HP Laserjet 3000 Driver Download. Print out faster with the colors black and white with print speeds up to 30 pages per minute (ppm) and fast color documents to 15 ppm.
In addition, Instant-on technology reduces warm-up time and allows the printer to consume less energy and realize keausa less on the machine.
Printer calibration with an easy to set up, so that you are not being raised when you need documents fast.
Share with multiple users and increase efficiency, improve teamwork even more productivity HP Jetdirect Fast Ethernet embedded print server delivers high performance and better compatibility, which is quite extensive, proven reliability and ease of use with advanced security features.
Printing complex that is not a problem for these printers. 533 MHz processor and up to 256 MB RAM standard (512 MB of RAM, the maximum) and allows to print faster and easier job processing, providing a quick return to work printing applications and increase uptime.
Optional 500-sheet input tray, providing a capacity of 850 sheet input, reducing user interaction. The printer can accommodate a wide variety of paper sizes, types and weights, and built-in automatic two-sided Printing lets you print more efficient paper.


text:
My Dad's Charity Work
March 24, 2018 mrjain	Leave a comment
Earlier this month was International Women's Day and an Indian news channel wanted to interview my dad for this charity work. His charity has given many loans to women and hence they wanted to highlight the work he has done to help women.
Once the news clip aired we shared it on YouTube, Facebook and WhatsApp and the outpouring of support was amazing. The video is below:
What most people don't know is that for the past 6 years my dad has slowly been working near his village of Shujalpur, Madhya Pradesh to help people in need. He has helped over 50 families and given out interest free loans of around Rs. 10 lakhs (USD $15,000). The default rate on these loans is something that not even a top banker at ICICI Bank can touch. He has had a single default of $60 because the person passed away before they could pay back the loan back!
My dad tells me it's not about the money for many people, they need guidance. A simple example is someone wanted to open a small provisions store and so may dad gave them a loan. After a month, my dad went back and asked about the sales. That person didn't keep track, so my dad said get a diary and write down all the sales. Once this person did this, he could see a pattern emerge and realized what sold and what didn't.
To most people this would seem standard to track daily sales but for many people that's not the case. This person was able to figure out that certain days were consistently slow, so he would close the shop and go into town on that day to buy more goods. A clear example of where money helped but the guidance and knowledge was much more useful.
For the past 6 years, my dad has kept a blog describing all the people he has given loans to. The stories are really interesting to read, visit ShantiSeva.org to read more.

redpajama_set_name: RedPajamaCommonCrawl


John Erik Fossum and Stéphane Roussel have published a special issue of International Journal called 'The Arctic is Hot - part II. Moving above and below the state: Actors and issues'.
Hans-Jörg Trenz has written the chapter 'Bottom-Up Europeanization: Civil Society Involvement and EU Governance in the New Member States' and Chris Lord has written the chapter 'The Quality of Democracy' in the volume 'Democracy, State and Society. European Integration in Central and Eastern Europe' edited by Magdalena Góra and Katarzyna Zielińska.
Marianne Riddervold has published the article "From reason-giving to collective action: Argument-based learning and European integration" in Cooperation and Conflict 46(4).
Hans-Jörg Trenz has published the chapter "Social theory and European integration" in the volume "Sociology of the European Union" edited by Adrian Favell and Virgine Guiraudon.
Helene Sjursen has edited a special issue of Journal of European Public Policy called "The EU's Common Foreign and Security Policy: the quest for democracy" with contributions from Christopher Lord and Erik O. Eriksen.
Erik Oddvar Eriksen and John Erik Fossum have edited the book 'Rethinking Democracy and the European Union' that draws upon leading scholars and practitioners from the RECON project.
The EEA as a model?
Ulf Sverdrup has written the chapter "Modes of association to the EU: The EEA as a model?" in the volume "Northern Europe and the Future of the EU" edited by Helge Høibraaten and Jochen Hille.
Erik O. Eriksen has written the chapter "Democracy lost. The EEA Agreement And Norway's Democratic Deficit" in the volume "Northern Europe and the Future of the EU" edited by Helge Høibraaten and Jochen Hille.
Marianne Riddervold has published the article "Finally flexing its muscles? Atalanta – The European Union's naval military operation against piracy" in European Security 20(3).
A Special Issue of the Journal of European Public Policy on 'Agency Governance in the European Union' edited by Berthold Rittberger and Arndt Wonka has been published.
Erik O. Eriksen has written the chapter "An emerging European public sphere" in the anthology "Jürgen habermas Volumes I and II" by Camil Ungureanu, Klaus Günther and Christian Joerges.
Christer Gulbrandsen has published the article "The EU and the implementation of international law: the case of 'sea-level bureaucrats'" in Journal of European Public Policy Vol. 18 No. 7.
Christopher Lord has published the article "The European Parliament and the legitimation of agencification" in Journal of European Public Policy Vol 18 No 6 2011.
EU-level agencies: new executive centre formation or vehicles for national control?
Morten Egeberg and Jarle Trondal have published the article "EU-level agencies: new executive centre formation or vehicles for national control?" in a special issue of the Journal of European Public Policy on "Agency Governance in the European Union".
Jarle Trondal has published the article 'Bureaucratic Structure and Administrative Behaviour: Lessons from International Bureaucracies' in West European Politics 34(4).
Review essay: A cosmopolitan constellation?
John Erik Fossum has published the review article 'A cosmopolitan constellation' in European Journal of Social Theory.
Agustín José Menéndez and John Erik Fossum have edited the volume "Law and Democracy in Neil MacCormick's Legal and Political Theory. The Post-Sovereign Constellation" published with the Springer series Law and Philosophy Library.
Agencification and Location: Does Agency Site Matter?
Morten Egeberg and Jarle Trondal have published the article 'Agencification and Location: Does Agency Site Matter?' in Public Organization Review 11(2).
Jarle Trondal has published the article 'Domestic Agencies in an Emergent European Executive Order' in Journal of European Integration vol. 33. no. 1.
Erik Oddvar Eriksen and John Erik Fossum have published the article "Bringing European Democracy Back In—Or How to Read the German Constitutional Court's Lisbon Treaty Ruling" in European Law Journal.
Christopher Lord has published the article 'Polecats, lions, and foxes: Coasian bargaining theory and attempts to legitimate the Union as a constrained form of political power' in European Political Science Review 3(1).
No effect, weapon of the weak or reinforcing executive dominance?
Pieter de Wilde has published the article 'No effect, weapon of the weak or reinforcing executive dominance? How media coverage affects national parliaments' involvement in EU policy-formulation' in Comparative European Politics 9(2).
Åse Gornitzka and Ulf Sverdrup have published the article 'Access of Experts: Information and EU Decision-making' in a special isssue of West European Politics 34(1).


text:
Sam Roddick, the founder of Coco de Mer, ushers us into the sitting room of her Hampstead home
Interview by Dominic Lutyens
Sat 11 Jul 2009 19.01 EDT
Sam Roddick's sitting room. Photograph: Katherine Rose
I've lived in this five-storey house for a year and a half with my 10-year-old daughter. It's Victorian but the architects converted it into a more open-plan house with 16 rather than 22 rooms. The work was completed in October 2007, two weeks after my mother [Anita] died. All the finishes inside, including wall tiles and stone floors, were supplied by London architectural salvage company Retrouvius - I wanted everything here to be reclaimed. They provided the mantelpiece, fire grate and 1950s side tables for my sitting room, which I call my "lazy space" because I lounge around here. I love sitting next to an open fire. I also entertain here. I see my home as a drop-in for friends. There's a big blackboard outside my front door where friends chalk up messages to say when they'd like to pop by. When they do come round, I cook for them - mainly Italian food.
I was brought up in an Italian household in Littlehampton, Sussex - my mother and her parents were Italian; my father is Scottish. The kitchen was the heart of the home, and that remains true for me.
I also use this space to screen films - mainly documentaries highlighting social and political issues - and I invite television journalists. I've screened films for the Burma Campaign UK and Josh Tickell's documentary Fuel, which promotes biodiesel made of algae as an alternative fuel to oil. On these occasions I sometimes hire celebrity chef Aine McAteer, who cooks macrobiotic meals.
My home is decorated with lots of nostalgic, old family ornaments.
I like to be surrounded with things that have a story, have value. My ideal environment would be minimalist, but I can't seem to manage that. This space is all about reflecting my character, and the more of myself I see reflected in my environment the more comfortable I feel.
• James Ostrer's show, Death, Sex and Re-Birth, is at Coco de Mer Gallery, 108 Draycott Avenue, London SW3 (020 7584 7615; www.coco-de-mer.com) until 29 August


Most photographers like to list out their equipment, and I am part of that sheep-like crowd!
Generic 0.42x wide-angle macro screw-in filter (screws apart to be macro only).
Car – in previous years, I travelled a lot for work and got to experience a lot of South East Queensland from behind the steering wheel.
Feet – I also walk a fair bit. In a short lunch break, the range is limited, but the city is rich with photographic opportunities. I spend a lot of my time walking around looking up. This often results in a photo, but even when it doesn't, there's a lot of beautiful things 10m and more above eye level.
It is awesome! I've done all of my portraits with it. You have to be lucky though – I've talked a few friends into buying one and they've all ended up with duds. I couldn't be happier with mine though!


