_t^{(i)}, a_t^{(i)} - V(s_t^{(i)})\right)</math>

===Other topics in RL===

* Inverse RL

* Multi-agent RL

* Model-based RL

==Summary of Course==

;What we covered

* Supervised DL

* Unsupervised DL (GANs, VAEs)

* Self-supervised DL

* Meta-Learning

* Learning with Attention (Transformers)

* Deep RL

* Optimization

* Generalization

* Robustness

* Interpretability

;What we didn't cover

* Fairness

* Privacy & Ethics

* Bayesian DL

* Federated Learning

* Graph NNs

;Things which may be on the final

* Transformers

* Wasserstein distance

* Kernel methods

==Resources==

@@ Line 2: / Line 2: @@
 * [http://www.cs.umd.edu/class/fall2020/cmsc828W/ Course Website]
+* [https://www.youtube.com/user/soheilfeiz/videos Lecture Videos]
 ==Basics==
-A refresher of [[Machine Learning]] and Supervised Learning.
+A refresher of [[Machine Learning]] and [[Supervised Learning]].
 ===Empirical risk minimization (ERM)===
@@ Line 93: / Line 94: @@
 \begin{aligned}
 \frac{1}{2}\Vert \nabla f(w) \Vert^2 &= \frac{1}{2}\Vert (F(w)-y)^T \nabla F(w)\Vert^2\\
-&=\frac{1}{2}(F(w)=y)^T \nabla F(w) \nabla F(w)^T (F(w)-y)\\
+&=\frac{1}{2}(F(w)-y)^T \nabla F(w) \nabla F(w)^T (F(w)-y)\\
 &\geq \frac{1}{2} \lambda_{\min}(K(w)) \Vert F(w)-y\Vert ^2\\
 &= \lambda_{\min}(K(w)) L(w)\\
@@ Line 502: / Line 503: @@
 ===Linear Regression===
-Assume we have a dataset:
+Assume we have a dataset:<br>
 <math>\{(x_i, y_i)\}_{i=1}^{n}</math>
-<math>y_i \in \mathbb{R}</math>
+<math>y_i \in \mathbb{R}</math><br>
-<math>x_i \in \mathbb{R}^d</math>
+<math>x_i \in \mathbb{R}^d</math><br>
 <math>f(w, x) = w^t x</math>
-<math>L(w) = \frac{1}{2} \sum_{i=1}^{n}(y_i - f(w, x_i))^2</math>
+<math>L(w) = \frac{1}{2} \sum_{i=1}^{n}(y_i - f(w, x_i))^2</math><br>
-<math>\min_{W} L(w)</math>
+<math>\min_{W} L(w)</math><br>
-GD: <math>w(t+1) = w(t) - \eta_{t} \nabla L(w_t)</math> where our gradient is:
+GD: <math>w(t+1) = w(t) - \eta_{t} \nabla L(w_t)</math> where our gradient is:<br>
 <math>\sum_{i=1}^{n}(y_i - f(w, x_i)) \nabla_{w} f(w_t, x_i) = \sum_{i=1}^{n}(y_i - f(w, x_i)) x_i</math>
@@ Line 532: / Line 533: @@
 </math>
-<math>f(w, x) = w^t \phi(x)</math>
+<math>f(w, x) = w^t \phi(x)</math><br>
-Is this model linear in w? Yes!
+Is this model linear in w? Yes!<br>
-Is this model linear in x? No!
+Is this model linear in x? No!<br>
-<math>\min \frac{1}{2} \sum_{i=1}^{n} (y_i - w^t \phi(x_i))^2</math>
+<math>\min \frac{1}{2} \sum_{i=1}^{n} (y_i - w^t \phi(x_i))^2</math><br>
 Apply GD or convex optimization.
@@ Line 542: / Line 543: @@
 * <math>\phi</math> is fixed!
 * <math>D = O(d^k)</math>
-** For ImageNet, d is approx <math>10^5</math> so <math>D=O(10^15)</math>
+** For ImageNet, <math display="inline">d</math> is approx <math display="inline">10^5</math> so <math display="inline">D=O(10^{15})</math>
 ;Kernel Trick:
-We may have a closed form solution for <math>\langle \phi(x_i), \phi(x_j) \rangle</math>.
+We may have a closed form solution for <math>\langle \phi(x_i), \phi(x_j) \rangle</math>.<br>
-This is called the kernel function <math>K(x_i, x_j)</math> or kernel matrix <math>K \in \mathbb{R}^{n \times n}</math>.
+This is called the kernel function <math>K(x_i, x_j)</math> or kernel matrix <math>K \in \mathbb{R}^{n \times n}</math>.<br>
-K is a PSD matrix.
+<math display="inline">K</math> is a PSD matrix.
 Idea: In many cases without "explicit" comp of <math>\phi(x_i)</math>, we can compute <math>K(x_i, x_j)</math>.
 ;Polynomial Kernels
-<math>K(x_i, x_j) = (x + x_i^t x_j)^k</math> with <math>\phi(x_i) \in \mathbb{R}^D</math>
+<math>K(x_i, x_j) = (x + x_i^t x_j)^k</math> with <math>\phi(x_i) \in \mathbb{R}^D</math><br>
 Here <math>D=O(d^k)</math> but <math>K(x_i, x_j)</math> is <math>O(d)</math>.
