commit 8dc33277c1db430818a7204578b09624ba4f059f
parent bbc978d01641de3832bee005101f3a782b1fa05c
Author: Silas Brack <s174433@student.dtu.dk>
Date: Wed, 18 Jan 2023 17:26:37 +0100
Removed mixture of Laplace and Higham sampling.
Diffstat:
13 files changed, 3636 insertions(+), 280 deletions(-)
diff --git a/chapters/appendix.tex b/chapters/appendix.tex
@@ -9,10 +9,10 @@
$\norm{\bm b}_2 \gets \left( \bm b \cdot \bm b \right)^{1/2}$\;
$\bm b \gets \bm b / \norm{\bm b}_2$\;
$\bm p \gets \bm K \bm b$\;
- $\bm z_{-1} = \bm 0$\;%Like \bm p
- $\bm z_{0} = \bm b$\;
- $\bm q_{0} = \bm P \bm z_{0}$\;
- $\beta_{0} = \left(\bm z_{0} \cdot \bm q_{0} \right)$\;
+ $\bm z_{-1} \gets \bm 0$\;%Like \bm p
+ $\bm z_{0} \gets \bm b$\;
+ $\bm q_{0} \gets \bm P \bm z_{0}$\;
+ $\beta_{0} \gets \left(\bm z_{0} \cdot \bm q_{0} \right)$\;
\For{$q \gets 1$ \KwTo $Q$}{
$\bm c^{(q)}_{0} \gets \bm 0$\;
$\bm d^{(q)}_{0}, \bm d^{(q)}_{-1} \gets \bm 0$\;%Like \bm b
@@ -27,7 +27,7 @@
$\bm z_j \gets \bm p - \alpha_j \bm z_{j-1} - \beta_{j-1} \bm z_{j-2}$\;
$\bm q_j \gets \bm P \bm z_j$\;
$\beta_j \gets \left({\bm z_j \bm q_j}\right)^{1/2}$\;
- % $\beta_j = \sqrt{\bm z_j \bm q_j}$\;
+ % $\beta_j \gets \sqrt{\bm z_j \bm q_j}$\;
$\bm z_j \gets \bm z_j / \beta_{j}$\;
$\bm q_j \gets \bm q_j / \beta_{j}$\;
\For{$q \gets 1$ \KwTo $Q$}{
diff --git a/chapters/experiments.tex b/chapters/experiments.tex
@@ -24,11 +24,11 @@ This experiment will then answer two questions:
\begin{marginfigure}
\centering
\includegraphics{hessian_profile.pdf}
- \caption{Top: comparison of the performance for Hessian-vector, GGN-vector, and manual matrix-vector products. Bottom: comparison of the performance of inverse-vector products for the same methods. Implicit inverse performed via conjugate gradient for Hessian-vector and GGN-vector products.}
+ \caption[Comparison of Hessian-vector product performance.]{Top: comparison of the performance for Hessian-vector, GGN-vector, and manual matrix-vector products. Bottom: comparison of the performance of inverse-vector products for the same methods. Implicit inverse performed via conjugate gradient for Hessian-vector and GGN-vector products.}
\label{fig:hessian-profile}
\end{marginfigure}
-We then compose a function \((g \circ f) (\bm x) := \sum^N_{i=1} (2 x_i + 0.5)^2\) in terms of
+We then compose the loss function \((g \circ f) (\bm x) := \sum^N_{i=1} (2 x_i + 0.5)^2\) as
\(f(\bm x) = 2 \bm x + 0.5, g(\bm x) := \sum^N_{i=1} x_i^2\).
This is equivalent to the sum-of-square-error loss \(g\) for a simple linear model \(f\).
Since \(f(x)\) is a linear function, the GGN matrix will exactly equal the actual Hessian.
@@ -55,7 +55,7 @@ comment on weird dip in inverse, possibly something to do with caching or someth
\begin{marginfigure}[-4cm]
\centering
\includegraphics{hessian_profile_ratios.pdf}
- \caption{Comparison of the speedup from computation of the Hessian-vector product (HVP) versus the GGN-vector product (GVP) (top) and inverse HVP versus inverse GVP using the conjugate gradient method (bottom). Speedup is calculated as the ratio of the wall-clock time of the HVP to the GVP. It can be seen that the two methods have the same time complexity (up to a linear factor).}
+ \caption[Comparison of speedup from HVP to GVP.]{Comparison of the speedup from computation of the Hessian-vector product (HVP) versus the GGN-vector product (GVP) (top) and inverse HVP versus inverse GVP using the conjugate gradient method (bottom). Speedup is calculated as the ratio of the wall-clock time of the HVP to the GVP. It can be seen that the two methods have the same time complexity (up to a linear factor).}
\label{fig:hessian-profile-ratios}
\end{marginfigure}
@@ -66,11 +66,11 @@ Thus, larger values suggest greater performance gains for the GVP, while values
These results can be seen in \cref{fig:hessian-profile-ratios}.
The number of parameters on the \(x\)-axis are exponentially increasing.
-This is in line with the theory, as noted in \cref{ssec:practical-ggn} (SHOULD IT BE MOVED HERE?)
+This is in line with the theory, as noted in \cref{sec:practical-ggn} (SHOULD IT BE MOVED HERE?)
\section{Sampling Ablation}
-The contour integral quadrature method (CIQ) is a Krylov subspace method which converges to the exact solution for \(\lim_{Q \to \infty} \bm s^{\mathrm{CIQ}}_{Q} = \bm K^{-1/2}\).
+Contour integral quadrature (CIQ) converges to the exact solution for \(\lim_{Q \to \infty} \bm s^{\mathrm{CIQ}}_{Q} = \bm K^{-1/2} \bm v\).
However, CIQ is typically extremely close to the exact solution for low values of \(Q\).\sidenote{\textcite{pleiss2020fast} find that \(Q=20\) is approximately sufficient for most problems.}
We then investigated the number of quadrature points \(Q\) required to determine an adequate approximate solution for the Laplace approximation.
@@ -84,19 +84,19 @@ What metrics can we use?
\section[Sampling for the Sine Function]{Sampling for the Sine Function}[The Sine Function]
-Note that the choice of \(\alpha\) is a non-trivial one
+The choice of \(\alpha\) is a non-trivial one
\begin{marginfigure}
\centering
\includegraphics{sin.pdf}
- \caption{Visualisation of the predicted output of multilayer perceptron (MLP) modelling a sine curve.}
+ \caption[Laplace predictions for a sine curve.]{Visualisation of the predicted output of multilayer perceptron (MLP) modelling a sine curve.}
\label{fig:sin}
\end{marginfigure}
\begin{figure}
\centering
\includegraphics{chisq_sin.pdf}
- \caption{Chi-squared sample plot for testing normality of Laplace approximation samples for a neural network modelling \(f(x) = \sin(x)\). The sample Mahalanobis distances are Chi-squared distributed, then the samples themselves are multivariate normal with correct (fixed) covariance. As can be seen by looking at the histogram and the q-q plot, samples are multivariate normal.}
+ \caption[Chi-squared sample plot for sine curve.]{Chi-squared sample plot for testing normality of Laplace approximation samples for a neural network modelling \(f(x) = \sin(x)\). The sample Mahalanobis distances are Chi-squared distributed, then the samples themselves are multivariate normal with correct (fixed) covariance. As can be seen by looking at the histogram and the q-q plot, samples are multivariate normal.}
\label{fig:chisq-sin}
\end{figure}
diff --git a/chapters/higham.tex b/chapters/higham.tex
@@ -1,60 +0,0 @@
-Consider a neural network function \(f: \reals^q \rightarrow \reals^k\) parameterized in \(\theta \in \reals^n\) with a Jacobian given by \(J = J_{\theta}(f)(x) \in \reals^{k \times n}\).
-As per the Laplace approximation, we compute its posterior precision as:
-\begin{align}
- \Sigma^{-1} = J\T J + \alpha \identity \label{eq:ggn-hessian}.
-\end{align}
-We can then sample from the Laplace posterior by taking the inverse square root of our precision matrix~\sidecite{fasi2022computing} and using the reparametrisation trick:
-\begin{align}
- \Sigma^{1/2} ={} & \alpha^{-1/2} \identity + J\T \left(J J\T\right)^{-1} \left(\left(J J\T + \alpha \identity\right)^{-1/2} - \alpha^{-1/2} \identity \right) J,\nonumber
- \\ \Sigma^{1/2} \epsilon ={}& \alpha^{-1/2} \epsilon + J\T \left(J J\T\right)^{-1} \left(\left(J J\T + \alpha \identity\right)^{-1/2} J \epsilon - \alpha^{-1/2} J \epsilon \right), \label{eq:higham-samples}
-\end{align}
-with \(\epsilon \in \reals^n\).
-Unfortunately, this requires a precision matrix in the form of \cref{eq:ggn-hessian}.
-In reality, for a dataset of \(N\) independent observations \(x_n \in \reals^q\), our precision matrix \(\Sigma^{-1}\) becomes
-\begin{align}\label{eq:sum-ggn-hessian}
- \Sigma^{-1} = \sum^N_n J_n\T J_n + \alpha \identity, \quad J_n = J_{\theta}(f)(x_n).
-\end{align}
-We can, however, stack the per-observation Jacobians and reframe this problem as
-\begin{align*}
- \Sigma^{-1} = {J^\prime}\T J^\prime + \alpha \identity, \quad {J^\prime}\T = [J_1\T, J_2\T, ..., J_N\T] \T, \quad J^\prime \in \reals^{k N \times n},
-\end{align*}
-effectively reformulating \cref{eq:sum-ggn-hessian} into the form of \cref{eq:ggn-hessian}.
-However, we are trying to only store matrices of size \(k \times k\), which would now become \(kN \times kN\), which is infeasible.