A real wow present for all gin loving bakers!
Cupcakes with a difference using a Michel Roux recipe.
All ingredients included - just add butter, eggs and limes!
Gin is definitely in! And now you can bake some Gin and Tonic cupcakes! Very unusual and great fun for all gin lovers out there.
With a recipe by Michel Roux, the Michelin chef from Le Gavroche restaurant in London, this gin and tonic cupcake kit is a top star present. The kit includes all the dry ingredients (caster sugar, self-raising flour, plain flour, baking powder, natural vanilla flavouring and icing sugar) and the all important liquid ingredients of a 50 ml miniature gin bottle and 150 ml tin of tonic water. All you need to provide is the muffin tin, unsalted butter, eggs and limes. The mixture included in the beautifully packaged kit is for 12 cupcakes. There's nothing stopping you from baking more - you will just have to buy more gin etc!
The recipe gives 4 stage instructions, preparation time is approximately 30 minutes and baking time is 15-20 minutes - time to pour yourself a G&T in anticipation! Cupcake cases, piping bag and nozzle are also included.
Made in Britain this present is a real hit with bakers, chefs, gin drinking friends and was a winner in the Great Taste Awards 2018. The kit makes a great gift for birthdays, celebrations and big parties.
L24, W23, H9 cms approx.
Included in the kit: caster sugar, self-raising flour, plain flour, baking powder, natural vanilla flavouring, icing sugar, gin, tonic water.


text:
Our House: The Madness Musical
Fri 20 Nov '20, 7:30pm – 10:30pm
Sat 21 Nov '20, 7:30pm – 10:30pm
Sun 22 Nov '20, 3:00pm – 6:00pm
Thu 3 Dec '20, 7:30pm – 10:30pm
Fri 4 Dec '20, 7:30pm – 10:30pm
Sat 5 Dec '20, 7:30pm – 10:30pm
View all sessions
Coasters Musical Theatre, 15 Hinemoa Street, Paraparaumu
Featuring our life in the house Our House band made up of fabulous local talent!
The story follows Camden lad Joe Casey who, on the night of his 16th birthday, makes a decision that will change his life. Trying to impress Sarah, the girl of his dreams, Joe breaks into a building development overlooking his home on Casey Street. But things take a turn for the worse as the police turn up. Joe's life splits into two; the Good Joe who stays and gives himself up and Bad Joe who flees and leaves Sarah to run from the police.
Our House follows the two paths that Joe's life could take after that fateful night; one path means a criminal record and social exclusion, while the other will lose him the girl that he loves.
Over a period of seven years and two alternative lives, Joe deals with the consequences of that night. Whilst one Joe fights to keep Sarah, the other is marrying her in a glitzy Vegas wedding and, ultimately, while Good Joe fights to save his house on Casey Street, Bad Joe is determined to demolish it with tragic consequences.
Spookers for Kids - Taste of Terror
Spookers, Karaka, Auckland
R13 Nights
Coastal Blues
Te Raukura ki Kāpiti, Paraparaumu, Kapiti Coast
Gowns, Beautiful Gowns
Phoenix Cabaret, Newton, Auckland
The Web Genius Kapiti Run for Youth 2021
Paraparaumu Beach, Paraparaumu, Kapiti Coast
Sun 21 Mar


text:
Q: You need to build spark before running this program I downloaded the pre-built spark 1.2.1 for hadoop 2.4. I then attempted to launch the spark shell using ./bin/spark-shell, and I got the following error message:
Failed to find Spark assembly in /Users/XXXX/Software Libraries/Java Libraries/General/spark-1.2.1-bin-hadoop2.4/lib
You need to build Spark before running this program.
Why is it failing to launch with this error even though I downloaded the pre-built version?

A: Bizarre though it may seem, moving into my home directory solves the problem. Perhaps something to do with that fact that my directory names have spaces in them, but I can't be sure.

A: build/mvn -Pyarn -Phadoop-2.4 -Dhadoop.version=2.4.0 -DskipTests clean package

redpajama_set_name: RedPajamaStackExchange


It was disorienting to new users, they were used to how folders work in Windows® Explorer and other tools, and they found the Indent/Outdent model of SpiraTest non-intuitive.
It led to strange behavior--for new and old customers alike - filtering and expanding/collapsing folders resulted in some atypical interactions. It also meant that dragging a test case into a folder was a non-trivial operation.
It had major performance impacts on the system - if you frequently saw 'database timeouts', 'deadlocks' and 'refresh the database index messages', this was the culprit.
It was the cause of a number of bugs that frustrated some customers - for example having to fill out a required custom property on a folder that was really only intended for a test case.
Finally, this folder model constrained us being able to further develop a number of improvements to test cases, test sets and tasks - including support for workflows.
These folders acted more like traditional 'containers' that had just a name and hierarchy with no other properties. They were displayed to the left of the main item list and were not filterable and did not ever appear in the main item list.
Not integrated into the main list view - consequently they were often ignored by users and there was no prominent view of the current child folders in the main list.
No bulk operations available - since they did not appear in the main list, there was no way to perform bulk operations on a folder.
Folders are distinct objects to the artifacts they contain. Folders have a name, a description, and a place in a folder tree, but they have no other properties. They are containers containing artifacts only. This is similar to how folders work in most other systems.
So what happened to the old test case and test set folders during the upgrade process?
The old folders will have been removed, replaced with the new simple container equivalent. That does mean that the old artifact data associated with each folder (history, custom properties, comments, etc.) will have been discarded during the upgrade.
List views for test cases, test sets, documents, and tasks have a folder tree on the left for navigating between folders.
Once a folder is selected on the left, it shows its contents in a list on the right, including any folders it contains.
If you want to move an item into a folder, you drag it from the right into the relevant folder on the left.
First, deleting a folder does not delete its contents. Instead the contents are moved to the 'root' folder.
We think this new model is much easier to understand and work with. It's faster and more robust. You can sort any list by any field--no more needing to manually order things.
For most of our customers, the transition should be seamless and quick and brings lots of immediate benefits. Some customers--notably those that made extensive use of the different attributes you could attach to a folder--will want to plan their migration more carefully. We recommend only upgrading to version 5.0 when you have transitioned your 4.2 data to no longer rely on the data attached to a folder.
This was a large architectural change for SpiraTeam but there is more that we can do. Currently the new folder system behaves mostly the same way as an application like Windows Explorer, but not completely. For instance, you can only drag an item into a folder on the left, not onto a folder on the right hand list view.
Over the next few months we hope to roll out refinements to the new model to make the experience more intuitive. Once the new system is as solid and easy to use as possible we may start exploring other ways that list data could be displayed to help your work in SpiraTeam be even faster.
We hope this overview of the changes to folders was helpful. The next post about SpiraTeam 5.0 will discuss improvements to the Instant Messenger.


text:
\subsubsection*{Acknowledgments}

Use unnumbered third level headings for the acknowledgments. All
acknowledgments, including those to funding agencies, go at the end of the paper.
\fi

\newpage
{\footnotesize
\bibliographystyle{iclr2018_conference}

\section{Meta-Learning Formulation}
\label{sec:background}

The goal of a meta-learner is to extract task-general knowledge through the experience of solving a number of related tasks.
By using this learned prior knowledge, the learner has the potential to quickly adapt to novel tasks even in the face of limited data or limited computation time.

Formally, we consider a dataset $\mathscr{D}$ that defines a distribution over a family of tasks $\mathcal{T}$.
These tasks share some common structure such that learning to solve a single task has the potential to aid in solving another.
Each task $\mathcal{T}$ defines a distribution over data points $\bvar{\x}$, which we assume in this work to consist of inputs and either regression targets or classification labels $\bvar{\y}$ in a supervised learning problem~\citep[although this assumption can be relaxed to include reinforcement learning problems; \textit{e.g.,}\  see][]{finn2017model}.
The objective of the meta-learner is to be able to minimize a task-specific performance metric associated with any given unseen task from the dataset given even only a small amount of data from the task; \textit{i.e.,}\  to be capable of \gls{fa} to a novel task.

In the following subsections, we discuss two ways of formulating a solution to the \gls{ml} problem: \gls{gb} \gls{ho} and probabilistic inference in a \gls{hbm}.
These approaches were developed orthogonally, but, in \secref{method_mamlashb}, we draw a novel connection between the two.

\subsection{\Gls{ml} as \Gls{gb} Hyperparameter Optimization}
\label{sec:background_mlopt}

A parametric meta-learner aims to find some shared parameters $\bm{\theta} $ that make it easier to find the right task-specific parameters $\bm{\phi} $ when faced with a novel task.
A variety of meta-learners that employ gradient methods for task-specific \gls{fa} have been proposed~\citepeg{andrychowicz2016learning,li2017alearning,li2017blearning,wichrowska2017learned}.
\Gls{maml}~\citep{finn2017model} is distinct in that it provides a \gls{gb} \gls{ml} procedure that employs a single additional parameter (the \gls{ml} rate) and operates on the same parameter space for both \gls{ml} and \gls{fa}.
These are necessary features for the equivalence we show in \secref{method_mamlashb}.