 Many classical techniques can be ''kernelized'':
-SVM to Kernel SVM
+* SVM to Kernel SVM
-Ridge regression to Kernel ridge regression
+* Ridge regression to Kernel ridge regression
-PCA to Kernel PCA
+* PCA to Kernel PCA
 ===Neural Networks===
-Consider a two-layer neural network.
+Consider a two-layer neural network.<br>
-We can write the output as:
+We can write the output as:<br>
-<math>y = f(w, x) = \frac{1}{\sqrt{m}} \sum_{i=1}^{m} b_i \sigma(a_i^t x)</math>
+<math>y = f(w, x) = \frac{1}{\sqrt{m}} \sum_{i=1}^{m} b_i \sigma(a_i^t x)</math><br>
-We use quadratic loss: <math>L(w) = \frac{1}{2} \sum_{i=1}^{n} (f(w, x_i) - y_i)^2</math>
+We use quadratic loss: <math>L(w) = \frac{1}{2} \sum_{i=1}^{n} (f(w, x_i) - y_i)^2</math><br>
 GD: <math>w(t+1) = w(t) - \eta_{t} \sum_{i=1}^{n} (f(w, x_i) - y_i) \nabla_w f(w_t, x_i)</math>
-Init N(0,1)
+# Init N(0,1)
-Our weights update along a trajectory: w(0), w(1), ...
+# Our weights update along a trajectory: w(0), w(1), ...
-Each <math>w</math> is a weight matrix.
+# Each <math>w</math> is a weight matrix.
 Empirical Observation: When the width of the network <math>m</math> is large, the trajectory of the gradient descent is ''almost'' static.
 This is called ''lazy'' training.
 * Not always the case! Especially for small <math>m</math>.
-Since the change in the model weights are not large, we can write the first-order taylor approximation:
+Since the change in the model weights are not large, we can write the first-order taylor approximation:<br>
-<math>f(w, x) \approx f(w_0, x) + \nabla_{w} f(w_0, x)^t (w - w_x) + ...</math>
+<math>f(w, x) \approx f(w_0, x) + \nabla_{w} f(w_0, x)^t (w - w_x) + ...</math><br>
-This model is linear in <math>w</math>.
+This model is linear in <math>w</math>.<br>
-<math>\phi(x) = \nabla_{w} f(w_0, x)</math>
+<math>\phi(x) = \nabla_{w} f(w_0, x)</math><br>
 The kernel <math>K = \langle \phi(x_i), \phi(x_j) \rangle</math> is called the ''Neural Tangent Kernel'' (NTK).
-Go back to our 2-layer NN:
+These features will not change during the optimization process because we use <math display="inline">w_0</math>
-<math>f_m(w, x) = \frac{1}{\sqrt{m}} \sum b_i \sigma(a_i^t x)</math>
-<math>\nabla_{a_i} f_m(w, x) = \frac{1}{\sqrt{m}} b_i \sigma'(a_i^t x) x</math>
+Go back to our 2-layer NN:<br>
+<math>f_m(w, x) = \frac{1}{\sqrt{m}} \sum b_i \sigma(a_i^t x)</math><br>
+<math>\nabla_{a_i} f_m(w, x) = \frac{1}{\sqrt{m}} b_i \sigma'(a_i^t x) x</math><br>
 <math>\nabla_{b_i} f_m(w, x) = \frac{1}{\sqrt{m}} \sigma(a_i^t x)</math>
-<math>K_{m}(x, x') = K_{m}^{(a)}(x, x') + K_{m}^{(b)}(x, x')</math>
+<math>K_{m}(x, x') = K_{m}^{(a)}(x, x') + K_{m}^{(b)}(x, x')</math><br>
-<math>K_{m}^{(a)}(x, x') = \frac{1}{m} \sum_{i=1}^{m} b_i^2 \sigma'(a_i^tx) \sigma'(a_i^tx) (x x')</math>
+<math>K_{m}^{(a)}(x, x') = \frac{1}{m} \sum_{i=1}^{m} b_i^2 \sigma'(a_i^tx) \sigma'(a_i^tx) (x x')</math><br>
 <math>K_{m}^{(b)}(x, x') = \frac{1}{m} \sum_{i=1}^{m} \sigma(a_i^t x) \sigma(a_i^t x')</math>
 * <math>a_i</math> and <math>b_i</math> are independent samples at initialization.
-Based on law of large numbers, as m goes to infinity,
+Based on law of large numbers, as m goes to infinity,<br>
-<math>K_{m}^{(a)}(x, x') \to K^{(a)}(x, x') = E \left[ b^2 \sigma'(a^t x) \sigma'(a^t x') (x x') \right]</math>
+<math>K_{m}^{(a)}(x, x') \to K^{(a)}(x, x') = E \left[ b^2 \sigma'(a^t x) \sigma'(a^t x') (x x') \right]</math><br>
 <math>K_{m}^{(b)}(x, x') \to K^{(b)}(x, x') = E \left[ \sigma(a^t x) \sigma(a^t x') \right]</math>
-<math>K^{(a)}(x, x') = \frac{(x x') E[b^2]}{2 \pi} (\pi - \theta(x, x))</math>
+<math>K^{(a)}(x, x') = \frac{(x x') E[b^2]}{2 \pi} (\pi - \theta(x, x))</math><br>
