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De Flying Tigers of de First American Volunteer Group (AVG) opereerde als deel van de Chinese Luchtmacht in 1941-1942. AVG stond onder leiding van Claire Lee Chennault. De piloten waren eerst civiele vrijwilligers, maar na presidentiële goedkeuring mochten piloten van de Amerikaanse luchtmacht, marine en mariniers ook deelnemen.

In april 1937 accepteerde Claire Lee Chennault een opdracht voor drie maanden om in China de luchtmacht door te lichten. Hij was net vertrokken bij de United States Army Air Corps, waar hij de rang van kapitein had. Voor Chiang Kai-shek had hij veel kennis van luchtmachtzaken die van belang waren in de strijd tegen het Japans Keizerlijk Leger waarmee China sinds 1937 in oorlog verkeerde. Zijn verblijf werd verlengd en in de zomer van 1938 vertrok hij naar Kunming om een nieuwe Chinese luchtmacht vorm te geven.

Door de aanhoudende Japanse aanvallen werd Chennault in het najaar van 1940 naar de Verenigde Staten gestuurd met de opdracht gevechtsvliegtuigen en bommenwerpers te kopen en piloten te werven. Er waren plannen voor drie groepen, twee met gevechtsvliegtuigen en een uitgerust met bommenwerpers. Alleen de plannen voor de eerste groep zijn uitgevoerd. Curtiss was bereid 100 vliegtuigen van het type Curtiss P-40 Warhawk te leveren. Deze waren besteld door de Britten, maar Curtiss bood de Britten een nieuwer model aan waardoor Chennault alsnog het oudere model vliegtuig kon kopen. Dit oudere model had nadelen, maar de vliegtuigen waren duidelijk opgewassen tegen de Japanse gevechtsvliegtuigen.

De groep bestond uit drie squadrons van elk ongeveer 30 gevechtsvliegtuigen. De vliegtuigen werden naar Rangoon in Birma getransporteerd. Hier werden ze in gereedheid gebracht en trainde de bemanning. In 1941 stond Birma nog onder Brits bestuur en de Verenigde Staten was nog niet in staat van oorlog met Japan.

De groep vrijwilligers maakte officieel deel uit de Chinese luchtmacht. De vrijwilligers sloten contracten af met salarissen variërend van US$ 250 per maand voor een monteur tot US$ 750 voor een squadroncommandant, ongeveer drie keer meer dan ze konden verdienen in het Amerikaanse leger. Hoewel AVG civiele vrijwilligers accepteerde bestond het grootste deel van haar personeel uit vliegeniers van het Amerikaanse leger.

Van de drie squadrons vertrokken er twee naar China en een bleef achter om de Britten te helpen met de verdediging van de stad. China had hier ook belang bij, in de haven van Rangoon kwam veel militair materiaal aan bestemd voor de Chinese strijdkrachten. Deze goederen kwamen over de Birmaweg naar China. Op 20 december 1941 kwam de AVG voor het eerst in actie boven China. Ze vielen een groep van 10 Japanse bommenwerpers aan, ze vernietigden er negen terwijl een eigen toestel neerstortte. Op 23 december kwam de AVG samen met de Royal Air Force in actie bij Rangoon. Tot en met nieuwjaarsdag schoten de vliegtuigen van de AVG boven Birma zeker 75 Japanse toestellen neer, tegen een eigen verlies van zes toestellen en twee piloten. De reputatie van de Flying Tigers werd hiermee gevestigd. Ondanks deze successen werd Birma door Japanse troepen bezet.

De piloten van de AVG bleven in actie in het westen en oosten van China. Op 4 juli 1942 werd de AVG ontbonden en werd onderdeel van de reguliere Amerikaanse luchtmacht. In de zeven maanden van haar bestaan had ze 299 Japanse vliegtuigen zeker vernietigd en nog eens 153 waarschijnlijk. De eigen verliezen bestonden uit 73 P-40's, waarvan 61 op de grond waren vernietigd. Er kwamen in totaal 23 piloten om en werden er drie gevangen genomen.

Chennault kwam weer in dienst van het Amerikaanse leger en bleef verantwoordelijk voor de luchtoperaties in China. Veel vrijwilligers vertrokken, ze kregen andere functies in Azië, gingen terug naar de Verenigde Staten of vochten verder in andere regio's.

Flying Tigers

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Ian Chee
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Mandhana among four players nominated for ICC Women's Cricketer of the Year award
Dubai : India women's team cricketer Smriti Mandhana is among four players who have been nominated for the prestigious Rachael Heyhoe Flint Trophy for ICC Women's Cricketer of the Year 2021 on Friday.
In a field dominated by opening batters, the other three cricketers are England's Tammy Beaumont, Australia's Lizelle Lee and Ireland's Gaby Lewis.
With 855 runs in 22 internationals, at an average of 38.86 with one century and five half-centuries, the year gone by has seen the 25-year-old Mandhana touch new heights despite the country's cricketers enduring difficult circumstance.
In the limited-overs series against South Africa where India won just two of the eight matches at home, Mandhana played a major role in both of the wins. She scored an unbeaten 80 as India chased down 158 in the second ODI that helped them level the series and scored an unbeaten 48 in the win in the final T20I.
Mandhana scored 78 in the first innings of the one-off Test against England that ended in a draw. She played an important knock of 49 in India's only win in the ODI series. Her 15-ball 29 and fifty in the T20I series went in vain, though, as India fell short in both the matches and lost the series 2-1.
Mandhana was in good touch in the series against Australia, starting with the ODI series where she scored 86 in the second ODI. She scored a brilliant century in the only Test, the first of her career, and was awarded the 'Player of the Match'. She scored her second T20I fifty of the year in the final T20I, though India fell short and lost the series 2-0.
During the pink-ball Test Down Under, Mandhana made the game memorable by smashing her maiden century in the longest format of the game. She played her natural game, and scored a run-a-ball half-century but played with caution under the lights.
She was caught when she was 80 but Ellyse Perry had overstepped, giving her a reprieve. She made the most of the lifeline and got her first Test century in style with a boundary. Her innings finally came to an end at 127, after putting India in a strong position. The match ended in a draw and Mandhana was declared the Player of the Match.
With 872 runs in 21 internationals at an average of 48.44, England's Tammy Beaumont finished as the highest run-getter in the calendar year and is a leading contender for the prestigious award. Lizelle Lee of Australia — with 864 runs in 19 internationals at an average of 57.6 with one century and seven half-centuries — and Ireland's Gaby Lewis — 624 runs in 15 internationals at an average of 52 with one century and four half-centuries – are the other contenders.
Tags: Smriti Mandhana
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Creating a financial plan after divorce is one action item I strongly encourage people to do. It's a plan for supporting themselves and their children and it's a plan because it's more than just a budget. It's about setting financial goals, taking stock of assets and debts, figuring out a budget, and knowing your income. And because divorce often changes the financial picture dramatically, a financial plan needs several variations: a now, a right after the divorce and a five year time frame.
I didn't really ask for spousal support because it didn't make a bit of difference to me, I was still going to get my life on track, with or without any spousal support. My attorney asked for it and to me, it's a little extra money. Not much, and I've already decided that money is going to go into a savings account. When the two years are up, that total amount, I'm going to do something nice with it. I'm not even going to be spending it now. The child support, that's going to be used to take care of me and my daughter to enjoy ourselves, when she's with me.
I'm still moving forward as I did before any of this happened, with building my business and my dream. I've got to make a life for myself and not everybody is going to be able to have spousal support or have child support, so no matter what, I've still got to move forward and do what I've got to do.
My spousal support isn't anything huge, that's why I said I would and save it for now, because I can make more than that in my own business. As far as half the retirement, my ex wanted to keep it in his 401(k) fund, and then I would get to draw on my half of it when he draws on his half of it, and I said, "No." I said I wanted to be in control of my money.
I do have a good friend who's a financial planner, and she's going to help me set up something to put my money in. I'm having to draw a little bit out of it but for the most part, I'm setting up my own retirement fund.
All of this goes back to what I'm trying to teach single mothers: when you become financially independent in your life, first of all you don't quickly get married for the wrong reason, and second of all, you can go into a marriage with that part of your life being whole. I think a part of that independence is knowing that you can invest your money, and how to put your money to good use.
To be honest, I've been able to take from this experience so much more knowledge that I am able to share now with my group, to say, "These are the steps that you need to start taking, this is what you need to start doing." In a way, all of this that's happened has been a gift. It hasn't been pretty, it hasn't been fun in the least, but it has been a gift, and I've seen how it's been a gift. I've already developed a workshop and I'm writing the last chapter of my book because everything's done now, it has an ending. I see a lot of good fruit coming out of this in the end, so in some ways I thank him, but at the same time, I wouldn't have wanted to go through it.
Marjorie has her feet firmly on the ground here – while both spousal support and child support are legal obligations, the reality is that payments are frequently late, not paid in full or even not paid at all. If you can build your life without being dependent on this money, you'll still be able to manage when payments are missed. That may not be realistic to start with but if you know you'll only be receiving spousal support for two years, then you have two years to increase your income or reduce your expenses so won't be short when the spousal support period is over.
I also agree with Marjorie about removing her share of the retirement assets from her ex's 401(k). Leaving the money in her ex's fund, leaves her exposed to future risks over which she has no control … what if he changes employer and moves the money somewhere else? What if he takes a loan against the fund? What if the company he works for goes into bankruptcy and that somehow impacts the payout from the fund? Would he consult with her on where her funds were invested? What happens to the growth in the intervening years? Too many what-ifs.
Not only does it make sense to take direct control over the monies, it's also a cleaner psychological break. Given the level of conflict in Marjorie's divorce, the fewer connections Marjorie has to maintain with her ex, the better it's going to be.
What do you think? Are there circumstances where you would wait for a payout?
This is the last segment in Marjorie's update and again, I'd like to thank her for bravely sharing her story.
I agree – the more financially independent you can become, the more control you will have over how you choose to spend your money. Therefore, you should start divorce planning seeking the advice of both an attorney and a certified financial planner before you engage in the divorce process.
I also recommend that anyone considering divorce should consult with an attorney to find out their legal rights and potential issues but know that it is always necessary to meet with a financial adviser. I would imagine it would depend on the level of assets involved. If you and your spouse already have a financial adviser I would caution against discussing the potential for divorce with him/her – I'd be concerned that it may inadvertently get back to your spouse.
The first few articles were somewhat helpful however the info given to find help I found as well….help from these places is not as forthcoming as the reader is led to believe.
That she so easily found an attorney the was willing to represent her is an anomaly…a fortuitous one for (the author of this article & the other articles).
I have had experience with Legal Aide in San Diego & all I received was false hope, wasted 8 hours of my time & immense frustration. The staff was uninformed, required information that concerned me due to it compromising personal security that would have had no bearing on the issue I sought help for. And they spoke to me as if I were mentally disabled, until I realized it was not my ignorance of legal matters but the staff members utter ignorance regarding the legal process that consisted of basic comprehension of legal terminology, what kind of legal representation would best fit my needs, & his shocking lack of how to fill out the forms HE claimed all applicant MUST fill out. He & the other 3 staff members, 2 of which claimed to be attorneys currently licensed in California!
This woman was no where near any of that and to elicit sympathy from her readers drawing them in with those title that grab you but when you read the actual article it deflates the intrigue then causes mind numbing anger that insults far too many strong, talented, beautiful women whose only crime was to have married a narcissist creep that used her for years until her resources dried up for one reason & one reason only: he EMBEZZLED every dime she worked to earn, support her family, invest, & save. His mask came off the day he stole her last dollar, then he goes on a smear campaign to claim she's crazy, he's been abused by her, & he is her victim so he had to run for safety. He plays his victim role perfectly while she is literally destitute, homeless can get NO HELP from any organization that Ms. Bostwicks' gives.
Plus, this average woman would not dramatize her situation to grab the attention of REAL VICTIMS of those narcissists AND employees of those crappy free clinics, legal aide offices,or social services. The ONLY people who recurve help are women that have access to the entitled wealthy relatives or accounts with funds they are not able to access easily or family members & friends of our government agencies that are SUPPOSED TO provide free assistance to American CITIZENS at a time in their life things have become unmanageable or people they trust & love have used & abused her so badly for so long she has given up on herself, others, & life because of so much deception & greed.
I also give up. Unless you can adjust your moral code, integrity, honesty, compassionate heart & living with the ability to help others, love unconditionally, you end up being maimed & sacrificed because you are unwilling to become like the masses have & you will not take advantage of those you love.
Ms. Bostwicks', why don't you research REAL WOMAN who have good hearts, who are in REAL DIRE STRAITS WITH NO HOPE OF RECOVERING? I for one have given up after giving & caring for & loving unconditionally, only to find out 3 of my loved ones took off their masks 2 years ago & despite having nothing left other than a small suitcase & a fraction of my household belongings in storage, they come around occasionally to torment me, steal more, & tell whoever will listen how much they "love" me. 3 of my adult children are traumatized, & we are fractured, adrift, & have little hope left.
I cray when I awaken each morning, I hate seeing the sun I cannot stand knowing this time I can't get up.


