text:
#ifndef FRACTION_TEMPLATE_CLASS
#define FRACTION_TEMPLATE_CLASS
/*

	Template class for doing fractions

	Assumes an integer-like class that support all operations on integers,
	and that interprets 0 as its additive identity
	and 1 as its multiplicative identity.
	Assignment (=) makes a new copy, not varying with the old one.
*/

template<class N> class Fraction
{
private:
	N num, den;
	void Normalize();

public:
	// Accessors
	N get_num() {return num;}
	N get_den() {return den;}

	// Constructors
	Fraction(); // Zero
	Fraction(const N &num_); // Integer
	Fraction(const N &num_, const N &den_); // Numerator, denominator
	Fraction(const Fraction<N> &frac);

	// Assignment
	template<class NI> Fraction<N> &operator = (const Fraction<NI> &val);
	Fraction<N> &set(const N &num_ = N(0), const N &den_ = N(1));

	// Arithmetic
	Fraction<N> &operator + ();
	Fraction<N> operator - ();

	template<class NX> friend Fraction<NX> operator + (const Fraction<NX> &a, const Fraction<NX> &b);
	template<class NX> friend Fraction<NX> operator - (const Fraction<NX> &a, const Fraction<NX> &b);
	template<class NX> friend Fraction<NX> operator * (const Fraction<NX> &a, const Fraction<NX> &b);
	template<class NX> friend Fraction<NX> operator / (const Fraction<NX> &a, const Fraction<NX> &b);

	Fraction<N> &operator += (const Fraction<N> &a);
	Fraction<N> &operator -= (const Fraction<N> &a);
	Fraction<N> &operator *= (const Fraction<N> &a);
	Fraction<N> &operator /= (const Fraction<N> &a);

	// Comparison

	template<class NX> friend bool operator == (const Fraction<NX> &a, const Fraction <NX> &b);
	template<class NX> friend bool operator != (const Fraction<NX> &a, const Fraction <NX> &b);
	template<class NX> friend bool operator > (const Fraction<NX> &a, const Fraction <NX> &b);
	template<class NX> friend bool operator < (const Fraction<NX> &a, const Fraction <NX> &b);
	template<class NX> friend bool operator >= (const Fraction<NX> &a, const Fraction <NX> &b);
	template<class NX> friend bool operator <= (const Fraction<NX> &a, const Fraction <NX> &b);
};

// Greatest common denominator, using Euclid's algorithm
// Returns a nonnegative result even if one or both inputs are negative
template<class N> N GCD(const N &a, const N &b)
{
	// Avoid dividing by zero
	if (a == N(0)) return N(1);
	if (b == N(0)) return N(1);

	// Make positive
	N ax = a;
	if (ax < N(0)) ax = - ax;
	N bx = b;
	if (bx < N(0)) bx = - bx;

	// Main algorithm
	N cx;
	while (bx != N(0))
	{
		cx = ax % bx;
		ax = bx;
		bx = cx;
	}
	return ax;
}

// Least common multiple
// Also nonnegative even if one or both inputs are negative
template<class N> N LCM(const N &a, const N &b)
{
	N res = (a/GCD(a,b))*b;
	if (res < N(0)) res = - res;
	return res;
}

// Absolute value
template<class N> N abs(const N &a)
{
	return Fraction<N>(abs(a.get_num()),a.get_den());
}

// Reduces a fraction to lowest terms,
// makes the denominator nonnegative
template<class N> void Fraction<N>::Normalize()
{
	// Handle the special cases with zeros first
	if (num == N(0) && den == N(0)) {}
	else if (num == N(0) && den != N(0)) den = N(1);
	else if (num != N(0) && den == N(0)) num = N(1);
	else
	{
		// No zeros
		N div = GCD(num,den);
		if (den < N(0)) div = - div;
		num /= div;
		den /= div;
	}
}

// Constructors

template<class N> Fraction<N>::Fraction():
	num(0), den(1) {}

template<class N> Fraction<N>::Fraction(const N &num_):
	num(num_), den(1) {}

template<class N> Fraction<N>::Fraction(const N &num_, const N &den_):
	num(num_), den(den_) {Normalize();}

template<class N> Fraction<N>::Fraction(const Fraction<N> &frac):
	num(frac.num), den(frac.den) {}

// Assignment

template<class N> template<class NI> Fraction<N> &Fraction<N>::operator = (const Fraction<NI> &val)
	{num = val.num; den = val.den; return *this;}

template<class N> Fraction<N> &Fraction<N>::set(const N &num_, const N &den_)
	{num = num_; den = den_; Normalize(); return *this;}

// Arithmetic

template<class N> Fraction<N> &Fraction<N>::operator + ()
	{return *this;}

template<class N> Fraction<N> Fraction<N>::operator - ()
	{Fraction<N> res(-num,den); return res;}

template<class N> Fraction<N> operator + (const Fraction<N> &a, const Fraction<N> &b)
{
	N div = GCD(a.den,b.den);
	N num = a.num*(b.den/div) + b.num*(a.den/div);
	N den = (a.den/div)*b.den;
	Fraction<N> res(num,den);
	return res;
}

template<class N> Fraction<N> operator - (const Fraction<N> &a, const Fraction<N> &b)
{
	N div = GCD(a.den,b.den);
	N num = a.num*(b.den/div) - b.num*(a.den/div);
	N den = (a.den/div)*b.den;
	Fraction<N> res(num,den);
	return res;
}

template<class N> Fraction<N> operator * (const Fraction<N> &a, const Fraction<N> &b)
{
	N div1 = GCD(a.num,b.den);
	N div2 = GCD(a.den,b.num);
	N num = (a.num/div1)*(b.num/div2);
	N den = (a.den/div2)*(b.den/div1);
	Fraction<N> res(num,den);
	return res;
}

template<class N> Fraction<N> operator / (const Fraction<N> &a, const Fraction<N> &b)
{
	N div1 = GCD(a.num,b.num);
	N div2 = GCD(a.den,b.den);
	N num = (a.num/div1)*(b.den/div2);
	N den = (a.den/div2)*(b.num/div1);
	Fraction<N> res(num,den);
	return res;
}

template<class N> Fraction<N> &Fraction<N>::operator += (const Fraction<N> &a)
{
	N div = GCD(den,a.den);
	num = num*(a.den/div) + a.num*(den/div);
	den = (den/div)*a.den;
	Normalize();
	return *this;
}

template<class N> Fraction<N> &Fraction<N>::operator -= (const Fraction<N> &a)
{
	N div = GCD(den,a.den);
	num = num*(a.den/div) - a.num*(den/div);
	den = (den/div)*a.den;
	Normalize();
	return *this;
}

template<class N> Fraction<N> &Fraction<N>::operator *= (const Fraction<N> &a)
{
	N div1 = GCD(num,a.den);
	N div2 = GCD(den,a.num);
	num = (num/div1)*(a.num/div2);
	den = (den/div2)*(a.den/div1);
	Normalize();
	return *this;
}

template<class N> Fraction<N> &Fraction<N>::operator /= (const Fraction<N> &a)
{
	N div1 = GCD(num,a.num);
	N div2 = GCD(den,a.den);
	num = (num/div1)*(a.den/div2);
	den = (den/div2)*(a.num/div1);
	Normalize();
	return *this;
}

// Comparison

template<class N> bool operator == (const Fraction<N> &a, const Fraction <N> &b)
{
	N div = GCD(a.den,b.den);
	return a.num*(b.den/div) == b.num*(a.den/div);
}

template<class N> bool operator != (const Fraction<N> &a, const Fraction <N> &b)
{
	N div = GCD(a.den,b.den);
	return a.num*(b.den/div) != b.num*(a.den/div);
}

template<class N> bool operator > (const Fraction<N> &a, const Fraction <N> &b)
{
	N div = GCD(a.den,b.den);
	return a.num*(b.den/div) > b.num*(a.den/div);
}

template<class N> bool operator < (const Fraction<N> &a, const Fraction <N> &b)
{
	N div = GCD(a.den,b.den);
	return a.num*(b.den/div) < b.num*(a.den/div);
}

template<class N> bool operator >= (const Fraction<N> &a, const Fraction <N> &b)
{
	N div = GCD(a.den,b.den);
	return a.num*(b.den/div) >= b.num*(a.den/div);
}

template<class N> bool operator <= (const Fraction<N> &a, const Fraction <N> &b)
{
	N div = GCD(a.den,b.den);
	return a.num*(b.den/div) <= b.num*(a.den/div);
}

#endif

redpajama_set_name: RedPajamaGithub


WASHINGTON, June 7 (Reuters) - A U.N. peace plan for Yemen calls on the Houthi movement to give up its ballistic missiles in return for an end to a bombing campaign against it by a Saudi-led coalition and a transitional governance agreement, according to a draft of the document and sources.
The plan, which has not yet been made public and could still be modified, is the latest effort to end Yemen's three-year-old civil war, which has spawned one of the world's worst ongoing humanitarian calamities.
The conflict pits the Iran-aligned Houthis, who took control of the capital Sanaa in 2014, against other Yemeni forces backed by a coalition led by U.S. allies Saudi Arabia and the United Arab Emirates. The coalition fears the Houthis are part of a regional power grab by Tehran.
"The intention is to link security and political aspects starting with a cessation of fighting...then to move towards a withdrawal of forces and the formation of a national unity government. This last objective could possibly be the hardest," one of the sources said.
"Politically, there is a necessity to back the U.N. effort. It will ultimately mean a transition, to a new political order in Yemen. Clearly with the U.N. effort, the military and political process will see the Houthis pull out of urban centres," he told UAE English-language newspaper The National.
The Emirati-backed Yemeni forces, who are loyal to exiled Yemeni President Abd-Rabbu Mansour Hadi have moved to within 10 km (six miles) of Hodeidah, a lifeline for humanitarian supplies.


