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PPP Loans Provided Lots Of Cash Assistance With Few Questions Asked : The NPR Politics Podcast
KATIE: Hi. This is Katie (ph). I'm driving with my family up I-39 North because we're moving from Louisiana back to our home state of Wisconsin. This podcast was recorded at…
TAMARA KEITH, HOST:
12:42 p.m. on Thursday, January 19.
KATIE: Things may have changed by the time you hear this. But we'll still be looking forward to celebrating Mardi Gras in the middle of Wisconsin winter. Go Tigers, and go Badgers.
(SOUNDBITE OF THE BIG TOP ORCHESTRA'S "TEETER BOARD: FOLIES BERGERE (MARCH AND TWO-STEP)")
KEITH: I love me a good road trip.
SUSAN DAVIS, BYLINE: She seems in good spirits, but, man, I do not like moving.
KEITH: I don't like moving. But I do love the road trip part of it. I'm just – I'm a sucker for road trips. Hey there. It's the NPR POLITICS PODCAST. I'm Tamara Keith. I cover the White House.
DAVIS: And I'm Susan Davis. I cover politics.
KEITH: In the early days of the pandemic, when the economy was in freefall and businesses were closing and uncertainty was off the charts, the federal government stepped in to help out. The Paycheck Protection Program, PPP, loaned businesses money so they could keep paying their employees. Those loans, in theory, could be forgiven, and most of them have been. But a new NPR investigation shows that some of the loans that have been forgiven maybe shouldn't have been. Sacha Pfeiffer is here from NPR's Investigations Desk to tell us more. Hi, Sacha.
SACHA PFEIFFER, BYLINE: Hi.
KEITH: So, Sacha, PPP was created by Congress. The thought was businesses aren't making money. Rather than employees being laid off and then having to apply for unemployment insurance, just send the businesses money to pay their workers. Keep the economy afloat. So how did the program end up working? How did businesses apply to get this money?
PFEIFFER: So businesses had to go to banks. And the two of you probably remember that banks just got deluged. They were inundated by people who wanted Paycheck Protection Program loans, so deluged that they ended up allowing financial technology companies also to give loans. These are companies like Kabbage and Womply, these…
KEITH: I've never heard of those.
PFEIFFER: Yeah – so, like, these high-tech companies that basically automate financial services. And a lot of the big banks at first were just serving really established customers. And a lot of small businesses and businesses run by people of color felt like they couldn't get the attention of the banks. So they went to fintechs. The application was relatively simple. You basically had to say, I think the economic threat of the pandemic could hurt my business. And then you could get a loan. It was a 1% interest rate, but these were potentially forgivable. And a lot of people were assuming and hoping when they applied that the money would eventually get forgiven.
KEITH: And I think the idea was mainly to help small businesses. And full disclosure – the small business that my husband owned at the time took out a PPP loan. But your reporting indicated, you know, a little bit of a flaw with that. You know, there is a distinction between small and in need, and there ended up being a lot of people who maybe didn't seem so needy. Like, for instance, Tom Brady, the football player, got a PPP loan.
PFEIFFER: Yes. And so did Khloe Kardashian. And what is so interesting about this is that need ultimately really wasn't part of the criteria or the application process. As I mentioned, you just had to say you thought you were going to need money because of the pandemic. Officially, you were supposed to have no more than 500 workers, although the rules changed over time. And there were ways for companies that had more than 500 workers to get a loan. But really, almost anyone who applied could get them.
And so we saw people like private jet companies, country clubs, publicly traded companies. The LA Lakers got loans. This caused a huge uproar. People like the Lakers and other companies actually gave their loans back because the public relations of this was so bad. But this was part of the fallout – is that the money ended up going to companies that didn't seem truly needy or that ultimately thrived during the pandemic yet still didn't have to pay it back, which is a lot of the reporting I did.
KEITH: Right. So, Sue, take us back to these early days. There were these negotiations when the Trump administration and Congress – and it was a Democratic Congress. It ended up being very bipartisan. But they were trying to figure out, how do we deal with a collapsing economy?
DAVIS: Yeah. I mean, this was born out of panic. This was not a program that had months of hearings and deliberation and structure for how it would be implemented. It was really a blunt instrument. And I think that there are a lot of people in Congress who survived the 2008 financial crisis and also had to vote to approve, at that time, hundreds of billions of dollars to pump into the U.S. economy. And there was just a broad willingness to just throw money at it. There was not a ton of check and balance put into the system by design.
I think that there was a recognition that the federal government is pretty clunky as an operator, so you have to smooth the passageway to the small business owner and that, you know, bigger businesses, businesses with lawyers or business connections would probably be able to navigate a system more easily than the little guy. And the system was designed to make it easy to get to that little guy. And I would say that I think even with hindsight being 20/20 in a lot of the problems that Sacha raised in her reporting, most members of Congress look at this as a very successful government intervention in the economy, that it was a vast amount of money but that it was necessary. And I think the question that always comes up in these debates is, what would the alternative have looked like?
PFEIFFER: And that is exactly what I heard when I spoke with officials in both the Biden and Trump administrations at SBA, the Small Business Administration, in Treasury. They feel like they accomplished something really monumental. They were a little thin-skinned, honestly, when I asked them a lot of questions about why weren't you more strict at the back end when you were forgiving applications. But one of the things they said is we had to move quickly. And here is Michael Faulkender – he was a senior Treasury official in the Trump administration – talking about what might have been if they hadn't acted quickly.
MICHAEL FAULKENDER: Because PPP got up and running, we did not realize the catastrophe that could have taken place had we failed. What would bread lines during a pandemic have looked like? Do we want to know? I didn't. And so we were going to get that program up and running.
PFEIFFER: I want you to hear one more piece of tape from a Biden administration current SBA official named Patrick Kelley. And, you know, he told me that he gets really frustrated when his agency is criticized over how it handled PPP because it was just carrying out a law passed by Congress. And he pointed out that Congress voted repeatedly to issue more loans and make them increasingly easy to forgive, even when problems with the program became obvious.
PATRICK KELLEY: It's an easy sentiment to say, well, there goes the government again. Why didn't they do it right? But, to me, it ignores the awesomeness of what did get done right. I've met many, many, many, many, many more people who are thankful for their PPP loan.
PFEIFFER: Now, I will add, Michael Faulkender did say to me that, in the end, he felt like this program basically became a small business grant program, which was not its original intention. It was supposed to save jobs. So even he acknowledges that the program changed over time in a way that wasn't true to its original intent.
KEITH: Right. And so in the end, something like $800 billion in PPP loans were given out. We've alluded to this, but, you know, there's this phrase that we use a lot in government – waste, fraud and abuse. How much of that money that went out the door would fall into that category?
PFEIFFER: There is a group of University of Texas researchers who have calculated that, of the roughly 11.4 million loans, 1.4 million of them show signs of fraud, totaling about $64 billion. So 64 billion of about 800 billion could have been fraudulent. There's a separate category of people who really didn't need the money and maybe even profited during the pandemic. They did nothing illegal. They just basically got a free government grant when they were already prosperous. The fraud is a different category.
KEITH: Right, like the program wasn't actually designed, as you said, based on need.
PFEIFFER: It really wasn't. I mean, you had to fill out a form saying, I think I need the money. And in the end, the forgiveness process became so lenient that you had to fill out a form that said I used the money correctly. It was basically that much of an honor system. And because the SBA and Congress wanted to make this as easy as possible, they told the banks they could be really kind of lenient in their underwriting, meaning they didn't have to do a great deal of due diligence. They could essentially take people's applications at face value. And human nature – you know, some people took money they shouldn't have taken.
DAVIS: Sacha, can I say, as someone who has covered government and certainly focused on waste, fraud and abuse, I have to admit that there was a part of me when I was listening to your report where the estimates are about 1- to 12%, potentially, of the money that went out the door was fraudulent. And there's a part of my brain that was like, huh, not as bad as I thought it could have been.
DAVIS: You know, like, if it falls somewhere in that, like, one to 10 percentage zone, there is a more cynical argument to be made that that's pretty good by waste, fraud and abuse standards.
PFEIFFER: Although $64 billion is a lot of money…
DAVIS: It's a lot of money.
PFEIFFER: …And could solve a lot of problems, and so not insignificant for sure.
KEITH: All right. Well, we're going to take a quick break, and we'll have more in a second.
And we're back. And I'm wondering, is there a balance here between what was legal and what was moral? And I guess, like, if everyone's doing it, then is it wrong?
PFEIFFER: I'm so glad you asked that, because I spent a lot of time thinking about exactly that – was there a moral or ethical aspect of this? And I think the answer is yes. However, there is also a very practical, pragmatic aspect, which is the government was essentially giving out free money, and would you be a fool not to take it? You can stand on principle and not, but then you're going to watch your neighbor or your neighbor's business take money. And you won't, and then what have you gained? But, you know, I interviewed an art gallery owner on Cape Cod, and he said – I think his words were, what bothers me is people who took the money that didn't need the money. It should have been for people who really needed it. Now, how do you define need?
DAVIS: Yeah.
PFEIFFER: I mean, should we have expected Tom Brady to reach into his own pocket and cover his company, TB12, the sports nutrition company, if they were closed for a while? Some people might say yes. Some people might say no. So it's a really interesting ethical, moral debate.
KEITH: Well, and Sue, as Congress was creating it, and then later on, as Sacha reported, as Congress made the rules more and more lenient…
KEITH: …What was their intent? Did they care about whether businesses truly needed it or whether they could just get it?
DAVIS: You know, I think the intent from a really pure form was to save the economy. And I think, from the bipartisan nature of the initial design of this program, that is a factual statement. I also think that, in the loosening of the rules – and this is a dynamic that I thought was interesting about your story, too. It's like, yes, the urgency to get it out the door, but could there have been more accountability on the back end? And from a policy perspective, I think that's a really valid question. Like, shouldn't Washington and Congress be more accountable for the money that goes out the door? But politically, it raises a lot of red flags to me because, on the front end, you had politicians across the board encouraging people – take these loans. Take them. Take them. Take them. Take them. Keep your businesses open. Like, the messaging was that this was basically free money, and it was yours for the taking to help your businesses – and then to maybe feel like they were changing the terms on the back end. Well, maybe they won't get forgiven, or maybe you didn't exactly meet the circumstances that we've changed our minds about. I think there was a lot of hesitance to doing that. I just don't think politically, even if from a good governance – from a waste, fraud and abuse standpoint – that should be what happens, from a political standpoint, I don't think many of them felt that they could encourage people to take this money and then change the terms by which that they make it tougher for them to keep the money. It just wasn't going to happen.
PFEIFFER: I think that's true. And we all know that businesses are a very powerful lobby, and many of them were lobbying their elected representatives to not have to pay back their loans. I spent a lot of time listening and reading to congressional hearings on PPP oversight, and there were many members of Congress who were speaking to Treasury officials, like the former treasury secretary under Donald Trump, Steven Mnuchin. They wanted to know whether their constituents were likely to have to repay the money. They were hearing from constituents who hoped they wouldn't have to repay the money. People like Senator John Kennedy of Louisiana – we have an exchange between Kennedy and Mnuchin where we hear that exact debate.
JOHN KENNEDY: The small businesswomen and small businessmen think that the federal government is going to double-cross them on the forgiveness of these loans. You need to be mindful of that.
STEVEN MNUCHIN: In the recent bill, there is an extension from two years to five to 10 years, but I don't expect businesses are going to need to use that because I think the majority of this money is going to be forgiven in the next few months, and that's our intent. You know, I'd like to make this as easy as possible.
PFEIFFER: And that's Treasury Secretary Mnuchin himself at the end. You can hear him saying, we're going to try to make this easy. This is even as he was acknowledging that they knew they had companies that shouldn't have taken the money take the money.
DAVIS: Sacha, one of the things that you had in your report that I think is so critical to understanding the implementation of this – but it might seem kind of boring and technical – is just how bad the infrastructure of the federal government is, that it's operating. The operating systems of the federal government are outdated. They're old, and they weren't equipped to deal with this kind of an influx of a new program.