To address the \gls{ml} problem, \gls{maml} estimates the parameters $\bm{\theta} $ of a set of models so that when one or a few batch gradient descent steps are taken from the initialization at $\bm{\theta} $ given a small sample of task data
$\smash{\bvar{\x}_{j_1}, \dots, \bvar{\x}_{j_N} \sim p_{\mathcal{T}\taskidx{j}}(\bvar{\x})}$
each model has good generalization performance on another sample
$\smash{\bvar{\x}_{j_{N+1}}, \dots, \bvar{\x}_{j_{N+M}} \sim p_{\mathcal{T}\taskidx{j}}(\bvar{\x})}$ from the same task.
The \gls{maml} objective in a maximum likelihood setting is
\begin{align}
    \label{eq:maml}
    \mathcal{L}(\bm{\theta} ) =
    \frac{1}{J}\sum_j
    \Bigg[
    \frac{1}{M}\sum_m
    -\log p
    \big(\, \bvar{\x}\taskidx{j_{N+m}} \;\vert\; \underbrace{\bm{\theta}
    - \alpha
    \grad{\bm{\theta} }
    \frac{1}{N}\sum_n
    -\log \prob{\bvar{\x}\taskidx{j_n} \;\vert\; \bm{\theta} }}_{\bm{\phi} \taskidx{j}}
    \big)\Bigg]
\end{align}
where we use $\bm{\phi} \taskidx{j}$ to denote the updated parameters after taking a single batch gradient descent step from the initialization at $\bm{\theta} $ with step size $\alpha$ on the negative log-likelihood associated with the task $\mathcal{T}\taskidx{j}$.
Note that since $\bm{\phi} \taskidx{j}$ is an iterate of a gradient descent procedure that starts from $\bm{\theta} $, each $\bm{\phi} \taskidx{j}$ is of the same dimensionality as $\bm{\theta} $.
We refer to the inner gradient descent procedure that computes $\bm{\phi} \taskidx{j}$ as \textit{\gls{fa}}.
The computational graph of \gls{maml} is given in \figref{model} (left).

\subsection{\Gls{ml} as \Gls{hb}ian Inference}
\label{sec:background_mlhb}
\input{figures/graphical_model}

An alternative way to formulate \gls{ml} is as a problem of probabilistic inference in the hierarchical model depicted in \figref{model} (right).
In particular, in the case of \gls{ml}, each task-specific parameter $\smash{\bm{\phi} \taskidx{j}}$ is distinct from but should influence the estimation of the parameters $\smash{\{\bm{\phi} \taskidx{j'} \mid j' \neq j\}}$ from other tasks.
We can capture this intuition by introducing a meta-level parameter $\bm{\theta} $ on which each task-specific parameter is statistically dependent.
With this formulation, the mutual dependence of the task-specific parameters $\bm{\phi} _j$ is realized only through their individual dependence on the meta-level parameters $\bm{\theta} $
As such, estimating $\bm{\theta} $ provides a way to constrain the estimation of each of the $\bm{\phi} \taskidx{j}$.

Given some data in a multi-task setting, we may estimate $\bm{\theta} $ by integrating out the task-specific parameters to form the marginal likelihood of the data.
Formally, grouping all of the data from each of the tasks as $\bvar{\X}$ and again denoting by $\bvar{\x}\taskidx{j_1}, \dots, \bvar{\x}\taskidx{j_N}$ a sample from task $\mathcal{T}\taskidx{j}$, the marginal likelihood of the observed data is given by
\begin{align}
    \prob {
        \bvar{\X} \;\vert\; \bm{\theta}
    }
    &=
    \prod\range{j}{}
    \brac{
        \int
            \prob{
               \bvar{\x}\taskidx{j_1}, \dots, \bvar{\x}\taskidx{j_N}
                \;\vert\;
                \bm{\phi} \taskidx{j}
            }
            \prob {
                \bm{\phi} \taskidx{j}
            \;\vert\;
                \bm{\theta}
            }
        \mathop{}\!\mathrm{d}  \bm{\phi} \taskidx{j}
    }
    \enspace .
    \label{eq:maml_marginal_exact}
\end{align}
Maximizing (\ref{eq:maml_marginal_exact}) as a function of $\bm{\theta} $ gives a point estimate for $\bm{\theta} $, an instance of a method known as \gls{eb} \citep{bernardo2006bayesian,gelman2014bayesian} due to its use of the data to estimate the parameters of the prior distribution.

\Gls{hb}ian models have a long history of use in both \gls{tl} and \gls{da}~\citep[\textit{e.g.,}\ ][]{lawrence2004learning,yu2005learning,gao2008knowledge,daume2009bayesian,wan2012sparse}.
However, the formulation of \gls{ml} as \gls{hb} does not automatically provide an inference procedure, and furthermore, there is no guarantee that inference is tractable for expressive models with many parameters such as deep neural networks.
\section{Introduction}
\label{sec:intro}

A remarkable aspect of human intelligence is the ability to quickly solve a novel problem and to be able to do so even in the face of limited experience in a novel domain.
Such fast adaptation is made possible by leveraging prior learning experience in order to improve the efficiency of later learning.
This capacity for \textit{\gls{ml}} also has the potential to enable an artificially intelligent agent to learn more efficiently in situations with little available data or limited computational resources~\citep{schmidhuber1987evolutionary,bengio1991learning,naik1992meta}.

In machine learning, \gls{ml} is formulated as the extraction of domain-general information that can act as an inductive bias to improve learning efficiency in novel tasks~\citep{caruana1998multitask,thrun1998learning}.
This inductive bias has been implemented in various ways:
as learned hyperparameters in a \gls{hbm} that regularize task-specific parameters~\citep{heskes1998solving},
as a learned metric space in which to group neighbors~\citep{bottou1992local},
as a trained \gls{rnn} that allows encoding and retrieval of episodic information~\citep{santoro2016meta},
or as an optimization algorithm with learned parameters~\citep{schmidhuber1987evolutionary,bengio1992optimization}.

The \gls{maml} of \cite{finn2017model} is an instance of a learned optimization procedure that directly optimizes the standard gradient descent rule.
The algorithm estimates an initial parameter set to be shared among the task-specific models; the intuition is that gradient descent from the learned initialization provides a favorable inductive bias for \gls{fa}.
However, this inductive bias has been evaluated only empirically in prior work~\citep{finn2017model}.

In this work, we present a novel derivation of and a novel extension to \gls{maml}, illustrating that this algorithm can be understood as inference for the parameters of a prior distribution in a \gls{hbm}.
The learned prior allows for quick adaptation to unseen tasks on the basis of an implicit predictive density over task-specific parameters.
The reinterpretation as \gls{hb} gives a principled statistical motivation for \gls{maml} as a \gls{ml} algorithm, and sheds light on the reasons for its favorable performance even among methods with significantly more parameters.
More importantly, by casting gradient-based \gls{ml} within a Bayesian framework, we are able to improve \gls{maml} by taking insights from Bayesian posterior estimation as novel augmentations to the gradient-based \gls{ml} procedure. We experimentally demonstrate that this enables better performance on a few-shot learning benchmark.
\section{Conclusion}
\label{sec:conc}

We have shown that \glsfirst{maml} estimates the parameters of a prior in a \glsfirst{hbm}.
By casting \glsfirst{gb} \gls{ml} within a Bayesian framework, our analysis opens the door to novel improvements inspired by probabilistic machinery.

As a step in this direction, we propose an extension to \gls{maml} that employs a Laplace approximation to the posterior distribution over task-specific parameters.
This technique provides a more accurate estimate of the integral that, in the original \gls{maml} algorithm, is approximated via a point estimate.
We show how to estimate the quantity required by the Laplace approximation using \glsfirst{kfac}, a method recently proposed to approximate the quadratic curvature of a neural network objective for the purpose of a second-order gradient descent technique.

Our contribution illuminates the road to exploring further connections between \gls{gb} \gls{ml} methods and \gls{hbm}ing.
For instance, in this work we assume that the predictive distribution over new data-points is narrow and well-approximated by a point estimate.
We may instead employ methods that make use of the variance of the distribution over task-specific parameters in order to model the predictive density over examples from a novel task.

Furthermore, it is known that the Laplace approximation is inaccurate in cases where the integral is highly skewed, or is not unimodal and thus is not amenable to approximation by a single Gaussian mode.
This could be solved by using a finite mixture of Gaussians, which can approximate many density functions arbitrarily well \citep{sorenson1971recursive,alspach1972nonlinear}.
The exploration of additional improvements such as this is an exciting line of future work.
\section{Linking \Gls{gb} \Gls{ml} \& \Gls{hb}}
\label{sec:method}

In this section, we connect the two independent approaches of \secref{background_mlopt} and \secref{background_mlhb} by showing that \gls{maml} can be understood as \gls{eb} in a hierarchical probabilistic model.
Furthermore, we build on this understanding by showing that a choice of update rule for the task-specific parameters $\bm{\phi} \taskidx{j}$ (\textit{i.e.,}\  a choice of inner-loop optimizer) corresponds to a choice of prior over task-specific parameters, $\smash{\prob{\bm{\phi} _j \;\vert\; \bm{\theta} }}$.