-Ideally, we would like to be able to sum over the whole pseudo-inverse term and compute something along the lines of\footnote{From this point on, we use \(J = J^\prime\) in our equations.}
-\begin{align*}
- \sum^N_n \left( J_n\T \left(J_n J_n\T\right)^{-1} \left(\left(J_n J_n\T + \alpha \identity\right)^{-1/2} J_n \epsilon - \alpha^{-1/2} J_n \epsilon \right) \right),
-\end{align*}
-since this would allow us to only compute one summation term at a time, and cumulatively sum them.
-However, this does not return the correct value.
-Let us analyse the shapes of the different terms in \cref{eq:higham-samples} as
-\begin{align*}
- \Sigma^{1/2} \epsilon = \underbrace{\alpha^{-1/2} \epsilon}_{n} + \underbrace{J\T}_{n \times kN} \underbrace{\left(J J\T\right)^{-1}}_{kN \times kN} \left( \underbrace{\left(J J\T + \alpha \identity\right)^{-1/2}}_{kN \times kN} \underbrace{J \epsilon}_{kN} - \underbrace{\alpha^{-1/2} J \epsilon}_{kN} \right).
-\end{align*}
-Now, which matrix multiplications involve summing over two \(kN\)-shaped dimensions?\footnote{Both \(J J\T\) and \(J \epsilon\) sum over \(n\).}
-Why is this relevant?
-These summations are the only thing which prevents us from simply summing over the whole term!
-These multiplications are bolded in the following equation:
-\begin{align*}
- \Sigma^{1/2} \epsilon = \alpha^{-1/2} \epsilon + J\T \bullet \left(J J\T\right)^{-1} \bullet \left(\left(J J\T + \alpha \identity\right)^{-1/2} \bullet J \epsilon - \alpha^{-1/2} J \epsilon \right)
-\end{align*}
-In each of these terms, we sum over the output size \(k\) naturally by performing the matrix multiplication, and that we would like to sum over \(N\) as well.
-Let us look at the individual terms which get summed over their \(kN\)-sized dimension.
-We separate these terms into \(A\) and \(B\):
-\begin{align*}
- \Sigma^{1/2} \epsilon = \alpha^{-1/2} \epsilon + J\T \underbrace{\left(J J\T\right)^{-1} \biggl( \underbrace{\left(J J\T + \alpha \identity\right)^{-1/2} J \epsilon }_{A} - \alpha^{-1/2} J \epsilon \biggr)}_{B}.
-\end{align*}
-If we isolate the \(A\) and \(B\) terms and look at them per observation, we see that
-\begin{align*}
- A_i ={} & \sum^N_n \left( J_i J_n\T + \alpha \identity \right)^{-1/2} J_n \epsilon,
- \\ B_i ={}& \sum^N_n \left( J_i J_n\T \right)^{-1} A_n - \alpha^{-1/2} J_n \epsilon,
- \\ \Sigma^{1/2} \epsilon ={}& \alpha^{-1/2} \identity + \sum^N_n J_n\T B_n.
-\end{align*}
-We can then sample as per
-\begin{align*}
- \Sigma^{-\frac{1}{2}} \epsilon = \alpha^{-\frac{1}{2}} \identity + \sum^N_i J_i\T \left( \sum^N_j \left( J_i J_j\T \right)^{-1} \left( \sum^N_k \left( J_j J_k\T + \alpha \identity \right)^{-\frac{1}{2}} J_k \epsilon \right) - \alpha^{-\frac{1}{2}} J_j \epsilon \right).
-\end{align*}
-
-Unfortunately, this leads to a cubic dependency on the dataset size.\footnote{:-(}
-Why is this the case?
-Is this unexpected?
-If so, what aspect of this formulation causes this?
diff --git a/chapters/introduction.tex b/chapters/introduction.tex
@@ -14,13 +14,12 @@ interpretability and assessment of model predictive confidence.
\section{Current Methods}
Currently, approximate Bayesian methods are either expensive to compute
-(Markov Chain Monte Carlo), are significantly more difficult to implement (such as
-variational inference), or simply perform poorly and are limited in
-their Bayesian interpretation (MC dropout). As such, there is demand for
-a method which exhibits the same computational cost as the optimization
-algorithms for deterministic neural networks while providing accurate
-posterior approximations and working out-of-the-box for any given
-architecture.
+(Markov Chain Monte Carlo), are significantly more difficult to implement (such as variational inference), or simply perform poorly and are limited in
+their Bayesian interpretation (MC dropout).
+Gaussian processes
+Deep ensembles
+Beyond these, there are also domain-specific models which aim to provide
+As such, there is demand for a method which exhibits the same computational cost as the optimization algorithms for deterministic neural networks while providing accurate posterior approximations and working out-of-the-box for any given architecture.
% Laplace approximations are simple, yet prin-
% cipled, posterior approximations suitable for
@@ -33,12 +32,44 @@ architecture.
% storing the Hessian implicitly as means to avoid
% the fuzz.
+The Laplace approximation~\sidecite{daxberger2021laplace,laplace1774memoire,} is a simple, yet principled, posterior approximation suitable for Bayesian modeling.
+In the Laplace approximation, to compute the posterior distribution of the model parameters, we need to compute the Hessian of the loss with respect to the model parameters.
+
+Typically, this is done by storing the Hessian matrix explicitly, and then computing the inverse of this matrix.
+Since the Hessian matrix is a \(D \times D\) square matrix with \(D^2\) elements, where \(D\) is the number of parameters, this is intractable for large models, which can contain millions or billions of parameters.
+To overcome this, we can use a crude approximation, such as only storing the Hessian matrix's diagonal~\sidecite{lecun1989optimal,denker1990transforming}, its Kronecker factorisation~\sidecite{heskes2000natural,martens2015optimizing,botev2017practical}, and other low-rank approximations.%~\sidecite{}.
+All of these Hessian approximations have successfully been applied to the Laplace approximation~\sidecite{ritter2018scalable,ritter2018online}
+
+% To address this, we propose a method which approximates the Hessian of the loss with respect to the model parameters without requiring the Hessian to be stored explicitly.
+
% \section{My Solution}
\section{Large-Scale Laplace}
-The ideas presented in this article essentially constitute a compilation of advice given by academics from multiple fields and institutions,~\sidecite{miani2022laplacian}
+Large-Scale Laplace (LSL) is a method which approximates the Hessian of the loss with respect to the model parameters without requiring the Hessian to be stored explicitly.
+This is achieved by using a sampling-based training procedure, where a sample is first drawn from a Gaussian weight prior, and then the loss is evaluated on this sample.
+Sampling from the posterior requires computing the covariance matrix as the inverse of the Hessian of the loss with respect to the model parameters, and then taking the square root of this covariance matrix.
+However, for large models, which can contain millions or billions of parameters, this is intractable using conventional methods, since the Hessian will be a \(D \times D\) square matrix with \(D^2\) elements, where \(D\) is the number of parameters.
+
+\sidetextcite{pleiss2020fast} propose a contour integral quadrature (CIQ), a method which approximates the square root of a square matrix multiplied by a vector without requiring this matrix to be stored explicitly.
+While they use it for computing the square root of the kernel matrix in Gaussian processes, we use it for computing the square root of the covariance matrix in Bayesian neural networks using the Laplace approximation.
+To run CIQ, we therefore only require the ability to compute matrix-vector products.
+Since, to sample from the Laplace posterior distribution we need to compute the square root of the covariance matrix, we need be able to compute Hessian-vector products (i.e., matrix-vector products with the Hessian matrix) without explicitly storing the Hessian matrix.
+% he Hessian-vector product, which is a crucial step in the Laplace approximation.
+% This method is used to approximate the Hessian-vector product, which is used to update the posterior variance of the weights in a Bayesian neural network.
+% This method is used in conjunction with the Laplace approximation, which is a simple, yet principled, posterior approximation suitable for Bayesian modeling.
+% The Laplace approximation relies on access to the Hessian matrix of the loss with respect to the model parameters.
+% For large models, it is even intractable to store this matrix, and crude approximations (e.g. only storing the Hessian diagonal) are called upon.
+% We investigate storing the Hessian implicitly as means to avoid the fuzz.
+
+\section{JAX and XLA}
+
+JAX~\sidecite{jax2018github} is a Python library for GPU-accelerated automatic differentiation and compilation of computational graphs into efficient machine code.
+To perform this compilation, it uses XLA, a compiler for linear algebra~\sidecite{sabne2020xla}.
+JAX enables us to efficiently compute Hessian-vector products without explicitly storing the Hessian matrix by compiling the Hessian-vector product computation using XLA.
+% JAX is designed to be a drop-in replacement for NumPy, and is compatible with most of the Python scientific stack. JAX is built on top of XLA, a domain-specific compiler for linear algebra. XLA is able to compile Python code into efficient machine code, and is used by Google to accelerate TensorFlow and PyTorch. JAX is able to compile Python code into efficient machine code, and is used by Google to accelerate TensorFlow and PyTorch.
+% It is designed to be a drop-in replacement for NumPy, and is compatible with most of the Python scientific stack. JAX is built on top of XLA, a domain-specific compiler for linear algebra. XLA is able to compile Python code into efficient machine code, and is used by Google to accelerate TensorFlow and PyTorch. JAX is able to compile Python code into efficient machine code, and is used by Google to accelerate TensorFlow and PyTorch.
-% \section{Research Objectives}
+\section{Research Objectives}
This should be discussed in a bit more detail in a thesis, since there
are certain objectives discussed in the project plan.
@@ -56,7 +87,3 @@ network, and the variance of the weight-posterior is updated with an
approximate Hessian. Approximating the Hessian is the most time-consuming and painful-to-engineer step of this training procedure. In
this project, we tap into the potential of modern machine learning
frameworks to efficiently approximate the Hessian.