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⟨Theatre|Science⟩
New Plays
Theatre Awards
Theatre Fesitvals
Actors and Performers
Science Plays
Q&A with Director Michael Longhurst
Written by Science Centre Stage in Biology, Ethics, Genetics, Interviews, Medicine, New Science-in-Theatre Production, Philosophy of Science, Physics, Politics of Science
A Number, Caryl Churchill, Constellations, John Shrapnel, Lee Curran, Lex Shrapnel, Michael Longhurst, Nick Payne, Royal Court Theatre, Tom Scutt, Young Vic
Michael Longhurst has directed many critically acclaimed productions, including Nick Payne's Constellations in the West End and Broadway. Longhurst is an Associate Director at Southampton's Nuffield Theatre and his new version of Caryl Churchill's A Number for Nuffield recently transferred to London's Young Vic Theatre. Science Centre Stage spoke to Michael Longhurst about his views on science in theatre and on how he worked with designer Tom Scutt on the remarkable stage set for A Number.
Michael Longhurst
A Number was first produced in 2002 at a time when cloning was very much in the public conscience (not least because of 'Dolly the Sheep'). What stimulated you to direct A Number again now?
It's a play that I had read at drama school and had fallen in love with at that moment. I think the line-by-line writing that Carol makes the scenes out of is extraordinary. I studied philosophy at university and I think the thematic ideas of the play are really interesting. They go beyond and above the idea of cloning and into the idea of personal identity, and beyond and above the idea of nature and nurture and into the idea of freewill and determinism. She is so erudite in packing in so many ideas into a very short, punchy play and it's an incredibly exciting theatrical premise.
Science is a key enabler in understanding who we are, who we are now and who we are becoming.
Fundamentally, we watch an actor pretend to be more than one version of himself, which is absolutely the most basic part of acting and theatre. It allows us to access the idea of a clone with the same genetics but who behaves differently. I love the simplicity, theatricality and complexity of the ideas it draws on. We were looking for an exciting project for Southampton University (where The Nuffield is based). It has a lot of specialization in biomedicine, so we hoped that it would appeal to some of the audience there. I was working with designer Tom Scutt, who also designed Constellations, and we were excited about the design opportunity to really push a conceptual version of this play in how we staged it. The strength of reaction is one of the major factors that has brought it to life again and brought it into town (at the Young Vic).
John and Lex Shrapnel in A Number, Young Vic Theatre 2015, Credit: Johan Pehrsson
In rehearsals we were looking at the progress of the genetic world and interestingly as science advances there's nothing that makes the play obsolete. The ideas that Caryl Churchill puts forward in the play are still pressing and they become more pertinent the more familiar we become with genetic advances and possibilities. Crudely, we haven't yet cloned a human so the play's not yet out of date. It is still a play that is asking- 'what if ?' The fact that we are more aware of the advances, such as being able to edit the human genome means that out idea of it as something that is in the dim and distant future in some dystopian world is being eroded. Our world is getting closer to this and I think that makes the ethical questions of the play present more pressing.
In developing the production did you talk to scientists in the similar manner to the way that you and Nick Payne did with Constellations?
Interestingly, the play never mentions the word cloning. The characters, certainly the clones, aren't aware of this process. So in some sense it wasn't something that the characters in the play necessarily understood. It was an opportunity that was offered to the father and that he benefits from and it's a complete shock and surprise to the son. We did work in rehearsals to try to understand how cloning works, but actually we didn't go out to meet scientists who are experts in their field because the characters themselves have a lay understanding of cloning, which we supplemented by research in the rehearsals.
How do you go about the process of putting together ideas with stage designer Tom Scutt?
Whenever I enter a design process it feels like there's what I call top-down and bottom-up work. Top down work asks what are the themes that we are trying to capture? How can we use metaphor? How can we create an environment that has an emotional resonance with what we're trying to say about the play? And then there's the practical stuff- the bottom-up stuff – which is what does the play need? A Number actually needs nothing. With two bodies and no props, it frees you from many stage constraints. We decided at Nuffield that we wanted to give the audience a new experience, so we created an installation. We weren't doing a traditional proscenium arch production of it. We were allowed to play with capacity, and make it an incredibly intimate experience. Tom hit on the idea of using mirrors, which felt like he had very cleanly, profoundly and simply hit upon both the ideas of identity – who are we? – and the idea of a multiplicity of reflections. Often as a director your job is to have these thematic discussions with a designer, and then be brave when they offer you an exciting solution. In a piece of live theatre the actors and audience are sharing a space, and in this instance I put them behind a glass wall and used microphones. But I hope that in addition to the ideas of reflection and identity, it also gives the audience a feeling of watching an interrogation. It's not dissimilar to when you go to a police line-up through a two-way mirror. I think what that does is tie into the ethical issues, the idea of responsibility, the idea of our agency, the idea of guilt, blame and consequence. I think that all of those were useful social ideas to bring up in a stage design.
Often as a director your job is to have these thematic discussions with a designer, and then be brave when they offer you an exciting solution.
Do you think there is complementarity with Jennifer Haley's play 'The Nether' which literally depicts an interrogation in a near-future science fiction scenario?
Yes, absolutely. Both are taking out technological capabilities and pushing them a little bit further, looking at how humanity will respond if we are able to do those things. And do we like how humanity could respond? And therefore do we want our society to go in that direction? I think that's the value of all "sci-fi work" is that it allows you to examine the society that we are in through looking at a society we're not in.
There are productions of A Number that have used test tubes a lot and very heavy scientific aesthetic and I think potentially that might have been interesting when the play first came out – when there was a sort of hype and horror around cloning. Actually what we wanted to do was push our aesthetic into a slightly different place. I put the idea of an interrogation room, which is more about responsibility and 'blame'.
Did recent public debate and legislative changes surrounding mitochondrial donation influence this new production of A Number?
These advances are having huge gestalt shifts in our thinking, the idea of a three parent family, or the idea of being able to edit our genes are huge. The fact that we can't achieve it at a certain level doesn't mean conceptually we're not on a certain pathway. It seems like as soon as we acknowledge the possibility of editing the genome, then we're able to correct genetic diseases but we're also a step nearer to eugenics. That is an important thing that we need to be thinking about.
Do you think the success of plays such as Constellations and A Number indicates a growing place for science in the cultural life?
I have directed predominantly new writing. I love theatre that is exploring who we are and science is a key enabler in understanding who we are, who we are now and who we are becoming. I think a writer who uses science to explore and answer that question can often tap into hugely exciting, revelatory and challenging areas of our humanity. I think as a theatre maker when you have a play that is exploring science, often formally – as is in the case of Constellations and actually in the case of A Number, both of those plays have the form of the play, or at least the theatrical experience dictated by the science. In A Number you've got one actor playing several clones. He literally embodies the act of cloning – but it also makes very good theatre. And in Constellations the form of the play was repeated versions of the multiverse. It's an exciting theatrical provocation for an audience. In the design there's lots of potential for metaphor. All of these things appeal to me as I'm interested in analytical ways of thinking. I guess it's a personal preference but I think it's a really valuable branch of theatre.
Would you argue that science in theatre works best when the science informs the structure rather than the didactic content?
Absolutely. If we're trying to understand scientific principles there are probably much deeper lectures that one could go to, essays or journals that one could read or documentaries that one could watch. But I think what theatre can do is dramatize emotional consequences of what the science is. It can help us have an emotional understanding and ask the ethical question. Constellations poses what does it feel like to be in a multiverse? Actually, what that does is it makes you realize that I can't access these other parallel universes, so my choices in this one are all the more precious. And that is the emotional feeling of the play.
I think A Number asks if we could do these things then what would be the ramifications be? What does it mean – this idea that man is born equally? Well genetically he's not. If we are given these opportunities, how can they be used or abused?
A Number, directed by Michael Longhurst, runs at London's Young Vic until 15th August.
The Royal Court Theatre tour of Constellations is at Trafalgar Studios until 1st August.
Science Centre Stage on Twitter
Michael Frayn Speaking to Kate Mosse at Chichester Festival Theatre
Certain to Please? – The Uncertainty Principle in Theatre 2017
Interview: Director Matthew Xia on Blue/Orange at the Young Vic
Highlights from a Year of Science in Theatre 2015
Director Poppy Burton-Morgan Serves Up Science-Inspired Food for Thought
Actor Edward Bennett on Playing Francis Crick in Photograph 51
A Dramatic Experiment: Science on Stage – Video of Panel Discussion
Latest Science Theatre News
Leading Big Science: Oppenheimer Fuses History and Office Politics
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Evolutionary Biology in The Hard Problem
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Oppenheimer and a New Stoppard Play for 2015
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A House Divided: Menagerie Theatre's Latest Science Collaboration
Australia, New Zealand and South Africa – Science in the Sun
A Payne In The Head Draws To An End
HeLa Success Continues to Replicate
Incognito is a Conspicuous Hit
Henrietta Lacks Perfomance on World Tour
Copernicus Stage Premiere
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Best Actor Nomination for BBC Radio Copenhagen
Festival of Science Plays in Florida
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Life of Galileo to Tour in 2014
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text:
New genes causing ALS recently discovered by Project MinE researchers
An international research team including Slovenian scientists has discovered new genes, whose pathological variants increase the risk of developing amyotrophic lateral sclerosis (ALS). This discovery opens up new possibilities for finding drugs that would be focused on researching the revealed biological processes, said the Jožef Stefan Institute representative.
Amyotrophic lateral sclerosis (ALS) is a fatal neurodegenerative disease with a lifetime risk of one in 350 people and an unmet need for disease-modifying therapies. Project MinE is a large-scale research initiative devoted to discovering the genetic cause of ALS. As it is almost certain that the disease has a genetic basis, the ultimate goal of the researchers for the last eight years is to identify genes that are associated with ALS. In 2016, a new C21orf2 gene was reported to increase the risk of developing this disease. A group of researchers from 22 countries has recently conducted a cross-ancestry genome-wide association study (GWAS) including 29,612 patients with ALS and 122,656 controls, which identified 15 risk loci while also confirming previously identified risk genes. Being the largest genome-wide association study (GWAS) of ALS so far, the study points toward a framework for drug discovery in ALS and also explores similarities between ALS and more common neurodegenerative diseases. Moreover, the paper reveals an elevated-risk loci expression in relevant tissue and activation of integrated stress response pathways, such as autophagy. It has also provided evidence that ALS glutamatergic neuronal damage is cell autonomous. Finally, a role for cholesterol in ALS was causally implicated.
A paper describing their research was published on December 6 in Nature Genetics. The study was led by Jan Veldink and Leonard van den Berg from the University Medical Center Utrecht in the Netherlands. Boris Rogelj from the Jožef Stefan Institute and the Faculty of Chemistry and Chemical Technology of the University of Ljubljana, Metka Ravnik-Glavač, Damjan Glavač and Blaž Koritnik from the Faculty of Medicine, University of Ljubljana and Janez Zidar from the Institute of Clinical Neurophysiology, University Medical Center Ljubljana are all the study co-authors.
More than 200,000 people worldwide are living with Amyotrophic Lateral Sclerosis (ALS), otherwise known as Motor Neuron Disease (MND) or Lou Gehrig's disease. Relatively little is known about the cause of this progressively degenerative neurological disease. The average life expectancy of ALS patients is three years. There is still no cure and still no effective treatment for ALS. The Project MinE dates back to 2013 when it was initiated by ALS patients in order to financially support researchers to discover the different genetic mutations that can be connected with ALS.
Congratulations to all the authors!