 <math>K^{(b)}(x, x') = \frac{\Vert x \Vert \Vert x' \Vert E[\Vert a \Vert^2]}{2 \pi d} ((\pi - \theta(x, x')) \cos(\theta) + \sin \theta)</math>
-;Q: When is this taylor approximation good?
+;Q: When is this taylor approximation good?<br>
 If the Hessian has bounded eigenvalues. (Hessian Control)
 ;Analyze GD:
-<math>\eta \to 0</math> Gradient-flow
+<math>\eta \to 0</math> Gradient-flow<br>
-<math>w(t+1) = w(t) - \eta \nabla_{w} L(w(t)) \implies \frac{w(t+1) - w(t)}{\eta} = - \nabla_{w} L(w(t))</math>
+<math>w(t+1) = w(t) - \eta \nabla_{w} L(w(t)) \implies \frac{w(t+1) - w(t)}{\eta} = - \nabla_{w} L(w(t))</math><br>
 <math>\to \frac{dw(t)}{dt} = -\nabla_{w} L(w(t))</math>
-<math>\frac{dw(t)}{dt} = -\nabla_{w} \hat{y}(w) (\hat{y}(w) - y)</math>
+<math>\frac{dw(t)}{dt} = -\nabla_{w} \hat{y}(w) (\hat{y}(w) - y)</math><br>
 <math>
 \begin{aligned}
@@ Line 615: / Line 618: @@
 </math>
-If we let <math>u = \hat{y} - y</math>, then <math>\frac{du}{dt} \approx -K(w_i) u</math>.
+If we let <math>u = \hat{y} - y</math>, then <math>\frac{du}{dt} \approx -K(w_i) u</math>.<br>
-This ODE implies <math>u(t) = u(0)\exp(-K(w_i)t)</math>.
+This ODE implies <math>u(t) = u(0)\exp(-K(w_i)t)</math>.<br>
 In the over-parameterized case, <math>K(w_0) > 0 </math> (positive definite).
@@ Line 1,724: / Line 1,727: @@
 We need pairs of similar images and dissimilar images.
-SimCLR [Chen ''et al.'' 2020]
+;SimCLR [Chen ''et al.'' 2020]
 # Create two correlated views of an image <math>x</math>: <math>\tilde{x}_i</math> and <math>\tilde{x}_j</math>.
 #* Random cropping + resize
@@ Line 1,745: / Line 1,748: @@
 Training is <math>\min_{f,g} L = \frac{1}{N} \sum_{k=1}^{N} \frac{l(2k-1,2k) + l(2k, 2k-1)}{2}</math>.
-==Misc==
+Practical observations:
-[[Visible to::users]]
+* Composition of data augmentation is important.
+* Larger batch sizes and longer training helps self-supervised learning!
+* Optimization over the MLP projection head <math>g</math> helps!
+** f is able to keep some information useful for the classification (e.g. color or orientation)
+Empirical results:
+* For image net, top-1 accuracy increases as number of parameters increases.
+* After learning embeddings <math>f</math>, you don't need much labeled data for supervised-training.
+===Theory of self-supervised learning===
+;[Arora ''et al.'' 2019]
+Modeling of similar pairs:
+* <math>x \sim D_{C1}(x)</math>
+* <math>x^+ \sim D_{C1}(x)</math> is semantically similar
+* <math>x^- \sim D_{C2}(x)</math> negative samples
+Downstream task:
+* pairwise classification
+** nature picks two classes C1, C2
+** generate samples from C1 & C2 and evaluate the classification loss
+* Assume <math>m \to \infty</math> so just look at population loss
+Notations:
+* <math>L_{sup}(f)</math> is the supervised learning loss with optimal last layer
+* <math>l=E_{(x, x^+) \sim D_{sim}, x^- \sim D_{net}}[\log(1+\exp(f(x)^T f(x^-) - f(x)^Tf(x^+)]</math> is a logistic loss
+Result 1:
+* <math>L_{sup}^{mean}(f) \leq \frac{1}{1-\tau}(L_{un}(f)-\tau)</math> for all <math>f \in F</math>
+** <math>\tau</math> is the collision probability for pair of random classes.
+Idea result:
+* <math>L_{sup}(f^*) \leq \alpha L_{sup}(f)</math> forall <math>f</math>.
+* In general it is impossible to get this.
+;[Tosh et al. 2020]
+This work connects self-supervised learning with multi-view representation theory.
+Start with (x, z, y) where x,z are two views of the data and y is the label.
+* x and z should share redundant information w.r.t. y. I.e. predicting y from x or z individually should be equivalent to predicting y from both.
+* <math>e_x = E(E[Y|X] - E[Y|X,Z])^2</math> and <math>e_z = E(E[Y|Z] - E[Y|X,Z])^2</math> should be small
+* Formulate contrastive learning as an artificial ''classification'' problem:
+** Classify between <math>(x, z, 1)</math> and <math>(x, \tilde{z}, 0)</math>.
+** <math>g^*(x,z) = \operatorname{argmin}(\text{classification loss}) = \frac{P_{X,Z}(x,z)}{P_X(x) P_Z(z)}</math>.
+* x,z have redundant info from Y so if we first predict z from x, then use the result to predict Y, we should be good
+<math>
+\begin{aligned}
+\mu(x) &= E[E[Y|Z]|X=x] \\
+&= \int E[Y|Z=z] P_{Z|X}(z|x) dz\\
+&= \int E[Y|Z=z] g^*(x,z) P_Z(z) dz
+\end{aligned}
+</math>
+Lemma: <math>E[(\mu(x) - E[Y | x,z])^2] \leq e_x + e_z + 2\sqrt{e_x e_z}</math>
+Landmark embedding: <math>g^*</math> is computed via contrastive learning.
+How to embed x? <math>\phi(x) = (g^*(x, z_1),...,g^*(x,z_m))</math>
+z's are randomly generated from <math>P_Z</math>.
+Each <math>g^*</math> output is a single number.
+* <math>\phi()</math> is good embedding if a linear classifier of <math>\phi</math> is approx <math>\mu(x)</math>.
+** I.e. <math>\exists w : w^t \phi(x) \approx \mu(x)</math>
+** <math>w^t \phi(x) = \frac{1}{m} \sum_{i=1}^{m} E[Y|Z_i] g^*(x, z_i) \stackrel{m \to \infty}{\rightarrow} \int E[Y|Z=z] g^*(x,z)P_{Z}*z dz = \mu(x)</math>
+Direct embedding
+Instead of learning a bivariate <math>g(x,z)</math>, learn <math>\eta(x)^T \psi(z)</math>.
+Lemma: For every <math>\eta: x \to \mathbb{R}^m</math> and <math>\psi:z \to \mathbb{R}^m</math>, there exists <math>w \in \mathbb{R}^m</math> such that <math>E[(w^t \eta(x) - \mu(x))^2] \leq E[Y^s \epsilon_{direct}(\eta, \psi)</math> where <math>\epsilon_{direct} = [(\eta(x)^t \psi(x) - g^*(x,z))^2</math>.
+Can we write <math>g^*(x,z)</math> as <math>\eta(x)^T \psi(z)</math>?
+* If there is a hidden variable H s.t. X and Z are conditionally independent given H.
+# Case 1: H is a discrete variable then <math>g^*(x, z) = \eta^*(x)^t \psi(z^*)</math>
+# Case 2: There exists <math>\eta,\psi</math> such that <math>E(\eta(x)^t \psi(z) - g^*(x,z))^2 \leq o(\frac{1}{m})</math>.
+==Meta Learning==
+In many applications, we don't have a large training dataset. However, humans can adapt and learn ''on the fly''.
+The key is to use prior knowledge in performing new tasks.
+How can we train AI/ML models similarly?
+Goal of meta learning: Train a model on different learning tasks such that it can solve new tasks using only a small number of training samples.
+Few shot classification problem:
+Inputs: <math>D_{metatrain} = \{(D_i^{train}, D_i^{test})\}_{i=1}^{n}</math>.
+<math>
+\begin{cases}
+D_{i}^{train} = \{(x_j^i, y_j^i) \}_{j=1}^{k}\\
+D_{i}^{test} = \{(x_j^i, y_j^i) \}_{j=1}^{k'}
+\end{cases}
+</math>
+* <math>i</math> is the index of the task
+* <math>j</math> is the index of the sample
+K-shot classification means k training samples per label. Some papers use k as the size of the whole training set.
+Q: How to use <math>D_{metatrain}</math> to solve meta-test tasks more effectively?
+* Use <math>D_{metatrain}</math> to learn meta parameters <math>\theta</math> such that:
+* Base learner <math>A</math> outputs task-specific model parameters <math>\phi_i = A(D_{i}^{train}, \theta)</math> good for <math>D_{i}^{test}</math>.
+* Training procedure:
+** Loss: <math>\min_{\theta} \sum_{i=1}^{n} loss(D_{i}^{test}, \phi_i)</math> where <math>\phi_i = A(D_{i}^{train}, \phi)</math>.
+* Test time: given a new task <math>(D^{train}, D^{test})</math>. Apply <math>A(D^{train}, \theta^*)</math> to get <math>\phi</math>.
+[Fin et al. 2017]
+* Idea: train a model that can be easy to fine-tune on new tasks
+* one-step fine tuning: <math>\theta - \alpha \nabla L(\theta, D_{i}^{train})</math> gives us <math>\phi_{i}</math>
+* Evaluate <math>\phi_i</math> on <math>D_i^{test}</math>
+* <math>\min_{\theta} \sum_{i=1}^{n} L(\phi_i, D_i^{test}) = L(\theta - \alpha \nabla L(\theta, D_i^{train}), D_i^{test})</math>