PFEIFFER: Yes. And I spent a lot of time talking with is very smart, funny, big-thinking MIT economist named David Autor. And he says a lot of people are conspiratorially concerned about the deep state. He says we should be concerned about the shallow state, and that is our antiquated, archaic, obsolete government technological systems, just like the recent flight grounding we saw where an FAA system wasn't able to function correctly because it was so old. And he says part of the reason our targeting of the PPP money was so inaccurate is because we didn't have the systems that could be accurate. In Europe and in Canada, their governments often know exactly what companies are paying people, exactly how many people were laid off. So if they want to compensate those companies exactly, they can do that. We can't. So in a way, this MIT economist David Autor, would argue, even if we wanted to have gotten the money out in a better way, our systems weren't designed to be able to do it.
KEITH: So a Republican House is coming in. They're in, and they want to cut spending across the board. Is this the kind of thing that they could cite and say, you know, here's an example of waste, fraud and abuse? Or are the politics not quite right to do that?
DAVIS: It's a little tricky on this particular issue because so many of them voted for the program to begin with, right? It's like kind of slapping yourself on the wrist. I do think the sort of broader, macro legacy here is that when push comes to shove when it comes to the economy, the animating impulse of government is to throw money at the problem. And we've seen that in different types of crisis. And also, I think that on the whole, the lesson for lawmakers and the economic lesson is that it's helped, worked and that these are very imperfect, blunt instruments. But when you're dealing with an outdated, clunky, slow federal bureaucracy, sometimes the only way to do it is to do it with the fire hose approach. And so while I do think Republicans are especially right now very focused on wasteful spending, on government overreach when it comes to specifically the pandemic response, spending in relation to crisis, there still doesn't seem to be much of a reckoning over how to do things differently. Now, when you talk about tax cuts or any other kind of spending, it's a different debate. But here, I still think most lawmakers look at this program and think that was a win.
KEITH: Like, it worked as designed.
DAVIS: It worked as well as it could work given the system it had to work with.
PFEIFFER: Well, and I do want to add because I do think we're cutting the government a lot of slack, and of course, I want to be very fair. And as we said, we've acknowledged it was an unprecedented economic and health catastrophe. Our systems are antiquated. However, even when Congress saw that there was a lot of abuse of the system and the wrong types – undeserving businesses were getting money, they kept giving out money. They kept making it easier to get the money, and they kept making it easier to get the money forgiven. They gave money to people on the Treasury's do-not-pay list. It essentially was an honor system. The government guaranteed the loans 100%, which meant the banks took on no risk. They told the banks – do the most superficial level of underwriting. So the banks were essentially rubber-stamping applications. And even the SBA's own inspector general and other government oversight agencies have said there were no controls. There could have been more oversight and controls, and we didn't. So I do think there is some fault here, and this system could have been designed with more oversight and control.
KEITH: All right. Well, we are going to leave it there for today. Sacha Pfeiffer from NPR's Investigations Desk, thank you so much for bringing this reporting to our pod.
PFEIFFER: You're welcome.
KEITH: I'm Tamara Keith. I cover the White House.
KEITH: And thank you for listening to the NPR POLITICS PODCAST.
(SOUNDBITE OF THE BIGTOP ORCHESTRA'S "TEETER BOARD: FOLIES BERGERE (MARCH AND TWO-STEP)")
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No Visible Bruises: What We Don't Know About Domestic Violence Can Kill Us (Paperback)
By Rachel Louise Snyder
WINNER OF THE HILLMAN PRIZE FOR BOOK JOURNALISM, THE HELEN BERNSTEIN BOOK AWARD, AND THE LUKAS WORK-IN-PROGRESS AWARD * A NEW YORK TIMES TOP 10 BOOKS OF THE YEAR * NATIONAL BOOK CRITICS CIRCLE AWARD FINALIST * LOS ANGELES TIMES BOOK PRIZE FINALIST * ABA SILVER GAVEL AWARD FINALIST * KIRKUS PRIZE FINALIST
NAMED ONE OF THE BEST BOOKS OF 2019 BY: Esquire, Amazon, Kirkus, Library Journal, Publishers Weekly, BookPage, BookRiot, Economist, New York Times Staff Critics
"A seminal and breathtaking account of why home is the most dangerous place to be a woman . . . A tour de force." —Eve Ensler
"Terrifying, courageous reportage from our internal war zone." —Andrew Solomon
"Extraordinary." —New York Times ,"Editors' Choice"
"Gut-wrenching, required reading." —Esquire
"Compulsively readable . . . It will save lives." —Washington Post
"Essential, devastating reading." —Cheryl Strayed, New York Times Book Review
The book that changed the conversation about domestic violence—an award-winning journalist's intimate investigation of the abuse that happens behind closed doors, now with a new afterword by the author.
We call it domestic violence. We call it private violence. Sometimes we call it intimate terrorism. But whatever we call it, we generally do not believe it has anything at all to do with us, despite the World Health Organization deeming it a "global epidemic." In America, domestic violence accounts for 15 percent of all violent crime, and yet it remains locked in silence, even as its tendrils reach unseen into so many of our most pressing national issues, from our economy to our education system, from mass shootings to mass incarceration to #MeToo. We still have not taken the true measure of this problem.
In No Visible Bruises, journalist Rachel Louise Snyder gives context for what we don't know we're seeing. She frames this urgent and immersive account of the scale of domestic violence in our country around key stories that explode the common myths—that if things were bad enough, victims would just leave; that a violent person cannot become nonviolent; that shelter is an adequate response; and most insidiously that violence inside the home is a private matter, sealed from the public sphere and disconnected from other forms of violence. Through the stories of victims, perpetrators, law enforcement, and reform movements from across the country, Snyder explores the real roots of private violence, its far-reaching consequences for society, and what it will take to truly address it.
Rachel Louise Snyder is the author of Fugitive Denim, the novel What We've Lost is Nothing, and No Visible Bruises, winner of the Lukas Work-in-Progress Award from the Columbia School of Journalism, the Hillman Prize for Book Journalism, and the Helen Bernstein Book Award. Her work has appeared in the New Yorker, the New York Times, Slate, and others. A 2020-2021 Guggenheim Fellow, Snyder is a Professor of Creative Writing and Journalism at American University. She lives in Washington, DC.
Social Science / Sociology / Marriage & Family
Political Science / Public Policy / Social Policy
Social Science / Violence in Society
Publisher: Bloomsbury Publishing
Publication Date: June 9th, 2020
"Extraordinary . . . [No Visible Bruises] takes apart the myths that surround domestic violence. . . In its scope and seriousness--its palpable desire to spur change--this book invites reflection not only about violence but about writing itself . . . [Snyder] brings all of fiction's techniques to this new book--canny pacing, an eye for the animating detail and bursts of quick, confident characterization. There is a fullness and density to every one of her subjects . . . She glides from history to the present day, from scene to analysis, with a relaxed virtuosity that filled me with admiration. This is a writer using every tool at her disposal to make this story come alive, to make it matter." —Parul Sehgal, New York Times, "Editors' Choice"
""[Snyder] has written a book about everything: about men who beat and kill their wives or girlfriends; about people who work to predict murder, and those who try to heal the abusers; and also, deeply, about gender, poverty, depression, despair, privilege, law enforcement, incarceration, justice, mental health, and politics . . . It takes a writer of uncommon talent and confidence to pull this off. Snyder's stories are about people, every single one of whom is drawn empathically. Her investigation is intellectual and unsparingly complex."" —Masha Gessen, New Yorker's "Page-Turner" blog
"Compulsively readable . . . In a writing style that's as gripping as good fiction, as intimate as memoir and deeply informed, [Snyder] takes us into the lives of the abused, the abusers and the survivors. . . The stories are devastating, but Snyder keeps us reading by pointing us toward possible solutions . . . After a few chapters, I was telling a prosecutor friend that everyone in her office--no, everyone in the state who deals with family violence--had to read this book. Because it will save lives." —The Washington Post
"Powerful . . . Snyder exposes this hidden crisis by combining her own careful analysis with deeply upsetting and thoughtfully told accounts of the victims . . . [An] important book." —New York Times Book Review
"Gut-wrenching, required reading." —Esquire,"Best of the Year"
"Snyder [goes] both wide and deep . . . her empathy for the victims is powerful, and infectious. But so is her interest in the perpetrators, some of whom may be able to recover, to change and atone. And as she makes very clear, those who undertake reform -- studying and quantifying risk, asking smart questions about whether women's shelters help or hurt, counseling survivors and getting them the support they need -- are heroes." —Los Angeles Times
"A brilliant work . . . what makes it move with the suspense of a beach novel is Snyder's passionate storytelling, [which] often soars to the point of lyricism but remains unfailingly honest." —Ms. Magazine
"Should be required reading for lawmakers across the country." —Minneapolis Star Tribune
"Snyder's in-depth reporting and vivid writing imbue the book with drama and tension . . . A welcome addition to the efforts that bring this brutal crime out from behind closed doors and provide hope for the future." —Los Angeles Review of Books
"A powerful exploration of the sinister, insidious nature of domestic violence in America… Bracing and gut-wrenching, with slivers of hope throughout, this is exemplary, moving reportage on an important subject that often remains in the dark due to shame and/or fear." —Kirkus Reviews, starred review
"[A] powerful investigation into intimate partner abuse . . . with closely observed, compassionate portraits of victims, advocates, abusers and police. Penetrating and wise, and written in sometimes novelistic prose, Snyder's sobering analysis will reward readers' attention." —Publishers Weekly, "Best of the Year," starred review
"A searing examination of domestic violence in the U.S." —The Huffington Post
"Snyder's willingness to tell the intimate stories of domestic violence sheds light on an often neglected subject. All of us have a stake in becoming more aware of and responsive to private violence, and this book proves why." —BookPage, starred review
"This sympathetic look at victims, perpetrators, and intervention efforts by law enforcement and social agencies makes for compelling reading. . . Snyder's chilling body of evidence shows that domestic abuse is a pervasive epidemic that can and does happen everywhere." —Booklist
"We can't afford not to be talking about domestic violence. Snyder argues that it has reached epidemic proportions in the country -- it accounts for 15 percent of all violent crimes -- with devastating effect. She combines her analysis with interviews with survivors, advocates, and occasionally, the perpetrators themselves." —New York Times, "Books To Watch For In May"
"An incredible piece of reporting" —BookRiot, "Best of the Year So Far"
"By focusing on case studies--individuals' stories--Snyder returns humanity to the horrifying larger issue . . . placing domestic violence in relationship to issues of economics, education, employment, the criminal justice system and other, more 'public' types of violence . . . No Visible Bruises speaks with urgency about solving a problem that, however invisible, affects us all." —Shelf Awareness
""No Visible Bruises is a seminal and breathtaking account of why home is the most dangerous place to be a woman. Through brilliant insights and myth-busting research, compelling storytelling, and a passionate focus on truth-telling, Rachel Louise Snyder places domestic violence exactly where it should be, smack in the center of everything. A tour de force."" —Eve Ensler, author of THE VAGINA MONOLOGUES and THE APOLOGY
"This is terrifying, courageous reportage from our internal war zone, a fair and balanced telling of an unfair and unbalanced crisis in American family life. Snyder writes with stark lucidity and great compassion, and tells stories of utmost urgency with considerable narrative skill." —Andrew Solomon, National Book Award-winning, bestselling author of THE NOONDAY DEMON, FAR FROM THE TREE, and FAR AND AWAY
"Snyder's singular achievement is that she illuminates the dark corners of this specter as a way to understand it and thus eliminate it." —J. Anthony Lukas Prize, Judges' Citation for NO VISIBLE BRUISES
"I cannot imagine how Rachel Louise Snyder had the strength to write this book-it's like the journal of a war correspondent. By witnessing the toll of family violence, she wants to take public this private horror. No Visible Bruises is a keening for the battered and a shout of outrage for the lost, a case for the higher awareness that could make us better humans." —Ted Conover, author of NEWJACK, and director of NYU's Arthur L. Carter Journalism Institute
"No Visible Bruises snapped open my eyes to the direct link between patriarchal entitlement and violence against women, between the way men are raised to the way women are treated. From her dismantling of the term 'domestic violence,' which not only couches a pervasive public menace in homey, private terms, but echoes a sick culture in denial, to her connecting the dots between acts of terror and acts of domestic terror, Snyder's is an indispensable, important book." —Carina Chocano, author of YOU PLAY THE GIRL
"There is a compelling determination to Snyder's writing as well as to her suggestion that if we tell these stories enough, that if we find the right way to tell them, then maybe someday we won't have to tell them anymore." —Ploughshares