text:
19 Dec Cherian George: Media Scholar
Posted at 13:04h in Reflections	by	The Center
Annenberg School for Communication doctoral student and Center steering committee member Muira McCammon sat down with Center for Media at Risk visiting scholar Cherian George to discuss the various kinds of pressure that cartooning faces around the world.
You recently organized a symposium with the Center called "Red Lines: Cartooning at Risk." How did it go? What were some of the highlights for you?
I should first clarify that I wasn't attempting any comprehensive study of cartooning in the world. The book I'm working towards is primarily a study of censorship, rather than a study of cartooning. Cartooning offers a very illuminating window on censorship, and so my intent with the symposium, as it will be with the book, was to find a stimulating range of cases that illustrates the various kinds of pressure that cartooning faces around the world. Although the cartoonists that we invited were certainly very accomplished stars in their own right, some of the interviews I will be doing won't necessarily be with famous or accomplished cartoonists. It really is the stories of censorship that interest me more than the art form as such, and that's what we got at the symposium.
For me, the symposium got at the complexity of political cartoons and by extension the nuances that the medium brings.
I hope what came through the eight speakers' presentations was the sheer range of restrictions that are brought to bear on cartooning, whether they're from government or from publishers, or from the community at large.
What I find most interesting is there are several unresolved or maybe even unresolvable issues surrounding cartoon censorship. Because we all know that there are cases of cartoons that are used for anti-democratic purposes, and are used for ill. There are bad actors in the cartooning world. Not all cartoonists are heroes. Not all cartoonists act in the public interest. Cartoons have been used for hate propaganda. They have been used to promote war and genocide. Between the extremes of cartoons used for liberating, democratic purposes on one hand and cartoons used to incite war and genocide on the other, there is a vast gray area of cartoons that some see as provoking very healthy democratic debate, while others see as wanton cruelty towards the targets of those cartoons. And in some cases, it is quite difficult to make that judgment call as to whether the cartoon did cross a line that society should more stringently police. At what point does mockery cross the line into sadism? At what point does ridicule get egregious? These are genuinely difficult issues.
Your book is focusing on cartoonists who have been censored and I'm wondering if everyone that you've reached out to has wanted to reflect on the atmosphere of censorship that they've experienced.
As with any qualitative research project that involves in-depth interviews, there will always be reluctance, quite understandable reluctance, on the part of some subjects. In this case, I expect, although I haven't yet come across such cases, that some cartoonists simply prefer to keep a low profile because they might be working underground and don't want to draw attention to themselves. There are other cartoonists who, because their work has been so controversial and has created such intense public debate, are just tired of talking about it and see no benefit in talking about it anymore. For example, I was in Paris in August, mainly to look at the Charlie Hebdo case, but did not succeed in meeting any Charlie Hebdo cartoonists — I think because they just don't want to debate it anymore.
One of the panelists, Rob Rogers, talked about his controversial firing as a cartoonist at the Pittsburgh Post-Gazette and discussed cartoons he opted not to publish as well as those his editor refused to print. The theme of self-censorship emerged multiple times during the symposium. How has your understanding of self-censorship evolved, and what have you learned about the ways in which editors apply pressure to cartoonists?
Well, what has always been quite clear to me is that our language needs to distinguish between, on the one side, very legitimate and necessary editorial judgment, which often results in voluntary self-restraint for reasons of taste, accuracy and so on, and on the other side, what I would call self-censorship, where power and fear are involved. This is restraint that is not quite voluntary. It's restraint to preempt punishment, or in anticipation of reward from people with power over you.
The killing of Rob Rogers' cartoons was really a classic case of silencing of a political viewpoint, and a highly legitimate political viewpoint, by a media owner who did not agree with that viewpoint. This, of course, doesn't end the debate, because the question then arises: don't media owners have the legitimate right to decide what should go in the outlets they've invested in? Conventionally, at least in American thinking, press freedom belongs significantly to media owners, as much as or even more so than it belongs to media workers. One standard answer to a case like Rob Rogers, among those who don't think it counts as censorship or self-censorship, is that, "Well, you know, Rogers is not shut out of the public sphere. He's free to publish his work elsewhere. So, why should anyone complain that his publisher chose to restrict access to his newspaper?" I think what's interesting is that the context has changed. Because we used to be in a situation where there was an abundance of newspapers, and an abundance of cartooning jobs within the newspaper industry. But because of the financial crisis that the industry has gone through, that is certainly no longer the case.
So, now, we are almost at the stage where the city newspaper that is able to speak to an entire community is almost as scarce a resource as broadcast airwaves — and, remember, trusteeship of airwaves has been associated with some public service obligations. The net effect of Rob Rogers's firing is that Pittsburgh residents now don't have access to a high-quality, critical diet of editorial cartoons in the course of their print news consumption. And for all the wrong reasons. So that, to me, does sound like censorship.
A good number of people at the symposium seemed to take the opinion that the real way to protest would have been to refuse to let anyone take Rogers' job. That really got at the heart of the political economy of our current moment. It goes beyond just cartooning and also gets at the core of the current plight of modern journalism.
It's not unique to cartooning, and it's not unique to the US, that one of the challenges for press freedom has to do with solidarity. Professional solidarity. Anywhere in the world you look, it is extremely difficult to find the kind of solidarity that you would need to mount a vigorous, collective defense of press freedom. The powers that be, whether political or economic, everywhere in the world, have been very, very successful at the divide-and-rule of media workers. And of course, this is part of a larger story that goes beyond the media. It's about labor generally. The function of the media unions, for example, has changed over the decades. It's rare to find a journalistic union that places the defense of press freedom near the top of its agenda. Most unions are simply fighting for better terms of employment, better pay, and so on, and hardly think about press freedom issues.
To what extent has being a journalist with The Straits Times in Singapore impacted how you engage with the world of journalism, as a scholar?
First of all, it's given me a very deep love for journalism that has never gone away. I love journalists as well. I'm married to one. Throughout my adult life, I've had this unshakable conviction that journalism should and can be a force for good in the world. So it upsets me when it fails to live up to that promise.
Second, I spent my career in a setting that was soft-authoritarian, semi-democratic, nothing close to the First Amendment context that's enjoyed in the US, but neither is it as repressive as many other societies. I've always been intrigued by these in-between societies like Singapore.
Singapore's not the kind of society where the repression is so visible and outrageous that a lot of people get upset about it. People know that the media are not particularly free, and don't particularly care, because it's not as if journalists are being arrested or killed. It's not as if Singaporeans lack choice in their general media diet. They enjoy many, many different ways to entertain themselves. And what I've seen over the last ten years or so is that such contexts are multiplying. We have situations where journalists and cartoonists don't enjoy the freedoms that they should. But restrictions are often only semi-visible. Most people don't know about them and don't care about the fact that they have been denied the complete diet of information and ideas that they deserve. So, one common theme in my research has been to try and understand this. How do power structures apply censorship in ways that have been quite effective, but under the radar?
What advice do you have for the next generation, my generation that is, of researchers who are trying to grapple with structures that are not always seen?
It's a challenge to expose the way power works. Especially the workings of power that are not immediately obvious, because those workings are embedded in the structures through which society operates, and because they are wrapped in ideology. It's always been the role of critical scholarship to try to strip away that ideology and to expose structures for what they are.
Personally, one thing that I've found stimulating is to look at the margins, alternative media, marginalized groups, dissident cartoonists, and so on. Because they're not only interesting in themselves; they are the evidence that different ways of thinking and expression are possible. And if it is possible to think in these different ways, then why are those alternative ways of expression, those alternative worlds, not part of mainstream discourse? Is it simply that they deserve to be on the margins because they're not very good ideas? Or is it because these ideas have merit but are excluded for other reasons? Excluded because they're not in the interest of those at the center of power to have them circulated?
At the symposium "Red Lines," Ted Rall was talking about his own legal battles. What advice do you have for cartoonists who are at risk but specifically facing legal battles? There are so many instances of censorship that are happening globally and in spheres where the First Amendment is not applicable.
You know, there's a political economy of communication. There's also a political economy of dissent and media protection. So, unfortunately, it's only certain cases that get flagged by free speech groups and international human rights groups, or get adopted by pro bono lawyers and activist groups and so on. We've got to assume that for every battle that gets our attention there are probably many more cases that we just never know about.
Sometimes it is for self-preservation that people keep quiet about censorship. To make an issue out of it might compromise their ability to find their next job. Certainly in more controlled societies, that dynamic is routine. Fortunately, there is a small number of organizations that are dedicated to helping cartoonists, journalists, human rights workers, at risk. As citizens we can support their work.
Within the media sector, cartoonists have been neglected more than other media professionals, more than writers. It's more likely that we would know of reporters in trouble than cartoonists in trouble. The reporters probably work in brand-name news organizations – with lawyers on retainer. But most cartoonists are freelancers. It's the other side of the coin of the simplicity of their work. What makes satirical cartooning such a potentially radical and powerful medium, and a centuries-old art form, is that you don't need much to do it. Just a piece of paper or a wall and something to scratch it with. You don't need a lot of capital or an organization behind you. That makes cartoonists the guerrillas of the media — but it also makes them more vulnerable.
Cherian George is Professor of Media Studies at the Journalism Department of Hong Kong Baptist University, where he also serves as Director of the Centre for Media and Communication Research. He researches media freedom, censorship and hate propaganda. He is the author of five books, including Hate Spin: The Manufacture of Religious Offense and its Threat to Democracy (MIT Press, 2016). He received his Ph.D. in Communication from Stanford University. Before joining academia, he was a journalist with The Straits Times in Singapore.
Cartooning Censorship Cherian George

redpajama_set_name: RedPajamaCommonCrawl


Play lively math, vocabulary, and geography games for free on this helpful site for first through fifth graders. Many games feature shooting and kid-friendly characters such as elephants and aliens.
Site based on popular PBS math cartoon is online learning at its best. For younger elementary schoolers new to reading, some elements may be too challenging, but others can be played with a simple knowledge of basic numbers and good listening skills.
A hand- and finger-exercise app for very young kids or for kids with special needs. The three modes -- Write It, Tap It, and Pinch It -- are exercises, not games. Mostly for therapy and practice.
Math practice focuses on kids' skill levels, then adapts. They'll play a variety of games to practice multiplication, division, integer operations, combining like terms, and solving equations and have fun personalizing their own alien avatar named Digi.
In this exciting geometry game with more than 100 challenging puzzles, kids learn the fundamental, as well as the more advanced, concepts related to shapes and geometric proofs.


Bomana (Agenzia Fides) - A conference to revive the mission and take stock of the transformation of the educational communities promoted within the initiatives for the celebrations of the three hundredth anniversary of the death of Saint John Baptist de la Salle recently concluded in Rome. Among the various international initiatives that of the Lasallian community of Papua New Guinea stands out: on February 22 they celebrated the anniversary at the De La Salle school in Bomana, Port Moresby, which was attended by the various members of the Lasallian Family, the staff, students and former students.
"The De La Salle brothers touched the hearts of millions of children all over the world", said Archbishop Kurian Mathew Vayalunkal, Apostolic Nuncio of Papua New Guinea and Solomon Islands. "Brothers of different nationalities have offered their time and continue to use their energies for the education of hundreds of thousands of young people in Papua New Guinea".
"As a member of this institute, learn to become responsible and teach future generations what you have learned. Do not forget to find time for the poor and the marginalized", said the Nuncio, addressing the participants in Bomana.


To book online, first select the course you want to take via our courses menu. Next pick the date and venue you wish to attend and click 'book'. Fill in the online form and click submit.
To book by telephone please call us on: 0115 948 7400. Alternatively you may email or download a booking form for manual completion or book online.


text:
Author: Jason Reynolds
Series: 14 thought-provoking structures
ASIN: B06ZZLYTK9
This tense and tautly-written novel-in-verse takes place in the short span of sixty seconds. Fifteen-year-old Will gets on the elevator with his brother's gun tucked into his waistband. His brother Shawn is dead, and he wants revenge. The elevator stops on the sixth floor, and Buck enters. He tells Will to check the gun; one bullet is missing. Did Shawn ever use his gun? And then Will remembers: Buck is dead. Another figure from Will's past enters a few floors later, and then another, all connected to Shawn. Each one reveals pieces of Shawn's story, and Will has a decision to make as the elevator reaches the ground floor.
"An intense snapshot of the chain reaction caused by pulling a trigger." —Booklist (starred review)
"Astonishing." —Kirkus Reviews (starred review)
"A tour de force." —Publishers Weekly (starred review)
A Newbery Honor Book
A Coretta Scott King Honor Book
A Printz Honor Book
A Los Angeles Times Book Prize Winner for Young Adult Literature
Longlisted for the National Book Award for Young People's Literature
Winner of the Walter Dean Myers Award
An Edgar Award Winner for Best Young Adult Fiction
Parents' Choice Gold Award Winner
An Entertainment Weekly Best YA Book of 2017
A Vulture Best YA Book of 2017
A Buzzfeed Best YA Book of 2017
An ode to Put the Damn Guns Down, this is New York Times bestselling author Jason Reynolds's electrifying novel that takes place in sixty potent seconds–the time it takes a kid to decide whether or not he's going to murder the guy who killed his brother.
A cannon. A strap.
A piece. A biscuit.
A burner. A heater.
A chopper. A gat.
A hammer
A tool
for RULE
Or, you can call it a gun. That's what fifteen-year-old Will has shoved in the back waistband of his jeans. See, his brother Shawn was just murdered. And Will knows the rules. No crying. No snitching. Revenge. That's where Will's now heading, with that gun shoved in the back waistband of his jeans, the gun that was his brother's gun. He gets on the elevator, seventh floor, stoked. He knows who he's after. Or does he?
As the elevator stops on the sixth floor, on comes Buck. Buck, Will finds out, is who gave Shawn the gun before Will took the gun. Buck tells Will to check that the gun is even loaded. And that's when Will sees that one bullet is missing. And the only one who could have fired Shawn's gun was Shawn. Huh. Will didn't know that Shawn had ever actually USED his gun. Bigger huh. BUCK IS DEAD. But Buck's in the elevator?
Just as Will's trying to think this through, the door to the next floor opens. A teenage girl gets on, waves away the smoke from Dead Buck's cigarette. Will doesn't know her, but she knew him. Knew. When they were eight. And stray bullets had cut through the playground, and Will had tried to cover her, but she was hit anyway, and so what she wants to know, on that fifth floor elevator stop, is, what if Will, Will with the gun shoved in the back waistband of his jeans, MISSES.
And so it goes, the whole long way down, as the elevator stops on each floor, and at each stop someone connected to his brother gets on to give Will a piece to a bigger story than the one he thinks he knows. A story that might never know an END…if Will gets off that elevator.
Told in short, fierce staccato narrative verse, Long Way Down is a fast and furious, dazzlingly brilliant look at teenage gun violence, as could only be told by Jason Reynolds.
Other books in "14 thought-provoking structures"
Einstein's Dreams
The Engagements: A Novel
Forty Ways to Look at Winston Churchill: A Brief Account of a Long Life
Life After Life: A Novel
When Women Were Birds: Fifty-four Variations on Voice
Silver Sparrow
The Warmth of Other Suns: The Epic Story of America's Great Migration
Nobody Will Tell You This But Me: A true (as told to me) story
The Yellow House: A Memoir
14 books with thought-provoking structures