More groups have hooked up with the Senatorial candidacy of the Senatorial candidate of the Action Democratic Party ADP, for Osun West Senatorial District, Omooba Dotun Babayemi come February 16th election.
This came to fore at the final of the Second edition of Omooba Dotun Babayemi Youth Football Competition among football teams in Osun West senatorial district held at St. Anthony School, Oke-Bode, Iwo, where Rovet FC of Iwo defeated Hon.Sakor FC of Ede by 3goals to nil.
The event witnessed the presence of prominent personalities including the national leader of the Action Democratic Party, Alh.Moshood Adeoti,Vice Chairman of the SDP in Osun State,Mr.Folaranmi Hamzat,as well many dignitaries in the senatorial district.
Addressing the event, Mr. Folaranmi Hamzat said that members of the SDP in Osun West Senatorial District decided to pitched their tent with Omooba Dotun Babayemi's senatorial ambition because of their strong conviction that he is the only candidate in the senatorial district capable of representing the people effectively.
While enjoining electorate in the West Senatorial District to support the Gbongan,Ayedaade Local Government born prince, Hamzat and other speakers at the programme commended the philanthropic action of Omooba Dotun Babayemi and advice others to emulate him.
The final of the Dotun Babayemi football competition witnessed splendid display of football artistry by the two finalists.However,Rovet FC proofed to be the better side by dictating the phase thereby leaving Hon.Sakor FC of Ede scampering for breathing space.
All the three goals scored by Rovet FC of Iwo came via the boot of the teams prolific striker, Nurudeen Aremu, who went home with a cash prize, courtesy of Alh.Moshood Adeoti, for the yeoman's job.
Earlier in the categorisation play-off, Al Imam FC of Iwo walloped TJ FC of Ede by 3 goal to 1.
The winner, Rovet FC went home with a cash prize of half a million naira, while the runners up,Hon.Sakor FC of Ede were #300,000 richer. The third place winner went home with #250,000.
Highlight of the well attended event was the presentation of a 'MERITORIOUS GOLD AWARD OF EXCELLENCE IN GRASSROOTS SPORTS DEVELOPMENT ' to Omooba Dotun Babayemi by the Sports Writers Association of Nigeria SWAN, Osun State chapter for his unrelented and support for the development of sports at the grassroot.


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redpajama_set_name: RedPajamaGithub


text:
\subsubsection*{\bibname}}

\input{macros}

\usepackage{multirow}
\usepackage{graphicx}

\begin{document}

\title{Learning from Multiple Corrupted Sources, \\ with Application to \\
Learning from Label Proportions}
\author{ Clayton Scott and Jianxin Zhang \\
University of Michigan \\ Electrical Engineering and Computer Science }
\maketitle

\begin{abstract}
We study binary classification in the setting where the learner is presented with multiple corrupted training samples, with possibly different sample sizes and degrees of corruption, and introduce an approach based on minimizing a weighted combination of corruption-corrected empirical risks. We establish a generalization error bound, and further show that the bound is optimized when the weights are certain interpretable and intuitive functions of the sample sizes and degrees of corruptions. We then apply this setting to the problem of learning with label proportions (LLP), and propose an algorithm that enjoys the most general statistical performance guarantees known for LLP. Experiments demonstrate the utility of our theory.
\end{abstract}

\section{INTRODUCTION}

We study the problem of binary classification in the setting where the learner does not have access to a conventional training data set with correctly labeled instances. Instead, the learner has access to several data sets for which the true labels have been randomly corrupted, with each data set having possibly different sample size and degree of corruption. Previous work has considered learning from a single corrupted data set, but the problem considered here raises the natural question of how best to aggregate and weight the information from these multiple corrupted data sets according to the sample size and degree of corruption. We extend the method of corruption corrected losses \citep{natarajan18jmlr} to this setting and establish a generalization error bound for kernel-based predictors. By optimizing this bound, we obtain a precise and interpretable scheme for aggregating the various corrupted sources according to the degree of corruption.

We then apply our framework to the problem of learning from label proportions (LLP), which is another weak supervision setting for binary classification. In this problem, training data come in the form of bags. Each bag contains unlabeled feature vectors (patterns) and is annotated with the proportion of patterns arising from class 1. We argue that this problem can be reduced to the first problem studied, and apply our results to obtain the most general theoretical analysis of this problem to date.

\subsection{Related Work and Contributions}

Our work is in the setting of classification with label-dependent label noise.
Prior work has focused on a single noisy dataset. \citet{blanchard16ejs} introduce a way to estimate the label noise proportions, while \citet{natarajan18jmlr} introduce a loss-correction framework when the noise proportions are known (or can be estimated). There have also been recent efforts to develop  noise-robust losses for training neural networks \citep{patrini17dnn,sabuncu18nips}.

\citet{rooyen18jmlr}, who extend the method of \citet{natarajan18jmlr} to other weakly supervised settings, considered the question of learning from multiple independent sources. Like us, they also prove a generalization error bound for the weighted empirical risk based on multiple corruption-corrected losses. However, their weights are fixed and depend only on the sample sizes, and they assume the corrected losses are bounded, which precludes many common losses. Relative to this work, our contribution is to prove a bound that holds for Lipschitz losses and arbitrary weights, with the optimized weights depending on both the sample sizes and degrees of corruption in an intuitive way.

The problem of learning a classifier from multiple corrupted datasets arises in some applications such as crowdsourcing and learning from multiple weak labeling functions \citep{ratner16nips}. In these applications, the corrupted datasets often arise from different annotators assigning labels to the same unlabeled datasets. This leads the datasets to be dependent, whereas our analysis assumes independence. However, if unlabeled data are very abundant as envisioned by \cite{ratner16nips}, independent unlabeled datasets could be provided to the different annotators, thus fitting our assumption.

LLP is motivated by a number of real-world applications including particle physics \citep{komiske18}, vote prediction \citep{sun17}, and image classification \citep{ding2017}. Several algorithms have been proposed in recent years, and we refer the reader to the recent summary in \citet{arnold19tr}.