-
-\section{JAX and XLA}
-
-\sidecite{jax2018github,sabne2020xla}
diff --git a/chapters/literature.tex b/chapters/literature.tex
@@ -1,7 +1,6 @@
\setchapterpreamble[u]{\margintoc}
\chapter{Literature}
\labch{literature}
-% http://saul.cpsc.ucalgary.ca/pmwiki.php/GradTips/GradTipsLiteratureReview
\section{Adam}
@@ -61,7 +60,7 @@ Normalising it yields
\\={}& \normal(\bm \theta_{\textsc{map}}, \bm \Lambda^{-1}).
\end{align}
-\marginnote{Note that \cref{eq:laplace-normal} only holds when the precision matrix \(\bm \Lambda\) is positive definite.}We have now shown that approximating the log-posterior with a second degree Taylor expansion around the \(\bm\theta_{\textsc{map}}\) corresponds to approximating the posterior with a Gaussian distribution given by \(\normal(\bm \theta_{\textsc{map}}, \bm \Lambda^{-1})\) where \(\bm \Lambda = - \left.\nabla^2_{\bm \theta} \log p(\bm \theta \given \bm y) \right|_{\bm \theta_{\textsc{map}}}\).
+\marginnote{\cref{eq:laplace-normal} only holds when the precision matrix \(\bm \Lambda\) is positive definite.}We have now shown that approximating the log-posterior with a second degree Taylor expansion around the \(\bm\theta_{\textsc{map}}\) corresponds to approximating the posterior with a Gaussian distribution given by \(\normal(\bm \theta_{\textsc{map}}, \bm \Lambda^{-1})\) where \(\bm \Lambda = - \left.\nabla^2_{\bm \theta} \log p(\bm \theta \given \bm y) \right|_{\bm \theta_{\textsc{map}}}\).
In practice, we can determine the Hessian of the loss with respect to the parameters, where the loss is given by the negative log-posterior.
In this case, we get \(\bm \Lambda = \left. - \nabla^2_{\bm \theta} \log p(\bm \theta \given \bm y) \right|_{\bm \theta_{\textsc{map}}} = \left. \nabla^2_{\bm \theta} \mathcal{L} \right|_{\bm \theta_{\textsc{map}}}\) and the posterior is approximated by \(\normal(\bm \theta_{\textsc{map}}, \bm \Lambda^{-1})\).
@@ -128,7 +127,7 @@ minimum eigenvalue determined by the perturbation \(\alpha\) which guarantees po
% Furthermore, the likelihood term in the ELBO can be decomposed further by assuming independence in the observations, calculated as the term given by \(\E{q}{\log \lik} = \frac{1}{N} \sum^N_i \E{q}{\log p(y_i \given \bm \theta)}\).
% Overall, BNNs have been found to be relatively ineffective unless the number of observations is greater than the number of model parameters.
-% \section{Methods}\label{ssec:methods}
+% \section{Methods}\label{sec:methods}
% Multiple approximate inference methods were implemented.
% Specifically, \emph{maximum a posteriori} estimation, Laplace approximation, VI with mean-field, full-rank, low-rank and radial variational families, deep ensembles, and MultiSWAG.
@@ -168,7 +167,7 @@ minimum eigenvalue determined by the perturbation \(\alpha\) which guarantees po
-% \subsection{Mean-Field Variational Approximation}\label{ssec:mfvi}
+% \subsection{Mean-Field Variational Approximation}\label{sec:mfvi}
% One common variational family used to approximate the real posterior is a product of independent Gaussian distributions (the mean-field approximation) such that each model parameter is sampled from a normal distribution and is independent of all other model parameters, yielding a Gaussian with a diagonal covariance matrix.
% In mean-field VI, model weights are sampled from an approximate posterior \(q(\bm \theta) \approx p(\bm{\theta} \given \bm y)\) as
@@ -221,7 +220,7 @@ minimum eigenvalue determined by the perturbation \(\alpha\) which guarantees po
% \end{align}
% where \(\bm{\varepsilon} \sim \normal(\bm{0}, \identity)\), \(r \sim \normal(0, 1)\).
% The entropy term of the Radial approximate posterior is given
-% \(\entropy{q(\theta)} = - \sum_i \log \sigma_i + \mathrm{const}\) (and is therefore approximately equal to the entropy of the mean-field approximation up to a constant) and the cross-entropy term is calculated via Monte Carlo simulation as for mean-field VI (\cref{ssec:mfvi}), but by sampling from a radial posterior \(\bm{\theta}^{(s)} \sim \mathrm{Radial}(\bm{\mu}, \bm{\sigma})\) as in \cref{eq:radial}.
+% \(\entropy{q(\theta)} = - \sum_i \log \sigma_i + \mathrm{const}\) (and is therefore approximately equal to the entropy of the mean-field approximation up to a constant) and the cross-entropy term is calculated via Monte Carlo simulation as for mean-field VI (\cref{sec:mfvi}), but by sampling from a radial posterior \(\bm{\theta}^{(s)} \sim \mathrm{Radial}(\bm{\mu}, \bm{\sigma})\) as in \cref{eq:radial}.
% \subsection{Deep Ensembles}
diff --git a/chapters/mixture.tex b/chapters/mixture.tex
@@ -1,31 +0,0 @@
-\setchapterpreamble[u]{\margintoc}
-\chapter{Future Work: Mixture of Laplace Approximations}
-\labch{mixture}
-
-We could then frame the Laplace posterior as a mixture of per-sample Laplace approximations as
-\begin{equation}
- p(\bm \theta \given \bm x) = \sum^N_i w_i \normal(\bm \mu_i, \bm \Sigma_i)
-\end{equation}
-such that \(\sum^N_i w_i \bm \mu_i = \bm \mu_{\textsc{map}}\) and \(\sum_i^N w_i = 1\).
-Sampling from this model would then involve sampling an index
-\begin{align}
- i \sim{} & \mathrm{Cat}(N, \bm w)
- \\ \bm \varepsilon \sim{}& \normal(\bm \mu_i, \bm \Sigma_i)
-\end{align}
-where \(\bm \mu_i\) is computed by shifting the MAP solution based on ??? and \(\bm \Sigma_i\) is computed as the inverse precision matrix from the Laplace approximation, as per \cref{eq:shifted-map}.
-\begin{align}\label{eq:shifted-map}
- \bm \mu_i ={} & \bm \mu_{\textsc{map}} - \nabla_{\bm\theta} F_n (\bm \theta) \nabla_{\bm\theta}^2 F_n (\bm \theta)
- \\ \bm \Sigma^{-1}_i ={}& - \nabla^2_{\bm \theta} \log p(\bm \theta \given x_i)
-\end{align}
-The reparametrisation trick can then be performed using only a single observation to determine the covariance matrix
-\begin{align}
- \bm \varepsilon ={} & \bm \mu_i + \bm \Sigma^{-1/2}_i \bm \varepsilon_0,
-\end{align}
-where, since \(\bm \Sigma^{-1}_i = J_i\T H_i J_i + \alpha / N \identity\) is low-rank with rank \(o\), the Woodbury identity from \cref{sec:woodbury} can be used.
-This method depends on the conditioning number of the outer product \(J H J\T\), which is significantly lower than that of the inner product \(J\T H J\).
-
-However, this method poses some major issues when it comes to computing the evidence of the model.
-\begin{align}
- p(\bm x) ={} & \int_{\bm \theta} p(\bm x \given \bm \theta) p(\bm \theta)\,d \bm \theta
- \\ ={}& \prod^N_n \exp F_n (\bm \theta_\textsc{map}) \int_{\bm \theta} \exp\left\{ -\frac{1}{2} (\bm\theta - \bm\theta_\textsc{map} )\T \Lambda_n (\bm\theta - \bm\theta_\textsc{map} )\,d\bm\theta \right\} \nonumber
-\end{align}
diff --git a/chapters/sampling.tex b/chapters/sampling.tex
@@ -66,7 +66,7 @@ it is then also trivial to compute the square root product by multiplying \(\bm
\begin{marginfigure}[-7cm]
\centering
\input{figures/cauchy_integral.tex}
- \caption{Representation of integration circle \(\Gamma\) contained within the domain of \(f\) as defined according to the Cauchy integral formula.}
+ \caption[Representation of integration circle for the Cauchy integral formula.]{Representation of integration circle \(\Gamma\) contained within the domain of \(f\) as defined according to the Cauchy integral formula.}
\label{fig:cauchy-integral-formula}
\end{marginfigure}
@@ -80,7 +80,7 @@ Cauchy's integral formula is a central theorem in complex analysis which states
f(\bm K) ={} & \frac{1}{2 \pi i} \oint_\Gamma f(\tau) \left(\tau \identity - \bm K\right)^{-1}\,d\tau.
\end{align}
Since, in our case, the eigenvalues of \(\bm K\) will all be real-valued, a circle centred on the real axis which encloses the minimum and maximum eigenvalues of \(\bm K\) will suffice.\marginnote{If a real-valued matrix is symmetric, its eigenvalues are real. In our case, \(\bm K\) will be the precision matrix obtained from the Laplace approximation.}
-To estimate these values, we can use the Lanczos algorithm (\cref{ssec:lanczos}) to compute a lower bound on the maximum eigenvalue and an upper bound on the minimum eigenvalue of \(\bm K\).
+To estimate these values, we can use the Lanczos algorithm (\cref{sec:lanczos}) to compute a lower bound on the maximum eigenvalue and an upper bound on the minimum eigenvalue of \(\bm K\).