\subsection{Model-Agnostic Meta-Learning as Empirical Bayes}
\label{sec:method_mamlashb}

In general, when performing \gls{eb}, the marginalization over task-specific parameters $\bm{\phi} \taskidx{j}$ in (\ref{eq:maml_marginal_exact}) is not tractable to compute exactly.
To avoid this issue, we can consider an approximation that makes use of a point estimate $\smash{\hat{\bm{\phi} }\taskidx{j}}$ instead of performing the integration over $\bm{\phi} $ in (\ref{eq:maml_marginal_exact}).
Using $\smash{\hat{\bm{\phi} }\taskidx{j}}$ as an estimator for each $\bm{\phi} \taskidx{j}$, we may write the negative logarithm of the marginal likelihood as
\begin{align}
    \label{eq:maml_marginal_point}
    - \log \prob{\bvar{\X} \;\vert\; \bm{\theta} }
    \approx \sum_j \sbrac{- \log \prob{\bvar{\x}\taskidx{j_{N+1}}, \dots \bvar{\x}\taskidx{j_{N+M}} \;\vert\; \hat{\bm{\phi} }\taskidx{j}}}
    \enspace .
\end{align}
Setting
$\hat{\bm{\phi} }\taskidx{j} = \bm{\theta}   + \alpha  \grad{\bm{\theta} }
\log \prob{\bvar{\x}\taskidx{j_{1}}, \dots, \bvar{\x}\taskidx{j_{N}} \;\vert\; \bm{\theta} }$
for each $j$ in (\ref{eq:maml_marginal_point}) recovers the unscaled form of the one-step MAML objective in (\ref{eq:maml}).
This tells us that the MAML objective is equivalent to a maximization
with respect to the meta-level parameters $\bm{\theta} $
of the marginal likelihood $\smash{\prob{\bvar{\X} \;\vert\; \bm{\theta} }}$,
where a point estimate for each task-specific parameter $\smash{\bm{\phi} \taskidx{j}}$ is computed via one or a few steps of gradient descent.
By taking only a few steps from the initialization at $\bm{\theta} $, the point estimate $\smash{\hat{\bm{\phi} \taskidx{j}}}$ trades off minimizing
the \gls{fa} objective $\smash{- \log \prob{\bvar{\x}\taskidx{j_{1}}, \dots, \bvar{\x}\taskidx{j_{N}} \;\vert\; \bm{\theta} }}$ with staying close in value to the parameter initialization $\bm{\theta} $.

\input{figures/alg_maml-hb}
\input{figures/alg_ml-point}

We can formalize this trade-off by considering the linear regression case.
Recall that the \gls{map} estimate of $\bm{\phi} \taskidx{j}$ corresponds to the global mode of the posterior
$\smash{\prob{\bm{\phi} _j \;\vert\; \bvar{\x}\taskidx{j_1}, \dots \bvar{\x}\taskidx{j_N}, \bm{\theta} }
\propto \prob{\bvar{\x}\taskidx{j_1}, \dots \bvar{\x}\taskidx{j_N} \;\vert\; \bm{\phi} _j}
\prob{\bm{\phi} _j \;\vert\; \bm{\theta} }}$.
In the case of a linear model, early stopping
of an
iterative gradient descent procedure to
estimate $\bm{\phi} \taskidx{j}$ is exactly equivalent to \gls{map} estimation of $\bm{\phi} \taskidx{j}$
under the assumption of a prior that depends on the number of descent steps as well as the direction in which each step is taken.
In particular, write the input examples as $\bvar{\X}$ and the vector of regression targets as $\bvar{\y}$, omit the task index from $\bm{\phi} $,
and consider the gradient descent update
\begin{align}
    \bm{\phi} \iteridx{k}
    &= \bm{\phi} \iteridx{k-1} - \alpha \grad{\bm{\phi} }
    \left[\| \bvar{\y} - \bvar{\X}  \bm{\phi}  \|_2^2\right]_{\bm{\phi} =\bm{\phi} \iteridx{k-1}} \nonumber\\
    &= \bm{\phi} \iteridx{k-1} - \alpha \bvar{\X}^\mathrm{T}\brac{\bvar{\X} \bm{\phi} \iteridx{k-1} - \bvar{\y}}

    \label{eq:grad_update}
\end{align}
for iteration index $k$ and learning rate $\alpha \in \mathbb{R}^+$.
\citet{santos1996equivalence} shows that,
starting from $\bm{\phi} \iteridx{0} = \bm{\theta} $,
$\bm{\phi} \iteridx{k}$ in \eqrefshort{grad_update}
solves the regularized linear least squares problem
\begin{align}
    \min\brac{\| \bvar{\y} - \bvar{\X}  \bm{\phi}  \|_2^2
    +
    \|\bm{\theta}  - \bm{\phi}  \|_{\bvar{Q}}^2}
    \label{eq:santos.3.18.1}

\end{align}
with $\bvar{Q}$-norm defined by $\|\bvar{\z}\|_{\bvar{Q}} = \bvar{\z}^\mathrm{T}\bvar{Q}^{-1} \bvar{\z}$ for a symmetric positive definite matrix $\bvar{Q}$ that depends on the step size $\alpha$ and iteration index $k$ as well as on the covariance structure of $\bvar{\X}$. We describe the exact form of the dependence in \secref{method_prior}.
The minimization in \eqrefshort{santos.3.18.1} can be expressed as a posterior maximization problem given a conditional Gaussian likelihood over $\bvar{\y}$
and a Gaussian prior
over $\bm{\phi} $.
The posterior takes the form
\vspace{0.1cm}
\begin{align}
    \prob{\bm{\phi}  \;\vert\; \bvar{\X}, \bvar{\y}, \bm{\theta} }
    &\propto
    \mathcal{N}(\bvar{\y} \, ;\, \bvar{\X} \bm{\phi} , \mathbb{I})
    \;
    \mathcal{N}(\bm{\phi}  \, ;\, \bm{\theta} , \bvar{Q})
    \enspace .
    \label{eq:posterior}
\end{align}

Since $\bm{\phi} \iteridx{k}$ in \eqrefshort{grad_update} maximizes \eqrefshort{posterior},
we may conclude that $k$ iterations of gradient descent in a linear regression model with squared error exactly computes the \gls{map} estimate of $\bm{\phi} $, given a Gaussian-noised observation model and a Gaussian prior over $\bm{\phi} $ with parameters $\smash{{\boldsymbol {\mu }}_0 = \bm{\theta} }$ and $\smash{{\boldsymbol {\Sigma }}_0 = \bvar{Q}}$.
Therefore, in the case of linear regression with squared error, \gls{maml} is exactly \gls{eb} using the \gls{map} estimate as the point estimate of $\bm{\phi} $.

In the nonlinear case, \gls{maml} is again equivalent to an \gls{eb} procedure to maximize the marginal likelihood that uses a point estimate for $\bm{\phi} $ computed by one or a few steps of gradient descent.
However, this point estimate is not necessarily the global mode of a posterior.
We can instead understand the point estimate given by truncated gradient descent as the value of the mode of an implicit posterior over $\bm{\phi} $ resulting from an empirical loss interpreted as a negative log-likelihood, and regularization penalties and the early stopping procedure jointly acting as priors~\cite[for similar interpretations, see][]{sjoberg1995overtraining,bishop1995regularization,duvenaud2016early}.

The exact equivalence between early stopping and a Gaussian prior on the weights in the linear case, as well as the implicit regularization to the parameter initialization the nonlinear case, tells us that every iterate of truncated gradient descent is a mode of an implicit posterior.
In particular, we are not required to take the gradient descent procedure of \gls{fa} that computes $\smash{\hat{\bm{\phi} }}$ to convergence in order to establish a connection between \gls{maml} and \gls{hb}.
\Gls{maml} can therefore be understood to approximate an expectation of the marginal \gls{nll} for each task $\mathcal{T}\taskidx{j}$ as
\begin{align*}
\expectationsubscr{\bvar{\x}\sim p_{\mathcal{T}\taskidx{j}}\brac{\bvar{\x}}}{-\log\prob{\bvar{\x}\;\vert\;\bm{\theta} }}
&\approx \frac{1}{M} \sum_m - \log \prob{\bvar{\x}\taskidx{j_{N+m}} \;\vert\; \hat{\bm{\phi} }\taskidx{j}}
\end{align*}
using the point estimate
$\hat{\bm{\phi} }\taskidx{j} = \bm{\theta}   + \alpha\grad{\bm{\theta} } \log \prob{\bvar{\x}\taskidx{j_{n}} \;\vert\; \bm{\theta} }$
for single-step \gls{fa}.

The algorithm for \gls{maml} as probabilistic inference is given in \algref{maml-hb}; \subref{ml-point} computes each marginal \gls{nll} using the point estimate of $\smash{\hat{\bm{\phi} }}$ as just described.
Formulating \gls{maml} in this way, as probabilistic inference in a \gls{hbm}, motivates the interpretation in \secref{method_prior} of using various meta-optimization algorithms to induce a prior over task-specific parameters.