text:
Chetco to Close with Assets and Shares Absorbed by Rogue and Coast Central
Member Deposits Protected up to $250,000 by the Share Insurance Fund
ALEXANDRIA, Va. (Dec. 28, 2012) – The National Credit Union Administration (NCUA) today announced the liquidation of Chetco Federal Credit Union (Chetco) effective Dec. 31, and the selection of bids from credit unions in California and Oregon for the purchase and assumption of the remaining assets and shares of Chetco.
The two credit unions are Coast Central Credit Union (Coast Central), of Eureka, Calif., and Rogue Federal Credit Union (Rogue), of Medford, Ore. Rogue will purchase and assume Chetco's five Oregon branches and memberships, and Coast Central will purchase and assume the Crescent City branch and California memberships. Rogue and Coast Central will reopen Chetco's former Oregon and California branches, respectively, Jan. 2, 2013.
The accounts of the new Rogue and Coast Central members will remain federally insured by the National Credit Union Share Insurance Fund up to $250,000. Administered by NCUA, the fund has the backing of the full faith and credit of the U.S. Government.
Rogue is a federal credit union with nearly 56,000 members and approximately $583 million in assets. Coast Central is a state-chartered, federally insured credit union with almost 55,000 members and $960 million in assets.
NCUA placed Chetco into conservatorship Sept. 23, 2011, to protect the credit union's financial stability and operations. NCUA made the decision to liquidate Chetco and discontinue operations after determining the credit union was insolvent with no prospect for restoring viable operations on its own. NCUA worked to find a solution in the best interests of Chetco's members that ensured continued credit union service in the affected counties in Oregon and California.
Chartered in 1957, Chetco was a community credit union serving people who live, work or worship in Coos and Curry counties in Oregon and Del Norte County in California. At the time of liquidation and subsequent purchase and assumption by Rogue and Coast Central, Chetco served 24,926 members and had approximately $259 million in deposits.
Chetco is the fourteenth federally insured credit union liquidation in 2012.
NCUA is the independent federal agency created by the U.S. Congress to regulate, charter and supervise federal credit unions. With the backing of the full faith and credit of the United States, NCUA operates and manages the National Credit Union Share Insurance Fund, insuring the deposits of account holders in all federal credit unions and the overwhelming majority of state-chartered credit unions. At MyCreditUnion.gov (opens new window), NCUA also educates the public on consumer protection and financial literacy issues.
"Protecting credit unions and the consumers who own them through effective regulation"
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The Double Diamond Adventure is a Limited Edition Hydro type weapon that can be upgraded from the Limited Edition Explosive Toboggan Ride.
Nudging the carefully balanced bait on this toboggan begins an amazing skiing adventure filled with perilous fun! After traversing a slope filled with landmines and sharpened moguls mice can let their aches literally melt away in the magma hot tub.
The Double Diamond Adventure is a Limited Edition weapon and can no longer be acquired by Hunters. It can only be crafted given the proper ingredients.
The Double Diamond Adventure acquires a 20% Power Bonus when armed with Snowball Charms.
Explosive Toboggan Ride - Source of Toboggan Ride Parts required for crafting the Double Diamind Adventure.
23 December 2010: The Double Diamond Adventure was introduced as part of the Great Winter Hunt 2010.
The sign with two black diamonds beside the ski sled is a reference to the difficulty ratings in Alpine skiing, where the double black diamond symbol indicates the most difficult ski trails intended only for the most experienced skiers.
13 December 2016: The Double Diamond Adventure gained a 20% power bonus when armed with Snowball Charms.


I was talking to a friend recently who mentioned that someone "loved the gospel." We were talking about a church with a pastor in which the Gospel is not consistently preached and the authority of Scripture not upheld. The current pastor won't mind anyone thinking whatever they want religiously so long as they don't think other people need to agree with them. I told my friend that when he wanted to get the others who loved the Gospel to agree together that they would make sure (as best they could) that there was a person who was truly preaching the Gospel, then they could be a true church. No Gospel, No Church. Gospel, but with no covenant, structure, way to assure Biblical teaching, then No [stable] Church.
As I talked with another friend tonight, counselling him about living as a Christian, I told him that what he needed was to be a member of a local church, to have his discipleship congregationally-shaped.
Seal the Tupperware! Practice church membership! Keep the Gospel witness fresh!
That was D. Martyn Lloyd-Jones in a sermon around 1929 (cited in Iain Murray, Lloyd-Jones, volume 1, page 228).


Good weekend of MLB betting action for this Friday around the diamond as we close in on the All-Star break. The marquee series of the weekend looks to be a battle for first in the NL West between the streaking Colorado Rockies and Los Angeles Dodgers. And with that in mind, here's a look at three series openers on Friday with the latest MLB odds projected.
When: Friday, June 23, 7:10 PM ET Where: Fenway Park, Boston MLB Betting Odds: TBA The Sox will be solid MLB betting favorites here behind reigning Cy Young winner Rick Porcello, but there's going to be a total likely of 10 or higher because Porcello is really struggling and the Angels are starting young Alex Meyer. This game is also on the MLB Network. Porcello (3-9, 5.05) looks nothing like he did last year. Last time out, he was shelled again, this time for seven runs and 10 hits in six innings in Houston that dropped him to 3-9.
The first Cy Young Award was awarded following the 1956 season and since then only three pitchers lost nine games in their first 15 starts of the next season and that has occurred in each of the last three seasons: Corey Kluber in 2015 (3-9, 3.65 ERA), Dallas Keuchel in 2016 (3-9, 5.32) and Porcello this season. Porcello has allowed five or more earned runs in his last three starts, 10 hits or more in four of his last six games and no fewer than five hits in any of his 15 starts.
What's Going on With Alex Meyer?
"To me, the very specific goal is to achieve being down in the bottom of the zone more consistently, then it opens it to be able to do more other things," Manager John Farrell said. This is the first time Meyer (3-3, 3.52) has ever faced Boston. Meyer won after allowing two hits and one walk with nine strikeouts over six scoreless innings Saturday vs. the Royals. Meyer entered Saturday"s start having walked 29 in 40 innings, so the improvement in control with one free pass and 62 of 94 pitches thrown for strikes is a welcome sign. Go over whatever total here.
When: Friday, June 23, 10:10 PM ET Where: Dodger Stadium, L.A. MLB Betting Odds: TBA I believe the National League MVP has a good chance of coming from these teams, either Rockies third baseman Nolan Arenado or Dodgers rookie Cody Bellinger, who has been simply unbelievable and leads the NL in homers despite not being called up until late April. Earlier this week Bellinger became the first player in MLB history with 5 multi-HR games in his first 51 games. The previous fastest to 5 was Mark McGwire (84g). The Dodgers have shifted around their rotation for this big series. They will go with lefty Alex Wood and start fellow lefty Hyun-Jin Ryu on Thursday vs. the Mets.
Is Alex Wood a Good Choice for the Dodgers?
Wood allowed five hits over six scoreless innings when he last faced the Rockies, May 13 at Coors Field, and earned the win. The southpaw finished with 10 strikeouts, his second-most in a start this season. Conversely, Ryu is 0-3 with a 6.75 ERA in three starts against the Rockies this year. He failed to complete five innings in two of the three outings. Wood is 6–0 with a 2.08 ERA in 10 starts this season (7-0 overall), making him the fourth Dodgers pitcher since the team relocated to Los Angeles in 1958 to compile a record of 6-0 or better in his first 10 starts of a season. Colorado rookie Kyle Freeland (8-4, 3.42) is 1-0 with a 1.80 ERA in two starts this year vs. the Dodgers. His incredible 58.7 percent groundball rate now leads the National League. Take L.A. at home as the MLB betting pick here.
When: Friday, June 23, 10:10 PM ET Where: Petco Park, San Diego MLB Betting Odds: TBA Interleague matchup, and the Tigers will be without the designated hitter. Their usual DH, Victor Martinez, is hurt anyway. Detroit started with an even 6-6 record over their first 12 road games, but they've lost 16 of their last 24 away from home (.333). They are 1-5- 2 in their last seven road series. The Padres are also without a regular in 2B Yangervis Solarte, who was placed on the 10-day disabled list with a right oblique strain. It came with unfortunate timing for Solarte, who was batting .317/.394/.667 with seven home runs and 15 RBIs in 16 games this month. Catcher Austin Hedges suffered a bad bruise on his right thigh in a nasty home-plate collision with Anthony Rizzo in the sixth inning of Monday's series opener vs. the Cubs. Hedges hasn't played since but the Friars think he will Friday.
Who Will Pitch in Both Teams?
For Detroit, it's Michael Fulmer (6-5, 3.45). He allowed three runs on six hits in 6.2 innings during last Saturday's loss to the Rays. He struck out three batters and walked four. After allowing 10 runs over his previous two starts (11.2 innings), Fulmer bounced back to toss a quality start despite getting tagged with his fifth loss of the season. He's 5-for-5 in quality starts for his career on the West Coast. It's rookie Dinelson Lamet (2-2, 7.50) for the Padres. Lamet came close to making franchise history in a no-decision against the Brewers on Saturday. The righty struck out 12 batters using just 75 pitches, marking the second-highest single-game strikeout total for a Padres rookie. Neither pitcher has faced the opposing team so I like the under total here, which should be 8.5-9.
In conclusion, it's going to be a good friday to get some MLB betting action going. So, be sure to check out for MLB odds as they come along.


text:
\section{Introduction}
We study the Multi-Armed Bandit (MAB) problem with mean bounds. Here, an agent is presented with a set of $K$ arms, along with side-information in the form of upper and lower bounds on the average reward for each of the arms. The agent is then asked to successively choose arms based on previous observations in order to maximize the cumulative reward. As is standard in MABs, the agent competes with a genie that always chooses the arm that gives the largest reward in expectation. This problem was studied for Bernoulli rewards by~\cite{zhang2017transfer}, where the authors design B-kl-UCB that uses for each arm its kl-UCB index~\citep{garivier2011kl} truncated at the corresponding {\em upper bound}. Here, UCB stands for Upper Confidence Bound after the UCB algorithm of \cite{auer2002finite}.
Our setting also fits the Structured Bandit framework of~\cite{combes2017minimal} where the authors develop OSSB, a non-optimistic algorithm to balance exploration across arms in structured spaces. However, in both the above, the information of lower bounds on arm means is only used initially to prune away arms that can be identified as suboptimal while the upper bounds are used to clip the arm indices at each time.