redpajama_set_name: RedPajamaCommonCrawl


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Thu., May 16, 2019
Cedar Rapids police cruisers sporting a new look
All-black look saves money, disguises dings and scratches
The new Cedar Rapids Police Department cruisers have black door panels, a change from the previous white wraps that showed scuffs and added cost. The new SUVs cost about $28,000 each and require another $28,000 in upgrades for law enforcement purposes. (Liz Martin/The Gazette)
The white vehicle wraps previously used on Cedar Rapids police vehicles shows scuffs, as shown above, more than an all-black vehicle, which is what police are now buying. (Liz Martin/The Gazette)
The new Cedar Rapids Police Department cruisers will have black door panels, a change from the previous white vehicle wraps. (Liz Martin/The Gazette)
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All articles by Kat
CEDAR RAPIDS — You may have noticed some new Cedar Rapids police cruisers sporting a new look.
That's because the Cedar Rapids Police Department is swapping out its classic black-and-white look for new, all-black cruisers.
Capt. Craig Furnish, who is involved in vehicle procurement, said the new look costs less.
When the SUVs are manufactured for the department, they are initially all black, with the white wrap added at extra cost.
"It's basically a cost-saving measure," Furnish said. "To put that white wrap on our cars is an $840 upgrade. Also the black looks better — it doesn't show cosmetic flaws like dents, scratches, as obviously as the white does."
The department has 65 marked cruisers, Furnish said. This past fiscal year, the department bought 13 new, all-black vehicles.
Each new vehicle costs about $28,000, Furnish said. The upgrades needed to make the vehicles functional for law enforcement — a "cage," lighting equipment, computers, radios and so on — add another $28,000 to the cost.
"We get about 13 or 14 new vehicles a year, and that depends on what we need that year — it could be 10 squad cars and 3 unmarked cars, or it could be 13 squad cars — it just depends on what vehicles need to be rotated out," he said.
Each vehicle has an expected life span of about five years, Furnish said, but that is dependent on how the vehicle holds up mechanically and whether it's been in any wrecks.
"The police department works with the city's Fleet Services Department, and they developed a replacement model that is based on analysis for each vehicle," he said.
"In addition to looking at mileage and age, they also look at how many fixes have been necessary," he said. "How often do they have to repair a vehicle? There's no set timeline. ...
"We could get a new vehicle and lose it right away because of an accident, or we could get one that lasts a few years and then starts showing significant mechanical issues," he said. "You have to take it on a car-by-car basis."
Six of the new all-black SUVs have already hit the streets. The remaining seven are expected to be in use over the next six months.
Furnish said he expects the department's entire fleet of marked vehicles will be all black within the next five or six years.
For fiscal year 2020, which spans from July 1 to June 31, 2020, the department has ordered 14 new vehicles — 10 patrol cruisers, one K-9 vehicle and three unmarked cars, Furnish said.
"Really, we're just trying to be more fiscally responsible," he said. "Why spend $840 on a wrap? You multiply that times 65 squad cars, and you're saving about $54,000, which is roughly the price of two squad cars."
l Comments: (319) 398-8238; kat.russell@thegazette.com
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So when it comes to this journey what words do you use? I believe this matters, it is very important. You just joined Weight Watchers, or Gold's Gym – what words come to mind? I don't pretend to know how you might answer this question. It doesn't matter what I think, the only person who matters here is you.
I can only tell you this when I got serious about getting to goal I became quite clear about what I wanted. why I want to do this, and how I was going to make it happen. Having these three ideas crystal clear in my mind has helped me to tell a better story about getting to goal. So I wonder what's your story? Think about your words carefully because they pave the road to success.


text:
\section{Introduction}

	We consider a system of $n$ agents communicating through a network to collaboratively solve the following optimization problem:
	\begin{equation} \label{problem}
	\min\limits_{x\in\mathbb{R}^p}~f(x):=\frac{1}{n}\sum\limits_{i=1}^n f_i(x),
	\end{equation}
	where $x$ is the global decision variable and each function $f_i: \mathbb{R}^p\rightarrow \mathbb{R}$ is known by agent $i$ only.
	The objective
	is to obtain an optimal and consensual solution through local updates and local neighbor communications and information exchange. Such local exchange is desirable when sharing a large amount of data is prohibitively expensive due to limited communication resources, or when privacy needs to be preserved for individual agents. Scenarios in which problem (\ref{problem}) is considered include distributed machine learning \cite{nedic2017fast,cohen2017projected,forero2010consensus}, multi-agent target seeking \cite{pu2016noise,chen2012diffusion}, and wireless networks \cite{cohen2017distributed,mateos2012distributed,baingana2014proximal}, among many others.

	To solve problem~\eqref{problem} in a multi-agent system,
	many distributed first-order algorithms have been proposed under various assumptions on the objective functions
	and the underlying network topology~\cite{shi2015extra,seaman2017optimal,uribe2017optimal,xu2015augmented,di2016next,li2019decentralized,nedic2015distributed,xi2017add,zeng2017extrapush,xu2017convergence,pu2018swarming,pu2020distributed,nedic2017achieving,qu2017harnessing,xi2018linear}. Most works, including \cite{shi2015extra,seaman2017optimal,uribe2017optimal,xu2015augmented,di2016next,li2019decentralized,pu2020distributed},
	often restrict the network connectivity structure to undirected graphs, or more commonly require doubly stochastic mixing matrices.
	For strongly convex and smooth objective functions, EXTRA~\cite{shi2015extra} first uses a gradient difference structure to achieve the typical linear convergence rates of a centralized gradient method.
	Recently, the gradient tracking technique has been employed to develop
	decentralized algorithms which are able to track the average of the gradients~\cite{xu2015augmented,di2016next,qu2017harnessing,nedic2017achieving,xi2017add,tian2020achieving} and enjoy linear convergence under possibly time-varying graphs.

	In this work, we are interested in the scenario where agents interact in a general directed network that includes undirected network as a special case.
	For directed graphs, constructing a doubly stochastic matrix needs weight balancing which requires an independent iterative process over the network. To avoid such a procedure, most efforts adopted the push-sum technique introduced in \cite{kempe2003gossip} for reaching average consensus over directed graphs. The work in \cite{tsianos2012push} first proposed a push-sum based distributed optimization algorithm for directed graphs. In~\cite{nedic2015distributed}, a push-sum based decentralized subgradient method was proposed and analyzed for time-varying directed graphs. For smooth objective functions, the work in~\cite{zeng2017extrapush,xi2017dextra} modifies the algorithm from~\cite{shi2015extra} with the push-sum technique, thus providing a new algorithm which converges linearly for smooth and strongly convex objective functions.

