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\begin{document}
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\chapter{Lecture 15 - 04-05-2020}
|
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||||
\section{Regret analysis of OGD}
|
||||
We introduce the \bred{Regret}.
|
||||
\\
|
||||
$$
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||||
\frac{1}{m} \ \sum_{t=1}^{T} \ell_t(w_t) - \frac{1}{T} \ \sum_{t=1}^{T} \ell_t(u^*_t)
|
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$$
|
||||
$$
|
||||
(x_1,y_1) ...(x_t,y_t) \qquad \ell_t(w) = \left( w^T \, x_t - y_t\right)^2
|
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$$
|
||||
we build a loss function for example with the square loss.
|
||||
\\
|
||||
The important thing is that $\ell_1 , \ell_2, ...$ is a sequence of \textbf{convex losses}.
|
||||
\\\\
|
||||
In general we define the regret in this way:
|
||||
$$
|
||||
R_T(u) \ = \ \frac{1}{m} \ \sum_{t=1}^{T} \ell_t(w_t) - \frac{1}{T} \ \sum_{t=1}^{T} \ell_t(u_t)
|
||||
$$\\
|
||||
The Gradiant descent is one of the simplest algorithm for minimising a convex function. We recall the iteration did by the algorithm:
|
||||
$$
|
||||
w_{t+1} \leftarrow w_t - \eta_t \nabla f(w_t) \qquad \eta_t > 0 \textbf{ learning rate} \quad \textit{f convex}
|
||||
$$
|
||||
$$
|
||||
f: \barra{R}^d \rightarrow \barra{R} \textit{ that's why use the gradiand instead of the derivative}
|
||||
$$
|
||||
Learning rate can depend on time and we approach the region of the function f where the region is 0. We keep on moving in the X axes in the direction where the function is decreasing.
|
||||
\newpage
|
||||
\subsection{Projected OGD}
|
||||
|
||||
2 parameters: $\eta > 0$ and $U > 0$
|
||||
\\
|
||||
Initialisation: $ w_1 = (0,...,0)$
|
||||
\\
|
||||
For $t = 1,2,...$
|
||||
|
||||
1) \blue{Gradiant step}:
|
||||
$$w'_{t+1} = w_t - \frac{\eta}{\sqrt[]{t}} \ \nabla \ell_t(w_t) \qquad (x_t, y_t) \rightsquigarrow \ell_t
|
||||
$$
|
||||
\begin{figure}[h]
|
||||
\centering
|
||||
\includegraphics[width=0.3\linewidth]{../img/lez15-img1.JPG}
|
||||
\caption{}
|
||||
%\label{fig:}
|
||||
\end{figure}\\
|
||||
|
||||
2) \blue{Projection step}: $$
|
||||
w_{t+1} = arg \min_{w: \| w \| \leq U} \| w- w'_{t+1} \| $$ \textbf{Projection of $w'_{t+1}$ onto the ball of radius $U$}.
|
||||
\begin{figure}[h]
|
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\centering
|
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\includegraphics[width=0.3\linewidth]{../img/lez15-img2.JPG}
|
||||
\caption{}
|
||||
%\label{fig:}
|
||||
\end{figure}\\
|
||||
Now we define the Regret:
|
||||
$$
|
||||
U^*_T = arg \min_{U \in \barra{R}^d \ \|U \| \leq U} \frac{1}{T} \ \sum_{t=1}^{T} \ell_t(U)
|
||||
$$
|
||||
We are interested in bounding the regret $R_T\left(U^*_T\right) $
|
||||
\\\\
|
||||
I will Fix $\ell_1, ... \ell_t$ \qquad let $U= U^{\divideontimes}_T$ for $U$.
|
||||
\\
|
||||
Taylor's theorem for multivariate functions\\
|
||||
Let's look a univariate first \quad $f: \barra{R} \rightarrow \barra{R}$ \textit{( has to be twice differentiable) }\qquad $w, u \in \barra{R}$
|
||||
$$
|
||||
f(u) = f(w) + f'(w) \, (u-w) + \frac{1}{2} \, f^"(x) \, (u-w)^2
|
||||
$$
|
||||
\\\\
|
||||
For the multivariate case:
|
||||
\\
|
||||
$f: \barra{R}^d \rightarrow \barra{R} \quad$ twice differentiable \ $ \forall u, w \in \barra{R}^d$
|
||||
$$
|
||||
f(u) = f(w) + \nabla f(w)^T \, (u-w) + \frac{1}{2} \left(u-w\right)^T \nabla^2 f(\xi) \, (u-w)
|
||||
$$
|
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where $\xi$ is some point on the segment goining $u$ and $w$.
|
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We have the Hessian matrix of $f$:
|
||||
$$
|
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\nabla^2 f(x)_{ij} \ = \ \frac{\partial^2 f(x)}{\partial x_i \, \partial x_j} \vert x=x_i
|
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$$
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If $f$ is convex then, $\nabla^2 f$ is positive and semidefinite.\\
|
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$
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\forall x \in \barra{R}^d \quad \forall z \in \barra{R}^d \qquad z^T \, \nabla^2 f(x) \, z \geq 0
|
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$
|
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\begin{figure}[h]
|
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\centering
|
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\includegraphics[width=0.3\linewidth]{../img/lez15-img3.JPG}
|
||||
\caption{}
|
||||
%\label{fig:}
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||||
\end{figure}\\
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|
||||
min. 38:38
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\end{document}
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Chapter 16.
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Chapter 17.
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Output written on main.pdf (105 pages, 2454637 bytes).
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Output written on main.pdf (106 pages, 2473242 bytes).
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938 PDF objects out of 1000 (max. 8388607)
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351 words of extra memory for PDF output out of 10000 (max. 10000000)
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@ -69,10 +69,12 @@
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\contentsline {subsection}{\numberline {14.2.1}Online Learning }{93}%
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\contentsline {subsection}{\numberline {14.2.2}Online Gradiant Descent (OGD)}{95}%
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\contentsline {chapter}{\numberline {15}Lecture 15 - 04-05-2020}{96}%
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\contentsline {chapter}{\numberline {16}Lecture 16 - 05-05-2020}{97}%
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\contentsline {chapter}{\numberline {17}Lecture 17 - 11-05-2020}{98}%
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\contentsline {chapter}{\numberline {18}Lecture 18 - 12-05-2020}{99}%
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\contentsline {section}{\numberline {18.1}Kernel functions}{99}%
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\contentsline {subsection}{\numberline {18.1.1}Feature expansion}{99}%
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\contentsline {subsection}{\numberline {18.1.2}Kernels implements feature expansion (Efficiently}{100}%
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\contentsline {section}{\numberline {18.2}Gaussian Kernel}{101}%
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\contentsline {section}{\numberline {15.1}Regret analysis of OGD}{96}%
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\contentsline {subsection}{\numberline {15.1.1}Projected OGD}{97}%
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\contentsline {chapter}{\numberline {16}Lecture 16 - 05-05-2020}{98}%
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\contentsline {chapter}{\numberline {17}Lecture 17 - 11-05-2020}{99}%
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\contentsline {chapter}{\numberline {18}Lecture 18 - 12-05-2020}{100}%
|
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\contentsline {section}{\numberline {18.1}Kernel functions}{100}%
|
||||
\contentsline {subsection}{\numberline {18.1.1}Feature expansion}{100}%
|
||||
\contentsline {subsection}{\numberline {18.1.2}Kernels implements feature expansion (Efficiently}{101}%
|
||||
\contentsline {section}{\numberline {18.2}Gaussian Kernel}{102}%
|
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