\subsection{The Prior Over Task-Specific Parameters}
\label{sec:method_prior}

From \secref{method_mamlashb}, we may conclude that
early stopping during \gls{fa} is equivalent to a specific choice of a prior over task-specific parameters,
$\smash{\prob{\bm{\phi} \taskidx{j} \;\vert\; \theta}}$.
We can better understand the role of early stopping in defining the task-specific parameter prior
in the case of a quadratic objective.
Omit the task index from $\bm{\phi} $ and $\bvar{\x}$, and
consider a second-order approximation  of the \gls{fa} objective $\smash{\ell (\bm{\phi} )=-\log \prob{\bvar{\x}_1 \dots, \bvar{\x}_N \;\vert\; \bm{\phi} }}$ about a minimum $\bm{\phi} ^{*}$:

\begin{align}
    \label{eq:newton_approx}
    \ell(\bm{\phi} )
    \approx
    \tilde{\ell}(\bm{\phi} )
    :=& \,
    \tfrac{1}{2} \| \bm{\phi}  - \bm{\phi} ^* \|^2_{\mathbf{H}^{-1}} +  \ell(\bm{\phi} ^{*})
\end{align}

where the Hessian $\smash{\mathbf{H} = \grad{\bm{\phi} }^2  \ell \brac{ \bm{\phi} ^*}}$ is assumed to be positive definite so that $\tilde{\ell}$ is bounded below.
Furthermore, consider using a curvature matrix $\mathcal{B}$ to precondition the gradient in gradient descent, giving the update

\begin{align}
    \bm{\phi} \iteridx{k}
    = \bm{\phi} \iteridx{k-1} - \mathcal{B} \grad{\bm{\phi} } \tilde{\ell}(\bm{\phi} \iteridx{k-1})
    \enspace .
    \label{eq:curvature_update}
\end{align}

If $\mathcal{B}$ is diagonal, we can identify \eqrefshort{curvature_update} as a Newton method with a diagonal approximation to the inverse Hessian; using the inverse Hessian evaluated at the point $\bm{\phi} \iteridx{k-1}$ recovers Newton's method itself.
On the other hand, meta-learning the matrix $\mathcal{B}$ matrix via gradient descent provides a method to incorporate task-general information into the covariance of the \gls{fa} prior,
$\smash{\prob{\bm{\phi}  \;\vert\; \theta}}$.
For instance, the meta-learned matrix $\mathcal{B}$ may encode correlations between parameters that dictates how such parameters are updated relative to each other.

Formally, taking $k$ steps of gradient descent from $\bm{\phi} \iteridx{0} = \bm{\theta} $ using the update rule in \eqrefshort{curvature_update} gives a $\bm{\phi} \iteridx{k}$ that solves
\begin{align}
    \min\brac{
    \| \bm{\phi}  - \bm{\phi} ^* \|^2_{\mathbf{H}^{-1}}
    +
    \|\bm{\phi} \iteridx{0} - \bm{\phi}  \|_{\bvar{Q}}^2}
    \enspace .
    \label{eq:santos.3.18.2}

\end{align}
The minimization in \eqrefshort{santos.3.18.2} corresponds to taking a Gaussian prior
$\prob{\bm{\phi}  \;\vert\; \bm{\theta} }$ with mean $\bm{\theta} $ and covariance $\bvar{Q}$
for $\smash{\bvar{Q} = \bvar{O}\mathbf{\Lambda}^{-1}((\mathbb{I} - \bvar{B}\mathbf{\Lambda})^{-k} - \mathbb{I})\bvar{O}^\mathrm{T}}$~\citep{santos1996equivalence}
where $\bvar{B}$ is a diagonal matrix that results from a simultaneous diagonalization of $\mathbf{H}$ and $\mathcal{B}$ as
$\smash{\bvar{O}^\mathrm{T} \mathbf{H} \bvar{O} = \diag{\lambda_1, \dots, \lambda_n} = \mathbf{\Lambda}}$
and
$\smash{\bvar{O}^\mathrm{T} \mathcal{B} ^{-1} \bvar{O} = \diag{b_1, \dots, b_n} = \bvar{B}}$
with $b_i, \lambda_i \ge 0$ for $i = 1, \dots, n$~\cite[Theorem 8.7.1 in][]{golub1983matrix}.
If the true objective is indeed quadratic, then, assuming the data is centered, $\mathbf{H}$ is the unscaled covariance matrix of features, $\smash{\bvar{\X}^\mathrm{T}\bvar{\X}}$.
\section{Related Work}
\label{sec:lit_rev}

\Gls{ml} and few-shot learning have a long history in hierarchical Bayesian modeling~\citep[\textit{e.g.,}\ ][]{tenenbaum1999bayesian,fei2003bayesian,lawrence2004learning,yu2005learning,gao2008knowledge,daume2009bayesian,wan2012sparse}.
A related subfield is that of transfer learning, which has used \gls{hb} extensively~\citep[\textit{e.g.,}\ ][]{raina2006constructing}.
A variety of inference methods have been used in Bayesian models, including exact inference \citep{lake2011one}, sampling methods~\citep{salakhutdinov2012one}, and variational methods~\citep{edwards2017towards}. While some prior works on hierarchical Bayesian models have proposed to handle basic image recognition tasks, the complexity of these tasks does not yet approach the kinds of complex image recognition problems that can be solved by discriminatively trained deep networks, such as the \emph{mini}ImageNet experiment in our evaluation~\citep{mansinghka2013approximate}.

Recently, the Omniglot benchmark~\cite{lake2016building} has rekindled interest in the problem of learning from few examples.
Modern methods accomplish few-shot learning either through the design of network architectures that ingest the few-shot training samples directly \citep[\textit{e.g.,}\ ][]{koch2015siamese,vinyals2016matching,snell2017prototypical,hariharan2017low,triantafillou2017few}, or formulating the problem as one of \textit{learning to learn}, or \textit{meta-learning}~\citep[\textit{e.g.,}\ ][]{schmidhuber1987evolutionary,bengio1991learning,schmidhuber1992learning,bengio1992optimization}.
Meta-learning has also recently been implemented with learned update rules \citep[\textit{e.g.,}\ ][]{andrychowicz2016learning,ravi2017optimization,li2017alearning,ha2017hypernetworks}) and memory-augmentation \cite[\textit{e.g.,}\ ][]{santoro2016meta}). In this work, we build on the gradient-based meta-learning method proposed by~\cite{finn2017model}.

Our work bridges the gap between gradient-based meta-learning methods and hierarchical Bayesian modeling.
Our contribution is not to formulate the \gls{ml} problem as a \gls{hb}ian model, but instead to formulate a gradient-based meta-learner as \gls{hb}ian inference, thus providing a way to efficiently perform posterior inference in a model-agnostic manner.
\section{Experimental Evaluation}
\label{sec:exp}

The goal of our experiments is to evaluate if we can use our probabilistic interpretation of MAML to generate samples from the distribution over adapted parameters, and futhermore, if our method can be applied to large-scale meta-learning problems such as \gls{mini}.

\subsection{Warmup: Toy Nonlinear Model}
\label{sec:exp_nonlinear}

\input{figures/toy_sinusoid}

The connection between \gls{maml} and \gls{hb} suggests that we should expect \gls{maml} to behave like an algorithm that learns the mean of a Gaussian prior on model parameters, and uses the mean of this prior as an initialization during \gls{fa}.
Using the Laplace approximation to the integration over task-specific parameters as in (\ref{eq:laplace_int}) assumes a task-specific parameter posterior with mean at the adapted parameters $\smash{\hat{\bm{\phi} }}$ and covariance equal to the inverse Hessian of the log posterior evaluated at the adapted parameter value.
Instead of simply using this density in the Laplace approximation as an additional regularization term as in (\ref{eq:laplace_obj}), we may sample parameters $\bm{\phi} \taskidx{j}$ from this density and use each set of sampled parameters to form a set of predictions for a given task.

To illustrate this relationship between \gls{maml} and \gls{hb}, we present a meta-dataset of sinusoid tasks in which each task involves regressing to the output of a sinusoid wave in \figref{toy}.
Variation between tasks is obtained by sampling the amplitude uniformly from $[0.1, 5.0]$ and the phase from $[0, \pi]$.
During training and for each task, $10$ input datapoints are sampled uniformly from $[-10.0, 10.0]$ and the loss is the mean squared error between the prediction and the true value.

We observe in \figref{toy} that our method allows us to directly sample models from the task-specific parameter distribution after being presented with $10$ datapoints from a new, previously unseen sinusoid curve. In particular, the column on the right of \figref{toy} demonstrates that the sampled models display an appropriate level of uncertainty when the datapoints are ambiguous (as in the bottom right).

\subsection{Large-Scale Experiment: \gls{mini}}
\label{sec:exp_miniimagenet}

We evaluate \gls{maml2} on the \text{mini}ImageNet~\cite{ravi2017optimization} 1-shot, 5-way classification task, a standard benchmark in few-shot classification.
\gls{mini} comprises 64 training classes, 12 validation classes, and 24 test classes.
Following the setup of \cite{vinyals2016matching}, we structure the $N$-shot, $J$-way classification task as follows: The model observes $N$ instances of $J$ unseen classes, and is evaluated on its ability to classify $M$ new instances within the $J$ classes.

We use a neural network architecture standard to few-shot classification~\cite[\textit{e.g.,}\ ][]{vinyals2016matching,ravi2017optimization}, consisting of 4 layers with $3 \times 3$ convolutions and $64$ filters, followed by \gls{bn}~\citep{ioffe2015batch}, a ReLU nonlinearity, and $2 \times 2$ max-pooling.
For the scaling variable $\beta$ and centering variable $\gamma$ of \gls{bn}~\citep[see][]{ioffe2015batch}, we ignore the \gls{fa} update as well as the Fisher factors for \gls{kfac}.
We use Adam~\citep{kingma2014adam} as the meta-optimizer, and standard batch gradient descent with a fixed learning rate to update the model during \gls{fa}.
\gls{maml2} requires the prior precision term $\tau$ as well as an additional parameter $\smash{\eta \in \mathbb{R}^+}$ that weights the regularization term $\smash{\log\det\hat{\mathbf{H}}}$ contributed by \gls{laplace}. We fix $\tau = 0.001$ and selected $\smash{\eta = 10^{-6}}$ via cross-validation; all other parameters are set to the values reported in \cite{finn2017model}.