In this paper, we observe that the lower bounds of the arms provide crucial information about the unknown optimal arm, which can be used to globally decrease exploration of {\em all} the sub-optimal arms. Surprisingly, such under exploration results in a lower regret (both theoretically and empirically) than that from B-kl-UCB and OSSB. We detail our contribution on bandits with mean bounds below.

\textbf{Contribution~1: Bandits with mean bounds:}
We develop GLobal Under-Explore (GLUE)---an index based policy which, unlike UCB and kl-UCB, is not optimistic; i.e. the indices do not serve as high probability upper bounds to true means of each arm. In particular, we use the key idea that as long as the index for the (unknown) optimal arm is a high probability upper bound to its true mean, violating the property for other arms does not affect the regret guarantees adversely. In fact, such violations allows suboptimal arms to be under-explored resulting in lower regret in specific settings. However, as the optimal arm is apriori unknown, the key is to infer a quantity for each arm, which we call ``\texttt{pseudo-variance}'', that can be strictly lesser than the true variance of the arm {\em only if} the arm is sub-optimal.

We show that for bandits with mean bounds and bounded support of rewards, the upper bound on variance of the optimal arm can be inferred from the {\em highest lower bound} of arm means. Therefore, we define the pseudo-variance of the arms to be the minimum of the individual arm variance and the aforementioned upper bound. We prove that while our use of pseudo-variances results in a regret that is never worse than standard UCB, it can also significantly improve over B-kl-UCB and OSSB in certain parameter regimes (see, Section~\ref{sec: comparison}). Interestingly, this shows that existing lower bounds on structural bandits in \cite{combes2017minimal} do not directly apply in our setting. We discuss this further in Section~\ref{sec: comparison} and Appendix~\ref{app:OSSB}.

\textbf{Motivation with Confounded Logs:} Suppose an agent is given access to a partially recorded (confounded) log of interventions and their corresponding effects and is tasked with the learning the best intervention. When the unrecorded variables affect the rewards, the best intervention is not readily identifiable. As a concrete example, in healthcare, it is well known that intravenous tissue plasminogen activators (tPA) are remarkably effective for acute ischemic strokes if administered within 3 hours of the onset of symptoms. Otherwise, a less-effective medical therapy is recommended, as tPA can have adverse side effects and higher chances of hemorrhage~\citep{powers2019guidelines}. If a log is then generated without recording the time since the onset of symptoms, it would present tPA to be better than the alternative. However, naively using such a log to infer tPA to be the best intervention could lead to unfavourable outcomes in areas with poor access to healthcare, where patients take longer to reach the hospital after symptoms appear. Our interest lies in the following question:

{\em Can we use such logged data to perform a non-trivial transfer of knowledge, despite the partial confounding?}

\textbf{Contribution~2: Mean bounds from confounded logs:} We develop techniques to extract upper and lower bounds on the means of arm rewards from partially confounded logged data. Specifically, we consider a data set that has been collected by an oracle that observes the full context, takes the optimal action and receives the corresponding rewards. However, the log only contains some parts of the context along with the corresponding (action, reward) pair, generalizing the work of \cite{zhang2017transfer}, where none of the contexts are recorded. Using bounds on the gap between the means of the best and second best arms {\em as observed by the oracle}, as well as the corresponding gap between best and worst arms, we derive upper and lower bounds on the mean rewards of arms to be used by an agent that acts only based {\em only} on the recorded parts of the context. We show that these are tight, i.e. there are instances that meet both the upper and lower bounds. We validate our work through semi-synthetic experiments with the Movielens 1M data set~\citep{harper2015movielens}.

\subsection{Related Work}\label{sec: related work}
Bandit problems have seen a lot of interest over the past few decades (see \cite{bubeck2012regret,lattimore2018bandit} for comprehensive surveys). A vein of generalization for the same has seen numerous advances in incorporating several forms of side information to induce further structure into this setting. Notable among them are graph-structures information \citep{buccapatnam2014stochastic,valko2014spectral,amin2012graphical}, latent models \citep{li2010contextual,bareinboim2015bandits,lattimore2016causal,sen2016contextual}, expert models \citep{auer1995gambling,mannor2011bandits}, smoothness of the search space \citep{kleinberg2008multi,srinivas2009gaussian,bubeck2011x}, among several others. We introduce side information in the form of mean bounds and study how such information can be inferred naturally from logged data. Learning from logged data has also been studied recently in \citep{zhang2017transfer,zhang2019warm,ye2020combining}. The first assumes that the log contains no information of the variables that affect reward generation, while the others assumes that all such variables are present. We consider the middle ground, by assuming that a fraction of these variables are included in the logs.

Our bandit problem has connections to the works by \cite{zhang2017transfer} and \cite{combes2017minimal}. An in-depth comparison with these can be found in Section \ref{sec: comparison}. Along another thread, \cite{bubeck2013bounded} provide algorithms with bounded regret if the mean of the best arm and a lower bound on the minimum suboptimality gap is known. These techniques, however, do not apply in our setting as the side information we consider does not allow us to extract such quantities in general. Recently, \cite{tennenholtz2020bandits} studied a related linear bandit problem where the agent is provided with partial observations collected offline according to a fixed behavioral policy which can be sampled from. These are then used to aid online decision making after the agent observes the full context. In contrast, we consider the case where the agent can only observe the {\em partial context} at each time and is provided with confounded logs collected from a policy (to which we do not assume sampling access) that is optimal under the full context.

The extraction of mean bounds from confounded logs has been studied in the context of estimating treatment effects in the presence of confounders. Here, actions are treatments, and the rewards capture the effects of this choice. A line of existing work performs sensitivity analysis by varying a model on the latents, measured variables, treatments and outcomes in a way that is consistent with the observed data \citep{robins2000sensitivity,brumback2004sensitivity,richardson2014nonparametric}. Recently in \citep{yadlowsky2018bounds,zhao2019sensitivity}, a universal bound on the ratio of selection bias due to the unobserved confounder is assumed. This means that the treatment choice has a bounded sensitivity on the unknown context (i.e., mostly irrelevant). We deal with the other extreme, where we assume that the outcomes in the log are recorded using an unknown {\em optimal policy} (under complete information), and that the knowledge of worst case sub-optimality gaps for the given latent context space is known. Our assumption allows for strong dependence on the unknown context.

\section{Bandits with Mean Bounds}\label{sec: bandits with mean bounds}

\underline{\em Problem Setting:} We consider a multi-armed bandit (MAB) instance with $K_0$ arms. Each arm is associated with a fixed bounded distribution with support $[a,b]$, common across all arms, from which rewards are drawn independently at each time. Arms are labeled as $\{1,2,...,K_0\}$ and the mean of arm $k\in [K_0]$ is denoted as $\mu_k$. Let $\mu^*$ be the reward of arm $i^* = \argmax_{k\in[K_0]} \mu_k$. At each time $t\geq 1$, an agent chooses arm $A_t \in [K_0]$ and observes the reward $Y_t$ associated with the chosen arm. The agent is allowed to use the accumulated rewards to inform future choices.

Additionally, for each arm $k\in [K_0]$, the agent is provided with quantities $l_k, u_k$ such that $\mu_k \in [l_k,u_k]$, i.e., for each $k$, $l_k,u_k\in [a,b]$ serve as lower and upper bounds (respectively) on the \emph{mean reward} of arm $k$. These upper and lower bounds can be \textit{different} for each arm. We note that the {\em observed rewards} can lie outside $[l_k, u_k]$ (but are supported over $[a,b]$).

An arm-selection policy is considered optimal if it plays the arm $i^*$ at each time. The goal of the agent is to minimize its average cumulative regret, defined at time $T$ as $R_T = \sum_{t=1}^T \mathbb{E}[\mu^* - Y_t]$. The cumulative regret is the expected difference in rewards accumulated by its policy from that of an optimal one.

Note that with $l_k = a, u_k = b$ for all $k\in [K_0]$, this reduces to the standard MAB problem, which is well studied. Optimism-based policies such as standard UCB \citep{auer2002finite} and KL-UCB \citep{garivier2011kl} achieve provably optimal performance for specific families of reward distributions. However, with non-trivial mean bounds, it is shown in \cite{zhang2017transfer} that one can outperform these methods by leveraging the information that is provided by these bounds.

\subsection{GLUE for MAB with Mean Bounds}

\begin{algorithm}[tb]
	\begin{algorithmic}[1]
		\STATE{{\bf Inputs:} Upper and lower bounds for each arm $k\in [K_0]: u_k,l_k$. }
		\STATE{{\bf Pruning Phase:}}
		\STATE{Define $l_{max} = \max_{k\in K_0} l_k$ and eliminate all arms $i\in [K_0]$ with $u_i<l_{max}$.}
		\STATE{Rename the remaining arms to be in $\{1,2,...,K\}$.}
		\STATE{{\bf Learning Phase:}}
		\STATE{Set UCB index scaling parameters $c_k$ for each arm as in Equation \eqref{eq:ck}.}
		\FOR{$t = 1,2,3,...$}
		\STATE{Play arm $A_t = \argmax_{k\in [K]} U_k(t-1)$ and observe reward $Y_t$}.
		\STATE{Increment $T_{A_t}(t)$ and update $\hat{\mu}_{A_t}(t)$.}
		\STATE{Update $U_k(t)$ as in Equation \eqref{eq:Uk}.}
		\ENDFOR
	\end{algorithmic}
	\caption{\textbf{GL}obal \textbf{U}nder-\textbf{E}xplore (GLUE) for MABs with mean bounds}
	\label{Algo1}
\end{algorithm}

We now present the Global Under-Explore (GLUE) algorithm for MABs with mean bounds, which is an index policy that sets arm indices as a function of the side information from all arms. GLUE is summarized in Algorithm \ref{Algo1} and consists of two phases described below.

\textbf{1. Pruning Phase: } In this phase, the mean bounds are used to discard deterministically suboptimal arms. In particular, let $l_{max} = \max_{k\in [K_0]} l_k$. Then, all arms $i$ such that $u_i < l_{max}$ are discarded. Such a strategy can also be seen in \cite{zhang2017transfer}.

\textbf{2. Learning Phase:} Let $\{1,2,..,K\}$ be the remaining number of arms after the pruning stage ($K\leq K_0$). For each arm $k\in [K]$, we define the variance bound $\sigma_k^2$ and the ``\texttt{pseudo-variance}" $c_k$ as follows:
\small
\begin{gather}\label{eq:ck}
	\sigma_k^2 = \begin{cases}
		{\scriptstyle B_{a,b}(a+b-l_k)(l_k-a)(b-l_k)}& \text{if } {\scriptstyle l_{k} \geq 0.2178a + 0.7822b}\\
		{\scriptstyle B_{a,b}(u_k)(u_k -a)(b-u_k)} & \text{if }{\scriptstyle u_k \leq 0.7822a + 0.2178b}\\
		\frac{(b-a)^2}{4} & \text{otherwise}
	\end{cases},\notag\\
	c_k = \begin{cases}
		{\scriptstyle B_{a,b}(a+b-l_{max})(l_{max}-a)(b-l_{max})}&\text{if } {\scriptstyle l_{max}\geq 0.2178a+0.7822b}\\
		\sigma_k^2 &\text{otherwise}
	\end{cases}.
\end{gather}
\normalsize

Here, the function $B_{a,b}(x) := \frac{(b-a)\exp\left( \nicefrac{b-a}{b-x}\right)}{\left(b-x+(x-a)\exp\left( \nicefrac{b-a}{b-x}\right)\right)^2}$.