	Equipped with the gradient tracking technique, the algorithms developed in~\cite{xi2017add,xi2018linear,nedic2017achieving,xi2018linear,tian2020achieving} enjoy linear convergence for possibly time-varying directed graphs \cite{nedic2017achieving} or under asynchronous updates \cite{tian2020achieving}.
	Recently, the work in \cite{pu2018push,pu2020push,xin2018linear} introduced a modified gradient-tracking algorithm called Push-Pull/AB for distributed consensus optimization over directed networks. Unlike the push-sum protocol, Push-Pull uses a row stochastic matrix for the mixing of the decision variables, while it employs a column stochastic matrix for tracking the average gradients. It unifies different computational architectures and can be implemented asynchronously or over time-varying networks \cite{pu2020push,saadatniaki2020decentralized}. Accelerated version of the algorithm has also been developed \cite{xin2019distributed}.

	Despite that the aforementioned gradient tracking based algorithms are able to achieve linear convergence over directed networks under the smoothness and strong convexity condition, they are not robust to errors caused by noisy communication links or quantization \cite{srivastava2011distributed,cavalcante2013distributed,reisizadeh2018quantized,alistarh2017qsgd}. Considering noisy information exchange is extremely important in distributed optimization, for example, when one wishes to lower communication bandwidth costs among the agents by performing gradient compression techniques \cite{reisizadeh2018quantized,alistarh2017qsgd}, or when there is receiver-side noise corruption of signals in wireless networks \cite{cavalcante2013distributed}. As we will show both theoretically and through experiments, existing gradient tracking based methods fail in this scenario due to inaccurate tracking of the average gradients.

	To address the challenge, we propose a novel gradient tracking algorithm (R-Push-Pull) for distributed optimization that is robust to noisy information exchange. R-Push-Pull inherits the advantages of Push-Pull and achieves linear convergence to the optimal solution under noiseless communication links. It also allows for flexible network design and unifies different computational architectures.

	Our work is related to the literature in distributed stochastic optimization, where only stochastic gradient information is available (see e.g., \cite{nedic2016stochastic,pu2019sharp,pu2020asymptotic,koloskova2019decentralized,pu2020distributed,xin2019distributed2}) and distributed optimization using quantized information (see e.g., \cite{reisizadeh2018quantized,reisizadeh2019exact,taheri2020quantized}). For instance, the work in \cite{pu2020distributed,xin2019distributed} combined gradient tracking with stochastic gradient updates and show the iterates convergence linearly to a neighborhood of the optimal solution assuming strongly convex and smooth objectives. The paper \cite{reisizadeh2019exact} studied an exact quantized decentralized gradient descent algorithm which achieves a vanishing mean solution
	error under customary conditions for quantizers.
	A recent work \cite{taheri2020quantized} considered quantized push-sum for decentralized optimization over directed graphs and establishes subliear convergence to the optimal solution.

	\subsection{Main Contribution}

	Our main contribution of the paper is summarized as follows. Firstly, we introduce a novel gradient tracking method (R-Push-Pull) for distributed optimization over directed networks. The method achieves linear convergence to the optimal solution for minimizing the sum of smooth and strongly convex objective functions. Secondly, R-Push-Pull addresses the challenge of noisy information exchange.  It is shown to be more robust than the other linearly convergent, gradient tracking based algorithms such as Push-Pull/AB, in the sense that the solutions obtained by each agent running R-Push-Pull reach a neighborhood of the optimum in expectation exponentially fast under a constant stepsize policy, while the other competing algorithms lead to divergent solutions. Finally, we provide a numerical example that demonstrate the effectiveness of R-Push-Pull.

	\subsection{Notation}
	\label{subsec: notation}
	Vectors default to columns if not otherwise specified.
	Let each agent $i\in\{1,2,\ldots,n\}$ hold a local copy $x_i\in\mathbb{R}^p$ of the decision variable and an auxiliary variable $s_i\in\mathbb{R}^p$, where their
	values at iteration $k$ are denoted by $x_{i,k}$ and $s_{i,k}$, respectively. Denote
	\begin{eqnarray*}
		&\mathbf{x}:=[x_1, x_2, \ldots,x_n]^{\intercal}\in\mathbb{R}^{n\times p},\\
		&\mathbf{s}:=[s_1, s_2, \ldots,s_n]^{\intercal}\in\mathbb{R}^{n\times p}.
	\end{eqnarray*}
	Define $F(\mathbf{x})$ to be an aggregate objective function of the local variables,
	i.e., $F(\mathbf{x}):=\sum_{i=1}^nf_i(x_i)$, and let
	\begin{equation*}
	\nabla F(\mathbf{x}):=\left[\nabla f_1(x_1), \nabla f_2(x_2), \ldots, \nabla f_n(x_n)\right]^{\intercal}\in\mathbb{R}^{n\times p}.
	\end{equation*}

	\begin{definition}\label{def: norm n p}
		Given an arbitrary vector norm $\|\cdot\|$ on $\mathbb{R}^n$, for any $\mathbf{x}\in\mathbb{R}^{n\times p}$, we define
		\begin{equation*}
		\|\mathbf{x}\|:=\left\|\left[\|\mathbf{x}^{(1)}\|,\|\mathbf{x}^{(2)}\|,\ldots,\|\mathbf{x}^{(p)}\|\right]\right\|_2,
		\end{equation*}
		where $\mathbf{x}^{(1)},\mathbf{x}^{(2)},\ldots,\mathbf{x}^{(p)}\in\mathbb{R}^n$ are columns of $\mathbf{x}$, and $\|\cdot\|_2$ represents the usual $2$-norm.
	\end{definition}

	\begin{definition}
		Given a square matrix $\mathbf{M}$, its spectral radius is denoted by $\rho(\mathbf{M})$.
	\end{definition}

	\subsection{Organization}
		The rest of this paper is organized as follows. We state the problem of interest in Section \ref{sec: problem}. Then, we introduce the robust Push-Pull algorithm in Section~\ref{sec: method} along with the motivation.
	We establish the convergence property of R-Push-Pull in Section~\ref{sec: analysis}.
	In Section~\ref{sec: numerical} we provide a simple numerical example. Section \ref{sec: conclusions} concludes the paper.

	\section{Problem Formulation}
	\label{sec: problem}

	We consider agents interact with each other in a general directed network (graph). 	A directed graph is a pair $\mathcal{G}=(\mathcal{V},\mathcal{E})$, where $\mathcal{V}$ is the set of vertices (nodes) and $\mathcal{E}\subseteq \mathcal{V}\times \mathcal{V}$ denotes the edge set consisted of ordered pairs of vertices.

	Given a nonnegative matrix $\mathbf{M}=[m_{ij}]\in\mathbb{R}^{n\times n}$, the directed graph induced by $\mathbf{M}$ is
	denoted by $\mathcal{G}_\mathbf{M}=(\mathcal{N},\mathcal{E}_\mathbf{M})$, where
	$\mathcal{N}=\{1,2,\ldots,n\}$ and $(j,i)\in\mathcal{E}_\mathbf{M}$ if and only if $m_{ij}>0$.  For an arbitrary agent $i\in\mathcal{N}$, its in-neighbor set $\inneighbor{\mathbf{M},i}$ is defined as the collection of all individual agents that $i$ can actively and reliably pull data from in the graph $\mathcal{G}_\mathbf{M}$; we also define its out-neighbor set $\outneighbor{\mathbf{M},i}$ as the collection of all individual agents that can passively and reliably receive data from agent $i$.

	To solve Problem (\ref{problem}), assume each agent $i$ hold a local copy $x_i\in\mathbb{R}^p$ of the decision variable.
	Then Problem (\ref{problem}) can be written in the following equivalent form:
	\begin{equation} \label{problem_equiv}
	\begin{array}{c}
	\min\limits_{x_1,x_2\ldots,x_n\in\mathbb{R}^p}\sum\limits_{i=1}^n f_i(x_i)\\
	\text{s.t. }x_1=x_2=\cdots=x_n,
	\end{array}
	\end{equation}
	where the consensus constraint is imposed.

	Regarding the objective functions $f_i$ in problem~\eqref{problem}, we assume the following strong convexity and smoothness conditions.
	\begin{assumption}
		\label{asp; strconvex Lipschitz}
		Each $f_i$ is $\mu$-strongly convex with $L$-Lipschitz continuous gradients, i.e., for any $x,x'\in\mathbb{R}^p$,
		\begin{equation}
		\begin{split}
		& \langle \nabla f_i(x)-\nabla f_i(x'),x-x'\rangle\ge \mu\|x-x'\|^2,\\
		& \|\nabla f_i(x)-\nabla f_i(x')\|\le L \|x-x'\|.
		\end{split}
		\end{equation}
	\end{assumption}
	Under Assumption \ref{asp; strconvex Lipschitz}, problem~\eqref{problem} has a unique optimal
	solution $x^*\in\mathbb{R}^{1\times p}$.