We find that \gls{maml2} is practical enough to be applied to this larger-scale problem.
In particular, our TensorFlow implementation of \gls{maml2} trains for 60,000 iterations on one TITAN Xp GPU in $9$ hours,  compared to $5$ hours to train \gls{maml}.
As shown in~\tabref{miniimagenet}, \gls{maml2} achieves comparable performance to the state-of-the-art meta-learning method by~\cite{triantafillou2017few}.
While the gap between \gls{maml} and \gls{maml2} is small, the improvement from \gls{laplace} suggests that a more accurate approximation to the marginalization over task-specific parameters will lead to further improvements.

\input{figures/miniimagenet_results}
\section{Improving Model-Agnostic Meta-Learning}
\label{sec:novel}

Identifying \gls{maml} as a method for probabilistic inference in a hierarchical model allows us to develop novel improvements to the algorithm.
In \secref{novel_laplace}, we consider an approach from Bayesian parameter estimation to improve the \gls{maml} algorithm, and in \secref{novel_curvature}, we discuss how to make this procedure computationally tractable for high-dimensional models.

\subsection{Laplace's Method of Integration}
\label{sec:novel_laplace}

We have shown that the \gls{maml} algorithm is an \gls{eb} procedure that employs a point estimate for the mid-level, task-specific parameters in a \gls{hb}ian model.
However, the use of this point estimate may lead to an inaccurate point approximation of the integral in \eqrefshort{maml_marginal_exact} if the posterior over the task-specific parameters, $\smash{\prob{\bm{\phi} \taskidx{j} \;\vert\;\bvar{\x}\taskidx{j_{N+1}}, \dots, \bvar{\x}\taskidx{j_{N+M}},\bm{\theta} }}$, is not sharply peaked at the value of the point estimate.
\Gls{laplace}~\citep{laplace1986memoir,mackay1992evidence,mackay1992practical} is applicable in this case as it replaces a point estimate of an integral with the volume of a Gaussian centered at a mode of the integrand, thereby forming a local quadratic approximation.

We can make use of this approximation to incorporate uncertainty about the task-specific parameters into the \gls{maml} algorithm at \gls{fa} time.
In particular, suppose that each integrand in \eqrefshort{maml_marginal_exact} has a mode $\bm{\phi} ^*\taskidx{j}$ at which it is locally well-approximated by a quadratic function.
\Gls{laplace} uses a second-order Taylor expansion of the negative log posterior in order to approximate each integral in the product in \eqrefshort{maml_marginal_exact}
as
\begin{align}
    \int
        \prob{
            \bvar{\X}\taskidx{j}
            \;\vert\;
            \bm{\phi} \taskidx{j}
        }
        \prob {
            \bm{\phi} \taskidx{j}
            \;\vert\;
            \bm{\theta}
        }
    \mathop{}\!\mathrm{d}  \bm{\phi} \taskidx{j}
    \approx
    \prob{
        \bvar{\X}\taskidx{j}
        \;\vert\;
        \bm{\phi} \taskidx{j}^*
    }
    \prob {
        \bm{\phi} \taskidx{j}^*
        \;\vert\;
        \bm{\theta}
    }
    \det(\mathbf{H}\taskidx{j}/2\pi)^{-\frac{1}{2}}
    \label{eq:laplace_int}
\end{align}
where $\smash{\mathbf{H}\taskidx{j}}$ is the Hessian matrix of second derivatives of the negative log posterior

Classically, \gls{laplace} uses the \gls{map} estimate for $\smash{\bm{\phi} \taskidx{j}^*}$, although any mode can be used as an expansion site provided the integrand is well enough approximated there by a quadratic.
We use the point estimate $\smash{\hat{\bm{\phi} }\taskidx{j}}$ uncovered by \gls{fa}, in which case the \gls{maml} objective in \eqrefshort{maml} becomes an appropriately scaled version of the approximate marginal likelihood
\begin{align}
    \label{eq:laplace_obj}
    -\log
    \prob {
        \bvar{\X} \;\vert\; \bm{\theta}
    }
    &\approx
    \sum\range{j}{}
    \sbrac{
        -\log
        \prob{
            \bvar{\X}\taskidx{j}
            \;\vert\;
            \hat{\bm{\phi} }\taskidx{j}
        }
        - \log \prob { \hat{\bm{\phi} }\taskidx{j} \;\vert\;  \bm{\theta}  }
        + \tfrac{1}{2} \log \det(\mathbf{H}\taskidx{j}) }
    \enspace .
\end{align}
The term $\smash{\log \prob { \hat{\bm{\phi} }\taskidx{j} \;\vert\;  \bm{\theta} }}$ results from the implicit regularization imposed by \gls{es} during \gls{fa}, as discussed in \secref{method_mamlashb}.
The term $\smash{\sfrac{1}{2} \log \det(\mathbf{H}\taskidx{j})}$, on the other hand, results from the Laplace approximation and can be interpreted as a form of regularization that penalizes model complexity.

\input{figures/alg_ml-laplace}

\subsection{Using Curvature Information to Improve \gls{maml}}
\label{sec:novel_curvature}

Using \eqrefshort{laplace_obj} as a training criterion for a neural network model is difficult due to the required computation of the determinant of the Hessian of the log posterior $\mathbf{H}\taskidx{j}$, which itself decomposes into a sum of the Hessian of the log likelihood and the Hessian of the log prior as
\begin{align*}
\mathbf{H}\taskidx{j}
=  \grad{\bm{\phi} _j}^2  \sbrac{- \log \prob { \bvar{\X}\taskidx{j}  \;\vert\; \bm{\phi} \taskidx{j} } }
+ \grad{\bm{\phi} _j}^2  \sbrac{- \log \prob { \bm{\phi} \taskidx{j}  \;\vert\; \bm{\theta}  }}
\enspace .
\end{align*}
In our case of early stopping as regularization, the prior over task-specific parameters $\smash{\prob { \bm{\phi} \taskidx{j}  \;\vert\; \bm{\theta}  }}$ is implicit and thus no closed form is available for a general model.
Although we may use the quadratic approximation derived in \secref{method_prior} to obtain an approximate Gaussian prior, this prior is not diagonal and does not, to our knowledge, have a convenient factorization.
Therefore, in our experiments, we instead use a simple approximation in which the prior is approximated as a diagonal Gaussian with precision $\tau$.
We keep $\tau$ fixed, although this parameter may be cross-validated for improved performance.

Similarly, the Hessian of the log likelihood is intractable to form exactly for all but the smallest models, and furthermore, is not guaranteed to be positive definite at all points, possibly rendering the Laplace approximation undefined.
To combat this, we instead seek a curvature matrix $\smash{\hat{\mathbf{H}}}$ that approximates the quadratic curvature of a neural network objective function.
Since it is well-known that the curvature associated with neural network objective functions is highly non-diagonal~\citep[\textit{e.g.,}\ ][]{martens2016second}, a further requirement is that the matrix have off-diagonal terms.

Due to the difficulties listed above, we turn to second order gradient descent methods, which precondition the gradient with an inverse curvature matrix at each iteration of descent.
The \gls{fisher}~\citep{fisher1925theory} has been extensively used as an approximation of curvature, giving rise to a method known as natural gradient descent~\citep{amari1998natural}.
A neural network with an appropriate choice of loss function is a probabilistic model and therefore defines a \gls{fisher}.
Furthermore, the \gls{fisher} can be seen to define a convex quadratic approximation to the objective function of a probabilistic neural model~\citep{pascanu2013revisiting,martens2014new}.
Importantly for our use case, the \gls{fisher} is positive definite by definition as well as non-diagonal.

However, the \gls{fisher} is still expensive to work with.
\cite{martens2015optimizing} developed \gls{kfac}, a scheme for approximating the curvature of the objective function of a neural network with a block-diagonal approximation to the \gls{fisher}.
Each block corresponds to a unique layer in the network, and each block is further approximated as a Kronecker product \citep[see][]{van2000ubiquitous} of two much smaller matrices by assuming that the second-order statistics of the input activation and the back-propagated derivatives within a layer are independent.
These two approximations ensure that the inverse of the \gls{fisher} can be computed efficiently for the natural gradient.

For \gls{laplace}, we are interested in the determinant of a curvature matrix instead of its inverse.
However, we may also make use of the approximations to the \gls{fisher} from \gls{kfac} as well as properties of the Kronecker product.
In particular, we use the fact that the determinant of a Kronecker product is the product of the exponentiated determinants of each of the factors, and that the determinant of a block diagonal matrix is the product of the determinants of the blocks~\citep{van2000ubiquitous}.
The determinants for each factor can be computed as efficiently as the inverses required by \gls{kfac}, in $\smash{\mathcal{O}(d^3)}$ time for a $d$-dimensional \gls{kf}.