Statistical analysis of LLP has been studied by a handful of papers. \citet{quadrianto09jmlr} reduce LLP to estimation of a certain mean operator, and use Rademacher complexity to establish uniform convergence guarantees for this estimate. \citet{nolabelnocry} extend the work of \citet{quadrianto09jmlr} in several ways, including a generalization error bound for LLP based on a notion of bag-Rademacher complexity, and present two algorithms for approximately optimizing the associated bag-empirical risk. Neither of these works establish consistency guarantees for a classification performance measure, and both focus on linear methods. \citet{yu15tr} study the algorithm of minimizing the ``empirical proportion risk," which seeks a classifier that best reproduces the bag label proportions. Bounds on the accuracy of the resulting classifier are established, but only under two restrictive assumptions, namely, that the bags are all very pure (label proportions close to 0 or 1) or that the feature space is finite. Relative to these works, our contributions are to establish distribution-free generalization error analysis and universal consistency with respect to a classification performance measure.

Our work may also be viewed as a novel setting for multi-source domain adaptation, where the target is the clean distribution. Typically, domain adaptation methods are classified as supervised, semi-supervised, or unsupervised, depending on whether the feature vectors from the target distribution are fully labeled, partially labeled, or unlabeled \citep{pan:10:kde}. In our case, however, we do not even assume access to unlabeled data from the target (clean) distribution.

Finally, we note that our framework yields a generalization error bound and consistency for the problem of classification with feature dependent label noise \citep{menon18,cannings18,scott19alt}, which has not previously been addressed. This follows by taking each corrupted data set to consist of a single point. We elaborate below.

\subsection{Notation and Terminology}

A binary classification loss function, referred to simply as a {\em loss} in this work, is a function $\ell:\mathbb{R} \times \{-1,1\} \to [0,\infty)$. We say a loss $\ell$ is $L$-Lipschitz if for every $y \in \{-1,1\}$, and every $t, t' \in \mathbb{R}$, $|\ell(t,y)-\ell(t',y)| \le L|t - t'|$. We say a loss is Lipschitz if it is $L$-Lipschitz for some $L$. We say a loss is a {\em margin loss} if it has the form $\ell(t,y) = \varphi(yt)$ for some $\varphi:\mathbb{R} \to [0,\infty)$.

A decision function is a measurable function $f:{\mathcal X} \to \mathbb{R}$. The classifier induced by a decision function $f$ is the function $x \mapsto \mathop{\mathrm{sign}}(f(x))$. We will only consider classifiers induced by a decision function. In addition, we will often refer to a decision function as a classifier, in which case we mean the induced classifier.

Finally, define the probability simplex $\Delta^N := \{w \in \mathbb{R}^N \, | \, w_i \ge 0 \, \forall i, \text{ and } \sum_i w_i =1\}$.

\subsection{Outline}

Sec. \ref{sec:mult} presents our generalization error analysis for learning from multiple corrupted sources, and Sec. \ref{sec:llp} applies it to LLP. Discrimination rules are discussed in Sec. \ref{sec:discrim}, while Sec. \ref{sec:exp} present experiments illustrating our theory. Sec. \ref{sec:disc} offers some concluding comments.

\section{LEARNING FROM MULTIPLE CORRUPTED SOURCES}
\label{sec:mult}

We first review the work of \citet{natarajan18jmlr} before presenting two settings for learning from multiple corrupted sources.

\subsection{Learning from A Single Corrupted Source}

Let ${\mathcal X}$ be the feature space and $P$ a probability distribution on ${\mathcal X} \times \{-1,1\}$, which shall denote a clean/uncorrupted distribution. Let $\rho = (\rho^{+},\rho^{-}) \in [0,1]^2$ be such that $\rho^{-} + \rho^{+} < 1$, referred to as {\em label noise proportions}. Given $P$ and $\rho$, we define a new probability distribution $P^{\rho}$ on ${\mathcal X} \times \{-1,1\}$ as follows. To generate a realization $(X,\tilde{Y})$ from $P^{\rho}$, first draw a realization $(X,Y)$ from $P$, and then let the conditional distribution of $\tilde{Y}$ given $Y$ be
\begin{align*}
\tilde{Y} \, | \, Y = 1 \ & \sim \ \text{Ber}(1 - \rho^+) \\
\tilde{Y} \, | \, Y = -1 \ & \sim \ \text{Ber}(\rho^-).
\end{align*}
In words, the noisy label $\tilde{Y}$ is obtained by randomly corrupting the true label $Y$ in a manner depending on the value of $Y$, but not on $X$, such that $\rho^+$ is the probability that a $+1$ is flipped to a $-1$, and $\rho^-$ is the probability that a $-1$ is flipped to a $+1$. Below we use $p(\tilde{y} | y ; \rho)$ to denote this conditional distribution of $\tilde{Y}$ given $Y = y$, with noise proportions $\rho$.

Let $\ell:\mathbb{R} \times \{-1,1\} \to [0,\infty)$ be a loss. Following \citet{natarajan18jmlr}, given label noise proportions $\rho = (\rho^{+},\rho^{-})$, define the {\em corrected loss} $\ell^{\rho}$ by
\begin{align*}
\ell^{\rho}(t,1):= \frac{(1 - \rho^{-})\ell(t,1) - \rho^+ \ell(t,-1)}{1 - \rho^{-} - \rho^{+}} \\
\ell^{\rho}(t,-1):= \frac{(1 - \rho^{+})\ell(t,-1) - \rho^- \ell(t,1)}{1 - \rho^{-} - \rho^{+}}.
\end{align*}
The corrected loss (also referred to as an unbiased loss) debiases the affects of noisy labels as captured by the following result.

\begin{lemma}
\label{lem:corrected}
With $\ell, \rho$ and $\ell^\rho$ as in the preceding paragraph, for any $y \in \{-1,1\}$ and $t \in \mathbb{R}$,
$$
\mathbb{E}_{\tilde{Y} \sim p(\tilde{y} | y; \rho)}[\ell^\rho(t,\tilde{Y})] = \ell(t,y).
$$
Additionally, consider any distribution $P$ on ${\mathcal X} \times \{-1,1\}$ with associated corrupted distribution $P^\rho$. If $f$ is a decision function such that $\ell(f(\cdot),y)$ is bounded for all $y$, then
$$
\mathbb{E}_{(X,\tilde{Y})\sim P^\rho}[\ell^\rho(f(X),\tilde{Y})] = \mathbb{E}_{(X,Y)\sim P}[\ell(f(X),Y)].
$$
\end{lemma}

The first part of the lemma was establish by \citet{natarajan18jmlr}, and the second part follows as an immediate corollary, where the boundedness assumption ensures the existence of the expectation. Given noisy data $(X_1,\tilde{Y}_1),\ldots,(X_n,\tilde{Y}_n) \sim P^\rho$, this result immediately suggests learning a classifier by empirical risk minimization over the noisy data,
$$
\widehat{f} = \operatornamewithlimits{arg\ min}_{f \in {\mathcal F}} \ \frac1{n} \sum_{j=1}^n \ell^\rho(f(X_j),\tilde{Y}_j),
$$
where ${\mathcal F}$ is some class of decision functions. \citet{natarajan18jmlr} show that the generalization error, i.e., the difference between the  corrupted empirical risk and the uncorrupted true risk, is bounded uniformly over ${\mathcal F}$, in terms of the Rademacher complexity of ${\mathcal F}$.

Below, we extend the above framework to the setting where there are multiple contaminated sources with different degrees of corruption.

\subsection{Learning from Multiple Corruptions of a Common Source}
\label{sec:common}

We now consider the problem of learning from multiple corrupted sources. In this section, we assume observations are drawn from one of several distributions, all of which are corruptions of a common clean distribution $P$. Our goal is to learn a decision function $f$ to minimize the risk
\begin{equation}
\label{eqn:risk}
{\mathcal E}_{P}(f) := \mathbb{E}_{(X,Y)\sim P} [\ell(f(X),Y)].
\end{equation}
The clean distribution is only observed through several corrupted samples. Thus, let $\rho_i = (\rho_i^{+},\rho_i^{-})$, $i=1,\ldots,N$ be $N$ pairs of label noise proportions, and let $P^{\rho_1}, \ldots, P^{\rho_N}$ be the associated corrupted distributions. For each $i$, let  $S_i = ((X_{ij}, \tilde{Y}_{ij}))_{1 \le j \le n_i}$ be $n_i$ iid draws from $P^{\rho_i}$, and assume the samples $S_1,\ldots,S_N$ are themselves independent.

To learn from this data, we propose to minimize a weighted sum of empirical risks
$$
\sum_{i=1}^N w_i \left[\frac1{n_i} \sum_{j=1}^{n_i} \ell^{\rho_i}(f(X_{ij}),\tilde{Y}_{ij}) \right]
$$
where $w \in \Delta^N$ is a tuning parameter.

Intuitively, samples $S_i$ with less corruption should receive larger weights. We confirm this intuition by establishing a generalization error bound, and then showing that the bound is indeed optimized by assigning larger weights to less corrupted samples.

To obtain an explicit bound, we specialize to kernel algorithms. Thus, let $k$ be a symmetric, positive definite (SPD) kernel, and let $B_k(R)$ denote the ball of radius $R>0$, centered at the origin, in the reproducing kernel Hilbert space (RKHS) associated to the kernel. Assume the kernel is bounded by the constant $K$. For example, in the case of the Gaussian kernel $k(x,x') = \exp(-\gamma \|x - x'\|^2)$, $K=1$.