+. Model-agnostic meta learning (MAML)
+* Use GD to optimize over <math>\theta</math>
+** <math>\nabla_\theta = \sum_{i=1}^{n}(\nabla_{\theta} \phi_i) \nabla_{\phi} L(\phi_i, D_i^{test})</math>
+** <math>(\nabla_{\theta} \phi_i)</math> involves second-order derivatives which are expensive.
+* First-order MAML: just ignore <math>\nabla_{\theta} \phi_i</math> term. Replace it with the identity matrix.
+Reptile (Michal et al 2018)
+. <math>A</math> is a simple linear/non-parametric learning on data embeddings computed via <math>f_{\theta}</math>
+[Lee et al. 2019]
+* <math>f_{\theta}</math> is used to compute embeddings
+* <math>A</math> is a linear classifier (e.g. SVM)
+* Use dual form of SVM so # of optimization variables = # training samples * # classes
+. <math>A</math> is a non-parametric learning
+* Embedding: <math>\tilde{x} = f_{\theta}(x)</math>
+[Snell et al. 2017)
+* Define prototypes (cluster centers)
+** <math>c_k = \frac{1}{|D_i^{tr}|} \sum_{(x,y) \in S_k} f_{\theta}(x)</math>
+** <math>P_{\theta}(y=k|x) = \frac{\exp(-d(f_{\theta}(x), c_k))}{\sum_{k'} \exp(-d(f_{\theta}(x), c_k))}</math>
+: <math>A</math> is a black box (e.g. LSTM).
+==LSTMs and transformers==
+Lecture 23 (November 19, 2020)
+===Recurrent Neural Networks (RNNs)===
+Hidden state: <math>h_t = \tanh(W_{hh} h_{t-1} + W_{xh} x_t)</math>
+Prediction at t: <math>y_t = W_{hy} h_{t}</math>
+;Backpropagation through time
+If <math>W</math> has largest singular value < 1, then gradient vanishes.
+If <math>W</math> has largest singular value > 1, then gradient explodes.
+Typically, gradient vanishes because of initialization of <math>W</math>.
+===Long Short Term Memory (LSTMs)===
+Goal is to solve the vanishing and exploding gradient problem.
+LSTM has several gates:
+* Input gate: <math>i_t = \sigma(W_{xi}x_t + W_{hi}h_{t-1} + b_i)</math>
+* Forget gate: <math>f_t = \sigma(W_{xf}x_t + W_{hf}h_{t-1} + b_f)</math>
+* Output Gate: <math>o_t = \sigma(W_{xo}x_t + W_{ho}h_{t-1} + b_o)</math>
+* Cell state: <math>c_t = f_t \odot c_{t-1} + i_t \odot \tilde{c}_t</math>
+* Hidden state: <math>h_t = o_t \odot \tanh(c_t)</math>
+;Bidirectional RNNs
+First LSTM takes input in correct order.
+Second LSTM takes input in reverse order.
+Concatenate outputs from both LSTMs.
+===Attention===
+Goal: To help memorize long source sentences in machine translation.
+;Encoder-decoder attention
+;Self-Attention
+===Transformer===
+;Positional encoding
+;Self-Attention
+Have queries, keys, and values.
+Multiply queries with keys, pass through softmax. Then times values.
+Yields attention of every work with respect to another.
+Initially, transformers had n=8 heads giving 8 queries, keys, and values.
+;Architecture
+Stack encoders.
+==Interpretability==
+;Interpretability Methods
+* Built-in model interpretability
+* Feature level interpretability
+* Instance based explanations
+We will focus on feature level interpretability.
+===Feature Level Interpretability===
+These are given through saliency maps.
+* Perturbation-based: Perturb the input to get another output and compute the difference.
+* Gradient-based
+===Gradient-based Methods===
+Take the derivative of the output with respect to the input.
+;Limitations
+* Too local and sensitive to slight perturbations
+* Saturated outputs lead to unintuitive gradients
+* Discontinuous gradients are problematic
+;SmoothGrad
+* Add gaussian noise to input and average the gradient.
+;Integrated Gradients
+* Average the gradients along path from baseline to input.
+;DeepLift
+* We don't care about gradient but the slope relative to the ''reference'' state
+;Limitations
+* Models must be able to compute the gradient of the output with respect to the input