	\section{A Robust Gradient Tracking Method}
	\label{sec: method}

	We describe the robust Push-Pull Method in Algorithm 1, where $\epsilon_{i,k}\in\mathbb{R}^p$ and $\xi_{i,k}\in\mathbb{R}^p$ summarize the noise encountered by agent $i$ during the information exchange at step $k$. Sources of the noise include quantization \cite{alistarh2017qsgd} and/or noisy communication links \cite{reisizadeh2018quantized}.
		\smallskip
	\begin{table}
		\normalsize
		\centering
		{\textbf{Algorithm 1: A Robust Push-Pull Method}}

		\smallskip
		\begin{tabularx}{0.5\textwidth}{X}
			\hline
			Choose stepsize $\alpha> 0$ and $\gamma,\eta\in(0,1]$,\\
			\quad in-bound mixing/pulling weights $R_{ij}\geq0$ for all $j\in\inneighbor{\mathbf{R},i}$,\\
			\quad and out-bound pushing weights $C_{li}\geq0$ for all $l\in\outneighbor{\mathbf{C},i}$;\\
			Each agent $i$ initializes with any arbitrary $x_{i,0},s_{i,0}\in\mathbb{R}^p$;\\
			\textbf{for} $k=0,1,\cdots$, \textbf{do} \\
			\quad for each $i\in\mathcal{N}$,\\
			\qquad agent $i$ pushes (noisy) $C_{li}s_{i,k}$ to each $l\in\outneighbor{\mathbf{C},i}$;\\
			\qquad agent $i$ pulls (noisy) $x_{j,k}$ from each $j\in\inneighbor{\mathbf{R},i}$;\\
			\quad for each $i\in\mathcal{N}$,\\
			\parbox{3cm}{\begin{align*}
				\begin{array}{ll}
					& s_{i,k+1} =  (1-\gamma)s_{i,k} + \gamma\left(\sum_{j=1}^n C_{ij}s_{j,k}+\epsilon_{i,k}\right)\\
				& \qquad\qquad+\nabla f_i(x_{i,k})\\
				& x_{i,k+1} =  (1-\eta)x_{i,k}+\eta\left(\sum_{j=1}^n R_{ij}x_{j,k}+\xi_{i,k}\right)\\
				&\qquad\qquad -\alpha (s_{i,k+1}-s_{i,k})
				\end{array}
				\end{align*}}\\
			\textbf{end for}\\
			\hline
		\end{tabularx}
	\end{table}
	Denote
	\begin{equation*}
		\boldsymbol{\epsilon}_k:=[\epsilon_{1,k}, \epsilon_{2,k}, \ldots,\epsilon_{n,k}]^{\intercal},\quad
		\boldsymbol{\xi}_k:=[\xi_{1,k}, \xi_{2,k}, \ldots,\xi_{n,k}]^{\intercal}.
	\end{equation*}
	We make the following standing assumption.
	\begin{assumption}
		\label{asp: noises}
		\sp{Random sequences $\{\boldsymbol{\epsilon}_k\}$ and $\{\boldsymbol{\xi}_k\}$ are independent.\footnote{Note that at the same $k\ge 0$, $\epsilon_{i,k}$ (respectively, $\xi_{i,k}$) can be dependent among different agents.}} The matrices $\boldsymbol{\epsilon}_k$ and $\boldsymbol{\xi}_k$ have zero mean and bounded variance, i.e., $\mathbb{E}[\boldsymbol{\epsilon}_k]=\mathbb{E}[\boldsymbol{\xi}_k]=\mathbf{0}$, $\mathbb{E}[\|\boldsymbol{\epsilon}_k\|^2]\le \sigma_{\boldsymbol{\epsilon}}^2$, $\mathbb{E}[\|\boldsymbol{\xi}_k\|^2]\le \sigma_{\boldsymbol{\xi}}^2$ for some $\sigma_{\boldsymbol{\epsilon}},\sigma_{\boldsymbol{\xi}}>0$.
	\end{assumption}
	Assumption \ref{asp: noises} holds true, for example, when unbiased quantization is performed \cite{alistarh2017qsgd}.

	Denote
	\begin{equation}
	\begin{split}
	& \mathbf{C}:=[C_{ij}],\, \mathbf{C}_{\gamma}:=(1-\gamma)\mathbf{I}+\gamma\mathbf{C},\\
	& \mathbf{R}:=[R_{ij}],\, \mathbf{R}_{\eta}:=(1-\eta)\mathbf{I}+\eta\mathbf{R}.
	\end{split}
	\end{equation}
	We can rewrite Algorithm 1 in the following matrix form:
	\begin{subequations}
		\label{R_pushpull}
		\begin{align}
		&\mathbf{s}_{k+1} =  \mathbf{C}_{\gamma}\mathbf{s}_k+\gamma\boldsymbol{\epsilon}_k+\nabla F(\mathbf{x}_k), \label{R_pushpull_s}\\
		&\mathbf{x}_{k+1} =  \mathbf{R}_{\eta}\mathbf{x}_k+\eta\boldsymbol{\xi}_k-\alpha(\mathbf{s}_{k+1}-\mathbf{s}_k). \label{R_pushpull_x}
		\end{align}
	\end{subequations}

	The matrices $\mathbf{R}$ and $\mathbf{C}$ and their induced graphs $\mathcal{G}_\mathbf{R}$ and $\mathcal{G}_{\mathbf{C}}$ (respectively) satisfy the same conditions as for Push-Pull \cite{pu2020push}.
	\begin{assumption}
		\label{asp: stochastic}
		The matrix $\mathbf{R}\in\mathbb{R}^{n\times n}$ is nonnegative row-stochastic and $\mathbf{C}\in\mathbb{R}^{n\times n}$ is nonnegative column-stochastic, i.e., $\mathbf{R}\mathbf{1}=\mathbf{1}$ and $\mathbf{1}^{\intercal} \mathbf{C}=\mathbf{1}^{\intercal}$. In addition, the diagonal entries of $\mathbf{R}$ and $\mathbf{C}$ are positive. The graphs $\mathcal{G}_\mathbf{R}$ and $\mathcal{G}_{\mathbf{C}^{\intercal}}$ each contain at least one spanning tree. Moreover, there exists at least one node that is a root of spanning trees for both $\mathcal{G}_\mathbf{R}$ and $\mathcal{G}_{\mathbf{C}^{\intercal}}$.
	\end{assumption}

    The readers are referred to Section II of \cite{pu2020push} for the motivation of Assumption \ref{asp: stochastic}, which differs from most existing works on the assumption of network topology. As a result, R-Push-Pull is flexible in network design and unifies different computational architectures, including (semi-)centralized and decentralized architecture.

\begin{lemma}
	\label{lem: eigenvectors u v}
	Under Assumption~\ref{asp: stochastic}, the matrix $\mathbf{R}$ has a nonnegative left eigenvector $u^{\intercal}$ (w.r.t.\ eigenvalue $1$) with $u^{\intercal}\mathbf{1}=n$, and the matrix $\mathbf{C}$ has a nonnegative right eigenvector $v$ (w.r.t. eigenvalue $1$) with $\mathbf{1}^{\intercal}v=n$ (see \cite{horn1990matrix}).
\end{lemma}

   \subsection{Algorithm Development}

   To motivate the development of Algorithm 1, we first take a look at a variant of the Push-Pull algorithm in its matrix form (without noise):
   \begin{subequations}
   	\label{Push_Pull}
   	\begin{align}
   	&\mathbf{x}_{k+1} =  \mathbf{R}\mathbf{x}_k-\alpha\mathbf{y}_k,\\
   	& \mathbf{y}_{k+1} =  \mathbf{C}\mathbf{y}_k+\nabla F(\mathbf{x}_{k+1})-\nabla F(\mathbf{x}_k),
   	\end{align}
   \end{subequations}
   where $\mathbf{x}_0$ is arbitrary and $\mathbf{y}_0=\nabla F(\mathbf{x}_0)$. The matrices $\mathbf{R}$ and $\mathbf{C}$ satisfy Assumption \ref{asp: stochastic}. As a result of $\mathbf{C}$ being column-stochastic,  we have by induction that
   \begin{equation}
   \label{oy_k_tracking}
   \mathbf{1}^{\intercal}\mathbf{y}_k = \mathbf{1}^{\intercal}\nabla F(\mathbf{x}_{k}), \qquad \forall k.
   \end{equation}
   Relation (\ref{oy_k_tracking}) is critical for (a subset of) the agents to track the average gradient $\mathbf{1}^{\intercal}\nabla F(\mathbf{x}_{k})/n$ through the $\mathbf{y}$-update. However, this gradient tracking property is not robust. For example, if $y_{i,0}$ are not properly initialized such that $\mathbf{1}^{\intercal}\mathbf{y}_0\neq \mathbf{1}^{\intercal}\nabla F(\mathbf{x}_0)$, then relation (\ref{oy_k_tracking}) will not hold for all $k>0$.\footnote{In fact, if $\mathbf{1}^{\intercal}\mathbf{y}_K\neq\mathbf{1}^{\intercal}\nabla F(\mathbf{x}_{K})$  for any $K>0$, then relation (\ref{oy_k_tracking}) will not hold for all $k>K$.} Under the scenario of noisy information exchange, instead of relation (\ref{oy_k_tracking}), we have
   \begin{equation*}
   \mathbf{1}^{\intercal}\mathbf{y}_k = \mathbf{1}^{\intercal}\nabla F(\mathbf{x}_{k})+\sum_{l=0}^{k-1}\boldsymbol{\xi}_l, \; \forall k,
   \end{equation*}
   which suggests that the gradient tracking will incur noise whose variance goes to infinity as $k$ grows.

   We remark here that although Push-Pull is not robust to noisy information exchange, it works well with stochastic gradient information under noiseless communication \cite{xin2019distributed2,pu2020distributed}. This is due to the fact that
   \begin{equation}
   \label{oy_k_stochastic}
   \mathbf{1}^{\intercal}\mathbf{y}_k = \mathbf{1}^{\intercal}G(\mathbf{x}_{k}), \; \forall k,
   \end{equation}
   where $G(\mathbf{x}_{k})$ stands for an unbiased estimate of $\nabla F(\mathbf{x}_k)$. As a result, the (stochastic) gradient tracking is still effective given that $G(\mathbf{x}_{k})$ has bounded variance. We will see below that after a variable transformation, R-Push-Pull resembles Push-Pull while ensuring gradient tracking in the form of (\ref{oy_k_stochastic}).

   Let $\mathbf{y}_k:=\mathbf{s}_{k+1}-\mathbf{s}_k,\; \forall k$. We have
   \begin{subequations}
   	\label{R_pushpull_transform}
   	\begin{align}
   	&\mathbf{x}_{k+1} =  \mathbf{R}_{\eta} \mathbf{x}_k+\eta\boldsymbol{\xi}_k-\alpha\mathbf{y}_k, 	\label{R_pushpull_transform_x}\\
   	& \mathbf{y}_{k+1} =  \mathbf{C}_{\gamma}\mathbf{y}_k+\tilde{\nabla} F(\mathbf{x}_{k+1})-\tilde{\nabla} F(\mathbf{x}_k), \label{R_pushpull_transform_y}
   	\end{align}
   \end{subequations}
   where
\begin{equation*}
\tilde{\nabla}F(\mathbf{x}_k):= \nabla F(\mathbf{x}_k)+\gamma\boldsymbol{\epsilon}_k,\; \forall k.
\end{equation*}
   This is exactly the Push-Pull update (\ref{Push_Pull}) if we ignore the noise and let $\gamma=\eta=1$.