Let $T_k(n) = \sum_{t=1}^n \mathbbm{1}\{A_t = k\}$ be the number of times arm $k$ is played up to time $n$, and let $\hat\mu_k(n) = \frac{1}{T_k(n)} \sum_{t=1}^n Y_t \mathbbm{1} \{A_t = k\}$ be its empirical mean at time $n$. Then, for each arm, we set the index as
\small
\begin{gather}\label{eq:Uk}
	U_k(t) = \min\left\{u_k, \hat\mu_k(t) + \sqrt{\frac{2 c_k \log(f(t+1))}{T_k(t)}}\right\},\notag\\
	\text{ for } f(t) = 1+ t\log^2(t).
\end{gather}
\normalsize

We observe two key things here: 1. For each arm $k$, $U_k(t)$ is set using the pseudo-variance $c_k$ rather than the true variance $\sigma_k^2$. In general, $c_k\leq \sigma_k^2$ and when $l_{max}\geq 0.2178a + 0.7822b$, $c_k$ is equal for all arms; 2. Arm indices are clipped at the respective upper bounds, which is sensible as these indices are a representation of our belief of true arm means. We also note that using $c_k = \nicefrac{(b-a)^2}{4}$ and setting $U_k(t)$ without clipping at $u_k$ recovers the standard Upper Confidence Bound (UCB) algorithm in \cite{auer2002finite}.

\subsection{Key Ideas}
We now describe the intuition for the two non-trivial ways in which we use the mean bounds (these will be made concrete in Section \ref{sec: Regret of UCB-SI}).

\textbf{1. Sharper variance bounds:} The arm rewards are supported over the interval $[a,b]$, which immediately imply a variance bound of $\nicefrac{(b-a)^2}{4}$ by way of the standard Chernoff-Hoeffding inequalities. However, when the provided mean bounds for arm $k$, ($l_k,u_k$) are informative, in the sense that they narrow down the uncertainty of the mean's location towards the boundaries of the interval $[a,b]$, we improve the Chernoff-Hoeffding inequalities (Theorem \ref{improved Hoeffding Inequality}) to obtain sharper estimates of the variance bound. This is reflected in our setting of $\sigma_k^2$ above.

\textbf{2. Pseudo-variances:} In order to understand our final use of the mean bounds, consider a standard MAB instance with $K$ arms where it is known that arm $k$ provides rewards with a variance upper bound of $\sigma_k^2$. The standard UCB algorithm would then explore arm $k$ at the rate dictated by the corresponding variance bound. Suppose additionally that it was known that the \emph{best arm} produced rewards with variance bound $\sigma_{best}^2$. Consider the following algorithm $\mathcal{A}$ which sets the index of arm $k$ analogous to UCB, but with exploration rate dictated by $\min\{\sigma_k^2,\sigma_{best}^2\}$. As a result, $\mathcal{A}$ potentially assumes incorrect variance bounds for suboptimal arms. Further, the indices of $\mathcal{A}$ are no longer high-probability upper bounds to the arm means, i.e., $\mathcal{A}$ is not optimistic. The key difference in the dynamics of UCB and $\mathcal{A}$ is that the latter potentially plays suboptimal arms {\em less frequently} than the former. This would mean that $\mathcal{A}$ incurs lesser regret than standard UCB.

In the above, $\min\{\sigma_k^2, \sigma_{best}^2\}$ is what we call pseudo-variance in this work. Specifically, when $l_{max}\geq 0.2178 a + 0.7822 b$, the side information allows us to estimate an upper bound on the variance of the best arm. We note that the arm that corresponds to $l_{max}$ need not be the best arm, and thus the problem of regret minimization is not trivial in this case. The above discussion motivates our setting of the pseudo-variance $c_k$ for arm $k$ and its use in $U_k(t)$.

\subsection{Regret Upper Bounds for GLUE}\label{sec: Regret of UCB-SI}
In this section, we study the regret performance of GLUE. We regard improvements from pruning as trivial and analyze regret for the set of arms that remain after pruning. We assume without loss of generality that \textit{after pruning} the arms are indexed in non-decreasing order of mean, i.e., $\mu^* = \mu_1 > \mu_2\geq ... \geq \mu_K$, where $K\leq K_0$. We also define the sub-optimality gap of an arm $k$ as $\Delta_k = \mu^*-\mu_k$. The following theorem bounds the regret of GLUE.

\begin{theorem}\label{UCB-SI theorem}
	Let $K_1(\delta) = \{k\in[K]: \mu^* > u_k + \delta, k > 1\}$ and $K_2(\delta) = \{k\in[K]: \mu^*\leq u_k + \delta, k > 1\}$, for any $\delta>0$. Then, the expected cumulative regret of Algorithm \ref{Algo1} satisfies $R_n \leq \inf_{\delta > 0} R_n(\delta)$ where
	\begin{align*}
		&R_n(\delta)\leq \sum_{k\in K_1(\delta)} \frac{5c_1\Delta_k}{\left(\max\{\delta , \mu^*-u_k\}\right)^2}+\sum_{k\in K_2(\delta)} \left(\Delta_k + \frac{5c_1}{\Delta_k} + \frac{2c_k\log(f(n))}{\Delta_k} + \Delta_k q(n)\right)
	\end{align*}
	for a function $q(n)\sim \Theta\left(\log(f(n))^\frac{2}{3}\right)$. Asymptotically, we have \small
	\begin{align}\label{eq: GLUE asymptotic}
		\limsup_{n\rightarrow\infty} \frac{R_n}{\log(n)} &\leq \sum_{k\in K_2(0)} \frac{2c_k}{\Delta_k}.
	\end{align}
	\normalsize
\end{theorem}

\begin{proof}[Proof Sketch]
	The proof follows the general structure of regret analysis for bandit problems, where we produce an upper bound on the number of times each suboptimal arm is played by some time $n$. While it is standard to use Chernoff-Hoeffding inequalities in this process, the mean bounds can be used to improve the rate of concentrations in certain cases. This is seen in our novel results that follow:
	\begin{theorem}[Chernoff-Hoeffding Bound with mean bounds]\label{improved Hoeffding Inequality}
		Let $\{X_i\}_{i=1}^n$ be a sequence of i.i.d random variables with support $[a,b]$ and mean $\mu$ such that $l\leq \mu\leq u$. Let $\hat\mu_n = \frac{1}{n}\sum_{i=1}^n X_i$. Then, for all $\epsilon \in [0, u-l]$,
		\small
		\begin{align*}
			&\mathbb{P}(\hat{\mu}_n - \mu\geq \epsilon) \leq \scriptstyle{\begin{cases}
					\exp\left(-\tfrac{n\epsilon^2}{2(l-a)(b-l)}\right), &l \geq \nicefrac{a+b}{2}\\
					\exp\left(-\tfrac{n\epsilon^2}{2B_{a,b}(u)(u-a)(b-u)}\right), &u \leq 0.7822a + 0.2178b\\
					\exp\left(-\tfrac{2n\epsilon^2}{(b-a)^2}\right), &\text{otherwise}
			\end{cases}},\\
			&\mathbb{P}(\hat{\mu}_n - \mu\leq -\epsilon) \leq \scriptstyle{\begin{cases}
					\exp\left(-\tfrac{n\epsilon^2}{2(u-a)(b-u)}\right), &u \leq \nicefrac{a+b}{2}\\
					\exp\left(-\tfrac{n\epsilon^2}{2B_{a,b}(a+b-l)(l-a)(b-l)}\right), &l \geq 0.7822b + 0.2178a\\
					\exp\left(-\tfrac{2n\epsilon^2}{(b-a)^2}\right), &\text{otherwise}
			\end{cases}}.
		\end{align*}
		\normalsize
	\end{theorem}

	The proof of Theorem \ref{improved Hoeffding Inequality} is motivated by the standard Hoeffding inequality~\citep{hoeffding63}. This involves analysis of the moment generating function(MGF) of the random variable. A key step is in upper bounding the exponent of MGF by a global constant using an appropriate Taylor expansion. However, with the mean bounds, we produce tighter upper bounds to this quantity in a specific regimes of side information. These tighter upper bounds translate into improvements in concentrations that are presented above.

	Recall that we only need to consider the set of $K$ arms that remain after pruning. Using Theorem \ref{improved Hoeffding Inequality}, we first show that the $\sigma_k^2$ is an upper bound on the true variance of arm $k$. We then partition the set $[K]$ into $K_1(\delta)$ and $K_2(\delta)$ and treat each of them separately. Specifically, we prove that arms in $K_1(\delta)$ only contribute a constant regret by any time $n$. We deem these arms to be ``\texttt{meta-pruned}''. For $K_2(\delta)$, we show that the use of pseudo-variances is sufficient to explore these arms at a logarithmic rate. The theorem then follows by combining these two results.
\end{proof}

Complete proofs of both the theorems above can be found in the appendix.