For all generalization error bounds in this paper, we assume that $\ell$ is a {\em margin loss}, that is, $\ell(t,y) = \varphi(yt)$ for some $\varphi: \mathbb{R} \to [0,\infty)$. This assumption could be dropped at the expense messier expressions.
\begin{thm}
\label{thm:gen}
Let $\ell$ be an $L$-Lipschitz margin loss and $k$ an SPD kernel bounded by $K$. Let $P$ be a distribution on ${\mathcal X} \times \{-1,1\}$, and for each $i$, let $\rho_i = (\rho_i^+, \rho_i^-) \in [0,1]^2$ such that $\rho_i^- + \rho_i^+ < 1$. For all $R > \varphi(0)/KL$, $w \in \Delta^N$, and $0 < \delta \le \frac14$, we have
with probability at least $1-\delta$ with respect to the draws of the corrupted samples $S_1, \ldots, S_N$,
\begin{equation}
\label{eq:gen}
\sup_{f \in B_{k}(R)} \abs{\sum_{i=1}^N w_i \left[ \frac1{n_i} \sum_{j=1}^{n} \ell^{\rho_i}(f(X_{ij}),\tilde{Y}_{ij}) \right] -  {\mathcal E}_{P}(f)} \\
\leq 4 K R L \sqrt{\sum_{i=1}^N \frac{w_i^2}{n_i} \left( \frac{1+|\rho_i^+ - \rho_i^-|}{1-\rho_i^- - \rho_i^+} \right)^2  \left(\frac{\log (2/\delta)}{2} \right) }.
\end{equation}
\end{thm}

To interpret the bound, first consider the case where $n_i = n$ for all $i$, and $w_i = 1/N$. The bound becomes
$$
\frac{4 K R L}{\sqrt{Nn}}\sqrt{\frac1{N}\sum_{i=1}^N \left( \frac{1+|\rho_i^+ - \rho_i^-|}{1-\rho_i^- - \rho_i^+} \right)^2 \frac{\log (2/\delta)}{2}}
$$
which tends to zero provided {\em either} of $N$ or $n$ tends to $\infty$ (but not necessarily both), and $1-\rho_i^- - \rho_i^+$ remains bounded away from zero.

However, uniform weights do not optimize the bound. Returning to the setting where the $n_i$ are arbitrary, introduce the notation
\begin{equation}
\label{eqn:nsr}
c_i = \frac1{\sqrt{n_i}} \cdot \frac{1 + |\rho_i^+ - \rho_i^-|}{1 - \rho_i^- - \rho_+}.
\end{equation}
Define $1/c_i^2$ to be the ``signal-to-noise" ratio (SNR) for the $i$th sample, where $n_i$ reflects the ``signal"  while $(1 + |\rho_i^+ - \rho_i^-|)^2/(1 - \rho_i^- - \rho_+)^2$ captures the overall amount of noise. This notion of overall noise is smaller whenever $\rho_i^+$ and $\rho_i^-$ are small and when the label noise is {\em symmetric} \citep{kamalaruban18tr}.

It can easily be shown using Lagrange multiplier theory that the $w$ minimizing
\begin{equation}
\label{eqn:quadbnd}
\sum_{i=1}^N w_i^2  c_i^{2}
\end{equation}
subject to $w \in \Delta^N$ is $w_i = c_i^{-2}/\sum_j c_j^{-2}$, and that the optimal value of (\ref{eqn:quadbnd}) is
$\frac1{N}H(c_1^2, \ldots, c_N^2),$
where $H$ denotes the harmonic mean. The harmonic mean is dominated by the smallest $c_i$, with large ones having considerably less impact. In contrast, if we selected uniform weights $w_i = 1/N$, the harmonic mean is replaced by the arithmetic mean, $A(c_1^2, \ldots, c_N^2) = \frac1{N} \sum_i c_i^2$, which is much more sensitive to large values of $c_i$ (samples with very low SNR). As an illustration, suppose $N=10$, $n_i = n = 100$, and for $i < N$, $\rho_i^+ = \rho_i^- = 0.01$, while $\rho_N^+ = \rho_N^- = 0.49$. Then the ratio of arithmetic mean to harmonic mean exceeds 100.

In summary, the optimal weight $w_i$ is proportional to the signal-to-noise ratio of the $i$th sample. Thus, samples with larger sample size, less total label noise, and more balanced label noise, are preferred.

{\bf Additional remarks:}
{\bf (1)} Theorem \ref{thm:gen} and the other generalization error bounds below are all special cases of a master theorem provided in the supplemental material. The proof of the master theorem is based on an extension of Rademacher complexity analysis to a novel weighted Rademacher complexity. The factor $L\frac{1 + |\rho_i^+ - \rho_i^-|}{1 - \rho_i^- - \rho_+}$ in Thm. \ref{thm:gen} is the Lipschitz constant of $\ell^{\rho_i}$. In the event of a single sample with no noise, we recover known bounds \citep{mohri12} up to minor differences.

{\bf (2)} In the problem of binary classification with feature dependent label noise, there is a single dataset where each label is corrupted with probability $\rho^+(x)$ (if $Y=1$) or $\rho^-(x)$ (if $Y=-1$), where $x$ is the feature vector. This problem can be viewed as the special case where $n_i = 1$ and $\rho^{\pm}_i = \rho^{\pm}(x_i)$. Thus, Theorem \ref{thm:gen} applies to this setting provided the corruption probability functions $\rho^{\pm}(x)$ are known which, unfortunately, is rarely the case.

{\bf (3)} For the generalization error bound to imply a consistent learning procedure, it must converge to zero. We note one asymptotic setting where the bound does converge to zero for the optimized weights, but not for uniform weights. This is the setting where $N$ is fixed, and some but not all of the sample sizes $n_1, \ldots, n_N$ tend to $\infty$. To see that the optimized bound vanishes, just realize that the optimized weights lead to a smaller bound than when the $w_i$ are uniform on the samples with diverging sizes.

\subsection{Learning from Multiple Corrupted Sources with Varying Class Priors}
\label{sec:ber}

We now assume each corrupted source is associated to its own clean distribution, where the different clean distributions differ only in their class prior probabilities. In particular, let $P_1, \ldots, P_N$ denote the $N$ clean distributions, and let $\pi_i = P_i(Y=1)$. All $P_i$ are assumed to have the same class-conditional distributions, $P_+$ and $P_-$, which are two fixed distributions on ${\mathcal X}$. This setting will be used to study LLP below.

In this setting it makes sense to look at a performance measure that depends only on the class-conditional distributions $P_+$ and $P_-$. Thus, we will adopt as our performance measure the balanced error rate (BER),
$$
{\mathcal E}_{\pm}(f):= \frac12 \left( \underset{X\sim P_+}{\mathbb{E}}[\ell(f(X),1)] + \underset{X\sim P_-}{\mathbb{E}}[\ell(f(X),-1)] \right)
$$
The BER may be related to a cost-sensitive risk as follows. Let $\a = (\a^+,\a^-) \in \mathbb{R}_+^2$. Given a loss $\ell$, define the associated cost-sensitive loss
$$
\ell_\a(t,y):= \a_+ \ind{y=1} \ell(t,1) + \a_- \ind{y=-1} \ell(t,-1).
$$
Given a distribution $P$ on ${\mathcal X} \times \{-1,1\}$, define the cost-sensitive risk with costs given by $\a$,
$$
{\mathcal E}_{P,\a}(f) := \mathbb{E}_{(X,Y)\sim P} \ell_\a(f(X),Y).
$$
If $\pi$ is the prior class probability of $P$, and $\a = (\frac1{2\pi},\frac1{2(1-\pi)})$, then it can easily be shown that
$$
{\mathcal E}_{P,\a}(f) = {\mathcal E}_{\pm}(f).
$$

We seek to learn a classifier minimizing ${\mathcal E}_{\pm}(f)$ given corrupted samples from each of $P_1, \ldots, P_N$. Thus, let $\rho_i = (\rho_i^{+},\rho_i^{-})$, $i=1,\ldots,N$ be $N$ pairs of label noise proportions, and let $P_1^{\rho_1}, \ldots, P_N^{\rho_N}$ be the associated corrupted distributions. For each $i$, let  $S_i = ((X_{ij}, \tilde{Y}_{ij}))_{1 \le j \le n_i}$ be $n_i$ iid draws from $P_i^{\rho_i}$, and assume  $S_1,\ldots,S_N$ are themselves independent.

We again propose to minimize a weighted sum of empirical risks
$$
\sum_{i=1}^N w_i \left[\frac1{n_i} \sum_{j=1}^{n_i} \ell_{\a_i}^{\rho_i}(f(X_{ij}),\tilde{Y}_{ij}) \right]
$$
where $\a_i = (\frac1{2\pi_i},\frac1{2(1-\pi_i)})$ and $\ell_{\a_i}^{\rho_i}$ denotes $(\ell_{\a_i})^{\rho_i}$, which is {\em not} the same as $(\ell^{\rho_i})_{\a_i}$. Adopting the same kernel setting as above, we have the following.
\begin{thm}
\label{thm:gen2}
Let $\ell$ be an $L$-Lipschitz margin loss, and let $k$ be an SPD kernel bounded by $K$. Let $P_+$ and $P_-$ be two probability distributions on ${\mathcal X}$, and for $i=1,\ldots, N$, let $P_i$ be the distribution on ${\mathcal X} \times \{-1,1\}$ having $P_+$ and $P_-$ as class-conditional distributions and $\pi_i$ as class prior. Set $\a_i = (\frac1{2\pi_i},\frac1{2(1-\pi_i)})$. For all $R > 2\varphi(0)/KL$, $w \in \Delta^N$, and $0 < \delta \le \frac14$, we have
with probability at least $1-\delta$ with respect to the draw of the corrupted samples $S_1, \ldots, S_N$,
\begin{equation*}
\sup_{f \in B_{k}(R)} \abs{
\sum_{i=1}^N w_i \left[ \frac1{n_i} \sum_{j=1}^{n} \ell_{\a_i}^{\rho_i}(f(X_{ij}),\tilde{Y}_{ij}) \right] -  {\mathcal E}_{\pm}(f)} \\
\le C \sqrt{\sum_{i=1}^N \frac{w_i^2}{n_i} \left( \frac{1+|\rho_i^+ - \rho_i^-|}{1-\rho_i^- - \rho_i^+} \right)^2  \frac1{\min(\pi_i,1-\pi_i)^2}  },
\end{equation*}
where $C = 2 K R L \sqrt{\frac{\log (2/\delta)}{2}}$.
\end{thm}

Similar remarks apply to this result as were made in the previous section. The optimized weights are still proportional to an appropriately defined SNR, updated to include the class priors, where now there is a preference for more balanced samples, i.e., $\pi_i \approx \frac12$.

\section{Learning from Label Proportions}
\label{sec:llp}

In learning from label proportions, the learner is presented with several  pairs $(B_1,\gamma_1), \ldots, (B_M, \gamma_M)$. Each $B_i$ is a {\em bag} of $m_i$ unlabeled instances, and each $\gamma_i \in [0,1]$ is the proportion of instances from class 1 in the bag. The goal is to learn an accurate classifier as measured by some performance measure. Our approach is to frame LLP in the setting of Section \ref{sec:ber}, and so we adopt the BER as our performance measure. This choice is already a departure from prior work on LLP, which adopts a risk as the performance measure. The BER has the advantage of not being sensitive to changes in class prior at test time.