% ------------------------------------------
% QUADRATURE
@@ -94,7 +94,7 @@ To estimate these values, we can use the Lanczos algorithm (\cref{ssec:lanczos})
\begin{marginfigure}
\centering
\input{figures/cauchy_quadrature.tex}
- \caption{The shifts \(\tau_1, \ldots, \tau_Q\) are sampled from a circle with its centre on the real axis which intersects the real axis at the points \(\lambda_{\mathrm{min}}\) and \(\lambda_{\mathrm{max}}\). Since we use Jacobi elliptic functions, we over-sample points that are closer to the minimum eigenvalue \(\lambda_{\mathrm{min}}\).}
+ \caption[Sampling of quadrature points for CIQ.]{The shifts \(\tau_1, \ldots, \tau_Q\) are sampled from a circle with its centre on the real axis which intersects the real axis at the points \(\lambda_{\mathrm{min}}\) and \(\lambda_{\mathrm{max}}\). Since we use Jacobi elliptic functions, we over-sample quadrature points that are closer to the minimum eigenvalue \(\lambda_{\mathrm{min}}\).}
\label{fig:ciq}
\end{marginfigure}
@@ -114,7 +114,7 @@ We could sample these quadrature points from the circle uniformly, which corresp
However, the convergence for this quadrature is linear with regard to the condition number of \(\bm K\)~\sidecite{pleiss2020fast,hale2008computing}.
Since the precision matrix (which will be computed from the Laplace approximation as the GGN matrix) can often have a very low rank and poor conditioning, uniform sampling would then require a very large number of quadrature points \(Q\) and would therefore be inadequate.
-Since our GGN precision matrix \(\bm \Lambda = \sum^N_{i=1} \bm J_i\T \bm H_i \bm J_i\) can be low-rank, we would instead prefer to oversample quadrature points which lie close to the minimum eigenvalue.
+Since our GGN precision matrix \(\bm \Lambda = \sum^N_{i=1} \bm J_i\T \bm H_i \bm J_i\) is often low-rank, we would instead prefer to oversample quadrature points which lie close to the minimum eigenvalue.
This is accomplished by applying a change of variable from \(\tau\) to \(\sigma = \tau^{1/2}\) and sampling the quadrature points from \(\sigma\).
To then account for this change of variable, we reformulate \cref{eq:ciq-integral-tau,eq:ciq-quadrature-tau} as
\begin{align}
@@ -171,7 +171,7 @@ We can thereby now compute the quadrature weights and shifts, as seen in \cref{a
\end{marginfigure}
The last step involves solving the \(Q\) linear systems defined in \cref{eq:ciq-sigma-product} to determine \(\bm c_q\).
-As mentioned above, this is done by using msMINRES (\cref{ssec:msminres,alg:msminres}).
+As mentioned above, this is done by using msMINRES (\cref{sec:msminres,alg:msminres}).
Since the Lanczos algorithm only requires the matrix-vector product \(\bm K \bm v\) with a random vector \(\bm v\), we can use the same random vector for all \(Q\) linear systems.
As such, the computational bottleneck of the contour integral quadrature is performing the msMINRES for the shifted systems.
Additionally, since msMINRES is an iterative Krylov method, it is also sensitive to the conditioning of the linear systems, i.e., the conditioning of \(\bm K\).
@@ -249,8 +249,8 @@ As such, if the preconditioner-vector product is similarly or more expensive to
However, an effective preconditioner will significantly improve the convergence speed of CIQ, and this effect can often outweigh the cost of computing the preconditioner.
\subsection[Adam-esque fully linear GGN]{Adam-esque fully linear GGN}[Fully Linear GGN]
-\label{ssec:adam-preconditioner}
-If we calculate the GGN approximation of the Hessian by linearizing over the whole model and loss (as described in \cref{ssec:practical-ggn}), we get
+\label{sec:adam-preconditioner}
+If we calculate the GGN approximation of the Hessian by linearizing over the whole model and loss (as described in \cref{sec:practical-ggn}), we get
\begin{align}
P ={} & \nabla \nabla\T + \alpha \identity
\\ P^{-1} ={}& \beta \cdot \nabla \nabla\T + \alpha^{-1} \identity
@@ -295,6 +295,7 @@ For the subsampled GGN preconditioner, the preconditioner-vector product will th
Since \(N \gg B\), then the cost of computing the preconditioner will be negligible compared to that of the GGN-vector product.
\subsection{Pivoted Cholesky}
+\label{sec:pivoted-cholesky}
\sidecite{harbrecht2012low,bach2013sharp,gardner2018gpytorch}
@@ -341,7 +342,7 @@ A common approach to sampling the largest eigenvalues is to sample vectors with
However, it is not possible to generally and efficiently determine the diagonal of a matrix via matrix-vector products.
Often, it is necessary to multiply a vector of zeros with a value of one at the index at which to find the diagonal.
-Alternatively, we can approximate the diagonal of the GGN matrix as the element-wise product of the gradient with itself, similarly to the preconditioner described in \cref{ssec:adam-preconditioner} and the GGN approximation computed in \cref{eq:fully-linear-ggn}.
+Alternatively, we can approximate the diagonal of the GGN matrix as the element-wise product of the gradient with itself, similarly to the preconditioner described in \cref{sec:adam-preconditioner} and the GGN approximation computed in \cref{eq:fully-linear-ggn}.
In conclusion, to find the optimal parameters for the pivoted Cholesky preconditioner (the number of observations \(B\) and the rank of the pivoted Cholesky factorisation \(k\)), there are some rules of thumb to consider.
Notably, the quality of the preconditioner depends largely on the ease of inverting the \(\identity + \alpha^{-1} L L\T\) matrix, which depends on its conditioning number.
@@ -352,18 +353,19 @@ Furthermore, you want enough observations \(B\) to be included in the preconditi
\subsection[Randomly Pivoted Cholesky]{Randomly Pivoted Cholesky}[Random Pivoted Cholesky]
-Unfortunately, similarly to the pivoted Cholesky preconditioner, the randomly pivoted Cholesky preconditioner requires the diagonal of the GGN matrix in order for the decomposition to be close to the GGN. (MENTION THIS IN THE PIVOTED CHOLESKY SECTION)
+Similarly to the pivoted Cholesky preconditioner, the randomly pivoted Cholesky preconditioner requires the diagonal of the GGN matrix in order for the decomposition to be close to the GGN.
+We can thus approximate this term as we do in the pivoted Cholesky preconditioner (see \cref{sec:pivoted-cholesky}).
\subsection[Other Laplace approximations]{Other Laplace approximations}[Other approximations]
-% Last-Layer Laplace
+% Last-Layer Laplace
+
% K-FAC Laplace
-
However, due to the speed, effectiveness, and theoretical properties of other preconditioners proposed in this section, these methods were not attempted.
\subsection[A Recap on Preconditioning]{A Recap on Preconditioning}[A Recap]
diff --git a/chapters/theory.tex b/chapters/theory.tex
@@ -1,6 +1,7 @@
\setchapterpreamble[u]{\margintoc}
-\chapter{Theory}
-\labch{theory}
+\chapter{Laplace}
+\labch{laplace}
+% http://saul.cpsc.ucalgary.ca/pmwiki.php/GradTips/GradTipsLiteratureReview
\section[Optimisation in Deep Learning]{Optimisation in Deep Learning}[Optimisation]
@@ -8,7 +9,7 @@ Suppose a neural network is a real-valued function \(f: \reals^n \times \reals^d
Our goal is to find the optimal parameters \(\bm\theta^*\) which best model the observed data.
From a frequentist perspective, we can define a loss function \(\mathcal{L}(\bm\theta): \reals^d \rightarrow \reals\) such that the optimal parameters \(\bm\theta^*\) minimise this loss function.
-Often the loss function is defined as the negative log-likelihood of the data under the model, \(\mathcal{L}(\bm\theta) = -\log \lik\).
+Often, we formulate the loss function as the negative log-likelihood of the data under the model, \(\mathcal{L}(\bm\theta) = -\log \lik\).
In this case, the goal of optimisation is to find the parameters \(\bm\theta^*\) which maximise the likelihood of the data.\sidenote[][5cm]{Maximising the likelihood is equivalent to minimising the negative log likelihood, since the log function is strictly monotonically increasing.}
From a Bayesian perspective, we define a prior distribution \(p(\bm\theta)\) and a likelihood function \(\lik\) such that the goal of optimisation is to find the parameters \(\bm\theta^*\) which maximise the posterior distribution \(p(\bm\theta \given \bm y)\).
@@ -44,11 +45,90 @@ The gradient is computed using the chain rule as
\nabla_{\bm\theta} \mathcal{L}(\bm\theta_t) = \nabla_{\bm\theta} \log \lik(\bm x_t, y_t \given \bm\theta_t) \nabla_{\bm\theta} f_{\bm\theta_t}(\bm x_t)
\end{equation*}
%
-where \(\nabla_{\bm\theta} f_{\bm\theta_t}(\bm x_t)\) is the gradient of the neural network with respect to its parameters \(\bm\theta_t\).
-
-Note that \(\eta_t\) can be interpreted as the step size of the gradient descent algorithm.
+where \(\nabla_{\bm\theta} f_{\bm\theta_t}(\bm x_t)\) is the gradient of the neural network with respect to its parameters \(\bm\theta_t\), and \(\eta_t\) can be interpreted as the step size of the gradient descent algorithm.
It can either be constant throughout training (\(\eta_t = \eta\)) or adaptive (if it varies throughout training).
+\subsection{Adam}
+
+% How does adam work?
+
+\sidecite{sutskever2013importance}
+\sidecite{kingma2014adam}
+
+% Integrate with the previous section
+Adam approximates the second moment of the gradient of the loss, that this corresponds to the variance of this gradient, and that this is the diagonal of the Fisher information matrix, as we defined in \cref{sec:fisher-information} (\cref{eq:gradient-covariance})!
+% This is a common approximation in practice, as the diagonal of the Fisher information matrix is often sufficient to approximate the curvature of the log-likelihood.
+
+\subsection[Natural Gradient Descent]{Natural Gradient Descent}[The Natural Gradient]
+% https://agustinus.kristia.de/techblog/2018/03/14/natural-gradient/
+
+Natural Gradient Descent is an approximate second-order optimisation method. It has an interpretation as optimizing over a Riemannian manifold using an intrinsic distance metric, which implies the updates are invariant to transformations such as whitening. By using the positive semi-definite (PSD) Gauss-Newton matrix to approximate the (possibly negative definite) Hessian, NGD can often work better than exact second-order methods.
