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\subtext{Introduction to Machine Learning (2026), p. 90} +\end{center} + +\newpage + +\subsection{Convergence Analysis} + +\textbf{Question}: When \& how fast does GD converge to $\hat{w}$? + +\definition \textbf{Linear Convergence}\\ +$\forall t:$ $\Vert w^{(t)} - \hat{w} \Vert \leq C \cdot \rho^t$ for some $C > 0, 0 < \rho < 1$.\\ +\subtext{Called linear, since we multiply with a fixed $\rho < 1$ at each step.} + +{\footnotesize + \remark Linear convergence is exponential in terms of $t$. +} + +\theorem \textbf{Linear Convergence} (GD)\\ +GD converges linearly to the optimal solution $\hat{w}$: +$$ + \hat{w} = \underset{w\in\R^d}{\text{arg min}} L(w) +$$ +\subtext{Where $\text{rank}(\mathbf{X}^\top \mathbf{X})$ is full and $\mu < \displaystyle\frac{2}{\lambda_\text{max}(\mathbf{X}^\top \mathbf{X})}$} + +Convergence \& convergence rate depend on $\eta$.\\ +The optimal $\eta$ depends on $\mathbf{X}$: +$$ + \eta_\text{opt} = \frac{2}{\lambda_\text{max} + \lambda_\text{min}} +$$ +\subtext{Where $\lambda_\text{max} = \lambda_\text{max}(\mathbf{X}^\top \mathbf{X})$ and $\lambda_\text{min} = \lambda_\text{min}(\mathbf{X}^\top \mathbf{X})$} + +\definition \textbf{Condition Number} $\kappa$ +$$ + \kappa := \frac{\lambda_\text{max}}{\lambda_\text{min}} +$$ +The optimal $\rho$ is $\rho_\text{min}=\frac{\kappa-1}{\kappa+1}$. Thus, the convergence rate is determined only be the eigenvalues of $\mathbf{X}^\top \mathbf{X}$, i.e. $\kappa$. + +\begin{center} + \begin{tabular}{l|l|l} + $\kappa \approx 1$ & well-conditioned & fast convergence \\ + \hline + $\kappa >> 1$ & ill-conditioned & slow convergence \\ + \end{tabular} +\end{center} + +{\footnotesize + \remark \textbf{Geometrical Interpretation of $\kappa$}\\ + Considering the ellipsoids $L^{-1}(c) = \{w \in \R^d: L(w) = c\}$, i.e. the contour lines of $L$: For a large $\kappa$, the ellipsoids are distorted (large axis ratio). Equivalently for small $\kappa$, the ellipsoids are nearly spherical.\\ +} + +\newpage +\subsection{Stochastic Gradient Descent} + +\method \textbf{Minibatch Stochastic Gradient Descent} + +The main cost of GD stems from calculating \& storing\\ +$\nabla_w\ l\Bigl(f_w(x_i), y_i\Bigr)$ for all $x_i$, at every step. + +We instead use only $\mathcal S \subset \{1,2,\ldots,n\}$, selected randomly: +$$ + \nabla L_\mathcal{S}(w) = \frac{1}{\mathcal{S}}\sum_{i\in\mathcal S} \nabla_w l\Bigl(f_w(x_i), y_i\Bigr) +$$ + +{\footnotesize + \remark if $|\mathcal{S}|=1$, this is called \textit{Stochastic Gradient Descent}. +} + +\begin{algorithm} + \caption{Stochastic Minibatch Gradient Descent} + $t \gets 0$ \; + $w^{(0)} \gets w_\text{initial}$ \; + \SetKwRepeat{Do}{repeat}{until} + \Do{\text{$\Vert w^{t} - w^{t+1} \Vert < \epsilon$}}{ + $\mathcal{S} \gets \mathcal{S} \subset \{1,\ldots,n\}$ randomly \; + $w^{(t+1)} \gets w^{(t)} - \eta \nabla L_\mathcal{S}(w^{(t)})$ \; + $t \gets t+1$ + } + \textbf{return} $w^{(t)}$ +\end{algorithm} + +Note how it no longer holds that the direction of SGD is a descent direction, however: + +\lemma \textbf{Descent Direction in Expectation}\\ +\smalltext{Consider $|\mathcal S|=k$ s.t. $\mathcal S \overset{\text{i.i.d.}}{\subset} \{1,\ldots,n\}$. (duplicate selections possible)}\\ +\subtext{$\mathcal{S}=\{I_1,\ldots, I_k\}$ s.t. $I_l = I_m$ is possible for $l\neq m$} +\begin{align*} + \E_\mathcal{S}\Bigl[ \nabla L_\mathcal{S}(w) \Bigr] &= \E_\mathcal{S} \Biggl[ \frac{1}{k}\sum_{j=1}^{k}\nabla_w l\bigl( f_w(x_{I_j}), y_{I_j} \bigr) \Biggr] & (\text{def. } \nabla L_\mathcal{S}) \\ + &= \frac{1}{k}\sum_{j=1}^{k} \E_{I_j}\Bigl[ \nabla_w l\bigl( f_w(x_{I_j}), y_{I_j} \bigr) \Bigr] & (\text{lin. } \E) \\ + &= \frac{1}{k}\sum_{j=1}^{k} \Biggl( \sum_{i=1}^{n}\frac{1}{n}\nabla_w l\bigl(f_w(x_i), y_i\bigr) \Biggr) & (\text{def. } \E_{I_j}) \\ + &= \sum_{i=1}^{n}\frac{1}{n}\nabla_w l\bigl(f_w(x_i), y_i\bigr) & (\text{arith.}) \\ + &= \nabla L(w) +\end{align*} + +\newpage + +\subsection{Other Gradient Methods} + +\method \textbf{Momentum} + +The GD Update is modified to consider previous updates: +$$ + w^{(t+1)} = w^{(t)} - \eta \nabla L(w^{(t)}) + \overbrace{\alpha (w^{(t)} - w^{(t-1)})}^\text{Momentum Term} +$$ +Where $\alpha = \eta \cdot \beta$ s.t. $\beta \in [0,1)$ is the \textit{momentum coefficient}. + +{\footnotesize + \remark \textbf{Intuition}: Dampens oscillations in the gradient direction, especially for ill-conditioned problems. The momentum term is a weighted average of previous updates. + + \remark \textbf{Previous Gradients}: The influence of prev. gradients decreases exponentially. This can be seen by expanding the recursion above. +} +\\ + +\method \textbf{Adaptive Methods} + +Adaptive methods use parameter-specific learning rates for each parameter $w_j$: +$$ + w^{(t+1)}_i = w^{(t)}_i - \frac{\eta}{\sqrt{\delta_i^{(t)} + \gamma}}\cdot\frac{\partial L}{\partial w_i}(w^{(t)}) +$$ +Where $\delta_i^{(t)} = (w_i^{(t)} - w_i^{(t-1)})^2$ and $\gamma>0$. + +{\footnotesize + \remark \textbf{Intuition}: The idea above is that parameters that have changed significantly already should have lower learning rates. + + \remark \textit{ADAM} is an adaptive method. +} +\\ + +\method \textbf{Second Order Methods} + +Methods which use the Hessian $\mathbf{H}_L(w^{(t)})$.\\ +Computationally expensive and usually not done directly. \ No newline at end of file diff --git a/semester6/iml/resources/gdConvergence.png b/semester6/iml/resources/gdConvergence.png new file mode 100644 index 0000000000000000000000000000000000000000..ebbb6388ab4041bfc6f82f944155005fa0f2fa5d GIT binary patch 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