Yuhan Wang

Python basics

Fri, 01 Jan 2021 00:00:00 +0000

Build a foundation in Python.

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Quiz

What is the difference between lists and tuples?

Lists

Lists are mutable - they can be changed
Slower than tuples
Syntax: a_list = [1, 2.0, 'Hello world']

Tuples

Tuples are immutable - they can’t be changed
Tuples are faster than lists
Syntax: a_tuple = (1, 2.0, 'Hello world')

Is Python case-sensitive?

Yes

Visualization

Fri, 01 Jan 2021 00:00:00 +0000

Learn how to visualize data with Plotly.

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Quiz

When is a heatmap useful?

Lorem ipsum dolor sit amet, consectetur adipiscing elit.

Write Plotly code to render a bar chart

import plotly.express as px
data_canada = px.data.gapminder().query("country == 'Canada'")
fig = px.bar(data_canada, x='year', y='pop')
fig.show()

Statistics

Fri, 01 Jan 2021 00:00:00 +0000

Introduction to statistics for data science.

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The general form of the normal probability density function is:

$$ f(x) = \frac{1}{\sigma \sqrt{2\pi} } e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2} $$

The parameter $\mu$ is the mean or expectation of the distribution. $\sigma$ is its standard deviation. The variance of the distribution is $\sigma^{2}$.

Quiz

What is the parameter $\mu$?

The parameter $\mu$ is the mean or expectation of the distribution.

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StyleInV: A Temporal Style Modulated Inversion Network for Unconditional Video Generation

Thu, 20 Jul 2023 00:00:00 +0000

Bilevel Programming Derivatives of LDS-GNN

Wed, 08 Dec 2021 00:00:00 +0000

Introduction

Now we have a bilevel programming problem:

$$\min_{\theta, w_{\theta}} F(w_{\theta},\theta)\mathrm{\ such\ that\ } w_{\theta}\in \arg \min_{w} L(w, \theta) \tag{1}$$

where for the inner problem, $w$ is learned by

$$ w_{\theta, T} = \Phi(w_{\theta, T-1}, \theta) = \Phi(\Phi(w_{\theta,T-2}, \theta), \theta) = \cdots \tag{2} $$

As is stated in the paper, we follow a process of $\tau$ times inner, once outer. In such process we can directly compute the gradients of the outer params, as $\theta$ in our problem.

Therefore, our major concern now is to compute $\nabla_{\theta}F(w_{\theta, T}, \theta)$, with which we can update $\theta$ using Adam or SGD optimizer.

Derivation

We have inner param $w \in \mathbb{R}^{N_w}$ and outer param $\theta \in \mathbb{R}^{N_{\theta}}$. $\nabla_{\theta}F(w_{\theta, T}, \theta)$ can be written as

$$ \frac{\partial F(w_{\theta, T}, \theta)}{\partial \theta} = F_w(w_{\theta, T}, \theta)\cdot \nabla_{\theta}w_{\theta, T} + F_{\theta}(w_{\theta,T},\theta) \tag{3} $$

Here $F_w$ and $F_{\theta}$ can both be easily computed using PyTorch, and $\nabla_{\theta}w_{\theta, T}$ builds the connection to inner updater $\Phi$. As the inner param $w$ undergoes $\tau$ times of update, we have to derive its recursive formula:

$$ \begin{aligned} \nabla_{\theta}w_{\theta,T} &= \frac{\partial \Phi(w_{\theta,T-1}, \theta)}{\partial \theta} \\
&= \Phi_w(w_{\theta,T-1}, \theta)\cdot \nabla_{\theta}w_{\theta,T-1}+\Phi_{\theta}(w_{\theta,T-1},\theta) \\
&= D_{T-1} \cdot \nabla_{\theta}w_{\theta,T-1} + E_{T-1} \end{aligned} \tag{4} $$

Here $D$ and $E$ denote the jacobian matrix of $\Phi$ w.r.t $w$ and $\theta$ respectively, who can be efficiently computed via Auto Grad. Therefore, $D \in \mathbb{R}^{N_w \times N_w}$, $E \in \mathbb{R}^{N_w \times N_{\theta}}$ and $\nabla_{\theta}w \in \mathbb{R}^{N_w \times N_{\theta}}$. The dimension of $(4)$ matches.