   To see why R-Push-Pull is robust in gradient tracking, note that from (\ref{R_pushpull_s}), we have by induction that
   \begin{equation*}
   \mathbf{1}^{\intercal}\mathbf{s}_k = \mathbf{1}^{\intercal}\mathbf{s}_0 + \sum_{l=0}^{k-1}\mathbf{1}^{\intercal}\tilde{\nabla} F(\mathbf{x}_{k}), \; \forall k.
   \end{equation*}
   Hence $\mathbf{s}_k$ tracks the aggregated gradients of the network over the history. Then from the definition of $\mathbf{y}_k$, we have
   	\begin{equation*}
   	\mathbf{y}_0 = \mathbf{s}_1-\mathbf{s}_0=\mathbf{C}_{\gamma} \mathbf{s}_0+\tilde{\nabla}F(\mathbf{x}_0)-\mathbf{s}_0,
   	\end{equation*}
   	indicating that $\mathbf{1}^{\intercal}\mathbf{y}_0=\mathbf{1}^{\intercal}\tilde{\nabla}F(\mathbf{x}_0)$
   	regardless of the initial choice $\mathbf{s}_0$. In addition,
   	\begin{equation}
   	\label{y_k_sum}
   	\mathbf{1}^{\intercal}\mathbf{y}_k = \mathbf{1}^{\intercal}\mathbf{s}_{k+1}-\mathbf{1}^{\intercal}\mathbf{s}_k = \mathbf{1}^{\intercal}\tilde{\nabla} F(\mathbf{x}_{k}), \; \forall k,
   	\end{equation}
   	whose variance is bounded just like in equation (\ref{oy_k_stochastic}).

   	It is worth noting that the robust gradient tracking technique employed by R-Push-Pull, i.e., using $\mathbf{s}_k$ and $\mathbf{s}_{k+1}-\mathbf{s}_k$ to track the aggregated history gradients and the average gradient at step $k$ respectively, can also be applied to other gradient tracking based methods such as Push-DIGing/ADDOPT \cite{nedic2017achieving,xi2018add}.

	\section{Convergence Analysis}
	\label{sec: analysis}

	In this section, we study the convergence properties of R-Push-Pull.
	We first define the following variables:
	\begin{eqnarray*}
		\bar{x}_k  :=  \frac{1}{n}u^{\intercal} \mathbf{x}_k,\ \
		\bar{y}_k  :=  \frac{1}{n}\mathbf{1}^{\intercal}\mathbf{y}_k.
	\end{eqnarray*}
	Our strategy is to bound $\mathbb{E}[\|\bar{x}_{k+1}-x^*\|_2^2]$, $\mathbb{E}[\|\mathbf{x}_{k+1}-\mathbf{1}\bar{x}_{k+1}\|_{\mathrm{R}}^2]$ and $\mathbb{E}[\|\mathbf{y}_{k+1}-v\bar{y}_{k+1}\|_{\mathrm{C}}^2]$ in terms of linear combinations of their previous values, where $\|\cdot\|_{\mathrm{R}}$ and $\|\cdot\|_{\mathrm{C}}$ are specific norms to be defined later. In this way we establish a linear system of inequalities which allows us to derive the convergence results. The proof technique is similar to that of \cite{xin2019distributed,pu2020push} and was inspired by earlier works \cite{qu2017harnessing,xi2018linear}.

	\subsection{Preliminary Analysis}

	From relation (\ref{R_pushpull_transform_x}) and Lemma \ref{lem: eigenvectors u v}, we have
	\begin{equation}
	\label{ox_k+1 pre}
	\bar{x}_{k+1} = \frac{1}{n}u^{\intercal} (\mathbf{R}_{\eta}\mathbf{x}_k+\eta\boldsymbol{\xi}_k-\alpha \mathbf{y}_k)=\bar{x}_k-\frac{\alpha}{n}u^{\intercal}\mathbf{y}_k+\frac{\eta}{n}u^{\intercal}\boldsymbol{\xi}_k.
	\end{equation}
	By relation \eqref{y_k_sum},
	\begin{equation}
	\label{oy_ave}
	\bar{y}_k = \frac{1}{n}\mathbf{1}^{\intercal}\tilde{\nabla} F(\mathbf{x}_{k}), \ \ \forall k.
	\end{equation}
	Let us further define
	\begin{align*}
	h_k := \frac{1}{n}\mathbf{1}^{\intercal}\nabla F(\mathbf{x}_k)=\frac{1}{n}\sum_{i=1}^n f_i(x_{i,k}),\\
	g_k := \frac{1}{n}\mathbf{1}^{\intercal}\nabla F(\mathbf{1}\bar{x}_k) = \frac{1}{n}\sum_{i=1}^n f_i(\bar{x}_k).
	\end{align*}
	Clearly, $\bar{y}_k=h_k+\frac{\gamma}{n}\mathbf{1}^{\intercal}\boldsymbol{\epsilon}_k$.
	Then, we obtain from relation (\ref{ox_k+1 pre}) that
	\begin{equation}
		\label{ox pre}
		\begin{array}{l}
		\bar{x}_{k+1} = \bar{x}_k-\frac{\alpha}{n}u^{\intercal}\left(\mathbf{y}_k-v\bar{y}_k+v\bar{y}_k\right)+\frac{\eta}{n}u^{\intercal}\boldsymbol{\xi}_k\\
		= \bar{x}_k-\frac{\alpha}{n}u^{\intercal}v\bar{y}_k-\frac{\alpha}{n}u^{\intercal}\left(\mathbf{y}_k-v\bar{y}_k\right)+\frac{\eta}{n}u^{\intercal}\boldsymbol{\xi}_k\\
		= \bar{x}_k-\tilde{\alpha} g_k-\tilde{\alpha} (h_k-g_k)-\frac{\alpha}{n}u^{\intercal}\left(\mathbf{y}_k-v\bar{y}_k\right)\\
		\quad -\tilde{\alpha}\frac{\gamma}{n}\mathbf{1}^{\intercal}\boldsymbol{\epsilon}_k+\frac{\eta}{n}u^{\intercal}\boldsymbol{\xi}_k,
		\end{array}
	\end{equation}
	where
	\begin{equation}
	\label{alpha'}
	\tilde{\alpha} :=\frac{\alpha}{n} u^{\intercal}v.\footnote{Assumption \ref{asp: stochastic} ensures $\tilde{\alpha} >0$ (see \cite{pu2018distributed}).}
	\end{equation}

	In view of \eqref{R_pushpull_transform_x} and Lemma \ref{lem: eigenvectors u v}, using \eqref{ox_k+1 pre} we have
	\begin{equation}
	\label{mx-ox pre}
	\begin{array}{ll}
	\mathbf{x}_{k+1}-\mathbf{1}\bar{x}_{k+1} \\
	= \mathbf{R}_{\eta}\mathbf{x}_k+\eta\boldsymbol{\xi}_k-\alpha \mathbf{y}_k-\mathbf{1}\bar{x}_k+\alpha\frac{\mathbf{1}u^{\intercal}}{n}\mathbf{y}_k-\frac{\mathbf{\eta}u^{\intercal}}{n}\boldsymbol{\xi}_k\\
	=  \mathbf{R}_{\eta}(\mathbf{x}_k-\mathbf{1}\bar{x}_k)-\alpha\left(\mathbf{I}-\frac{\mathbf{1}u^{\intercal}}{n}\right)\mathbf{y}_k+\eta(\mathbf{I}-\frac{\mathbf{1}u^{\intercal}}{n})\boldsymbol{\xi}_k\\
	=  \left(\mathbf{R}_{\eta}-\frac{\mathbf{1}u^{\intercal}}{n}\right)(\mathbf{x}_k-\mathbf{1}\bar{x}_k)-\alpha\left(\mathbf{I}-\frac{\mathbf{1}u^{\intercal}}{n}\right)\mathbf{y}_k\\
	\quad +\eta(\mathbf{I}-\frac{\mathbf{1}u^{\intercal}}{n})\boldsymbol{\xi}_k,
	\end{array}
	\end{equation}
	and from \eqref{R_pushpull_transform_y} and \eqref{oy_ave} we obtain
	\begin{equation}
	\label{my-oy pre}
	\begin{array}{ll}
	\mathbf{y}_{k+1}-v\bar{y}_{k+1} \\
	= \mathbf{C}_{\gamma}\mathbf{y}_k+\tilde{\nabla} F(\mathbf{x}_{k+1})-\tilde{\nabla} F(\mathbf{x}_k)-v(\bar{y}_{k+1}-\bar{y}_k)-v\bar{y}_k\\
	= \mathbf{C}_{\gamma}\mathbf{y}_k-v\bar{y}_k+\left(\mathbf{I}-\frac{v\mathbf{1}^{\intercal}}{n}\right)\left(\tilde{\nabla} F(\mathbf{x}_{k+1})-\tilde{\nabla} F(\mathbf{x}_k)\right)\\
	=\left(\mathbf{C}_{\gamma}-\frac{v\mathbf{1}^{\intercal}}{n}\right)(\mathbf{y}_k-v\bar{y}_k)\\
	\quad +\left(\mathbf{I}-\frac{v\mathbf{1}^{\intercal}}{n}\right)\left(\tilde{\nabla} F(\mathbf{x}_{k+1})-\tilde{\nabla} F(\mathbf{x}_k)\right).
	\end{array}
	\end{equation}

	Denote by $\mathcal{F}_k$ the $\sigma$-algebra generated by $\{\boldsymbol{\epsilon}_0,\boldsymbol{\xi}_0\ldots,\boldsymbol{\epsilon}_{k-1},\boldsymbol{\xi}_{k-1}\}$, and define $\mathbb{E}[\cdot \mid\mathcal{F}_k]$ as the conditional expectation given $\mathcal{F}_k$. We prepare a few useful supporting lemmas for our further analysis, deferred to Appendix \ref{subsec: supporting lemmas}.