+
+side{amari1998natural} introduced the natural gradient as a way to optimize a function \(f\) parameterised by \(\bm\theta\) by following the direction of the steepest descent in the Fisher information metric.
+It can be interpreted as
+% The algorithm can be seen in \cref{natural-gradient}.
+%
+% \begin{algorithm}
+% \caption{Natural Gradient Descent} \label{natural-gradient}
+% \begin{algorithmic}[1]
+% \For {$i = 1$ to $T$}
+% \State Calculate \(\mathcal{L}(\bm\theta)\)
+% \State Calculate the gradient of the loss \(\nabla_{\bm\theta} \mathcal{L}(\bm\theta)\)
+% \State Calculate the Fisher information matrix \(\mathcal{F}(\bm\theta)\)
+% \State Calculate the natural gradient \(\nabla_{\bm\theta} \mathcal{L}(\bm\theta) \mathcal{F}(\bm\theta)^{-1}\)
+% \State Update the parameters \(\bm\theta \gets \bm\theta - \eta \nabla_{\bm\theta} \mathcal{L}(\bm\theta) \mathcal{F}(\bm\theta)^{-1}\)
+% \EndFor
+% \State \Return {\(\bm\theta\)}
+% \end{algorithmic}
+% \end{algorithm}
+%
+In practice, deep learning models have millions (or billions) or parameters, and the Fisher information matrix is consequently often too large to be computed and stored explicitly.
+This therefore limits the applicability of the natural gradient.
+Instead, it is often approximated by the diagonal of the Hessian.
+
+% This brings us to the Adam optimizer, which is a popular choice for optimizing deep neural networks.
+\sidecite{kingma2014adam}
+\sidecite{martens2015optimizing}
+\sidecite{botev2017practical}
+
+For a recent, detailed discussion of the natural gradient, see \sidetextcite{martens2020new}.
+\sidecite{wu2019logan}
+
+\section{Bayesian Deep Learning}
+\label{sec:bayesian-deep-learning}
+
+\section{Laplace Approximation}
+\label{sec:laplace}
+
+In the Laplace approximation (LA)~\sidecite{laplace1774memoire,bishop1995neural,mackay2003information,daxberger2021laplace}, the posterior is approximated by a Gaussian, similarly to mean-field VI\@.
+However, instead of finding the optimal Gaussian distribution locations and scales by maximising the ELBO, the LA finds the location by computing the MAP solution \(\bm \theta_{\textsc{map}}\) and the scale by approximating the log posterior with a second degree Taylor expansion around this solution (\(\bm \theta_0 = \bm \theta_{\textsc{map}}\)) and determining the curvature via its Hessian matrix \(\bm \Lambda = - \left.\nabla^2_{\bm \theta} \log p(\bm \theta \given \bm y) \right|_{\bm \theta_\mathrm{MAP}}\).
+Since the Taylor expansion is performed around the MAP solution, the first order derivative is zero, and the expansion is given by
+\begin{align}
+ \ln p(\bm \theta \given \bm y)
+ \approx{} & \ln p(\bm \theta_{\textsc{map}} \given \bm y) + \nonumber
+ \\ & + \frac{1}{2} (\bm \theta - \bm \theta_{\textsc{map}})\T \left( \left.\nabla^2_{\bm \theta} \ln p(\bm \theta \given \bm y) \right|_{\bm \theta_\mathrm{MAP}} \right) (\bm \theta - \bm \theta_{\textsc{map}}) \label{eq:laplace-taylor}
+ \\ \Rightarrow \tilde{p}(\bm \theta \given \bm y) \approx{}& p(\bm \theta_{\textsc{map}} \given \bm y) \exp\left(- \frac{1}{2} (\bm \theta - \bm \theta_{\textsc{map}})\T \bm \Lambda (\bm \theta - \bm \theta_{\textsc{map}})\right) \propto p(\bm \theta \given \bm y) \nonumber
+\end{align}
+where \(\tilde{p}(\bm \theta \given \bm y)\) corresponds to the unnormalised posterior.
+Normalising it yields
+\begin{align}\label{eq:laplace-normal}
+ p(\bm\theta \given \bm y) \approx{} & \sqrt{\frac{\det (\bm \Lambda)}{(2 \pi)^D}} \exp\left(- \frac{1}{2} (\bm \theta - \bm \theta_{\textsc{map}})\T \bm \Lambda (\bm \theta - \bm \theta_{\textsc{map}})\right) \nonumber
+ % \\={}& \sqrt{\frac{\det (- \bm H)}{(2 \pi)^D}} \exp\left(- \frac{1}{2} (\bm \theta - \bm \theta_0)\T (- \bm H) (\bm \theta - \bm \theta_0)\right) \nonumber
+ \\={}& \normal(\bm \theta_{\textsc{map}}, \bm \Lambda^{-1}).
+\end{align}
+
+\marginnote{The equality in \cref{eq:laplace-normal} only holds when the precision matrix \(\bm \Lambda\) is positive definite.}We have now shown that approximating the log-posterior with a second degree Taylor expansion around the \(\bm\theta_{\textsc{map}}\) corresponds to approximating the posterior with a Gaussian distribution given by \(\normal(\bm \theta_{\textsc{map}}, \bm \Lambda^{-1})\) where \(\bm \Lambda = - \left.\nabla^2_{\bm \theta} \log p(\bm \theta \given \bm y) \right|_{\bm \theta_{\textsc{map}}}\).
+
+In practice, we can determine the Hessian of the loss with respect to the parameters, where the loss is given by the negative log-posterior.
+In this case, we get \(\bm \Lambda = \left. - \nabla^2_{\bm \theta} \log p(\bm \theta \given \bm y) \right|_{\bm \theta_{\textsc{map}}} = \left. \nabla^2_{\bm \theta} \mathcal{L} \right|_{\bm \theta_{\textsc{map}}}\) and the posterior is approximated by \(\normal(\bm \theta_{\textsc{map}}, \bm \Lambda^{-1})\).
+
+\cref{eq:negative-log-likelihood-ggn}
+
+GGN APPROXIMATION FOR POSITIVE DEFINITENESS
+
+EIGENVALUES?
+minimum eigenvalue determined by the perturbation \(\alpha\) which guarantees positive definiteness
+
\section{The Hessian}
The Hessian is a matrix of second-order partial derivatives of a scalar function.
@@ -116,9 +196,9 @@ For log-likelihood loss, we get
\subsection{Practical Considerations}
% \subsection{Practical Considerations for the GGN Approximation}
-\label{ssec:practical-ggn}
+\label{sec:practical-ggn}
-Note that the generalised Gauss-Newton approximation requires that the function for which we would like to compute the Hessian, \(h(x)\), can be framed as a composition of two functions \(h(x) = g(f(x))\).
+The generalised Gauss-Newton approximation requires that the function for which we would like to compute the Hessian, \(h(x)\), can be framed as a composition of two functions \(h(x) = g(f(x))\).
However, there are many possible choices of functions \(f\) and \(g\) which, when composed, yield \(h\), and these different compositions can yield wildly varying performance for the GGN approximation~\sidecite{kunstner2019limitations}.
Given that, in our case, \(h\) is the loss function with respect to the model parameters of a neural network, a natural choice of functions (and the choice we have assumed above) are \(g: \reals^o \rightarrow \reals\) and \(f: \reals^d \rightarrow \reals^o\) where \(g\) is the loss function with respect to the model output and \(f\) is a function which maps the model parameters to a model output, for a given model input \(x\).
This construction allows us to capture the curvature of the loss with respect to the model output, which can often be easily computed in a simple closed form, without requiring the computation of the curvature of the model function with respect to its parameters; this therefore corresponds to linearising the model function with a Taylor expansion as in \cref{eq:nn-taylor}.
@@ -138,7 +218,7 @@ This yields \(g(y) = -\log(y)\), and thus we obtain
\end{align}
The diagonal of this matrix is given by the element-wise square of the gradient \( \nabla_{\bm\theta} \mathcal{L} \odot \nabla_{\bm\theta} \mathcal{L} \).
-Note that the choice of application of the generalised Gauss-Newton approximation is not based on having a reduced computational cost --- both the Hessian and the GGN approximation require the same number of forward and backward passes.
+The choice of application of the generalised Gauss-Newton approximation is not based on having a reduced computational cost---both the Hessian and the GGN approximation require the same number of forward and backward passes.
However, the GGN approximation is better behaved than the exact Hessian, thereby making it more suitable for practical applications.
This is because the GGN approximation is designed to be positive semi-definite, while the exact Hessian can be vulnerable to negative curvature.
This feature can limit its use for optimisation algorithms that require a positive definite Hessian, such as conjugate gradient descent, or the Laplace approximation, which employs the inverse of the Hessian as the covariance matrix of a Gaussian distribution, thereby assuming a positive semi-definite covariance matrix.
@@ -148,7 +228,8 @@ The Levenberg-Marquardt heuristic~\sidecite{pascanu2013revisiting}
A GGN approximation is any approximation which fulfils \cref{eq:ggn-whole-hessian}.
\sidecite{kunstner2019limitations}
-\section{Fisher Information}\label{sec:fisher-information}
+\section[The Fisher Information Matrix]{The Fisher Information Matrix}[Fisher Information]
+\label{sec:fisher-information}
% https://agustinus.kristia.de/techblog/2018/03/11/fisher-information/
Consider the gradients of the log-likelihood optimisation objective, given by the expression \(\nabla_{\bm\theta} \log\lik\).
@@ -206,69 +287,45 @@ RELATIONSHIP WITH KL DIVERGENCE
LIMITATIONS OF FISHER INFORMATION (MAYBE ESPECIALLY EMPIRICAL?)