We make use of \gls{laplace} and \gls{kfac} to replace \subref{ml-point}, which computes the task-specific marginal \gls{nll}s using a point estimate for $\smash{\hat{\bm{\phi} }}$.
We call this method the \gls{maml2}, and give a replacement subroutine in \subref{ml-laplace}.

redpajama_set_name: RedPajamaArXiv


So, your machine now needs minutes to boot before you can ssh in where it used to be seconds before the Debian Buster update?
Linux 3.17 (2014-10-05) learnt a new syscall getrandom() that, well, gets bytes from the entropy pool. Glibc learnt about this with 2.25 (2017-02-05) and two tries and four years after the kernel, OpenSSL used that functionality from release 1.1.1 (2018-09-11). OpenSSH implemented this natively for the 7.8 release (2018-08-24) as well.
Now the getrandom() syscall will block1 if the kernel can't provide enough entropy. And that's frequenty the case during boot. Esp. with VMs that have no input devices or IO jitter to source the pseudo random number generator from.
I vividly remember not seeing my Alpine Linux VMs back on the net after the Alpine 3.5 upgrade. That was basically the same issue.
Systemd makes this behaviour worse, see issue #4271, #4513 and #10621.
Basically as of now the entropy file saved as /var/lib/systemd/random-seed will not - drumroll - add entropy to the random pool when played back during boot. Actually it will. It will just not be accounted for. So Linux doesn't know. And continues blocking getrandom(). This is obviously different from SysVinit times2 when /var/lib/urandom/random-seed (that you still have lying around on updated systems) made sure the system carried enough entropy over reboot to continue working right after enough of the system was booted.
#4167 is a re-opened discussion about systemd eating randomness early at boot (hashmaps in PID 0...). Some Debian folks participate in the recent discussion and it is worth reading if you want to learn about the mess that booting a Linux system has become.
While we're talking systemd ... #10676 also means systems will use RDRAND in the future despite Ted Ts'o's warning on RDRAND [Archive.org mirror and mirrored locally as 130905_Ted_Tso_on_RDRAND.pdf, 205kB as Google+ will be discontinued in April 2019].
Debian is seeing the same issue working up towards the Buster release, e.g. Bug #912087.
with delays up to tens of minutes on systems with very little external random sources.
Check dmesg | grep -E "(rng|random)" to see how your systems are doing.
If this is not fully solved before the Buster release, I hope some of the below can end up in the release notes3.
You need to get entropy into the random pool earlier at boot. There are many ways to achieve this and - currently - all require action by the system administrator.
From kernel 4.19 (Debian Buster currently runs 4.18 [Update: but will be getting 4.19 before release according to Ben via Mika]) you can set RANDOM_TRUST_CPU at compile time or random.trust_cpu=on on the kernel command line. This will make recent Intel / AMD systems trust RDRAND and fill the entropy pool with it. See the warning from Ted Ts'o linked above.
Update: Since Linux kernel build 4.19.20-1 CONFIG_RANDOM_TRUST_CPU has been enabled by default in Debian.
The Trusted Platform Module has an embedded random number generator that can be used. Of course you need to have one on your board for this to be useful. It's a hardware device.
to the kernel command line. 1000 means "trust", 0 means "don't use". So you can chose any value in between that works for you depending on how much you consider your TPM to be unbugged.
For Virtual Machines (VMs) you can forward entropy from the host (that should be running longer than the VMs and have enough entropy) via virtio_rng.
and within the VM newer kernels should automatically load virtio_rng and use that.
You can confirm with dmesg as per above.
The Fedora bugtracker has a bash / python script that replaces the systemd rnd seeding with a (better) working one. The script can also serve as a good starting point if you need to script your own solution, e.g. for reading from an entropy provider available within your (secure) network.
The wonderful Keith Packard and Bdale Garbee have developed a USB dongle, ChaosKey, that supplies entropy to the kernel. Hard- and software are open source.
Kernel 4.2 introduced jitterentropy_rng which will use the jitter in CPU timings to generate randomness.
apt install jitterentropy-rngd (available from Buster/testing).
The current version 1.0.8-3 installs nicely on Stretch. dpkg -i is your friend.
But - drumroll - that daemon doesn't seem to use the kernel module at all.
That's where I stopped looking at that solution. At least for now. There are extensive docs if you want to dig into this yourself.
Haveged is a user-space daemon that gathers entropy though the timing jitter any CPU has. It will only run "late" in boot but may still get your openssh back online within seconds and not minutes.
It is also - to the best of my knowledge - not verified at all regarding the quality of randomness it generates. The haveged design and history page provides and interesting read and I wouldn't recommend haveged if you have alternatives. If you have none, haveged is a wonderful solution though as it works reliably. And unverified entropy is better than no entropy. Just forget this is 2018 2019 .
Thorsten Glaser has posted newly developed early-rng-init-tools in a debian-devel thread. He provides packages at http://fish.mirbsd.org/~tg/Debs/dists/sid/wtf/Pkgs/early-rng-init-tools/ .
First he deserves kudos for naming a tool for what it does. This makes it much more easily discoverable than the trend to name things after girlfriends, pets or anime characters. The implementation hooks into the early boot via initrd integration and carries over a seed generated during the previous shutdown. This and some other implementation details are not ideal and there has been quite extensive scrutiny but none that discovered serious issues. Early-rng-init-tools look like a good option for non-RDRAND (~CONFIG_RANDOM_TRUST_CPU) capable platforms.
Stefan Fritsch, the Apache2 maintainer in Debian, OpenBSD developer and a former Debian security team member stumbled over the systemd issue preventing Apache libssl to initialize at boot in a Debian bug #916690 - apache2: getrandom call blocks on first startup, systemd kills with timeout.
The bug has been retitled "document getrandom changes causing entropy starvation" hinting at not fixing the underlying issue but documenting it in the Debian Buster release notes.
Unhappy with this "minimal compromise" Stefan wrote a comprehensive summary of the current situation to the Debian-devel mailing list. The discussion spans over December 2018 and January 2019 and mostly iterated what had been written above already. The discussion has - so far - not reached any consensus. There is still the "systemd stance" (not our problem, fix the daemons and the "ssh/apache stance" (fix systemd, credit entropy).
> etc. depending on the situation.
countless hours of debugging and useless bug reports.
This is exactly why I wrote this blog entry and keep it updated. We need to either fix this or tell everybody we can reach before upgrading to Buster. Otherwise this will lead to huge amounts of systems dead on the network after what looked like a successful upgrade.
Raphael Hertzog fixed the issue for Kali Linux by installing haveged by default. Michael Prokop did the same for the grml distribution within its December 2018 release.
Ben Hutchings pointed to an interesting thread on the debian-release mailing list he kicked off in May 2018. Multiple people summarized the options and the fact that there is no "general solution that is both correct and easy" at the time.
Sam Hartman identified Debian Buster VMs running under VMware as an issue, because that supervisor does not provide virtio-rng. So Debian VMs wouldn't boot into ssh availability within a reasonable time. This is an issue for real world use cases albeit running a proprietary product as the supervisor.
If that pool is used by the guest to generate long-term secrets because it appears to be well-initialized, that could be a serious problem.
I've just opened https://bugs.launchpad.net/qemu/+bug/1811758 to report a way to improve that situation in qemu by default.
So ... make sure that your host OS has access to a hardware random number generator or at least carries over its random seed properly across reboots. You could also delay VM starts until the crng on the host Linux is fully initialized (random: crng init done).
Otherwise your VMs may get insufficiently generated pseudo-random numbers and won't even know.
Ben Hutchings has enabled CONFIG_RANDOM_TRUST_CPU for Debian kernels from 4.19.20-1 so the problem is somewhat contained for recent CPU AMD64 systems (RDRAND capable) in Buster.
Thorsten Glaser developed early-rng-init-tools which combine a few options to try and get entropy carried across boot and generated early during boot. He received some scrutiny as can be expected but none that would discourage me from using it. He explains that this is for early boot and thus has initrd integration. It complements safer randomness sources or haveged.
The Debian installer for Buster is running into the same problem now as indicated in the release notes for RC1. Bug #923675 has details. Essentially choose-mirror waits serveral minutes for entropy when used with https mirrors.


Monroe County Health Department CERT Team, has begun a "Sleep Safe in Monroe County" campaign. Our goal is to install dual sensor smoke alarms into at-risk homes in Monroe County. We are targeting homes with elderly people, children or folks with disabilities, and where wood burning is the primary source of heat. If you meet this criteria and would like us to install a smoke alarm in your home please contact Kelly Shreve at the Monroe County Health Department at 772-3065, Ext. 108 or email her at kshreve@monroehealthcenter.com . Monroe CERT's are here to help, and we want everyone to Sleep Safe in Monroe County.


When more is more Anantara Dhigu is the place to be! Your tropical island paradise awaits, with its 50 shades of blue and one of the most beautiful lagoons in all of the Maldives. Weather you're seeking a typical "flop and drop" holiday experience or after something a little more adventure driven, Anantara Dhigu really does offer something for everyone. Learn to surf if the magical lagoon or take a ride out to some of the most famous surf breaks the Maldives has to offer!