	\subsection{Main Results}
	The following lemma establishes a linear system of inequalities that bound $\mathbb{E}[\|\bar{x}_{k+1}-x^*\|_2^2]$, $\mathbb{E}[\|\mathbf{x}_{k+1}-\mathbf{1}\bar{x}_k\|_{\mathrm{R}}^2]$ and $\mathbb{E}[\|\mathbf{y}_{k+1}-v\bar{y}_k\|_{\mathrm{C}}^2]$.
	\begin{lemma}
		\label{lem: important inequalities}
		Under Assumptions \ref{asp; strconvex Lipschitz}-\ref{asp: stochastic}, when $\tilde{\alpha} \le 1/(\mu+L)$, we have the following linear system of inequalities:
		\begin{equation}
		\label{main ineqalities}
		\begin{bmatrix}
		\mathbb{E}[\|\bar{x}_{k+1}-x^*\|_2^2]\\
		\mathbb{E}[\|\mathbf{x}_{k+1}-\mathbf{1}\bar{x}_{k+1}\|_{\mathrm{R}}^2]\\
		\mathbb{E}[\|\mathbf{y}_{k+1}-v\bar{y}_{k+1}\|_{\mathrm{C}}^2]
		\end{bmatrix}
		\le
		\mathbf{A}
		\begin{bmatrix}
		\mathbb{E}[\|\bar{x}_k-x^*\|_2^2]\\
		\mathbb{E}[\|\mathbf{x}_k-\mathbf{1}\bar{x}_k\|_{\mathrm{R}}^2]\\
		\mathbb{E}[\|\mathbf{y}_k-v\bar{y}_k\|_{\mathrm{C}}^2]
		\end{bmatrix}+\mathbf{B},
		\end{equation}
		where the inequality is to be taken component-wise. The transition matrix $\mathbf{A}=[a_{ij}]$ and the vector $\mathbf{B}$ are given by:
		\begin{equation}
		\mathbf{A} = \begin{bmatrix}
		1-\tilde{\alpha} \mu & c_1\alpha & c_2\alpha\\
		c_5\alpha^2 & \frac{1+\tau_{\mathrm{R}}^2}{2}+c_6\alpha^2 & c_7\alpha^2\\
		c_{10}\alpha^2 & c_{11} & \frac{1+\tau_{\mathrm{C}}^2}{2}+c_{12}\alpha^2
		\end{bmatrix}
		\end{equation}
		and
		\begin{equation}
		\mathbf{B}=\begin{bmatrix}
		c_3\alpha^2\\
		c_8\alpha^2\\
		c_{13}
		\end{bmatrix}
		\gamma^2\sigma_{\boldsymbol{\epsilon}}^2
		+
		\begin{bmatrix}
		c_4\\
		c_9\\
		c_{14}
		\end{bmatrix}
		\eta^2\sigma_{\boldsymbol{\xi}}^2,
		\end{equation}
		respectively, where constants $c_1-c_{13}$ are defined in (\ref{c1-c4}), \eqref{c5-c9}, and \eqref{c10-c13}.
	\end{lemma}
	\begin{proof}
		See Appendix \ref{proof lem: important inequalities}.
	\end{proof}
		The following theorem established the convergence properties for R-Push-Pull in \eqref{R_pushpull}.
	\begin{theorem}
		\label{Theorem1}
		Suppose Assumptions \ref{asp; strconvex Lipschitz}-\ref{asp: stochastic} hold and
		the stepsize $\alpha$ satisfies
		\begin{equation}
		\label{alpha_ultimate_bound}
		\alpha\le \min\left\{\sqrt{\frac{1-\tau_{\mathrm{R}}^2}{6c_6}},\sqrt{\frac{1-\tau_{\mathrm{C}}^2}{6c_{12}}},\sqrt{\frac{2d_3}{d_2+\sqrt{d_2^2+4d_1d_3}}}\right\},
		\end{equation}
		where $d_1-d_3$ are defined in \eqref{d1-d3}.
		Then $\sup_{l\ge k}\mathbb{E}[\|\bar{x}_l-x^*\|_2^2]$ and $\sup_{l\ge k}\mathbb{E}[\|\mathbf{x}_{l}-\mathbf{1}\bar{x}_{l}\|_{\mathrm{R}}^2]$, respectively, converge to $\limsup_{k\rightarrow\infty}\mathbb{E}[\|\bar{x}_k-x^*\|_2^2]$ and $\limsup_{k\rightarrow\infty}\mathbb{E}[\|\mathbf{x}_k-\mathbf{1}\bar{x}_k\|_{\mathrm{R}}^2]$ at the linear rate $\mathcal{O}(\rho(\mathbf{A})^k)$, where $\rho(\mathbf{A})<1$ is the spectral radius of the matrix $\mathbf{A}$. Furthermore,
		\begin{equation}
		\label{limsup_pre}
		\begin{split}
		& \limsup_{k\rightarrow\infty}\mathbb{E}[\|\bar{x}_k-x^*\|_2^2] \le [(\mathbf{I}-\mathbf{A})^{-1}\mathbf{B}]_1,\\
		& \limsup_{k\rightarrow\infty}\mathbb{E}[\|\mathbf{x}_k-\mathbf{1}\bar{x}_k\|_{\mathrm{R}}^2] \le [(\mathbf{I}-\mathbf{A})^{-1}\mathbf{B}]_2,
		\end{split}
		\end{equation}
		where $[(\mathbf{I}-\mathbf{A})^{-1}\mathbf{B}]_j$ denotes the $j$-th element of the vector $[(\mathbf{I}-\mathbf{A})^{-1}\mathbf{B}]$. Their specific forms are given in (\ref{opt_error_bound}) and (\ref{consensus_error_bound}) respectively.
	\end{theorem}
	\begin{proof}
		In light of Lemma \ref{lem: important inequalities}, by induction we have
		{\small\begin{eqnarray}
			\label{linear_system_bound}
			\begin{bmatrix}
			\mathbb{E}[\|\bar{x}_{k}-x^*\|_2^2]\\
			\mathbb{E}[\|\mathbf{x}_{k}-\mathbf{1}\bar{x}_{k}\|_{\mathrm{R}}^2]\\
			\mathbb{E}[\|\mathbf{y}_{k}-v\bar{y}_{k}\|_{\mathrm{C}}^2]
			\end{bmatrix}
			\le
			\mathbf{A}^k\begin{bmatrix}
			\mathbb{E}[\|\bar{x}_{0}-x^*\|_2^2]\\
			\mathbb{E}[\|\mathbf{x}_{0}-\mathbf{1}\bar{x}_{0}\|_{\mathrm{R}}^2]\\
			\mathbb{E}[\|\mathbf{y}_{0}-v\bar{y}_{0}\|_{\mathrm{C}}^2]
			\end{bmatrix}+\sum_{l=0}^{k-1}\mathbf{A}^l\mathbf{B}.
			\end{eqnarray}}\normalsize
		If the spectral radius of $\mathbf{A}$ satisfies $\rho(\mathbf{A})<1$, then $\mathbf{A}^k$ converges to $\mathbf{0}$ at the linear rate $\mathcal{O}(\rho(\mathbf{A})^k)$  (see \cite{horn1990matrix}), in which case $\sup_{l\ge k}\mathbb{E}[\|\bar{x}_l-x^*\|^2]$, $\sup_{l\ge k}\mathbb{E}[\|\mathbf{x}_l-\mathbf{1}\bar{x}_l\|^2]$ and $\sup_{l\ge k}\mathbb{E}[\|\mathbf{y}_l-\mathbf{1}\bar{y}_l\|^2]$ all converge to a neighborhood of $0$ at the linear rate $\mathcal{O}(\rho(\mathbf{A})^k)$.

		The next lemma provides conditions that ensure $\rho(\mathbf{A})<1$.
		\begin{lemma}
			\label{lem: rho_M}
			(Lemma 5 in \cite{pu2020distributed}) Given a nonnegative, irreducible matrix $\mathbf{M}=[m_{ij}]\in\mathbb{R}^{3\times 3}$ with $m_{11},m_{22},m_{33}<\lambda^*$ for some $\lambda^*>0$. A necessary and sufficient condition for $\rho(\mathbf{M})<\lambda^*$ is $\text{det}(\lambda^* \mathbf{I}-\mathbf{M})>0$.
		\end{lemma}

		In light of Lemma \ref{lem: rho_M}, it suffices to ensure $a_{11},a_{22},a_{33}<1$ and $\text{det}(\mathbf{I}-\mathbf{A})>0$, or more aggressively,
		{\begin{equation}
			\label{|I-A|>0 pre}
			\begin{split}
			& \text{det}(\mathbf{I}-\mathbf{A})=(1-a_{11})(1-a_{22})(1-a_{33})-a_{12}a_{23}a_{31}\\
			& \quad -a_{13}a_{21}a_{32}-(1-a_{22})a_{13}a_{31}-(1-a_{11})a_{23}a_{32}\\
			& \quad -(1-a_{33})a_{12}a_{21}>\frac{1}{2}(1-a_{11})(1-a_{22})(1-a_{33}),
			\end{split}
			\end{equation}}\normalsize
		which is equivalent to
			\begin{equation}
			\label{|I-A|>0}
			\begin{split}
			& \frac{1}{2}(1-a_{11})(1-a_{22})(1-a_{33})-c_1 c_7 c_{10}\alpha^5\\
			& \quad -c_2 c_5 c_{11}\alpha^3-(1-a_{22})c_2c_{10}\alpha^3-(1-a_{11})c_7 c_{11}\alpha^2\\
			& \quad -(1-a_{33})c_1 c_5 \alpha^3>0.
			\end{split}
			\end{equation}\normalsize
		We now provide some sufficient conditions under which $a_{11},a_{22},a_{33}<1$ and (\ref{|I-A|>0}) holds true.

		First, $a_{11}<1$ is ensured by choosing $\tilde{\alpha} \le 1/(\mu+L)$, and $a_{22},a_{33}<1$ is guaranteed by
		\begin{equation}
		\label{1-a22,a33ge}
		1-a_{22} \ge \frac{1}{3}(1-\tau_{\mathrm{R}}^2),\, 1-a_{33} \ge \frac{1}{3}(1-\tau_{\mathrm{C}}^2),
		\end{equation}
		which requires
		\begin{equation}
		\label{alpha loose condition}
		\alpha\le \min\left\{\sqrt{\frac{1-\tau_{\mathrm{R}}^2}{6c_6}},\sqrt{\frac{1-\tau_{\mathrm{C}}^2}{6c_{12}}}\right\}.
		\end{equation}

		Second, notice that $a_{22}>\frac{1+\tau_{\mathrm{R}}^2}{2}$ and $a_{33}>\frac{1+\tau_{\mathrm{C}}^2}{2}$. In light of \eqref{1-a22,a33ge}, a sufficient condition for $\text{det}(\mathbf{I}-\mathbf{A})>0$ is to substitute the first $(1-a_{22})$ (respectively, $(1-a_{33})$) in (\ref{|I-A|>0}) by $(1-\tau_{\mathrm{R}}^2)/3$ (respectively, $(1-\tau_{\mathrm{C}}^2)/3$), and substitute the second $(1-a_{22})$ (respectively, $(1-a_{33})$) by $(1-\tau_{\mathrm{R}}^2)/2$ (respectively, $(1-\tau_{\mathrm{C}}^2)/2$). We then have
		\begin{equation*}
		d_1\alpha^4+d_2\alpha^2-d_3<0,
		\end{equation*}
		where
		\begin{equation}
		\label{d1-d3}
		\begin{split}
		& d_1 := c_1 c_7 c_{10}\\
		& d_2 := c_2 c_5 c_{11}+\frac{(1-\tau_{\mathrm{R}}^2)}{2}c_2 c_{10}+\mu u^{\intercal}v c_7 c_{11}\\
		& \quad +\frac{(1-\tau_{\mathrm{C}}^2)}{2}c_1 c_5\\
		& d_3 := \frac{\mu u^{\intercal}v(1-\tau_{\mathrm{R}}^2)(1-\tau_{\mathrm{C}}^2)}{18}.
		\end{split}
		\end{equation}
		Hence
		\begin{equation}
		\label{alpha strict condition}
		\alpha^2\le \frac{2d_3}{d_2+\sqrt{d_2^2+4d_1d_3}}.
		\end{equation}
		Relations (\ref{alpha loose condition}) and (\ref{alpha strict condition}) yield the final bound on $\alpha$.