\subsection{Comparison and Discussions}\label{sec: comparison}
~~~~\textbf{Upper Bounds: GLUE vs Existing Algorithms:}
We now compare GLUE with existing algorithms for asymptotic performance below.

\textbf{1. Standard UCB and B-UCB:} The standard UCB of \cite{auer2002finite} is an optimistic index-based policy that sets arm indices to be high-probability upper bounds for each arm as a function of a universal bound on reward variances. As it does not use the mean bounds in any way, we also consider the B-UCB algorithm, which clips indices of standard UCB at the provided mean bounds.

\textbf{2. kl-UCB, B-kl-UCB:} kl-UCB~\citep{garivier2011kl} is an optimistic algorithm that is designed to match the lower bounds for stochastic bandit problems when the reward distributions are known. To set indices at each time, kl-UCB solves an optimization problem(designed for the corresponding reward distributions) for each arm. This problem does not present closed-form solutions in general. Extending this to the case of MABs with mean bounds, \cite{zhang2017transfer} introduce the B-kl-UCB algorithm, by clipping the kl-UCB indices using the side information. These algorithms are applicable for unknown bounded rewards by using their corresponding Bernoulli variants.

\textbf{3. OSSB for Structured Bandits:} Our setting of MABs with mean bounds can be translated to that of Structured Bandits in \cite{combes2017minimal}. Choosing arm parameters as the corresponding arm means and restricting the set of feasible instances using the provided mean bounds, one can apply the OSSB algorithm for the structured bandit problem. Specifics of this connection can be found in Appendix \ref{sec:OSSB opt solution}. The OSSB algorithm uses the solution to a semi-infinite optimization problem at each time to instruct the choice of arms. This optimization problem assumes the knowledge of reward distributions, and is carefully designed to balance exploration across arms. Analogous to the one in kl-UCB, this problem can not be solved in closed form for general distributions. We discuss the closed-form solution for {\em Bernoulli rewards}, and its asymptotic regret upper bound in Appendix \ref{sec:OSSB opt solution}.

We summarize the regret upper bounds of all the algorithms above in the case of {\em Bernoulli rewards} in Table \ref{table: GLUE vs others}. In particular, B-kl-UCB and OSSB both attain lowest asymptotic regret upper bound
($\limsup_{n\to \infty} \nicefrac{R_n}{\log(n)}$) among the existing algorithms. We observe that GLUE is no worse than B-UCB asymptotically. More importantly, for a range of parameters, GLUE attains lower asymptotic regret than OSSB/B-kl-UCB (see, Figure~\ref{fig: heatmaps}).

\begin{table}[t]
	\centering
	\caption{Comparing Performance of GLUE to Existing Algorithms for Bernoulli Rewards: Columns list the algorithm, if it uses information about reward distributions, displays meta-pruning, is optimal with respect to the lower bound in \cite{combes2017minimal}, and the {\bf asymptotic} regret upper bound}
	\label{table: GLUE vs others}

	\scriptsize
	\begin{tabular}{|l|l|l|l|l|}
\hline
{\bf Algorithm} & \textbf{Reward Distribution} & \textbf{Meta-pruning} & \textbf{``Optimal''} & \textbf{Regret Upper Bound} \\ \hline
UCB & No & No & No & $\sum\limits_{\scriptscriptstyle k\in[K],k\neq 1} \frac{2}{\Delta_k}\cdot \frac{(b-a)^2}{4}$ \\ \hline
B-UCB & No & Yes  & No & $\sum\limits_{\scriptscriptstyle k\in K_2(0)} \frac{2}{\Delta_k}\cdot \frac{(b-a)^2}{4}$ \\ \hline
kl-UCB & Yes & No  & No & $\sum\limits_{\scriptscriptstyle k\in [K], k\neq 1} \frac{\Delta_k}{d(\mu_k,\mu^*)}$ \\ \hline
B-kl-UCB & Yes & Yes  & Yes & $\sum\limits_{\scriptscriptstyle k\in K_2(0)} \frac{\Delta_k}{d(\mu_k,\mu^*)}$ \\ \hline
OSSB & Yes & Yes  & Yes & $\sum\limits_{\scriptscriptstyle k\in K_2(0)} \frac{\Delta_k}{d(\mu_k,\mu^*)}$ \\ \hline
{\bf GLUE} & {\bf No} & {\bf Yes}  & {\bf Can improve} & $\sum\limits_{\scriptscriptstyle k\in K_2(0)} \frac{2c_k}{\Delta_k}$  \\ \hline
\end{tabular}
\normalsize
\end{table}

\begin{figure*}[t]
	\centering
	\begin{center}
		GLUE vs B-kl-UCB/OSSB($\epsilon=\gamma=0$): Heatmaps for $\nicefrac{R_{\infty}^{GLUE}}{R_{\infty}^{B-kl/OSSB(\epsilon=\gamma=0)}}$
	\end{center}\par\medskip

	\begin{subfigure}[b]{0.25\textwidth}
		\centering
		\includegraphics[width = 1.4in]{Heatmap_0_92.png}
		\caption{\small $l = 0.92$}
	\end{subfigure}%
	\begin{subfigure}[b]{0.25\textwidth}
		\centering
		\includegraphics[width = 1.33in]{Heatmap_0_95.png}
		\caption{\small $l=0.95$}
	\end{subfigure}%
	\begin{subfigure}[b]{0.25\textwidth}
		\centering
		\includegraphics[width = 1.3in]{Heatmap_0_98.png}
		\caption{\small $l=0.98$}
	\end{subfigure}%
	\begin{subfigure}[b]{0.25\textwidth}
		\centering
		\includegraphics[width = 1.3in]{Heatmap_l_0_98_mean_0_985.png}
		\caption{\small Regret for $\mu_1 = 0.985, \mu_2 = 0.2, l_1 = 0.98$}
	\end{subfigure}%
	\caption{\small Heatmaps comparing the asymptotic upper bounds of GLUE with B-kl-UCB/OSSB($\epsilon=\gamma=0$) (both have the same asymptotic regret upper bound) on a 2-armed bandit system. Arm parameters are as follows: {\bf Arm 1:} $u_1 = 1, l_1 = l$; {\bf Arm 2: } $u_2 = 1, l_2 = 0$. $l$ is mentioned in the captions. The red regions indicate where GLUE outperforms B-kl-UCB/OSSB. Note the increase of the red regions with rising $l$. The last subplot contains empirical regret plot for $\mu_1 = 0.985$, $\mu_2 = 0.2, l_1 = 0.98$ averaged over $500$ independent runs.}
	\vspace{5pt}
	\label{fig: heatmaps}
\end{figure*}

\textbf{GLUE and Structural Bandits Lower Bound:} While OSSB is applicable when the reward distributions are known, the corresponding lower bounds do not apply to our setting. To argue this, we consider a 2-armed system where Arm 1's mean is known (upper and lower bounds coincide), whereas Arm 2 has uninformative mean bounds. This side information can be captured by a semi-infinite sequence of historical arm pulls and corresponding rewards for Arm 1, presented to the agent before it acts on the system.
These histories, however, can not be directly captured in the usual divergence decomposition result used in standard bandit lower bounds, as they lead to zero-probability events.
We discuss these issues at length in Appendix \ref{app:OSSB}.

Additionally in Appendix \ref{sec:OSSB opt solution}, we show that the lower bound for structured bandits that is matched by OSSB is matched by the upper bound of the B-kl-UCB algorithm (see Table \ref{table: GLUE vs others}). Importantly however, this lower bound can be {\em strictly greater} than the upper bound of GLUE in Equation \ref{eq: GLUE asymptotic}.
Thus, this indicates that the OSSB lower bound does not hold in our setting.
Furthermore, from numerical computations for a two armed case, when $\mu_1 \in [0.95,1]$ and $\mu_2 \in [0,1]$ with $\mu_1 = 0.96, \mu_2 = 0.2$, the lower bound of structured bandits computes to 0.36488, while the upper bound of GLUE is 0.29965. The regimes of improvement are also summarized in Figure~\ref{fig: heatmaps}.

\textbf{Empirical Comparisons:} We compare GLUE with the existing algorithms discussed above through synthetic experiments. We provide these on a 3-armed bandit instances, where rewards are in $[0,1]$ and are drawn independently from Bernoulli distributions with the corresponding means. All algorithms are only provided with the upper and lower bounds for each arm, and that all rewards are in $[0,1]$. Plots are averaged over 500 independent runs. These can be found in Figure \ref{fig: synthetic bernoulli}. The input parameter $\gamma$ for OSSB is always set to be 0, and the parameter $\epsilon$ used is mentioned in each plot. We observe empirically that OSSB can be highly unstable when we set $\epsilon=\gamma=0$. Examples for such behavior are deferred to the appendix.

\begin{figure*}[t]
	\centering
	\begin{subfigure}[b]{0.33\textwidth}
		\includegraphics[width = 1\textwidth,height = 0.9in]{Case1Bernoulli}
		\caption{\small Global UE, favorable SI}
		\label{fig:case1 bernoulli}
	\end{subfigure}%
	\begin{subfigure}[b]{0.33\textwidth}
		\includegraphics[width = 1\textwidth,height = 0.9in]{Case2Bernoulli}
		\caption{\small Global UE, adversarial SI}
		\label{fig:case2 bernoulli}
	\end{subfigure}%
	\begin{subfigure}[b]{0.33\textwidth}
		\includegraphics[width = 1\textwidth,height = 0.9in]{Case3Bernoulli}
		\caption{\small Meta-pruning only}
		\label{fig:case3 bernoulli}
	\end{subfigure}%

	\begin{subfigure}[b]{0.33\textwidth}
		\includegraphics[width = 1\textwidth,height = 0.9in]{Case4Bernoulli}
		\caption{\small Local UE, favorable SI}
		\label{fig:case4 bernoulli}
	\end{subfigure}%
	\begin{subfigure}[b]{0.33\textwidth}
		\includegraphics[width = 1\textwidth,height =0.9in]{Case5Bernoulli}
		\caption{\small Local UE, adversarial SI}
		\label{fig:case5 bernoulli}
	\end{subfigure}%
	\begin{subfigure}[b]{0.33\textwidth}
		\includegraphics[width = 1\textwidth,height = 0.9in]{Case6Bernoulli}
		\caption{\small Uninformative SI}
		\label{fig:case6 bernoulli}
	\end{subfigure}
	\caption{\small Synthetic Experiments for Bernoulli Rewards: Best arm is always Arm 1. Captions indicate the type of instance: UE = under-exploration, SI = Side Information. Global UE = $l_{max}\geq 0.7822$, Local UE = $\max_{k\in[K]}u_k \leq 0.2178$, Favorable SI = best arm has largest upper bound, Adversarial SI = suboptimal arm has largest upper bound, Uninformative SI = no improvement over UCB. \underline{Observations}: GLUE can outperform all other algorithms in case of Global UE, and outperforms UCB and B-UCB with Local UE. All algorithms that clip indices benefit from meta-pruning. GLUE is never worse than standard UCB.}
	\vspace{5pt}
	\label{fig: synthetic bernoulli}
\end{figure*}

We also compare these algorithms on a more general distribution, where the differences in performance are more pronounced, and discuss the interesting transient dynamics. These can be found in the appendices.

\subsection{Open Questions on Bandits with Mean Bounds:}
~~~~\textbf{1. Lower Bounds:} From our results and discussions above, we see that the lower bounds for Structured Bandits~\citep{combes2017minimal} do not apply in our setting and thus, the problem of lower bounds for our setting serves as an important avenue for future work.

\textbf{2. Uncertain mean bounds:} Our focus in this work has been the case when the provided mean bounds hold with probability $1$. The immediate follow-up would be the case where these bounds only hold for each arm $k$ with some probability $p_k<1$. Practically, such settings occur when we are given historical data about plays from each arm (this is the setting studied in \cite{shivaswamy2012multi}). In this case, we can readily translate our any-time regret analysis to a horizon-dependent bound as follows:
\begin{corollary}\label{corr: uncertain bounds}
	If $\mu_k\in [l_k,u_k]$ with probability $(1- p_k)$ for each arm $k\in [K]$, then, GLUE presents regret as in Theorem 1 with an additional regret of $n\sum_{k\in [K]}p_k$ by time $n$.
\end{corollary}
If we know the horizon to be $T$ and each arm is observed $\mathcal{O}(\log T)$ times in the provided history, using standard Chernoff-type arguments, we easily see that $\sum_{k\in[K]} p_k = O(1/T)$. Thus, the additional regret incurred by time $T$ is simply a constant. However, the cases when the number of samples for an arm are \emph{sub-logarithmic} in the horizon and that with an \emph{unknown horizon} are left open.

\section{A Use Case: Learning from Confounded Logs}\label{transfer section}
In this section, we consider the task of extracting non-trivial bounds on mean from partially confounded data from an optimal oracle. We show that under mild assumptions, this problem leads to the MAB problem with mean bounds organically. We begin by describing the setting:

\textbf{The Oracle Environment:} We consider a contextual environment, where nature samples a context vector $(z,u)\in \mathcal{C} = \mathcal{Z}\times \mathcal{U}$ from an unknown but fixed distribution $\mathbb{P}$. We assume that sets $\mathcal{Z}$ and $\mathcal{U}$ are both \emph{discrete}. At each time any of the $K_0$ actions from the set $\mathcal{A} = \{1,2,...,K_0\}$ can be taken. The reward of each arm $k\in \mathcal{A}$ for a context $(z,u)\in\mathcal{C}$ has mean $\mu_{k,z,u}$ and support $[0,1]$\footnote{By appropriate shifting and scaling, this can be generalized for any finite interval $[a,b]$.}. For each context $(z,u) \in \mathcal{C}$, let there be a {\em unique best arm} $k_{z,u}^*$ and let $\mu^*_{z,u}$ be the mean of this arm.

\textbf{Confounded Logs:} The oracle observes the complete context $(z_t,u_t)\in \mathcal{C}$ at each time $t$ and also knows the optimal arm $k^*_{z_t,u_t}$ for this context. She picks this arm and observes an independently sampled reward $y_t$ with mean value $\mu_{k^*_{z_t,u_t}, z_t, u_t} = \mu^*_{z_t,u_t}$. She logs the information in a data set while omitting the partial context $u_t$. In particular, she creates the data set $\mathcal{D} = \{(z_t, k_t, y_t): t\in \mathbb{N}\}$.