+* Interpretation of neural networks is fragile
+** Saliency maps are uninterpretable for adversarial examples on clean models and adversarially trained models.
+* Needs white-box gradient access to the model.
+===Evaluation of interpretability methods===
+* Human evaluation
+** Can humans evaluate saliency?
+* Accuracy drop after removing ''salient'' features
+* Sanity checks
+** Model parameter randomization test - compare output of saliency method on trained vs untrained method to make sure saliency depends on model parameters.
+* Synthetic Data
+* Data randomization test
+** Train on random labels and see if saliency depends on relationship between input & output.
+Temporal saliency Rescaling
+* If you remove this feature, how is the gradient going to change.
+==Reinforcement Learning==
+Lecture 27 (December 1, 2020)
+;Goal: Sequential decision making problems
+* Current decisions can affect future decisions
+Examples: games, robotics, self-driving, finance
+Agent takes an action <math>a_t</math> in the environment which leads to a new state <math>\delta_{t+1}</math>.
+;Definitions
+* <math>S</math> is the state space. This can be discrete or continuous.
+* <math>O</math> are the observations.
+* <math>A</math> is the set of actions which can be discrete or continuous.
+* <math>P</math> are the transition probabilities which model the environment.
+Example:
+* <math>|S|</math> = 2
+* <math>|A|</math> = 2
+In general, <math>S_t \sim P(S_t=s | s_0, a_0, s_1, a_1, ..., a_{t-1})</math>.
+The given variables are the trajectory <math>T_{t-1}</math>.
+Modeling this can be very complex for large <math>T</math>.
+Thus, we apply a markov assumption:
+* <math>S_t \sim P(S_t=s | a_{t-1}, s_{t-1})</math>
+Objective: To maximize the rewards
+* <math>R_t \stackrel{\triangle}{=} R(s_t, a_t)</math>
+Two regimes for rewards
+* Finite horizon; T is finite
+* Infinite horizon; T goes to infinity
+* Discount factor
+===Classical RL===
+Finite Markov Decision Process (MDP)
+* S: Finite number of states
+* A: Finite number of actions
+* <math>R_t = R(a_t, S_t)</math>
+* r discount factor
+Goal: Choose actions to maximize the total rewards.
+* Policy function <math>\pi</math> determines how actions should be taken.
+* The policy function could yield a dist on actions or can be deterministic. We will focus on deterministic actions.
+* Assume this is time invariant.
+* Value function determines the expected reward if we start at state s and follow policy <math>\pi</math>.
+* Q function (action-value function)
+** <math>Q_\pi (s, a) = E[R_0 + \gamma R_1+... | A_0=a, S_0=s, \pi]</math>
+* If we have <math>\pi</math>, can we evaluate <math>V_{\pi}(s)</math>?
+** <math>V_{\pi}(s) = E[R_0 + \gamma R_1 + ... | S_0 = s, \pi]</math>
+* Use Bellman's equation
+** <math>V_{\pi}(s) = R_0 + \gamma \sum_{s'} P_{\pi}(s' | s) V_{\pi}(s')</math>
+** <math>V_{\pi} = R + \gamma P_{\pi} V_{\pi}</math>
+** <math>V_{\pi} = (I - \gamma P_{\pi})^{-1}R</math>
+** <math>P_{\pi}</math> is a stochastic matrix. It is not symmetric so eigenvalues can be complex.
+All eigen values are <math>\Vert \lambda_i \Vert \leq 1</math> with one eigenvalue norm exactly 1.
+This implies <math>I - \gamma P_{\pi}</math> is always invertible.
+However, computing the inverse is computationally expensive since it scales with <math>|S|^3</math>.
+===Value-Iteration (Bellman Recursion)===
+<math>V_{\pi} = R + \gamma P_{\pi} V_{\pi}</math>
+Define an operator <math>L_{\pi}v = R + \gamma P_{\pi}v</math>
+<math>v_pi = L_{\pi} v_{\pi}</math> so <math>v_{\pi}</math> is a fixed point to <math>L_{\pi}</math>.
+;Claim: <math>L_{\pi}</math> is a contraction.
+Proof: Take <math>v_1, v_2 \in \mathbb{R}^d</math>.
+<math>
+\begin{aligned}