If we combine equation $(3)$ and $(4)$, we can derive the equation between $\nabla_{\theta}F(w_{\theta, T}, \theta)$ and $\nabla_{\theta}w_{\theta,T-1}$. Then if we repeat the process of $(4)$ we can get the equation between $\nabla_{\theta}F(w_{\theta, T}, \theta)$ and $\nabla_{\theta}w_{\theta,T-2}$, $\nabla_{\theta}w_{\theta,T-3}$, and so on. Therefore, as is done by LDS-GNN paper, we can try to derive the equation between $\nabla_{\theta}F(w_{\theta, T}, \theta)$ and jacobian matrix $\nabla_{\theta}w_{\theta,t}$ of any round as,

$$ \frac{\partial F(w_{\theta, T}, \theta)}{\partial \theta} = P_t \cdot \nabla_{\theta}w_{\theta, t} + G_t \tag{5} $$

From equation $(3)$ we know that $P_T = F_w(w_{\theta,T},\theta)$ and $G_T = F_{\theta}(w_{\theta,T}, \theta)$, which are both easy to compute. And if equation $(5)$ holds for $t$, we can derive that for $t-1$ as

$$ \begin{aligned} \frac{\partial F(w_{\theta, T}, \theta)}{\partial \theta} &= P_t \cdot \nabla_{\theta}w_{\theta, t} + G_t \\
&= P_t \cdot (D_{t-1} \cdot \nabla_{\theta}w_{\theta,t-1} + E_{t-1}) + G_t \\
&= (P_t \cdot D_{t-1}) \cdot \nabla_{\theta}w_{\theta,t-1} + (P_t \cdot E_{t-1} + G_t) \\
&= P_{t-1} \cdot \nabla_{\theta}w_{\theta, t-1} + G_{t-1} \end{aligned} \tag{6} $$

Hence, $P_{t-1} = P_{t} \cdot D_{t-1}$ and $G_{t-1} = P_{t} \cdot E_{t-1} + G_{t}$.

All we need to do is keep deriving it until $t=T-\tau$, where $\nabla_{\theta} w_{\theta, T-\tau} = \mathbf{0}$ as $w_{\theta, T-\tau}$ has nothing to do with current $\theta$.

$$ \begin{aligned} \frac{\partial F(w_{\theta, T}, \theta)}{\partial \theta} &= P_{T-\tau} \cdot \nabla_{\theta} w_{\theta, T-\tau} + G_{T-\tau} \\
&= P_{T-\tau} \cdot \nabla_{\theta} w_{\theta, T-\tau} + (P_{T-\tau+1} \cdot E_{T-\tau} + G_{T-\tau+1}) \\
&= P_{T-\tau+1} \cdot E_{T-\tau} + G_{T-\tau+1} \end{aligned} \tag{7} $$

Finally, Eq. $(7)$ gives the final formula of $\nabla_{\theta}F(w_{\theta, T}, \theta)$ in a recursion form. Derivations above exactly correspond to Algorithm 2 in the supplementary material of LDS-GNN.

HifiFace: 3D Shape and Semantic Prior Guided High Fidelity Face Swapping

Sun, 25 Jul 2021 00:00:00 +0000

CurricularFace: Adaptive Curriculum Learning Loss for Deep Face Recognition

Mon, 30 Mar 2020 00:00:00 +0000

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Tue, 05 Feb 2019 00:00:00 +0000

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Mon, 01 Jan 0001 00:00:00 +0000

Yuhan Wang

Python basics

Learn

Quiz

Visualization

Learn

Quiz

Statistics

Learn

Quiz

Recent Posts

Projects

Recent & Upcoming Talks

Featured Publications

Popular Topics

Example Talk

StyleInV: A Temporal Style Modulated Inversion Network for Unconditional Video Generation

Bilevel Programming Derivatives of LDS-GNN

Introduction

Derivation

HifiFace: 3D Shape and Semantic Prior Guided High Fidelity Face Swapping

CurricularFace: Adaptive Curriculum Learning Loss for Deep Face Recognition

Slides

Create slides in Markdown with Wowchemy

Features

Controls

Code Highlighting

Math

Fragments

Speaker Notes

Themes

Custom Slide

Custom CSS Example

Questions?

Example Project