\textbf{The Agent Environment:} A new agent is provided with the oracle's log. In this paper, we consider the {\em infinite data} setting. The agent makes sequential decisions about the choice of arms having observed the context $z_t\in \mathcal{Z}$ at each time $t = \{1,2,...\}$, while the part of the context $u\in \mathcal{U}$ is {\em hidden} from this agent. Let $a_t$ be the arm that is chosen, $z_t$ be the context, and $Y_t$ be the reward at time $t$. Define the average reward of arm $k\in \mathcal{A}$ under the {\em observed} context $z\in \mathcal{Z}$ as $\mu_{k,z} = \sum_{u\in \mathcal{U}} \mu_{k,z,u} \mathbb{P}(u|z)$.

The optimal reward of the agent under context $z \in \mathcal{Z}$ is defined as
$\mu^*_{z} = \max_{k\in \mathcal{A} }\mu_{k,z}$. The agent aims to minimize her cumulative regret for each context separately. The cumulative reward for each $z\in \mathcal{Z}$ at time $T$ is defined as:
$R_z(T) = \mathbb{E}\left[\sum_{t} \mathbb{I}(z_t = z)(\mu^*_z - Y_t)\right]$

\subsection{Transferring Knowledge through Bounds}
\label{sec:transfer-bounds}
The agent is interested in the quantities $\mu_{k,z}$, the mean reward of arm $k$ under the partial context $z$, for all arms $k\in \mathcal{A}$ and contexts $z\in \mathcal{Z}$, to minimize her cumulative regret. As the oracle only plays an optimal arm after seeing the hidden context $u$, the log provided by the oracle is biased and thus $\mu_{k,z}$ can not be recovered from the log in general.
However, it is possible to extract non-trivial upper and lower bounds on the average $\mu_{k,z}$. In a binary reward and action setting, similar observations have been made in \cite{zhang2017transfer}. Alternative approaches to this problem, that include assuming bounds on the inverse probability weighting among others, are discussed in Section~\ref{sec: related work}.

Our assumption below is different, and in a setting where there are {\em more than two arms}. We specify that the logs have been collected using a policy that plays an {\em optimal arm} for each $(z,u)\in \mathcal{C}$, but do not explicitly impose conditions on the distributions. Instead, in Assumption~\ref{asn:gap}, we impose a separation condition on the {\em means} of the arms conditioned on the full context. We define:
\begin{definition}
	\label{defn:gap}
	Let us define $\underline{\delta}_z,\overline{\delta}_z$ for each $z\in \mathcal{Z}$ as follows:
	\small
	\begin{align*}
		\underline{\delta}_z \leq \min_{u \in \mathcal{U}} \left[ \mu^*_{z,u} - \max_{k \in \mathcal{A}, k \neq k_{z,u}^* } \mu_{k,z,u} \right],\\
		\overline{\delta}_z \geq \max_{u \in \mathcal{U}} \left[ \mu^*_{z,u} - \min_{k\in \mathcal{A}, k \neq k^*_{z,u} }\mu_{k,z,u} \right].
	\end{align*}
	\normalsize
\end{definition}

Thus, for each observed context $z,$ these quantities specify the sub-optimality gaps between the best and second best arms, as well as the best and the worst arms, respectively and which hold uniformly over the hidden contexts $u \in \mathcal{U}$.

\begin{assumption}[Separation Assumption]\label{asn:gap}
	The vectors $\{\underline{\delta}_z, \overline{\delta}_z : z\in \mathcal{Z}\}$ are provided as a part of the log. Additionally, for each $z\in \mathcal{Z}$, we have that $\underline{\delta}_z>0$.
\end{assumption}
\noindent\textbf{Remarks on Model and Assumption~\ref{asn:gap}:}

$\bullet~$\noindent\underline{\em Relation to Gap in Agent Space:} We note that these gaps in the \emph{latent space} do not allow us to infer the gap in {\em the agent space}. However, due to optimal play by oracle, these gaps help in obtaining mean bounds for \emph{all} arms, even those which have not been recorded in the log (Theorem~\ref{transfer of knowledge theorem}).

$\bullet~$\noindent\underline{\em Interpretation:}
The gap $\underline{\delta}_z$ gives us information about the hardest arm to differentiate in the latent space when the full context was observed ($\overline{\delta}_z$ analogously is for the easiest arm to discredit).
We also note that our approach can still be applied when the trivial bound $\overline{\delta}_z=1$, or universal bounds not depending on context $\overline{\delta} \geq \overline{\delta}_z$, $ \underline{\delta} \leq \underline{\delta}_z $ are used, leading to bounds that are not as tight.

$\bullet~$\noindent\underline{\em Motivating Healthcare Example:}
Often, hospital medical records contain information $z_t$ about the patient, the treatment given $k_t$, and the corresponding reward $y_t$ that is obtained (in terms of patient's health outcome). However, a \textit{good doctor} looks at some other information $u_t$ (that is not recorded) during consultation and prescribes the best action $k_t$ under the full context $(z_t,u_t)$. If one is now tasked with developing a machine learning algorithm (an agent) to automate prescriptions given the medical record $z$, this agent algorithm needs to find the best treatment $k(z)$ on average over $\mathbb{P}(u|z)$. Furthermore, the gaps on treatments effects can potentially be inferred from other data sets like placebo-controlled trials.

$\bullet~$\noindent\underline{\em Existing alternate assumptions:} Studies in \cite{yadlowsky2018bounds,zhao2019sensitivity} impose conditions on bounded sensitivity of the effects with respect to the hidden/unrecorded context (effectively, that this context does not significantly alter the effect). In our work, we explore the other alternative where the treatment has been chosen optimally with respect to the hidden/unrecorded context, and allows strong dependence on this hidden context.

\textbf{Quantities computed from log data:}\label{sec: data from logs}
The following quantities can now be computed by the agent from the observed log for each arm $k\in [K_0]$ and each visible context $z\in \mathcal{Z}$:

\textbf{1.} $p_z(k):$ The probability of picking arm $k$ under each context $z$ is denoted as $p_z(k)$. Mathematically, $p_z(k) = \sum\limits_{ u\in\mathcal{U}: k= k^*_{z,u}} \mathbb{P}(u|z).$

\textbf{2.} $\mu_z:$ The average reward observed under each context $z$ is defined as $\mu_z$. It can be computed by averaging observed rewards for all the entries with context $z$. This can be written as $\mu_z = \sum\limits_{u\in \mathcal{U}} \mu^*_{z,u} \mathbb{P}(u|z)$.

\textbf{3.} $\mu_z(k)$: The contribution from arm $k$ to the average reward $\mu_z$. Thus we have that $\mu_z(k) = \sum\limits_{u\in\mathcal{U}:k=k^*_{z,u}} \mu^*_{z.u}\mathbb{P}(u|z).$

\textbf{4.} $K_>(k,z)$: Finally, we identify the set of arms with "large rewards" $K_>(k,z) = \{k'\in[K_0]: k' \neq k, \mu_z(k') > \overline{\delta}_z p_z(k')\}$ for each $k\in [K]$ and $z\in \mathcal{Z}$.

\textbf{Bounds in terms of computed quantities:}
Using the quantities defined above, the following theorem describes how one can compute lower and upper bounds on the arm rewards in the agent space.

\begin{theorem}\label{transfer of knowledge theorem}
	The following statements are true for each $k\in [K_0]$ and $z\in \mathcal{Z}$:

	1. \textbf{Upper Bound:} $\mu_{k,z} \leq u_{k,z} := \mu_z - \underline{\delta}_z(1-p_z(k))$.

	2. \textbf{Lower Bound:} $\mu_{k,z} \geq l_{k,z} := \mu_z(k) + \sum\limits_{k' \in K_>(k,z)} \left[\mu_z(k')-\overline{\delta}_z p_z(k')\right]$.
\end{theorem}
Note that these bounds can be provided for all arms $k\in [K_0]$ and contexts $z\in \mathcal{Z}$. Specifically, bounds can also be extracted for the arms that are never played in the log. This comes as a result of the gaps defined in Definition \ref{defn:gap}.

\noindent\textbf{Remarks:}\\
\noindent{\em 1. \underline{Finite Log}:} While we study the case of infnite logs, our approach can be readily extended to finite logs by replacing the quantities $\mu_z, \mu_z(k), p_z(k)$ with empirical estimates and using standard Chernoff bounds to augment the bounds in Theorem \ref{transfer of knowledge theorem} to hold with a probability which is a function of the size of the log. Then, the performance of the learning algorithm is as in Corollary \ref{corr: uncertain bounds} and the discussion that follows.\\
\noindent{\em 2. \underline{Distribution shifts}:} Our results can also be used in the case where under a fixed visible context, the conditional distribution of the hidden contexts changes from oracle to the learner. In particular, for some $z\in \mathcal{Z}$, suppose that $\psi_z \geq \max_{u\in \mathcal{U}} |\mathbb{P}_o(u|z) - \mathbb{P}_l(u|z)|$, where $\mathbb{P}_o$ and $\mathbb{P}_l$ are used to differentiate the oracle and learner environments respectively. Using $\psi_z$, we can readily modify our results in Theorem \ref{transfer of knowledge theorem} to produce upper and lower bounds for arm rewards under the context $z$ when the distributions are no longer the same.

Now, we show the existence of instances for which our bounds are tight. We say an instance is {\em admissible} if and only if it satisfies all the statistics generated by the log data. The following proposition shows all the bounds defined in Theorem \ref{transfer of knowledge theorem} are partially tight.
\begin{proposition}[Tightness of Transfer]\label{tightness of transfer}
	For any log with $u_{k,z},l_{k,z}$ as defined in Theorem \ref{transfer of knowledge theorem} the following statements hold:\\
	1. There exists an admissible instance where upper bounds $\mu_{k,z} = u_{k,z}$, for all $k\in [K_0]$ and $z\in \mathcal{Z}$.\\
	2. For each $k\in [K_0]$, there exists a (separate) admissible instance such that $\mu_{k,z} = l_{k,z}$ for each $z\in \mathcal{Z}$.
\end{proposition}

Proofs of Theorem \ref{transfer of knowledge theorem} and Proposition \ref{tightness of transfer} are provided in the appendix and use conditioning on the latent context $u\in \mathcal{U}$ and along with the definitions of the computed quantities above to establish the results. The learning procedure is then carried out by using the bounds of Theorem \ref{transfer of knowledge theorem} to instantiate GLUE (Algorithm \ref{Algo1}, with regret as in Theorem \ref{UCB-SI theorem}) for each partial context $z \in \mathcal{Z}$ with $u_k = u_{k,z}$ and $l_k = l_{k,z}$.

\begin{figure*}[p]
	\centering
	\begin{subfigure}[b]{0.5\textwidth}
		\centering
		\includegraphics[width = 0.95\textwidth,height = 1.1in]{Academic}
		\caption{ Occupation: Academic}
		\label{fig:academic}
	\end{subfigure}%
	\begin{subfigure}[b]{0.5\textwidth}
		\centering
		\includegraphics[width = 0.95\textwidth,height = 1.1in]{Student}
		\caption{Occupation: Student}
		\label{fig:student}
	\end{subfigure}%