text:
Administration | Operations |
michael.heffernan@jeffco.k12.co.us
Upon returning from Peace Corps Ecuador in 1991, Michael Heffernan began his career in education at the University of Michigan in Ann Arbor completing a master's degree in education and in Detroit Public Schools as an elementary school teacher. From 1991-95, he taught in a two-way Spanish/English immersion school and studied Waldorf education with faculty from two area independent schools. In July 1995, Michael and his family relocated to Flagstaff, AZ to assist in the opening of Pine Forest Charter School (PFS). From 1995-2000, he was the grade 4-8 lead teacher, teaching all subjects, as well as Spanish, games, music and health. He also acted as faculty chairperson and served as a faculty member on the PFS Governing Board.
After graduating his 8th grade class, Michael took a one year leave from teaching and worked with a local river rafting company as a guide and business manager. He returned to formal teaching in the fall of 2001 as an educator in the Coconino County Juvenile Detention Center and Coconino County Jail. Michael and his family moved to Tucson for a year and, while his wife completed her masters degree in education, he taught algebra and physical education in a grades 6-8 middle school with 1200+ students. During that time, Michael completed his administrative coursework and received his Arizona principal's certificate. In 2003, he and his family returned to Flagstaff and PFS where he served as the school's principal and the executive director from 2003-17.
In addition, Michael and his wife Rose, public-Waldorf educated adult children and school-aged son, is an avid outdoor adventurer and professional musician. As a family, they have embraced the lifestyle and principles of public Waldorf education for many years and serve the Mt. Phoenix community with excited enthusiasm and desire to support carrying the school's mission forward.
Maggie Payne
Director of Enrichment & Student Support
maggie.payne@jeffco.k12.co.us
Melissa Kacel
Assistant Director of Student Support
melissa.kacel@jeffco.k12.co.us
Suzanne Blette
Suzanne Blette@jeffco.k12.co.us
Joy Wegs
Program Director for Early Childhood
jwegs@jeffcoschools.us
As the Program Director for Early Childhood, "Auntie Joy" is here to help, hold and assist in anyway she can! She has been an integral part of Mountain Phoenix since the school moved to this campus in 2011. She is thrilled to wrap her arms around the entire Early Childhood Department and various committees throughout our campus. She proudly calls MPCS her home away from home!
Along with being an Early Childhood Director and Teacher, she also has a random degree in Outdoor Recreation (rafting anyone?), as well as a certificate in Waldorf Early Childhood Education. Outside of Mountain Phoenix, Joy loves spending time teaching and practicing yoga, hiking with her family and snuggling up her pups.
Suzy Kumley-Ross
School Secretary
susan.ross-kumley@jeffco.k12.co.us
Suzy was born in Arvada, Colorado. My children are 7th generation Arvada area residents. She attended Red Rocks College, Front Range, Benedictine College, and DU while trying to figure out what she wanted to be when she grew up. In the meantime, Suzy fell into the Real Estate business and got her brokers license when she was 20. After working in Real Estate for years, Suzy is happy to be in the role of Office Manager at MPCS.
"I love to spend time with my husband, two children (who both attended Mountain Phoenix), and step-daughter. I love riding my ATV, camping, playing with my chickens, cooking, and gardening. "
Julie Ortiz
Assistant Director of School Development
julie.ortiz@jeffco.k12.co.us
Julie came to Mountain Phoenix a year after the Wheat Ridge location opened to enroll her younger boys. As she was looking for a place where her boys could be boys, learning but playing in nature whenever possible, she felt MPCS was the perfect fit. Both of her children were blessed with a stress free, play and art oriented ECE and Elementary experience while only doing homework when necessary. What more could you ask for?
Julie is married with 4 children - all boys! Her husband was born in Colorado while she hailed from New Jersey. Having a large extended family and they spend most of their time visiting and looking out for each other. Her favorite pastimes are traveling, going for early morning walks or paddle board rides, camping, and hiking, spending time with family and friends, and solving fun puzzles.
"It's been a privilege to be part of this community. I am inspired by all people big and small, relationships, spirituality, being in nature, learning, working with people to realize a vision, problem solving, and going on an adventures."
Cathy Darrow
cathy.darrow@jeffco.k12.co.us
Cathy graduated from Red Rocks Community College with a degree in Early Childhood Education and Metro State College (1991) with a BA in Elementary Education and a Minor in Special Education.
She taught at Foothills Academy from 1994 to the closing of the school in 2011. When MPCS took over the campus, she joined the faculty as an Interventionist and Substitute Teacher until 2014 and then became a Middle School Office Assistant.
Cathy also was an MPCS parent. Her youngest daughter Chloe attended middle school at MPCS.
Nicole Rohnke
nicole.rohnke@jeffco.k12.co.us
Nicole grew up in Castle Rock, Colorado, back in the days when it was mainly open fields and roaming cattle. She then went on to college at the University of Montana in Missoula where she received a bachelors degree in Political Science with emphasis in Public Administration. After becoming a mother, Nicole started assistant teaching in a private Montessori school in Evergreen. Eventually she ventured on to receive her Upper Elementary Montessori Training at the Center for Montessori Training in New Rochelle, NY.
In her personal time, Nicole enjoys time with her family, as well as hiking, reading, horseback riding, gardening, snowboarding and any kind of home fix-it and design projects.
Vince Ruiz
vincent.ruiz@jeffco.k12.co.us
Adele Sakich
Student Health Coordinator
adele.sakich@jeffco.k12.co.us
Sonja Pierce
Backup Health Aide/Office Assistant
sonja.hill@jeffco.k12.co.us
Lynn Edone
Upper School Office Secretary
lynn.edone@jeffco.k12.co.us
Originally from Texas, Lynn and her family have been pioneers at Mountain Phoenix with her two girls (currently high schoolers) which attended MPCS when it first opened and continued through Grade 8. Lynn loves hiking, gardening and skiing!
Stephanie O'Friel
School Enrichment Program Coordinator
sjofriel@jeffcoschools.us
Stephanie found Mountain Phoenix 9 years ago with her now sophmore in high school and is blessed to continue the experience with her 3rd grader and kindergartener. She has become passionate about Waldorf education and has enjoyed working at the school for the last four years. Stephanie is excited to take on the position of Enrichment Coordinator this year!
Stephanie holds a Bachelor's degree in Sustainable Agriculture and a Master's Degree in Holistic Nutrition. Her favorite past times include spending time in nature - hiking camping, and paddle boarding, being a plant nerd, gardening, naturalist hikes, spending time at the botanical gardens and going to see live music.
"I love winter and snow so I feel even more blessed to be a Colorado native! I also enjoy spending time with my friends and family. I have three kids, two energetic australian shepards, and a very large extended family that keeps me busy!
Sondra Lockett-Cameron
Communication & Marketing Manager, Volunteer Coordinator
sondra.lockett-cameron@jeffcoschools.us
Mountain Phoenix graced Sondra's family over 6 years ago with the admission of her then 6th grader and Dandelion Preschooler. Over the years, the community has become family, the festivals have become tradition and the campus has become her office.
Sondra holds a Bachelor's degree in Public Management with a lifetime of experience in Marketing. Her other past lives include Real Estate and Property Management, Aveda Esthethician and Marine Corps Reservist. In her spare time (or scheduled free time), she loves to repurpose, craft sustainable goods, study holistic nutrition, and practice photography. Hiking, discovering new adventures, trying hidden gem restaurants, and attending live theatre/concerts with her 2 kids, husband and father are also some favorite past times!


text:
CX DICTIONARY
What is Quality Assurance?
Quality Assurance (QA) is the practice of maintaining consistent standards when delivering a product or a service to a customer.
QA is crucial for contact centers as it ensures a consistent level of customer service that matches predefined requirements. Customers increasingly view the experience they have when interacting with a company to be equally as important as its products. Therefore, ensuring both quality and consistency in customer experience is key to making customer care a competitive differentiator.
QA also improves contact center performance by enforcing consistent brand messaging, accelerating conflict resolution, and expediting identification of recurring customer issues.
What do current approaches to Quality Assurance look like?
Customers' expectations are higher than ever, and companies who successfully meet these rising expectations have a tremendous opportunity to differentiate themselves. Heightened customer expectations and increasing standards mean that QA must also be optimized accordingly. For example, "customer-obsessed" Zappos describes its purpose as "to live and deliver 'WOW!'" Therefore, a Zappos contact center representative who doesn't deliver "wow" to a customer is not acting in alignment with the brand's mission.
To maintain brand standards, most QA programs follow these main processes:
Establish Expectations:
Traditional rubrics or scorecards often evaluate performance by examining the presence of desired behaviors, some of which may be easier to consistently measure than others. For example, a rubric might evaluate the presence of a required disclosure statement or whether a representative showed soft skills like empathy. While a disclosure statement is relatively straightforward to both communicate and identify, soft skills requirements are often fuzzier and leave much up to the subjective interpretation of both the auditors and the representatives themselves.
Evaluate Interactions:
Contact centers evaluate a percentage of their representatives' calls. Auditors evaluate adherence to expected behaviors; however, when done manually, this process introduces the potential for subjective variation and human error. Extreme sampling can also make it harder to find significant trends that inform smart coaching.
Address Missteps:
Coaches may work with representatives to review customer interactions and correct undesired behaviors. However, this approach focuses on punitive corrections and does not reinforce desired behaviors. Additionally, coaches are limited by the sample size, hours in a day, and the audit timeframe. Coaches may not give feedback to an agent until a month after a call, thereby making it very difficult for the agent to remember the interaction and learn from the situation, Additionally, a lack of historical records makes it challenging for agents and supervisors to track improvement over time.
Clarabridge's approach to Quality Assurance:
With Clarabridge's Intelligent Scoring (IS) feature, you can move from manual sampling to automatically evaluating 100% of contact center interactions by leveraging powerful rubrics curated to your own organization's standards. By consistently analyzing each interaction in near-real time, you'll obtain a more holistic picture of your representatives' performance over time, as well as ensure transparency across your organization.
• Decrease QA staff expenses
• Reduce cost of complaint handling
• Ensure compliance and avoid regulatory fines
• Improve KPIs such as NPS, CSAT, and OSAT
Debunking Natural Language Processing
We're on a mission to demystify a complex term: Natural Language Processing. Natural Language Processing (NLP) is a field of study that focuses on a computer's ability to interpret human language in order to process, analyze, and extract meaning from large volumes of natural language text data.
The Pillars of Text Analytics: Sentiment, Categorization, Effort, and Emotion
Clarabridge Cited As A Strong Performer in The Forrester Wave™: AI-Fueled Speech Analytics Solutions Report Q2 2018
Quality Management Reimagined