\sidecite{kunstner2019limitations}
-\section{The Natural Gradient}
-% https://agustinus.kristia.de/techblog/2018/03/14/natural-gradient/
-
-Natural Gradient Descent is an approximate second-order optimisation method. It has an interpretation as optimizing over a Riemannian manifold using an intrinsic distance metric, which implies the updates are invariant to transformations such as whitening. By using the positive semi-definite (PSD) Gauss-Newton matrix to approximate the (possibly negative definite) Hessian, NGD can often work better than exact second-order methods.
-
-side{amari1998natural} introduced the natural gradient as a way to optimize a function \(f\) parameterised by \(\bm\theta\) by following the direction of the steepest descent in the Fisher information metric.
-It can be interpreted as
-% The algorithm can be seen in \cref{natural-gradient}.
-%
-% \begin{algorithm}
-% \caption{Natural Gradient Descent} \label{natural-gradient}
-% \begin{algorithmic}[1]
-% \For {$i = 1$ to $T$}
-% \State Calculate \(\mathcal{L}(\bm\theta)\)
-% \State Calculate the gradient of the loss \(\nabla_{\bm\theta} \mathcal{L}(\bm\theta)\)
-% \State Calculate the Fisher information matrix \(\mathcal{F}(\bm\theta)\)
-% \State Calculate the natural gradient \(\nabla_{\bm\theta} \mathcal{L}(\bm\theta) \mathcal{F}(\bm\theta)^{-1}\)
-% \State Update the parameters \(\bm\theta \gets \bm\theta - \eta \nabla_{\bm\theta} \mathcal{L}(\bm\theta) \mathcal{F}(\bm\theta)^{-1}\)
-% \EndFor
-% \State \Return {\(\bm\theta\)}
-% \end{algorithmic}
-% \end{algorithm}
-%
-In practice, deep learning models have millions (or billions) or parameters, and the Fisher information matrix is consequently often too large to be computed and stored explicitly.
-This therefore limits the applicability of the natural gradient.
-Instead, it is often approximated by the diagonal of the Hessian.
-
-% This brings us to the Adam optimizer, which is a popular choice for optimizing deep neural networks.
-\sidecite{kingma2014adam}
-\sidecite{martens2015optimizing}
-\sidecite{botev2017practical}
-
-For a recent, detailed discussion of the natural gradient, see \sidetextcite{martens2020new}.
-\sidecite{wu2019logan}
-
-\section[Inverting a low-rank matrix]{Inverting a low-rank matrix}[Woodbury-Morrison]
+\section[The Woodbury matrix identity]{The Woodbury matrix identity}[Woodbury]
\label{sec:woodbury}
-% Theorem 1.1 ([18, Thm. 1.35]). Let U, V ∈ Cn×k with k ≤ n and assume that V ∗U is nonsingular. Let f be defined on the spectrum of A = αIn + U V ∗, and if k = n let f be defined at α. Then (1.2) f (A) = f (α)In + U (V ∗U )−1(f (αIk + V ∗U ) − f (α)Ik )V ∗.
+In many methods, such as Natural Gradient Descent~\sidecite{martens2020new} and the Laplace approximation~\sidecite{daxberger2021laplace} (explained in \cref{sec:laplace}), we must invert the Hessian.
+Since, in neural networks, the number of parameters is often very large, this inversion is computationally expensive and/or cannot be stored in memory.
+Furthermore, the Hessian matrix is often ill-conditioned, which can lead to numerical instability.
+We typically approximate the Hessian matrix with the Gauss-Newton approximation, which is a low-rank approximation of the Hessian matrix.
+This also leads to many methods being inefficient (TODO: explain this more, cite, conditioning, iterative methods, etc.).
+The Woodbury matrix identity~\sidecite{woodbury1950inverting} is a formula that allows us to invert a matrix that is composed as the sum of a low-rank matrix and a diagonal matrix.
+If we have a matrix \(\bm P\) that is the sum of a low-rank matrix \(\bm L\T \bm L\) and a diagonal matrix \(\alpha \identity\), then we can invert \(\bm P\) using the Woodbury matrix identity as
+% Theorem 1.1 ([18, Thm. 1.35]). Let U, V ∈ Cn×k with k ≤ n and assume that V ∗U is nonsingular. Let f be defined on the spectrum of A = αIn + U V ∗, and if k = n let f be defined at α. Then (1.2) f (A) = f (α)In + U (V ∗U )−1(f (αIk + V ∗U ) − f (α)Ik )V ∗.
\begin{align}
- f\left(J\T J + \alpha \identity\right) ={} & f(\alpha) \identity + J\T \left(J J\T\right)^{-1} \left( f\left( J J\T + \alpha \identity \right) - f(\alpha) \identity \right) J \nonumber
- \\ f\left(J\T J + \alpha \identity\right) v ={}& f(\alpha) v + J\T \left(J J\T\right)^{-1} \left( f\left( J J\T + \alpha \identity \right) - f(\alpha) \identity \right) J v
+ \bm P ={} & \bm L\T \bm L + \alpha \identity
+ \\ \bm P^{-1} ={}& \alpha^{-1} \identity - \alpha^{-2} \bm L\T \left( \identity + \alpha^{-1} \bm L \bm L\T \right)^{-1} \bm L.
\end{align}
-
+We can then compute the matrix-vector product and inverse matrix-vector product as
\begin{align}
- \left(J\T J + \alpha \identity\right)^{-1} ={} & \alpha^{-1} \identity + J\T \left(J J\T\right)^{-1} \left( \left( J J\T + \alpha \identity \right)^{-1} - \alpha^{-1} \identity \right) J
+ \bm P \bm v ={} & \bm L\T ( \bm L \bm v ) + \alpha \bm v
+ \\ \bm P^{-1} \bm v ={}& \alpha^{-1} \bm v - \alpha^{-2} \bm L\T ( \underbrace{\identity + \alpha^{-1} \bm L \bm L\T}_{k \times k} )^{-1} ( \bm L \bm v )
\end{align}
+such that we do not need to instantiate the whole matrix.
+Since \(\identity + \alpha^{-1} \bm L \bm L\T\) is framed as an outer product \(\bm L \bm L\T\), we only need to compute the inverse of a \(k \times k\) matrix, which is much easier than computing the inverse of a \(n\times n\) matrix, since \(k \ll N\).
+Furthermore, the condition number of this matrix is given by the ratio of the largest to the smallest singular value of \(\bm L \bm L\T\), which is much smaller than the condition number of \(\bm P\) itself (since the smallest singular value of \(\bm L\T \bm L\) is zero).
-\begin{align}
- P ={} & L\T L + \alpha \identity
- \\ P^{-1} ={}& \alpha^{-1} \identity - \alpha^{-2} L\T \left( \identity + \alpha^{-1} L L\T \right)^{-1} L
-\end{align}
-\begin{align}
- P v ={} & L\T ( L v ) + \alpha v
- \\ P^{-1} v ={}& \alpha^{-1} v - \alpha^{-2} L\T ( \underbrace{\identity + \alpha^{-1} L L\T}_{k \times k} )^{-1} ( L v )
-\end{align}
+% For an arbitrary function \(f\),
+% \begin{align}
+% f\left(J\T J + \alpha \identity\right) ={} & f(\alpha) \identity + J\T \left(J J\T\right)^{-1} \left( f\left( J J\T + \alpha \identity \right) - f(\alpha) \identity \right) J \nonumber
+% \\ f\left(J\T J + \alpha \identity\right) v ={}& f(\alpha) v + J\T \left(J J\T\right)^{-1} \left( f\left( J J\T + \alpha \identity \right) - f(\alpha) \identity \right) J v
+% \end{align}
+% \begin{align}
+% \left(J\T J + \alpha \identity\right)^{-1} ={} & \alpha^{-1} \identity + J\T \left(J J\T\right)^{-1} \left( \left( J J\T + \alpha \identity \right)^{-1} - \alpha^{-1} \identity \right) J
+% \end{align}
-% https://mathoverflow.net/questions/121411/expectation-of-square-root-of-binomial-r-v
-\begin{align}
- \E{}{\sqrt{X}} \approx \sqrt{\E{}{X}} \left( 1 - \frac{\Var{X}}{8 \E{}{X}^2} \right)
-\end{align}
+% % https://mathoverflow.net/questions/121411/expectation-of-square-root-of-binomial-r-v
+% \begin{align}
+% \E{}{\sqrt{X}} \approx \sqrt{\E{}{X}} \left( 1 - \frac{\Var{X}}{8 \E{}{X}^2} \right)
+% \end{align}
\section{Krylov Subspace Methods}
@@ -277,7 +334,7 @@ A Krylov subspace of a square matrix \(\bm K \in \reals^{N \times N}\) and a vec
\mathcal{K}_j( \bm K, \bm b) = \mathrm{span}\left\{ \bm b, \bm K \bm b, \bm K^2 \bm b, \ldots, \bm K^{j-1} \bm b \right\}
\end{align}
-Because of this, matrix-vector products alone are sufficient to construct a Krylov subspace.
+Because of this, matrix-vector products alone are enough to construct a Krylov subspace.
Krylov subspace methods leverage this fact to estimate functions of matrices multiplied with vectors, \(f(\bm K) \bm b\).
These methods therefore identify approximate solutions which are contained within the \(j\)\textsuperscript{th} Krylov subspace, i.e., \(f(\bm K) \bm b \approx \bm c_j \in \mathcal{K}_j( \bm K, \bm b)\).
@@ -286,7 +343,7 @@ One major advantage Krylov subspace methods have is that they only require compu
As such, if allocation of memory to store a matrix is infeasible, but the computational graph for this matrix can be determined, then methods such as the conjugate gradient method or the minimum residual method can be used to approximate the inverse of this matrix~\sidecite{giordano2018covariances, nocedal1999numerical}.