To reduce LLP to the setting of Section \ref{sec:ber}, we pair the bags, and view each pair of bags as a corrupted sample. Thus, we re-index the bags and let $(B_i^-,\gamma_i^-)$ and $(B_i^+,\gamma_i^+)$ constitute the $i$th pair of bags, and assume $\gamma_i^- < \gamma_i^+$. Also assume the total number of bags is $M=2N$, so that the number of pairs of bags is $N$.
We associate to every $X_{ij}$ in $B_i^-$ the noisy label $\tilde{Y}_{ij} = -1$, and to every $X_{ij}$ in $B_i^+$ the noisy label $\tilde{Y}_{ij}=1$. Finally, let $S_i$ denote the set of all pairs $(X_{ij},\tilde{Y}_{ij})$, $j=1,\ldots,n_i$ from the $i$th pair of bags.

\subsection{Probabilistic model for paired bags}

We propose a probabilistic model that connects LLP to the problem of learning from corrupted sources as described in Section \ref{sec:ber}. Let $P_+$ and $P_-$ be two probability distributions on ${\mathcal X}$.

First, we define a joint distribution $Q_i$ on a realization $(X,\tilde{Y})$ from the $i$th pair of bags as follows:  Assume $\tilde{Y} \sim \text{Ber}(\frac12)$, and that the conditional distribution of $X$ given $\tilde{Y}$ is
\begin{align*}
X &\sim \gamma_i^+ P_+ + (1-\gamma_i^+) P_- \qquad \text{given $\tilde{Y} = 1$}\\
X &\sim \gamma_i^- P_+ + (1-\gamma_i^-) P_- \qquad \text{given $\tilde{Y} = -1$}.
\end{align*}
The conditional distributions of $X | \tilde{Y}$ are consistent with $\gamma_i^+$ and $\gamma_i^-$ being the label proportions for the two bags, making this a reasonable model for LLP. The label proportions themselves are viewed as fixed and nonrandom\footnote{However, it is easy to extend our analysis to the setting where the bag proportions are also random, in which case the above model is conditioned on the $\gamma_i^{\pm}$. In this case, the joint distribution of bag proportions can be arbitrary as long as $\gamma_i^- < \gamma_i^+$ almost surely. The results are proved by first conditioning on, and the integrating out, the $\gamma_i^{\pm}$.}.

Next, we will construct a clean distribution $P_i$, and label noise proportions $\rho_i = (\rho_i^+, \rho_i^-)$, such that $Q_i = P_i^{\rho_i}$. These label {\em noise} proportions are not to be confused with the label proportions $\gamma_i^+, \gamma_i^-$ that annotate the bags, and in fact the former will be defined in terms of the latter. Let $P_i$ be the joint distribution of $(X,Y)$ whose class conditional distributions are $P_+$ and $P_-$, and whose prior class probability is
$$
\pi_i := \frac{\gamma_i^+ + \gamma_i^-}2.
$$
Also, define the label noise proportions $\rho_i = (\rho_i^+, \rho_i^-)$ according to
$$
\rho_i^+ := \frac{\gamma_i^-}{\gamma_i^- + \gamma_i+}, \qquad
\rho_i^- := \frac{1 - \gamma_i^+}{2 - \gamma_i^- - \gamma_i^+}.
$$
A simple calculation shows
\begin{equation}
\label{eqn:rhogam}
1-\rho_i^- - \rho_i^+ = \frac{\gamma_i^+ - \gamma_i^-}{(\gamma_i^+ + \gamma_i^-)(2 - \gamma_i^+ -\gamma_i^-)}
\end{equation}
which ensures that $\rho_i^- + \rho_i^+ < 1$, because $\gamma_i^+ - \gamma_i^- > 0$ by assumption.

We claim that $Q_i = P_i^{\rho_i}$. First,  consider the distribution of $\tilde{Y}$ under $P_i^{\rho_i}$. We have
\begin{equation*}
\Pr(\tilde{Y} = 1) = \pi_i (1-\rho_i^+) + (1-\pi_i) \rho_i^- \\
= \frac{\gamma_i^- + \gamma_i^+}2 \cdot \frac{\gamma_i^+}{\gamma_i^- + \gamma_i^+} + \frac{2 -\gamma_i^- - \gamma_i^+}2 \cdot \frac{1 - \gamma_i^+}{2 - \gamma_i^- - \gamma_i^+},
\end{equation*}
which equals $\frac12$ in agreement with $Q_i$. Next, consider the conditional distribution of $X$ given $\tilde{Y}$ under both $Q_i$ and $P_i^{\rho_i}$. To show that these agree, it suffices to show $\Pr(Y=1 \, | \, \tilde{Y}=1) = \gamma_i^+$ and $\Pr(Y=1 \, | \, \tilde{Y}=-1) = \gamma_i^-$. This follows from
\begin{equation*}
\Pr(Y=1 \, | \, \tilde{Y} = 1) = \frac{\Pr(\tilde{Y} = 1 \, | \, Y=1) \Pr(Y=1)}{\Pr(\tilde{Y} = 1)} \\
= \frac{(1-\rho_i^+)\frac{\gamma_i^- + \gamma_i^+}{2}}{\frac12}
= \frac{\gamma_i^+}{\gamma_i^- + \gamma_i^+} (\gamma_i^- + \gamma_i^+)
= \gamma_i^+,
\end{equation*}
and the other case is similar.

\subsection{A generalization error bound for LLP}

From the preceding discussion, we may view each pair of bags as a corrupted sample, and since each $P_i$ has the same class-conditional distributions, we are in the setting of Section \ref{sec:ber} and may deduce the following result. The result is actually obtained by applying the master theorem in the appendix, which allows us to save a factor of 3 in the leading constant compared with applying Theorem \ref{thm:gen2} directly.
\begin{thm}
\label{thm:gen4}
Let $\ell$ be an $L$-Lipschitz margin loss, and let $k$ be a SPD kernel bounded by $K$. Let $P_+$ and $P_-$ be two probability distributions on ${\mathcal X}$. Let $S_1, \ldots, S_N$ be pairs of bags described by the construction and probabilistic model above. For each $i$, let (a) $\gamma_i^- < \gamma_i^+$ be the label proportions for the $i$th pair of bags, (b) $\pi_i = \frac{\gamma_i^+ + \gamma_i^-}2$, (c) $P_i$ be the distribution on ${\mathcal X} \times \{-1,1\}$ with $P_+$ and $P_-$ as class-conditional distributions and $\pi_i$ as the class prior (d) $\a_i = (\frac1{\gamma_i^+ + \gamma_i^-},\frac1{2-\gamma_i^+ - \gamma_i^-})$, and (e) $\rho_i = (\frac{\gamma_i^-}{\gamma_i^- + \gamma_i^+}, \frac{1 - \gamma_i^+}{2 - \gamma_i^- - \gamma_i^+})$. For all $R > \varphi(0)/KL$, $w \in \Delta^N$, and $0 < \delta \le \frac14$, we have
with probability at least $1-\delta$ with respect to the draw of the corrupted samples $S_1, \ldots, S_N$,
\begin{equation*}
\sup_{f \in B_{k}(R)} \abs{
\sum_{i=1}^N w_i \left[ \frac1{n_i} \sum_{j=1}^{n_i} \ell_{\a_i}^{\rho_i}(f(X_{ij}),\tilde{Y}_{ij}) \right] -  {\mathcal E}_{\pm}(f)} \\
\le 4 K R L \sqrt{\sum_{i=1}^N \frac{w_i^2}{n_i (\gamma_i^+ - \gamma_i^-)^2} \left(\frac{\log (2/\delta)}{2} \right)} .
\end{equation*}
\end{thm}

\subsection{Optimal pairing of bags}
\label{sec:pair}

The bound above is minimized by selecting weights
$$
w_i \propto n_i (\gamma_i^+ - \gamma_i^-)^2,
$$
which gives preference to pairs of bags where one bag is mostly +1's (large $\gamma_i^+$) and the other is mostly -1's (small $\gamma_i^-$). With these weights, the bound becomes
$$
4 K R L \sqrt{\left( \sum_{i=1}^N n_i (\gamma_i^+ - \gamma_i^-)^2 \right)^{-1} \frac{\log (2/\delta)}{2} }.
$$

We can also optimize the pairing of bags by further optimizing the bound.
Consider the unpaired bags $(B_i, \gamma_i)$, $i=1,\ldots, 2N$, and assume all unpaired bags have the same size\footnote{We make this assumption because for our probabilistic model, bags in a pair have the same (expected) size.}. We would like to pair each bag to a different bag, forming pairs $(\gamma_i^+, \gamma_i^-)$, such that
$$
\sum_{i=1}^N (\gamma_i^+ - \gamma_i^-)^2
$$
is maximized. For each $i < j$, let $u_{ij}$ be a binary variable, with $u_{ij}=1$ indicating that the $i$th and $j$th bags are paired. The optimal pairing of bags is given by the solution to the following integer program:
\begin{align*}
\max_{u}  & \ \ \sum_{1 \le i < 2N} \sum_{i < j \le 2N} (\gamma_i - \gamma_j)^2 u_{ij} \\
 \text{s.t.} & \ \ u_{ij} \in \{0,1\}, \forall i,j \\
& \ \  \sum_{i < j} u_{ij} + \sum_{j < i} u_{ji} = 1, \forall i
\end{align*}
The equality constraint ensures that every bag is paired with precisely one other distinct bag. This problem is known as the ``maximum weighted (perfect) matching" problem, and in our experiments we solve it using the algorithm of \citet{edmonds1965maximum}.

\section{Discrimination Rules}
\label{sec:discrim}

The bounds above all show that a weighted empirical risk of the form
$$
\widehat{\sE}(f) = \sum_i \frac{w_i}{n_i} \sum_j (\ell_{\a_i})^{\rho_i} (f(X_{ij}),\tilde{Y}_{ij})
$$
is, with high probability, uniformly close to an error quantity ${\mathcal E}(f)$ ($={\mathcal E}_P(f)$ or ${\mathcal E}_{\pm}(f)$ depending on context) over a ball in a RKHS. This suggests a discrimination rule based on regularized, weighted empirical risk minimization (ERM):
\begin{equation}
\label{eqn:discrule}
\widehat{f}:= \operatornamewithlimits{arg\ min}_{f \in {\mathcal H}_k} \ \widehat{\sE}(f) + \lambda \| f \|_{{\mathcal H}_k}^2,
\end{equation}
where $\lambda \ge 0$ and ${\mathcal H}_k$ is the RKHS associated to $k$. In this section we discuss statistical and computational aspects of this learning rule.