	\caption{\small Experiments on Movielens 1M: Arm rewards are normalized from [1,5] to [0,1]. True means and bounds are shown to the left and regret plots on the right. The captions show the Occupation (visible context). The top 17 movies (arms) were sub-sampled from over 3900 movies based on ratings, $l_{\max} \in [0.877,1]$ for all the plots.}
	\vspace{5pt}
	\label{fig: real}
\end{figure*}

\begin{table}[p]
	\centering
	\caption{Ground truth of instances in Figure \ref{fig: syn_transfer}. Mean rewards of arms and the probability of the hidden contexts are indicated. In the agent environment, Arm 2 is the best arm with mean 0.99145 and Arm 1 is the second best arm with mean 0.9593 (which is also the largest lower bound). }
	\label{table: synth stats}
	\begin{tabular}{|c|l|l|l|}
		\hline
		\textbf{Hidden Context} & {\bf Arm 1} & {\bf Arm 2} & {\bf Other arms} \\ \hline
		$u_1$,$\mathbb{P}(u_1)   = 0.475$ & 0.998 & 0.997 & 0.3 \\ \hline
		$u_2$,$\mathbb{P}(u_2)   = 0.475$ & 0.99  & 0.985 & 0.3 \\ \hline
		$u_3$,$\mathbb{P}(u_3)   = 0.05$  & 0.3   & 1     & 0.3 \\ \hline
	\end{tabular}
\end{table}

\begin{figure*}[p]
	\centering
	\begin{subfigure}[b]{0.5\textwidth}
		\centering
		\includegraphics[width = 0.95\textwidth,height = 1in]{Synth_trans_14_arms.png}
		\caption{14-armed instance}
		\label{fig: 14-armed synth transfer}
	\end{subfigure}%
	\begin{subfigure}[b]{0.5\textwidth}
		\centering
		\includegraphics[width = 0.95\textwidth,height = 1in]{Synth_trans_25_arms.png}
		\caption{25-armed instance}
		\label{fig: 25-armed synth transfer}
	\end{subfigure}%
	\caption{\small Synthetic Transfer Setup: Arm statistics are summarized in Table \ref{table: synth stats}. There is no visible context. The bounds on arm rewards as computed using Theorem \ref{transfer of knowledge theorem}. Arm 2 is the best arm in both cases, while Arm 1 is competitive. The benefits of GLUE can be seen as the number of suboptimal arms rise. }
	\vspace{5pt}
	\label{fig: syn_transfer}
\end{figure*}

\subsection{Experiments}
~~~~\textbf{1. Movielens 1M: }The Movielens 1M data set \citep{harper2015movielens} consists of over 1 million ratings of over 3900 movies by 6000 users. Each user is associated with attributes of age, gender and occupation. The reward matrix is completed using the SoftImpute algorithm \citep{mazumder2010spectral} using the fancyimpute package \citep{fancyimpute}. Occupation is treated as the visible context while age and gender are hidden. Users are then clustered based on attributes and confounded logs are then generated by dropping users who's ratings are too low for all movies. Upper and lower bounds are computed using Theorem \ref{transfer of knowledge theorem} after which the learning procedure is carried out. We omit OSSB in these experiments (from experience, the Bernoulli version of OSSB is extremely unstable). Regret is averaged over 100 independent runs. Additional information and plots are in the appendix.

\textbf{2. Synthetic Experiments: }We also present synthetic instances to show that the improvements we observed over B-kl-UCB in Section \ref{sec: bandits with mean bounds} are valid in the transfer setting as well. We present two instances in Figure \ref{fig: syn_transfer}. In both of these, there is no visible context, while there is one hidden context unknown to the learner. Arm 2 is optimal, while Arm 1 is highly competitive and displays a large upper bound. The remaining arms all have means set at $0.3$. The statistics of these instances are summarized in Table \ref{table: synth stats}. All rewards are Bernoulli and regret is averaged over 200 independent runs. More details about the setup are provided in the appendix.

\section{Discussions and Concluding Remarks}

In this work we studied a novel non-optimistic algorithm for bandit problems where bounds on arm means are provided as side information. Further, we also develop means to infer these bounds for contextual bandit problems in the presence of confounders, generalizing the results of \cite{zhang2017transfer}.

In section \ref{sec: bandits with mean bounds}, we introduce and analyze the GLUE algorithm and show (numerically) that asymptotically, our upper bounds can potentially improve over those of OSSB (and B-kl-UCB), which were thought to be optimal. We believe this to be a result of the manner in which the side information is leveraged by GLUE. The existing algorithms treat these bounds for each arm independently, while GLUE can improve exploration parameters of a chosen arm based on the bounds of another.

In section \ref{transfer section}, we study a practical setting of confounded logs, where a learner is presented historical data collected from an oracle. Here, non-trivial mean bounds can be extracted for all arms, even ones that have not been recorded in the log. We empirically validate our findings, both on the Movielens 1M data set as well as synthetic instances.

The improvements of GLUE over the existing state of the art can be realized in the contextual setting in specific instances. This is mainly because the lower bounds on a particular arm's rewards also depend on the statistics of other arms. As a result, our improvements are prominent when the number of arms grow. We also note that GLUE is more stable than OSSB (which can behave unpredictably even for Bernoulli rewards), and computationally more efficient than B-kl-UCB for general reward distributions. This makes GLUE an attractive alternative in applications with a large number of arms.

We also reiterate the two interesting avenues for future work: 1. To find valid regret lower bounds for the problem of bandits with mean bounds, 2. Algorithms to handle uncertain mean bounds. We show how GLUE can serve as a competitive baseline for the latter when there is sufficiently large certainty, but the former remains an important problem to address.

\section*{Acknowledgements}
This research was partially supported by NSF Grant 1826320, ARO grant W911NF-17-1-0359, US DOD grant H98230-18-D-0007, the Wireless Networking and Communications Group Industrial Affiliates Program, and the the US DoT supported D-STOP Tier 1 University Transportation Center.
\newpage
\bibliographystyle{plainnat}

redpajama_set_name: RedPajamaArXiv


text:
Live: Flight Brigade @ Notting Hill Arts Club
by Lauren Patel | Live Reviews, Photography
Flight Brigade @ Notting Hill Arts Club
Oliver Baines (Kalpesh Patel)
Hampshire-raised septet Flight Brigade chose the intimate Notting Hill Art Club's Nova presents…..Over & Over live music night to host the release party for their new EP Stealing Fire. Offering delicious cupcakes to those first through the door, the band ignited the party atmosphere at the packed famous west London venue on Wednesday evening.
Miriam Baines (Kalpesh Patel)
Formed in 2010, the folk/rock collective comprises a mixture of family and childhood friends. Lead singer Ollie Baines is married to Miriam who supports her husband with stunning harmonies. Miriam's sister Dorry Macaulay, who is a classically trained musician, exerts a special energy on vocals and violin. The remaining four members Tom Clay (bass), Thomas Pink (guitar), Neil Blandford (drums), and Jonny Barker (keyboard) help to fill the trendy basement venue's stage snuggly.
Dorry Macaulay (Kalpesh Patel)
The talented group of musicians' first four-track EP Graduation Day was released in 2013 and support spots with Dry the River, Stornaway and King Charles have followed.
Flight Brigade (Kalpesh Patel)
Their new EP Stealing Fire promises to raise the bar even higher. Released on the 23rd October to coincide with a 16-date UK autumn tour, the EP features four eclectic tracks When We Were Young, The Phantom, Housefire, and Our Friends Our Enemies which were mixed by Donald Clark (The Knife, Muse) and Brian Virtue (30 Seconds to Mars, Jane's Addiction).
The slot at Over & Over is the last gig of the tour before the EP release and the band are joined by friends to support the special moment. Ollie kicks off the 6-song set with old favourite 39 Steps, which creates an instant buzz before enthusing the crowd by declaring that "Wednesday's are the best day for gigs!"
The band then treat us to stand-out track Housefire from the new EP, which exerts a heavier rock influence than their earlier offerings. Dancing around the stage, the girls' vocal softens with feminine harmonies, whilst Dorry heightens the track's intensity with her talents on violin.
Before launching into the new single, Ollie thanks the crowd for joining the band at their EP launch show and then introduces The Phantom. The future disco anthem differs in style from the other tracks, but maintains the same blend of eccentricity that makes the band so unique. A sure pleaser, the synth-pop tune has everyone in the venue bopping around.
The new single When We Were Young takes the crowd from bop to bounce, showing off the fun side of the close-knit group, whilst maintaining their signature sound. A powerful chorus sees Ollie blasting out "giving it soul" from the tiny stage. BBC Radio 2's Janice Long has described them as 'astonishing' and featured When We Were Young as one of her Janice Loves tracks earlier this month.
Neil Blandford (Kalpesh Patel)
Baines announces the penultimate song, You Kill Me, as "something brand new and a world premiere!" A live first, the song delves into the difficulties of Crystal Meth addiction before concluding with a transcending instrumental from Dorry.
The band close their set with Our Friends Our Enemies, a new track about the bravery of World War II factory workers in Europe who sabotaged bomb detonators so they wouldn't go off. Starting off with radio recordings from the era whirring from the speakers, the crowd are transported back in time before returning forward to the present day with a spirited and energetic finale. Chants for "one more" are inevitable, and so the party continued.
The band are simply great to watch live. Their blend of musical tastes and styles come together to create an electric mix sitting somewhere between alternative folk and hard rock. These passionate performances earned them no less than 16 festival slots this summer, including Bestival and playing three stages at Glastonbury. The band's debut album, with tracks co-produced by Gavin Monaghan (Editors, Kings of Leon) will be out in early 2016. The future is very bright for the extremely talented Flight Brigade.
Flight Brigade play various UK venues between now and the end of November 2015.
Interview with the band by Lauren Patel to follow shortly.
Review by Lauren Patel and Photography by Kalpesh Patel. Flight Brigade at Notting Hill Arts Club 21st October 2015. Kalpesh has his great Flickr stream here: https://www.flickr.com/photos/somethingforkate


text:
Q: CASE statement based on SUM results in query I have a query where I am using SUM to total up bill detail values. I would like to create a CASE statement that works off the results of that sum, rather than the values being summed. Below is a partial example of what my query currently looks like.
SELECT
    bill.billnumber_id,
    billdetail.billclass,
    SUM(billdetail.amount) AS Amount
GROUP BY bill.billnumber_id, billdetail.billclass

Say this totals 4 lines of billdetail (2, 0, 5, 3) for a summed amount of 10. I would like to do a CASE statement based on the 10 amount, rather than the individual values. Thanks.

A: Is this what you want?
SELECT bill.billnumber_id, billdetail.billclass, SUM(billdetail.amount) AS Amount,
       (case when SUM(billdetail.amount) > 10 then 'BigOne'
             else 'Little'
        end)
GROUP BY bill.billnumber_id, billdetail.billclass

A: You need to use a sub-query:
SELECT CASE WHEN Amount = 0 THEN 'Nothing' ELSE 'somthing' END AmountComment
FROM(
  SELECT bill.billnumber_id, billdetail.billclass, SUM(billdetail.amount) AS Amount
  GROUP BY bill.billnumber_id, billdetail.billclass
)x

A: You can either make it a subquery:
SELECT billnumber_id, billclass, CASE WHEN Amount = 10 THEN ... ELSE ... END
FROM
(
  SELECT bill.billnumber_id, billdetail.billclass, SUM(billdetail.amount) AS Amount
  GROUP BY bill.billnumber_id, billdetail.billclass
) A

Or just repeat the aggregation:
SELECT bill.billnumber_id, billdetail.billclass, SUM(billdetail.amount) AS Amount
     , CASE WHEN SUM(billdetail.amount) = 10 THEN ... ELSE ... END
GROUP BY bill.billnumber_id, billdetail.billclass

A: SELECT bill.billnumber_id, billdetail.billclass, (Select SUM(billdetail.amount) from billdetail)
AS Amount
from bill inner join billdetail on bill.id = buildetail.kBillId
/* include from join (not added above)*/
GROUP BY bill.billnumber_id, billdetail.billclass

I believe this is what you are after... you might wanat to include the same from structure in your inner select statement as:
 SELECT bill.billnumber_id, billdetail.billclass, (Select SUM(billdetail.amount) from bill inner
join billdetail on bill.id = buildetail.kBillId ) AS Amount
from bill inner join billdetail on bill.id = buildetail.kBillId
/* include from join (not added above)*/
GROUP BY bill.billnumber_id, billdetail.billclass

Let me know if this is what you meant.

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