\subsection{The Lanczos Algorithm}
-\label{ssec:lanczos}
+\label{sec:lanczos}
One application known as the Lanczos algorithm adapts the power iteration method, which computes the greatest eigenvector of a square matrix \(\bm K\).
The power iteration method involves iteratively computing the normalised product of \(\bm K\) with a random vector, i.e.,
@@ -305,7 +362,7 @@ Further adjustments to the Lanczos algorithm yield the conjugate gradient method
\begin{align*}
\bm x = \bm K^{-1} \bm b,
\end{align*}
-where \(\bm K\) is a positive semi-definite matrix.\marginnote{Note that a positive semi-definite matrix is square.}
+where \(\bm K\) is a positive semi-definite matrix.\marginnote{A positive semi-definite matrix is square.}
The algorithm itself can be seen in \cref{alg:cg}.
% EXPLANATION
@@ -314,47 +371,28 @@ The algorithm itself can be seen in \cref{alg:cg}.
\begin{marginfigure}
\begin{algorithm}[H]
\caption{Conjugate Gradient (CG)}\label{alg:cg}
- \Input{$\bm K \succ 0$, $\bm b$, $\bm P \succ 0$, $J > 0$}
- \Output{$\bm x = \bm K^{-1} \bm b$, $\bm\alpha$, $\bm\beta$}
- $\bm x_0 \gets \bm 0$\;
- $\bm r_0 = \bm b - \bm K \bm x_0$\;
- $\bm p_0 = \bm z_0 = \bm P \bm r_0$\;
- $\gamma_0 = \bm r_0 \bm z_0$\;
- \For{$j \gets 1$ \KwTo $J$}{
- $\bm y_{j} \gets \bm A \bm p_{j-1}$\;
- $\alpha_j = \gamma_{j-1} / (\overline{\bm p_{j-1}} \bm y_{j})$\;
- $\bm x_{j} = \bm x_{j-1} + \alpha_j \bm p_{j-1}$\;
- $\bm r_{j} = \bm r_{j-1} - \alpha_j \bm y_{j}$\;
- $\bm z_{j} = \bm M \bm r_{j}$\;
- $\gamma_{j} = \overline{\bm r_{j}} \bm z_{j}$\;
- $\beta_{j} = \gamma_{j} / \gamma_{j-1}$\;
- $\bm p_{j} = \bm z_{j} + \beta_{j} \bm p_{j-1}$\;
+ \Input{\(\bm K \succ 0\), \(\bm b\), \(\bm P \succ 0\), \(J > 0\)}
+ \Output{\(\bm x = \bm K^{-1} \bm b\), \(\bm\alpha\), \(\bm\beta\)}
+ \(\bm x_0 \gets \bm 0\)\;
+ \(\bm r_0 = \bm b - \bm K \bm x_0\)\;
+ \(\bm p_0 = \bm z_0 = \bm P \bm r_0\)\;
+ \(\gamma_0 = \bm r_0 \bm z_0\)\;
+ \For{\(j \gets 1\) \KwTo \(J\)}{
+ \(\bm y_{j} \gets \bm A \bm p_{j-1}\)\;
+ \(\alpha_j = \gamma_{j-1} / (\overline{\bm p_{j-1}} \bm y_{j})\)\;
+ \(\bm x_{j} = \bm x_{j-1} + \alpha_j \bm p_{j-1}\)\;
+ \(\bm r_{j} = \bm r_{j-1} - \alpha_j \bm y_{j}\)\;
+ \(\bm z_{j} = \bm M \bm r_{j}\)\;
+ \(\gamma_{j} = \overline{\bm r_{j}} \bm z_{j}\)\;
+ \(\beta_{j} = \gamma_{j} / \gamma_{j-1}\)\;
+ \(\bm p_{j} = \bm z_{j} + \beta_{j} \bm p_{j-1}\)\;
}
- $\bm x \gets \bm x_{J}$\;
+ \(\bm x \gets \bm x_{J}\)\;
\end{algorithm}
\end{marginfigure}
-% \begin{algorithm}
-% \caption{Conjugate Gradient (CG)}\label{alg:cg}
-% \Input{$K \succ 0$, $b$, $P \succ 0$, $J > 0$}
-% \Output{$x = K^{-1} b$, $\bm\alpha$, $\bm\beta$}
-% $r = b - K x$\;
-% $p = z = P r$\;
-% $\gamma_0 = r z$\;
-% \While{$\norm{r}^2_2 > 10^{-4} \And j < J$}{
-% $y \gets K p$\;
-% $\alpha_j \gets \gamma_{j-1} / (p\T y)$\;
-% $x \gets x + \alpha_j p$\;
-% $r \gets r - \alpha_j y$\;
-% $z \gets P r$\;
-% $\gamma_j \gets {r}\T z$\;
-% $\beta_j \gets \gamma_j / \gamma_{j-1}$\;
-% $p \gets z + \beta_j p$\;
-% }
-% \end{algorithm}
-
\subsection[The Minimum Residual Method]{The Minimum Residual Method}[The Minimum Residual]
-\label{ssec:msminres}
+\label{sec:msminres}
Another method which is derived from the Lanczos algorithm is the minimum residual method (MINRES).
Given a certain number of iterations \(J\), the MINRES algorithm constructs a Krylov subspace by exploiting the three-vector recurrence (like conjugate gradient does) and solves a linear system \(\bm K \bm c = \bm b\), where \(\bm K\) is a square matrix (though not necessarily a symmetric one), by finding the least-squares solution which lies within this Krylov subspace of dimension \(J\), i.e., by solving the constrained optimisation problem characterised by
@@ -402,9 +440,9 @@ The MINRES algorithm can then be trivially extended to solve for multiple shifts
Complete elliptic integral of the first kind is given by
\begin{align}
- \mathcal{K}(p) = \int_0^{\pi/2} \left( 1 - m \sin(t)^2 \right)^{-1/2}\,dt, \quad p = 1 - m,
+ \mathcal{K}(p) = \int_0^{\pi/2} \left( 1 - m \sin(t)^2 \right)^{-1/2}\,dt,
\end{align}
-and is defined in the domain \(0 < p \leq 1\).\sidenote{Note that if \(p > 1\), then what should be computed is instead the identity \(\mathcal{K}(p) = \mathcal{K}(1/p) / \sqrt{p}\).}
+where \(p = 1 - m\), and is defined in the domain \(0 < p \leq 1\).\sidenote{If \(p > 1\), then what should be computed is instead the identity \(\mathcal{K}(p) = \mathcal{K}(1/p) / \sqrt{p}\).}
This integral is approximated by
\begin{align}
\mathcal{K}(p) \approx P(p) - \log(p) Q(p)
@@ -422,7 +460,7 @@ Honestly don't understand any of these whatsoever right now
\begin{marginfigure}
% \input{images/ellipk.tex}
\includegraphics[scale=1]{ellipk.pdf}
- \caption{Comparison of SciPy and JAX \texttt{ellipk} implementations. For values for which the complete elliptic integral \(K(p)\) is well-defined, both implementations yield the same results. Furthermore, due to its static nature, JAX implementation is compatible with \emph{just-in-time} compilation.}
+ \caption[Comparison of SciPy and JAX \texttt{ellipk} implementations.]{Comparison of SciPy and JAX \texttt{ellipk} implementations. For values for which the complete elliptic integral \(K(p)\) is well-defined, both implementations yield the same results. Furthermore, due to its static nature, JAX implementation is compatible with \emph{just-in-time} compilation.}
\labfig{fig:ellipk}
\end{marginfigure}
@@ -430,10 +468,8 @@ Honestly don't understand any of these whatsoever right now
\centering
% \input{images/ellipj.tex}
\includegraphics[scale=1]{ellipj.pdf}
- \caption{Comparison of SciPy and JAX \texttt{ellipj} implementations. For values for which the functions are well-defined, both implementations yield the same results. Functions plotted are $\sn(u \given m = 0.7)$, $\cn(u \given m = 0.7)$, and $\dn(u \given m = 0.7)$ Furthermore, due to its static nature, JAX implementation is compatible with \emph{just-in-time} compilation.}
+ \caption[Comparison of SciPy and JAX \texttt{ellipj} implementations.]{Comparison of SciPy and JAX \texttt{ellipj} implementations. For values for which the functions are well-defined, both implementations yield the same results. Functions plotted are $\sn(u \given m = 0.7)$, $\cn(u \given m = 0.7)$, and $\dn(u \given m = 0.7)$ Furthermore, due to its static nature, JAX implementation is compatible with \emph{just-in-time} compilation.}
\label{fig:ellipj}
\end{figure}
% \section{The Mahalanobis Distance and the Chi-Squared Distribution}
-
-
diff --git a/chapters/training.tex b/chapters/training.tex
@@ -6,9 +6,9 @@
According to Bayes' rule, the posterior probability for a given model is equal to
\begin{align}
- p(\theta \given x) = \frac{p(x \given \theta) p(\theta)}{p(x)},
+ p(\bm \theta \given \bm x) = \frac{p(\bm x \given \bm \theta) p(\bm \theta)}{p(\bm x)},
\end{align}
-where \(p(x)\) is known as the evidence.\sidenote{It's also known as the \emph{marginal likelihood}, since it's obtained by marginalising out \(\theta\) from the likelihood, as per \(p(x) = \int_\theta p(x \given \theta) p(\theta)\,d\theta\).}
+where \(p(\bm x)\) is known as the evidence.\sidenote{It's also known as the \emph{marginal likelihood}, since it's obtained by marginalising out \(\bm \theta\) from the likelihood, as per \(p(\bm x) = \int_{\bm \theta} p(\bm x \given \bm \theta) p(\bm \theta)\,d\bm \theta\).}
\sidetextcite{fong2020marginal} argue that maximising the marginal (and consequently the log-marginal) is equivalent to performing k-fold cross validation for all values \(k = 1, \ldots, \infty\) and choosing the model with the highest average posterior probability (across each of \(k\) folds and across all values of \(k\)).