\subsection{Consistency}

An algorithm is consistent if the sequence of random variables ${\mathcal E}(\widehat{f})$ (where the randomness arises from the training data) converges asymptotically to ${\mathcal E}^* := {\mathcal E}(f^*)$, where $f^*$ is a global minimizer of ${\mathcal E}(f)$.
To prove consistency using our generalization error bounds,
 two things must happen asymptotically: (1)
$B_k(R)$ can approximate $f^*$ to arbitrary accuracy as $R \to \infty$, and (2) the generalization error bound tends to zero. (1) is ensured by taking $k$ to be a universal kernel on a compact domain \citep{steinwart08}. (2) Requires balancing the sample sizes with a shrinking value of $\lambda$, which implicitly determines $R$.
These ideas are captured in the following general result that covers all three settings discussed above. We say that the problems in Secs. \ref{sec:common}, \ref{sec:ber}, and \ref{sec:llp} satisfy the {\em bounded noise condition} if $1-\rho_i^+ - \rho_i^-, (1-\rho_i^+ - \rho_i^-)\min(\pi_i,1-\pi_i)$, and $\gamma_i^+ - \gamma_i^-$ are bounded away from zero, respectively.
\begin{thm}
\label{thm:llpconsist}
Let ${\mathcal X}$ be compact and let $k$ be a universal kernel on ${\mathcal X}$. Let $\lambda$ be such that $\lambda \to 0$ and $\lambda (n_1 + \cdots + n_N)/\log(n_1 + \cdots + n_N) \to \infty$ as $n_1 + \cdots + n_N \to \infty$. For each of the settings in Secs. \ref{sec:common}, \ref{sec:ber}, \ref{sec:llp}, under the bounded noise condition, and with optimized weights,
\begin{equation}
    {\mathcal E}(\widehat{f}) \to {\mathcal E}^*
\end{equation}
in probability as $n_1 + \cdots + n_N \to \infty$.
\end{thm}
It is important to note that the condition $n_1 + \cdots + n_N \to \infty$ covers a wide range of asymptotic regimes. It essentially requires that at least one of $n_1, \ldots, n_N$ or $N$ tend to infinity. In particular, consistency is possible even though some sample sizes may remain bounded.

This consistency result is universal in the sense that it makes no assumptions on $P_-, P_+$ other than compact support. In the case of LLP, the only prior work to consider consistency makes much stronger distributional assumptions \citep{yu15tr}.

\subsection{Calibrated Losses}

The consistency discussion above applies to the loss function that is used in the algorithm. This loss is typically chosen for computational properties such as convexity, whereas the true loss of interest may be the 0-1 loss. Fortunately, we can ensure consistency wrt the 0-1 loss simply by choosing $\ell$ to be classification calibrated which, in the case of a margin loss $\varphi$, requires that it be differentiable at 0 and $\varphi'(0) < 0$ \citep{bartlett06}. This decoupling of calibration and noise correction is one nice feature of the method of \citet{natarajan18jmlr}.

\subsection{Convexity}

We say that the loss $\ell$ is {\em convex} if, for each $y$, $\ell(t,y)$ is a convex function of $t$. \citet{natarajan18jmlr} establish the following result, where $\ell''$ denotes the second derivative of $\ell$ with respect to its first variable.
\begin{prop}
\label{prop:conv1}
Let $\ell$ be a convex, twice differentiable loss satisfying
\begin{equation}
\label{eqn:second}
\ell''(t,1) = \ell''(t,-1)
\end{equation}
for all $t \in \mathbb{R}$. Then for any $\rho = (\rho^+,\rho^-) \in [0,1]^2$ satisfying $\rho^+ + \rho^- < 1$, $\ell^\rho$ is convex.
\end{prop}
This result follows easily from the definition of the corrected loss. Examples of losses satisfying the second order condition include the logistic, Huber, and squared error losses.

This result immediately implies that the weighted average of losses in Section \ref{sec:common}, $\sum_i \frac{w_i}{n_i} \sum_j \ell^{\rho_i}(t,\tilde{Y}_{ij})$ is a convex function of $t$, provided $\ell$ satisfies \eqref{eqn:second}. The result is less helpful in the case of Section \ref{sec:ber}, because cost-sensitive losses $\ell_\a$ do not generally satisfy \eqref{eqn:second}. In the setting of LLP, however, the choices of $\rho_i$ and $\a_i$ play together nicely and allow us to establish the following.

\begin{prop}
\label{prop:conv2}
Let $\ell$ be a convex, twice differentiable loss satisfying
\begin{equation}
\label{eqn:second2}
\ell''(t,1) = \ell''(t,-1)
\end{equation}
for all $t \in \mathbb{R}$. For $1 \le i \le N$, let $\gamma_i^+, \gamma_i^-, \a_i^+, \a_i^-, \rho_i^+$ and $\rho_i^-$ be as in the statement of Theorem \ref{thm:gen4}, and let $n_i^{\pm} := |\{j : \tilde{Y}_{ij} = \pm 1\}|$. Then
$$
J(t):=\sum_i \frac{w_i}{n_i} \sum_j (\ell_{\a_i})^{\rho_i} (t,\tilde{Y}_{ij})
$$
is a convex function of $t$ provided
\begin{equation}
\label{eqn:convsuff}
\sum_i \frac{w_i}{n_i(\gamma_i^+ - \gamma_i^-)} \Bigg[ n_i^+ \Bigg(\frac12 - \gamma_i^-\Bigg) + n_i^- \Bigg(\gamma_i^+ - \frac12\Bigg) \Bigg] \ge 0.
\end{equation}
If, in addition, $\ell''(t,y) > 0$ for all $t,y$, then \eqref{eqn:convsuff} is also necessary for convexity of $J(t)$.
\end{prop}

The condition in \eqref{eqn:convsuff} is satisfied if either (a) for all $i$, $\gamma_i^- \le \frac12$ and $\gamma_i^+ \ge \frac12$, or (b) for all $i$, $n_i^+ = n_i^-$. The condition in \eqref{eqn:convsuff} may still be satisfied even if neither (a) nor (b) are, and can be easily verified numerically for any instance of the problem.

\subsection{LLP Implementation}

We employ the logistic loss and use a Gaussian kernel.
To compute the minimizer of \eqref{eqn:discrule}, we appeal to the representer theorem \citep{scholkopf01} which implies that the solution has the form
$    f(x) = \sum_{i} \alpha_i k(x, x_i)$.
 In our LLP experiments, all (unpaired and therefore paired) bags have equal size. Prop. \ref{prop:conv2} thus implies that the objective is convex in the $\alpha_i$, which are learned via gradient descent. We also perform the optimal pairing of bags as described in Sec. \ref{sec:pair}, and use the weights $w_i \propto (\gamma_i^+ - \gamma_i^-)^2$ {\em as prescribed by our theory}. It is possible that parametrizing the weights in terms of $(\gamma_i^+ - \gamma_i^-)$, and tuning them by cross-validation, may lead to improved performance. If a bag pair has a weight of zero, i.e., $\gamma_i^+ = \gamma_i^-$, we drop the pair from the training data. Finally, we remark that the $\gamma_i$'s are fixed population level parameters in our model, whereas they are typically empirical quantities (computed from true but unobserved class labels) in practice. We take the latter to be estimates of the former.

\section{LLP EXPERIMENTS}
\label{sec:exp}

To illustrate our theory, we compare our approach against InvCal \citep{rueping10} and alternate-$\propto$SVM \citep{yu13}, which are the two most common reference methods in learning with label proportions.

\subsection{Datasets}
We employ datasets from the UCI repository (see table). For datasets with multiple classes, we convert them to binary by choosing one class as positive and the others are merged to form the negative class.

\begin{center}
\resizebox{0.5\textwidth}{!}{%
 \begin{tabular}{c || c || c || c}
 \hline
 Dataset & Size & Classes & Prior label distribution \\ [0.5ex]
 \hline\hline
 Banknote & 1372 & 2 & (55.54\%, 44.46\%) \\
 \hline
 Cardiotocography & 2126 & 3 & (22.15\%, 77.85\%) \\
 \hline
 CNAE & 1080 & 9 & (11.11\%, 88.89\%) \\
 \hline
 Contraceptive & 1473 & 3 & (42.70\%, 57.30\%) \\
 \hline
 German bank & 1000 & 2 & (30.00\%, 70.00\%) \\
 \hline
 Sports article & 1000 & 2 & (63.50\%, 36.50\%) \\
 \hline
 Wireless Indoor & 2000 & 4 & (25.00\%, 75.00\%) \\
 \hline
\end{tabular}
}
\end{center}

\subsection{Experiment Setup}
We first randomly split each dataset into training and test sets, using 80\% for training. The training set is then split into bags of fixed sizes. We consider the bag sizes of 2, 4, 8, 16, 32, 64, and select the parameters of each model by 5-fold cross validation. As we assume instance-level labels are inaccessible during training, the parameters of our method are selected by the corrected empirical risk, while the other two methods use the empirical proportion risk. The above process is repeated for 5 times per dataset per bag size. The mean accuracy and the mean balanced accuracy are reported with their standard deviations.

For InvCal, the parameters are tuned from $C_p \in \{0.1, 1, 10 \}$, and $\epsilon \in \{0, 0.01, 0.1\} $. For alter-$\propto$SVM, the parameters are tuned from $C \in \{0.1, 1, 10\}, C_p \in \{1, 10, 100\}.$ Following \citep{yu13}, The alternate $\propto$SVM is randomly initialized 10 times and the result with the smallest objective values is picked. In our approach, the parameters are tuned from learning rate $\in \{ 0.1, 0.01, 0.001 \}$, decay $\in \{0.01, 0.001, 0.0001 \}$, iterations $\in \{100, 800, 1600, 3200\}$, and regularization parameter $\in \{0.001, 0\}$.
For all approaches, the Gaussian kernel is used and the parameter $\gamma$ is tuned from \{0.001, 0.1, 1\}

\subsection{Results}
Performance is measured by balanced accuracy (1 - BER) with 0/1 loss, which is shown in Table 1 in the appendix. As a reference, the conventional accuracy is shown in Table 2. As shown by Table 1, corrected loss outperforms the other two methods 16 times, while for InvCal the number is 13 and for $\propto$SVM 15. We can conclude that our method is comparable to the other two methods for balanced accuracy.