+L_{\pi}v_1 - L_{\pi}v_2 &= (R + \gamma P_{\pi}v_1) - (R + \gamma P_{\pi}v_2) \\
+&= \gamma P_{\pi}(v_1 - v_2)
+\end{aligned}
+</math>
+<math>
+\begin{aligned}
+\Vert L_{\pi}v_1 - L_{\pi}v_2 \Vert &= \Vert \gamma P_{\pi}(v_1 - v_2) \Vert\\
+& \leq \gamma \Vert P_{\pi} \Vert \Vert v_1 - v_2 \Vert \\
+& \leq \Vert v_1 - v_2 \Vert
+\end{aligned}
+</math>
+By Banach's Fixed Point Theorem, we can converge to the fixed point iteratively by repeatedly applying <math>L_{\pi}</math>.
+===Optimal Policy===
+Elementwise max: <math>\max_{\pi} v_{\pi}(s)</math> leads to the optimal policy <math>\pi^*(s)</math>.
+Bellman (1957) showed that for an MDP, there exists an optimal policy that is deterministic and such that <math>v_{\pi^*}(s) \geq v_{\pi}(s)</math> for all <math>s, \pi</math>.
+The policy may not be unique but if two policies are equal, they have the same value function.
+Intermediate questions:
+* Given <math>v^*</math>, can we compute <math>Q^*</math>?
+<math>
+\begin{aligned}
+Q^*(s, a) &= \max_{\pi} Q(s,a)\\
+&= \max_{\pi} R(s) + \gamma \sum_{s'} p(s' | s,a) v_{\pi}(s')\\
+&= R(s) + \gamma \sum_{s'} P(s' | s, a) \max_{\pi}(v_{\pi}(s'))\\
+\end{aligned}
+</math>
+* Given <math>Q^*</math>, can we compute <math>v^*</math>.
+Bellman's optimality states that <math>v^*(s) = \max_{a} Q^*(s, a)</math>.
+<math>\pi^*(s) = \operatorname*{argmax}_{a} Q^*(s, a)</math>
+;How to compare optimal policies?
+First approach: Value Iteration
+<math>V^*(s) = \max_{a} Q^*(s, a) = \max_{a} [R(s) + \gamma \sum_{s'} p(s'|s,a) v^*(s')]</math>
+Define an operator: <math>L(v) = \max_{\pi}[R + \gamma P_{\pi} v]</math>
+<math>v^*</math> is a fixed point of <math>L(v)</math>.
+Claim: <math>L</math> is also a contraction.
+Thus <math>v^*</math> can be computed by repeated application of <math>L</math>.
+===Value Iteration===
+* Start with a random <math>v^{(0)} \in \mathbb{R}^d</math>
+* <math>v^{(r+1)}(s) = \max_{a} [R(s) + \gamma \sum_{s'} p(s'|s,a) v^{r}(s')]</math>
+* Repeat until <math>\Vert v^{(r+1)} - v^{(r)} \Vert \leq \epsilon</math>
+===Policy Iteration===
+Directly optimize policy.
+* Start with a random policy <math>\pi^{(0)}</math>
+* Evaluate the policy <math>v_{\pi}</math> using closed form or value iteration
+* Evaluate the Q-Function <math>Q_{\pi}(s, a) = R(s) + \gamma \sum_{s'} P(s'|s,a) V_{\pi}(s')</math>
+* Find the best action to be taken at state <math>s</math>:
+** <math>a^*(s) = \operatorname{argmax}_{a} Q_{\pi}(s, a)</math>
+* Update <math>\pi</math> using <math>\pi(s) = a^*(s)</math>
+* Repeat until convergence.
+Policy iteration is guaranteed to converge to an optimal policy.
+Oftentimes converges faster than value iteration.
+===Deep Reinforcement Learning===
+;Relaxing some unrealistic assumptions
+# Evaluate <math>v_{\pi}(s)</math>
+#* <math>Q_{\pi}(s, a) = R(s, a) + \gamma E_{s' \sim P(s'|s,a)} v_{\pi}(s')</math>
+# Improve the policy
+#* <math>\operatorname{argmax}_{a_t} Q_{\pi}(s_t, a_t)</math>
+#* Assumption <math>|S|=d</math>
+# How to represent <math>V(s)</math>?
+#* Can we use a neural network to represent <math>V(s)</math>? Yes
+;How to train <math>v_{\phi}</math>?
+* Start with an old <math>v_{\phi}</math>, compute <math>Q_{\pi}(s,a)</math>.
+** <math>Q_{\pi}(s,a) = R(s,a) + \gamma E[v_{\phi}^{old}(s')]</math>
+* Fit <math>v_{\phi}</math> to <math>\max_{a}Q_{\pi}(s,a)</math> using a quadratic loss:
+** <math>L(\phi) = \frac{1}{2} \Vert V_{\phi}(s) - \max_{a} Q_{\pi}(s,a) \Vert^2</math>
+* Iterate
+;Similarly we can parameterize the Q function
+Compare target <math>y_i \leftarrow R(s_i, a_i) + \gamma E[v_{\phi}(s_i')]</math>
+We need to know transition probabilities <math>P(s'|s,a)</math> to compute the expectation (model-based RL).
+With model-free RL:
+We can approximate as <math>E[v(s_i')] \approx v(s_i') = \max_a Q(s_i', a')</math>
+This is called ''Q-Learning''.
+;What if we have continuous actions:
+* Approach 1: Use a function class such that <math>\max_{a}Q(s,a)</math> is easy to solve
+** [Gu ''et al.'' 2016] use quadratic functions.
+* Approach 2: Learn another network to approximate the maximizer: <math>\max_{a} Q(s,a')</math>
+===Policy Gradient Method===
+Lecture 29 (Dec 8, 2020)
+Probability of observing a trajectory:
+<math>P_{\theta}(\tau) = P(s_1) \prod_{t=1}^{\tau} \pi_{\theta}(a_t | s_t) P(s_{t+1} | s_t, a_t)</math>
+Reward for a trajectory:
+<math>R(\tau_1) = R(s_1, a_1) + R(s_2, a_2) + ... + R(s_t, a_t)</math>
+The average reward is:
+<math>J(\theta) = E[R(\tau)] = \sum E[R(s_t, a_t)]</math>
+Our goal is to maximize the average reward: <math>\max_{\theta} J(\theta)</math>.
+Gradient of the average reward:
+<math>
+\begin{aligned}
+\nabla_{\theta} J(\theta) &= \nabla_{\theta} E[R(\tau)] \\
+&= \nabla_{\theta} \int P_{\theta}(\tau) R(\tau) d\tau \\
+&= \int \nabla_{\theta} P_{\theta}(\tau) R(\tau) d\tau \\
+&= \int P_{\theta}(\tau) \nabla_{\theta} \log P_\theta(\tau) R(\tau) d\tau\\
+&= E[\log P_\theta(\tau) R(\tau)]
+\end{aligned}
+</math>
+<math>
+\nabla_{\theta} \log P_{\theta}(\tau) = \sum_{t=1}^{\tau} \nabla_{\theta} \log \pi_{\theta}(a_t | s_t)
+</math>
+Implies,
+<math>
+\begin{aligned}
+\nabla_{\theta} J(\theta) &= E[...]\\
+&\approx \frac{1}{N} \sum_{i=1}^{N}(\sum \nabla_{\theta} \log \pi(a_t^{(i)} | s_t^{(i)}) ....
+\end{aligned}
+</math>
+;Summary
+* Sample trajectories
+* Approximate <math>\nabla_{\theta} J(\theta)</math>
+* <math>\theta \leftarrow \theta + \alpha \nabla J(\theta)</math>
+;Intuition
+<math>E[\nabla_{\theta} \log P_{\theta}(\tau) R(\tau)]</math>
+Formalizing ''trial & error''.
+[Finn & Levin, ICML]
+===Issues with Policy Gradient===
+;High variance of gradient estimation
+;Solutions
+* Subtract a baseline
+<math>b = \frac{1}{N} \sum_{i=1}^{N} R(\tau^{(i)})</math>
+* Reward-to-go
+<math>
+\begin{aligned}
+\nabla_{\theta} J(\theta) &\approx \frac{1}{N} \sum_{i=1}^{N} \nabla_{\theta} \log P_{\theta}(\tau) R(\tau)\\
+&= \frac{1}{N} \sum_{i=1}^{N} \left(\sum_{t=1}^{T} \nabla_{\theta} \pi_{\theta}(a_t^{(i)}| s_t^{(i)}) \right) \left(\sum_{t'=1}^T R(s_{t'}^{(i)}, a_{t'}^{(i)})\right)\\
+&= \frac{1}{N} \sum_{i=1}^{N} \sum_{t=1}^{T} \nabla_{\theta} \pi_{\theta}(a_t^{(i)}| s_t^{(i)}) \left(\sum_{t'=1}^T R(s_{t'}^{(i)}, a_{t'}^{(i)})\right)\\
+&\approx \frac{1}{N} \sum_{i=1}^{N} \sum_{t=1}^{T} \nabla_{\theta} \pi_{\theta}(a_t^{(i)}| s_t^{(i)}) \left(\sum_{t'=t}^T R(s_{t'}^{(i)}, a_{t'}^{(i)})\right)\\
+\end{aligned}
+</math>
+;Some parameters can change <math>\pi_{\theta}</math> more than others so it's hard to choose a fixed learning rate.
+Use natural policy gradient: <math>\theta' \leftarrow \theta - \eta F^{-1}\nabla L(\theta)</math>
+===Actor-critic algorithms===
+Have an actor <math>\pi_{\theta}</math>.
+Have a critic <math>V_{\phi}/Q</math>
+<math>\nabla_{\theta} J(\theta) \approx \frac{1}{N} \sum_{i=1}^{N} \sum_{t=1}^{T} \nabla_{\theta} \log \pi_{\theta}(a_t^{(i)} | s_t^{(i)}) \left(Q(s_t^{(i)}, a_t^{(i)} - V(s_t^{(i)})\right)</math>
+===Other topics in RL===
+* Inverse RL
+* Multi-agent RL
+* Model-based RL
+==Summary of Course==
+;What we covered
+* Supervised DL
+* Unsupervised DL (GANs, VAEs)
+* Self-supervised DL
+* Meta-Learning
+* Learning with Attention (Transformers)
+* Deep RL
+* Optimization
+* Generalization
+* Robustness
+* Interpretability
+;What we didn't cover
+* Fairness
+* Privacy & Ethics
+* Bayesian DL
+* Federated Learning
+* Graph NNs
+;Things which may be on the final
+* Transformers
+* Wasserstein distance
+* Kernel methods
 ==Resources==