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+\*!wO/l82V:L8%ea4LKޜ$:?_QW xŭ JTږ*li5 .}UC.
+X&pWNtK'[t讎Mٛ puVhAh:NVz5/5ԵI ~qwb!N16!A;$$xEBWk%)ւYMM0R' suͳY$T 1S
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+o[] Wy8Āі gՎ̟Կ;D~ wL>b}J;.mI:<1-)]^u\*v0LOb';AOfO~yɍ lFߜjc\]63(;J ^*) O. g ˪Ƙ/^h~X[>(ʿ?1 yۃz^pS
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+Xׂ{nV53)6jÚ
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+W l7 Fܯ ~EZ6aݼƗ;^T(%֮H5fR͢ b"/Xu^SO,@bu> 3χ:* 8)?Wȅo~鄤>#~Lζ}?(9Hivr[d ?a8Ed-ouͯWM/x4ڏ|ז s ̔|/N);AiCԄ&j/<Ơl9/Ym#!MqxƊ_X f%1Wڟ`_&h3KQFXweFv19d夕u3.v NݼSLۣG˪7fDNh
+ kK6b_SW c]Ʋ×!|Wq|ۖbn0
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+\ No newline at end of file
diff --git a/journals.bib b/journals.bib
@@ -1,4 +1,5 @@
-@String{colt="Conference on Learning Theory"}
-@String{icml="International Conference on Machine Learning"}
-@String{jmlr="Journal of Machine Learning Research"}
+@String{iclr="International Conference on Learning Representations"}
@String{nips="Advances in Neural Information Processing Systems"}
+@String{jmlr="Journal of Machine Learning Research"}
+@String{icml="International Conference on Machine Learning"}
+@String{colt="Conference on Learning Theory"}
diff --git a/main.bib b/main.bib
@@ -13,7 +13,7 @@
}
@inproceedings{botev2017practical,
- title = {Practical gauss-newton optimisation for deep learning},
+ title = {Practical {G}auss-{N}ewton optimisation for deep learning},
author = {Botev, Aleksandar and Ritter, Hippolyt and Barber, David},
booktitle = icml,
year = {2017}
@@ -26,6 +26,13 @@
year = {2021}
}
+@article{denker1990transforming,
+ title = {Transforming neural-net output levels to probability distributions},
+ author = {Denker, John and LeCun, Yann},
+ journal = nips,
+ year = {1990}
+}
+
@article{fong2020marginal,
title = {On the marginal likelihood and cross-validation},
author = {Fong, Edwin and Holmes, Chris C},
@@ -38,7 +45,7 @@
}
@inproceedings{gardner2018gpytorch,
- title = {GPyTorch: Blackbox Matrix-Matrix Gaussian Process Inference with GPU Acceleration},
+ title = {{GP}yTorch: Blackbox Matrix-Matrix Gaussian Process Inference with {GPU} Acceleration},
author = {Gardner, Jacob and Pleiss, Geoff and Weinberger, Kilian Q and Bindel, David and Wilson, Andrew G},
booktitle = nips,
year = {2018}
@@ -65,7 +72,7 @@
}
@article{harbrecht2012low,
- title = {On the low-rank approximation by the pivoted Cholesky decomposition},
+ title = {On the low-rank approximation by the pivoted {C}holesky decomposition},
author = {Harbrecht, Helmut and Peters, Michael and Schneider, Reinhold},
journal = {Applied numerical mathematics},
volume = {62},
@@ -75,6 +82,17 @@
publisher = {Elsevier}
}
+@article{heskes2000natural,
+ title = {On ``natural'' learning and pruning in multilayered perceptrons},
+ author = {Heskes, Tom},
+ journal = {Neural Computation},
+ volume = {12},
+ number = {4},
+ pages = {881--901},
+ year = {2000},
+ publisher = {MIT Press}
+}
+
@software{jax2018github,
author = {James Bradbury and Roy Frostig and Peter Hawkins and Matthew James Johnson and Chris Leary and Dougal Maclaurin and George Necula and Adam Paszke and Jake Vander{P}las and Skye Wanderman-{M}ilne and Qiao Zhang},
title = {{JAX}: composable transformations of {P}ython+{N}um{P}y programs},
@@ -83,29 +101,36 @@
year = {2018}
}
-@article{kingma2014adam,
- title = {Adam: A method for stochastic optimization},
- author = {Kingma, Diederik P and Ba, Jimmy},
- journal = {arXiv preprint arXiv:1412.6980},
- year = {2014}
+@inproceedings{kingma2014adam,
+ title = {Adam: A method for stochastic optimization},
+ author = {Kingma, Diederik P and Ba, Jimmy},
+ booktitle = iclr,
+ year = {2015}
}
@inproceedings{kunstner2019limitations,
- title = {Limitations of the empirical Fisher approximation for natural gradient descent},
+ title = {Limitations of the empirical {F}isher approximation for natural gradient descent},
author = {Kunstner, Frederik and Hennig, Philipp and Balles, Lukas},
booktitle = nips,
year = {2019}
}
@article{laplace1774memoire,
- title = {Mémoire sur la probabilité des causes par les événements.},
+ title = {M\'emoire sur la probabilit\'e des causes par les \'ev\'enements},
author = {Laplace, Pierre-Simon},
- journal = {M\'em. De Math. Et Phys. Pr\'esent\'es à l’Acad. Roy. Des Sci},
+ journal = {M\'em. De Math. Et Phys. Pr\'esent\'es à l'Acad. Roy. Des Sci},
volume = {6},
pages = {621--656},
year = {1774}
}
+@article{lecun1989optimal,
+ title = {Optimal brain damage},
+ author = {LeCun, Yann and Denker, John and Solla, Sara},
+ journal = nips,
+ year = {1989}
+}
+
@book{mackay2003information,
title = {Information theory, inference and learning algorithms},
author = {MacKay, David},
@@ -114,7 +139,7 @@
}
@inproceedings{martens2015optimizing,
- title = {Optimizing neural networks with kronecker-factored approximate curvature},
+ title = {Optimizing neural networks with {K}ronecker-factored approximate curvature},
author = {Martens, James and Grosse, Roger},
booktitle = icml,
year = {2015}
@@ -152,14 +177,30 @@
}
@inproceedings{pleiss2020fast,
- title = {Fast matrix square roots with applications to Gaussian processes and Bayesian optimization},
+ title = {Fast matrix square roots with applications to {G}aussian processes and Bayesian optimization},
author = {Pleiss, Geoff and Jankowiak, Martin and Eriksson, David and Damle, Anil and Gardner, Jacob},
booktitle = nips,
year = {2020}
}
+@article{ritter2018online,
+ title = {Online structured laplace approximations for overcoming catastrophic forgetting},
+ author = {Ritter, Hippolyt and Botev, Aleksandar and Barber, David},
+ journal = {Advances in Neural Information Processing Systems},
+ volume = {31},
+ year = {2018}
+}
+
+@inproceedings{ritter2018scalable,
+ title = {A scalable laplace approximation for neural networks},
+ author = {Ritter, Hippolyt and Botev, Aleksandar and Barber, David},
+ booktitle = iclr,
+ volume = {6},
+ year = {2018}
+}
+
@article{sabne2020xla,
- title = {XLA: Compiling Machine Learning for Peak Performance},
+ title = {{XLA}: Compiling Machine Learning for Peak Performance},
author = {Sabne, Amit},
year = {2020}
}
@@ -171,6 +212,13 @@
year = {2013}
}
+@book{woodbury1950inverting,
+ title = {Inverting modified matrices},
+ author = {Woodbury, Max},
+ year = {1950},
+ publisher = {Statistical Research Group}
+}
+
@article{wu2019logan,
title = {Logan: Latent optimisation for generative adversarial networks},
author = {Wu, Yan and Donahue, Jeff and Balduzzi, David and Simonyan, Karen and Lillicrap, Timothy},
diff --git a/main.tex b/main.tex
@@ -173,16 +173,16 @@
}
\makeatother
-% %----------------------------------------------------------------------------------------
-% % DEDICATION
-% %----------------------------------------------------------------------------------------
+%----------------------------------------------------------------------------------------
+% DEDICATION
+%----------------------------------------------------------------------------------------
-% \dedication{
-% % \blackout
-% % {It’s an incredibly exciting thing, this one, meaningless life of yours.}\\
-% {You don't have to have a dream.}\\
-% \flushright -- Tim Minchin
-% }
+\dedication{
+ % \blackout
+ % {It’s an incredibly exciting thing, this one, meaningless life of yours.}\\
+ {You don't have to have a dream.}\\
+ \flushright -- Tim Minchin
+}
%----------------------------------------------------------------------------------------
% OUTPUT TITLE PAGE AND PREVIOUS
@@ -192,18 +192,18 @@
\maketitle
-% %----------------------------------------------------------------------------------------
-% % PREFACE
-% %----------------------------------------------------------------------------------------
+%----------------------------------------------------------------------------------------
+% PREFACE
+%----------------------------------------------------------------------------------------
-% \input{chapters/preface.tex}
-% \index{preface}
+\input{chapters/preface.tex}
+\index{preface}
-% \input{chapters/acknowledgments.tex}
-% \index{acknowledgments}
+\input{chapters/acknowledgments.tex}
+\index{acknowledgments}
-% \input{chapters/abstract.tex}
-% \index{abstract}
+\input{chapters/abstract.tex}
+\index{abstract}
%----------------------------------------------------------------------------------------
% TABLE OF CONTENTS & LIST OF FIGURES/TABLES
@@ -243,10 +243,8 @@
\input{chapters/introduction.tex}
\input{chapters/theory.tex}
-\input{chapters/literature.tex}
\input{chapters/training.tex}
\input{chapters/sampling.tex}
-\input{chapters/mixture.tex}
\input{chapters/experiments.tex}
\input{chapters/results.tex}
\input{chapters/conclusion.tex}