\section{DISCUSSION}
\label{sec:disc}

We have presented one approach to learning from multiple corrupted sources, building on the method of unbiased losses  \citep{natarajan18jmlr}. At least two other approaches may be worth exploring. First, \citet{natarajan18jmlr} present a second general approach, the method of calibrated surrogates. With this method, it is no longer possible to decouple  calibration and noise correction, and different techniques would be needed. Second, one could use {\em symmetric} losses, which are immune to class-conditional label noise \citep{rooyen15tr,charoenphakdee19icml}.

We also point out that while our approach has focused on kernel methods for binary classification, it can be generalized. For example, unbiased losses for multiclass have be developed in recent years \citep{patrini17dnn, rooyen18jmlr}. Furthermore, our analysis is based on Rademacher complexity and should be extensible to other types of classifiers \citep{neyshabur15colt}.

\bibliographystyle{plainnat}

redpajama_set_name: RedPajamaArXiv


"The Sri Lankan government needs to go further and faster on human rights and reconciliation," David Cameron told a press conference.
AFP – Britain's David Cameron put Sri Lanka on notice Saturday to address allegations of war crimes within months or else he would lead a push for action at the UN.
Speaking at a troubled Commonwealth summit in Colombo, the British premier warned his hosts that pressure over alleged abuses at the end of Sri Lanka's ethnic conflict was not about to go away.
But a Sri Lankan minister said Colombo would "definitely" not allow international investigators to conduct a probe on its soil.
Cameron, who made an historic visit to the former war zone on Friday, also told of how he had "frank" exchanges with Sri Lankan President Mahinda Rajapakse on his return.
"The Sri Lankan government needs to go further and faster on human rights and reconciliation," he told a press conference.
"Ultimately all this is about reconciliation and closure and healing to this country, which now has the chance, if it takes it, of a much brighter future but that will only happen by dealing with these issues and not ignoring them.
The UN and rights group say as many as 40,000 civilians may have been killed in the final stages of the war in May 2009 when Tamil Tiger rebels were routed.
However Rajapakse has denied any civilians were killed and has also blocked all calls for an independent probe into claims of war crimes committed by government forces against the Tamil population in the Jaffna region.
Cameron said Rajapakse wanted more time to address the claims but put him on notice to deliver by March or else he would push for an international investigation through UN human rights bodies.
Cameron upstaged the first day of the three-day meeting Friday by travelling to Jaffna which bore the main brunt of the 37-year war, meeting local ethnic Tamils who lost loved ones or were left homeless.
He was the first foreign leader to visit Jaffna since Sri Lanka, a former British colony, gained independence in 1948.
While Sri Lanka had hoped the summit would showcase its revival since troops from the mainly Sinhalese government crushed the Tigers, Cameron's visit and boycotts have torpedoed its strategy.
In an interview with AFP, Rajapakse's brother Basil — who is also the economic development minister– thanked Cameron for at least attending but rejected any idea of an inquiry by foreign investigators.
Asked about the March deadline for the Sri Lankans to complete their own inquiry, the minister rejected any talk of a timetable.
"They can't give dates. It is not fair… Even in Northern Ireland it took a lot of time," he said.
Cameron received an emotional reception in Jaffna from locals who told him of their plight during the war and in its aftermath.
Their desperation was underlined when two women tried to hurl themselves in front of his convoy, clutching pictures of loved ones who were killed in a conflict which claimed more than 100,000 lives.
Cameron held talks late Friday with Rajapakse, acknowledging that "very strong views were expressed on both sides".
According to his office, Rajapakse asked Cameron to give Sri Lanka "more time to overcome all major challenges".
The talks would have added to Rajapakse's anger with his peers after the leaders of Canada, India and Mauritius stayed away.
Late Friday Mauritian Prime Minister Navin Chandra Ramgoolam said his country was withdrawing as host of the next gathering in 2015 as "human rights are more important than hosting a Commonwealth summit".
Since the war, Sri Lanka's economy has grown at rates of up to 8.2 percent and major infrastructure projects such as a new $350 million highway linking Colombo to the main airport bear testimony to its revival.
Basil Rajapakse said a Commonwealth Business Forum on the sidelines of the summit had helped showcase Sri Lanka's economic potential.
"In the last few days, we have finalised investments amounting to about 1.2 billion dollars. This comes from Australia and New Zealand into the mineral sands and dairy sectors," he said.
The summit is meant to focus on issues such as poverty reduction and climate change but press conferences have been dominated by questions over the war crimes issue.
The leaders shifted from the main conference venue in downtown Colombo Saturday, gathering behind closed doors at a deluxe hotel on the outskirts of the capital.


text:
AMIE was co-founded by Priya Mukherjee and Nishith Prakash with the vision to identify and address group-based or other outcome disparities for junior scholars in the economics profession. They come up with the idea to start this initiative informally while both were visiting scholars at Boston University's Institute for Economic Development during 2018-19. It wasn't until 2020, with the support and encouragement of students, peers, and advisors, that they decided to create AMIE.
Priya Mukherjee is an assistant professor at the Department of Agricultural and Applied Economics at the University of Wisconsin, Madison, and is an affiliate of the Center for South Asia.
Born in Wellington (Tamil Nadu) and raised in New Delhi, India, she earned a B.A. (honors) in mathematics from Lady Shri Ram College, Delhi University (India), an MSc in economics from the London School of Economics (London, UK), and a Ph.D. in economics from Cornell University (Ithaca, NY). She was previously an assistant professor at the William & Mary (Williamsburg, VA), and a Visiting Scholar at Boston University.
Prof. Mukherjee's primary research interests include development economics, education policy, political economy, and health.
Her interests lie in political economy, education, and health, and she utilizes both field experiments and non-experimental methods in her work. Her current research projects are based in India, Indonesia, and Bangladesh, and include studies on the impact of democratization and decentralization, improving state capacity, understanding education markets, and more recently, investigating the effects of the pandemic on the wellbeing of children and women.
Priya Mukherjee
Assistant Professor, University of Wisconsin, Madison
Associate Professor of Economics, University of Connecticut
Born and raised in Bihar, India, he earned a B.A. (honors) in economics from Shivaji College, an M.A. in economics from Delhi School of Economics, Delhi University (India), and a Ph.D. in economics from University of Houston, TX. He was a post-doctoral research associate at Cornell University, NY from July 2010 till December 2011. He has previously held Visiting Assistant Professor positions at Ohio University, and Dartmouth College, and he has been a Visiting Fellow at Yale University, Columbia University, Massachusetts Institute of Technology, and Boston University. Professor Prakash is currently a fellow with the Women and Public Policy Program at Harvard Kennedy School (2019-2021).
He is a Research Fellow at Centre for Research and Analysis of Migration (CReAM), The Institute for the Study of Labor (IZA), HiCN Households in Conflict Network (HiCN), Innovations for Poverty Action (IPA) Network Researcher, Global Labor Organization (GLO), and Member of Insights on Immigration and Development (INSIDE-SPAIN). He is also currently serving in the Editorial Board of the journal Economies.
Prof. Prakash's primary research interests include development, political economy, public policy, economics of education, and behavioral economics. Throughout his academic pursuit, Prof. Prakash has focused on the relationship between government policies and economic development and harnessed rigorous empirical evidence to study the impact of policies and institutions in enabling inclusive growth. He has experience in conducting surveys in developing countries, working with large-scale observational and administrative data sets, and conducting field experiments (RCTs). Prof. Prakash has worked closely with Department of Education (in Bihar, Nepal, Zanzibar, and Zambia), the State Police (in Bihar, and Telangana), the Ministry of Home Affairs (Government of India), and the World Bank. His teaching interests include Development Economics, Applied Econometrics and Economics of Gender and Inequality.
Professor Prakash's research has featured in The Economist, World Bank Development Impact Blog, World Economic Forum, Financial Times, Forbes, The Statesman, The Atlantic, The Indian Express, The Hindu, The Times of India, and other national and international newspapers.
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text:
, comumente conhecida como Tadano, é a maior empresa japonesa fabricante de guindastes e de plataformas aéreas. A empresa é a sétima maior fabricante de guindastes no mundo.

História
Masuo Tadano, o fundador da companhia, iniciou como fabricante de aço em 1919 na localidade de Fujitsuka-cho, Takamatsu. Em 1948, ele fundou a Tadano Ltd. e com ela a primeira planta industrial de manejo de carga, expandindo para a produção de vários tipos de equipamentos industriais nos anos seguintes. Em 1955, a Tadano introduziu o primeiro caminhão-guindaste hidráulico com capacidade de duas toneladas no mercado japonês. Em 1960, a empresa produziu o primeiro modelo de caminhão-guindaste hidráulico para o mercado externo, exportando-o para a Indonésia.

Posteriormente, a empresa veio a ter atuação mundial, estabelecendo participação sólida no mercado de guindastes hidráulicos móveis. Em 1972, a Tadano abriu seu capital na primeira seção da Bolsa de Valores de Tóquio.

A empresa também ficou conhecida pelo projeto de restauração das estátuas moai de Ahu Tongariki na Ilha de Páscoa entre 1992 e 1994.

Em 2014 a empresa adquiriu o distribuidor de seus produtos no Reino Unido. Essa empresa, até então conhecida como Cranes UK, mudou seu nome para Tadano UK.

Galeria

Ligações externas

 TADANO Ltd.
 TADANO Group global portal
 TADANO FAUN GmbH
 TADANO America Corp.
 TADANO (Beijing) Ltd.
 Worldwide Network of TADANO Group
 Article on RitchieWiki
 Member of JCEMA

Empresas listadas na Bolsa de Valores de Tóquio
Fabricantes de guindastes
!Artigos com texto em japonês
Empresas do Japão
Empresas fundadas em 1948


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