		Relation \eqref{limsup_pre} follows from (\ref{linear_system_bound}) directly given that $\rho(\mathbf{A})<1$. In light of \eqref{|I-A|>0 pre} and \eqref{1-a22,a33ge}, we obtain from (\ref{linear_system_bound}) that

		{\small\begin{equation}
			\label{opt_error_bound}
			\begin{split}
			&  [(\mathbf{I}-\mathbf{A})^{-1}\mathbf{B}]_1\\
			& = \left[((1-a_{22})(1-a_{33})-a_{23}a_{32})(c_3\alpha^2\gamma^2\sigma_{\boldsymbol{\epsilon}}^2+c_4\eta^2\sigma_{\boldsymbol{\xi}}^2)\right.\\
			& \quad +(a_{13}a_{32}+a_{12}(1-a_{33}))(c_8\alpha^2\gamma^2\sigma_{\boldsymbol{\epsilon}}^2+c_9\eta^2\sigma_{\boldsymbol{\xi}}^2))\\
			& \quad \left. +(a_{12}a_{23}+a_{13}(1-a_{22}))(c_{13}\gamma^2\sigma_{\boldsymbol{\epsilon}}^2+c_{14}\eta^2\sigma_{\boldsymbol{\xi}}^2))\right]\frac{1}{\text{det}(\mathbf{I}-\mathbf{A})}\\
			& \le \left[\left(\frac{(1-\tau_{\mathrm{R}}^2)(1-\tau_{\mathrm{C}}^2)}{4}-c_7 c_{11}\alpha^2\right)(c_3\alpha^2\gamma^2\sigma_{\boldsymbol{\epsilon}}^2+c_4\eta^2\sigma_{\boldsymbol{\xi}}^2))\right.\\
			& \quad +\left(c_2c_{11}\alpha+\frac{c_1(1-\tau_{\mathrm{C}}^2)\alpha}{2}\right)(c_8\alpha^2\gamma^2\sigma_{\boldsymbol{\epsilon}}^2+c_9\eta^2\sigma_{\boldsymbol{\xi}}^2))\\
			& \quad \left. +\left(c_1c_7\alpha^3+\frac{c_2(1-\tau_{\mathrm{R}}^2)\alpha}{2}\right)(c_{13}\gamma^2\sigma_{\boldsymbol{\epsilon}}^2+c_{14}\eta^2\sigma_{\boldsymbol{\xi}}^2))\right]\\
			& \quad \cdot\frac{18}{\tilde{\alpha} \mu(1-\tau_{\mathrm{R}}^2)(1-\tau_{\mathrm{C}}^2)},
			\end{split}
			\end{equation}}\normalsize
		and
		{\small\begin{equation}
			\label{consensus_error_bound}
			\begin{split}
			&  [(\mathbf{I}-\mathbf{A})^{-1}\mathbf{B}]_2\\
			& =\left[(a_{23}a_{31}+a_{21}(1-a_{33}))(c_3\alpha^2\gamma^2\sigma_{\boldsymbol{\epsilon}}^2+c_4\eta^2\sigma_{\boldsymbol{\xi}}^2)\right.\\
			& \quad  +((1-a_{11})(1-a_{33})-a_{13}a_{31})(c_8\alpha^2\gamma^2\sigma_{\boldsymbol{\epsilon}}^2+c_9\eta^2\sigma_{\boldsymbol{\xi}}^2)\\
			& \quad \left. +(a_{13}a_{21}+a_{23}(1-a_{11}))(c_{13}\gamma^2\sigma_{\boldsymbol{\epsilon}}^2+c_{14}\eta^2\sigma_{\boldsymbol{\xi}}^2)\right]\frac{1}{\text{det}(\mathbf{I}-\mathbf{A})}\\
			& \le \left[\left(c_7c_{10}\alpha^4+c_5\alpha^2\frac{(1-\tau_{\mathrm{C}}^2)}{2}\right)(c_3\alpha^2\gamma^2\sigma_{\boldsymbol{\epsilon}}^2+c_4\eta^2\sigma_{\boldsymbol{\xi}}^2)\right.\\
			& \quad +\left(\tilde{\alpha} \mu\frac{(1-\tau_{\mathrm{C}}^2)}{2}-c_2c_{10}\alpha^3\right)(c_8\alpha^2\gamma^2\sigma_{\boldsymbol{\epsilon}}^2+c_9\eta^2\sigma_{\boldsymbol{\xi}}^2)\\
			& \quad \left. +\left(c_2 c_5\alpha^3+c_7\alpha^2\tilde{\alpha} \mu\right)(c_{13}\gamma^2\sigma_{\boldsymbol{\epsilon}}^2+c_{14}\eta^2\sigma_{\boldsymbol{\xi}}^2)\right]\\
			& \quad \cdot\frac{18}{\tilde{\alpha} \mu(1-\tau_{\mathrm{R}}^2)(1-\tau_{\mathrm{C}}^2)}.\\
			\end{split}
			\end{equation}}\normalsize
	\end{proof}
	From Theorem \ref{Theorem1}, if $\sigma_{\boldsymbol{\epsilon}}=\sigma_{\boldsymbol{\xi}}=\mathbf{0}$ (no noise), we have $\mathbf{B}=\mathbf{0}$, then R-Push-Pull converges linearly to the optimal solution $x^*$. Specifically, when $\alpha$ is sufficiently small, it can be shown that the linear rate indicator $\rho(\mathbf{A})\simeq 1-\tilde{\alpha} \mu$.

	The upper bounds in (\ref{opt_error_bound}) and (\ref{consensus_error_bound}) are functions of $\alpha$, $\eta$, $\gamma$ and other problem parameters, and they are decreasing in the variances $\sigma_{\boldsymbol{\epsilon}}$ and $\sigma_{\boldsymbol{\xi}}$. It can also be numerically verified that $\limsup_{k\rightarrow\infty}\mathbb{E}[\|\bar{x}_k-x^*\|_2^2]$
	and $\limsup_{k\rightarrow\infty}\mathbb{E}[\|\mathbf{x}_k-\mathbf{1}\bar{x}_k\|_{\mathrm{R}}^2]$ decrease in the stepsize $\alpha$.
	Moreover, in order to mitigate the effect of noise $\boldsymbol{\epsilon}_k$ on the final optimization error, we may take $\eta$ to be in the order of $\mathcal{O}(\alpha)$.

	\section{Numerical Example}
	\label{sec: numerical}

	We provide a simple illustration example. Consider the Ridge regression problem, i.e.,
	\begin{equation}
	\label{Ridge Regression}
	\min_{x\in \mathbb{R}^{p}}f(x)=\frac{1}{n}\sum_{i=1}^nf_i(x)\left(=\left(u_i^{\intercal} x-v_i\right)^2+\rho\|x\|^2\right),
	\end{equation}
	where $\rho>0$ is a penalty parameter.
	Each agent $i$ has sample $(u_i,v_i)$ with $u_i\in\mathbb{R}^p$ representing the features and $v_i\in\mathbb{R}$ being the observed outputs. Let each $u_i\in[-1,1]^p$ be generated from uniform distribution, and $v_i$ is drawn according to $v_i=u_i^{\intercal} \tilde{x}_i+\varepsilon_i$, where parameters $\tilde{x}_i$ are  evenly located in $[0,10]^p$, and $\varepsilon_i\sim\mathcal{N}(0,25)$.\footnote{These randomly generated problem parameters have negligible effects on the simulation results.} Given these predetermined parameters,
	Problem (\ref{Ridge Regression}) has a unique solution $x^*=(\sum_{i=1}^n[u_iu_i^{\intercal}]+n\rho\mathbf{I})^{-1}\sum_{i=1}^n[u_iu_i^{\intercal}]\tilde{x}_i$.

	We compare the performance of R-Push-Pull against other two gradient tracking based algorithms Push-Pull/AB \cite{pu2018push,xin2018linear} and Push-DIGing/ADDOPT \cite{nedic2017achieving,xi2018add}.
	To model noisy information exchange, we assume the transmitted values such as $x_{i,k}$, $C_{li}y_{i,k}$ and $C_{li}s_{i,k}$ are corrupted with independent Gaussian noises $\mathcal{N}(0,0.01)$.
	Regarding the network topology, we generate a directed graph $\mathcal{G}$ of $15$ nodes by adding random links to a ring network, where a directed link exists between any two nonadjacent nodes with probability $0.3$. Then for R-Push-Pull and Push-Pull, we let $\mathcal{G}_{\mathbf{R}}=\mathcal{G}_{\mathbf{C}}=\mathcal{G}$ for simplicity.

	Design matrix $\mathbf{C}$ as follows: for any agent $i$,
	$C_{li}=\frac{1}{|\outneighbor{\mathbf{C},i}|+1}$ for all $l\in\outneighbor{\mathbf{C},i}$ and $C_{ii}=1-\sum_{l\in \outneighbor{\mathbf{C},i}}C_{li}$. Letting $\gamma=0.5$, the same $\mathbf{C}_{\gamma}$ is used for all the algorithms.
	In R-Push-Pull and Push-Pull, for any agent $i$, $R_{ij}=\frac{1}{|\inneighbor{\mathbf{R},i}|+1}$ for all $j\in \inneighbor{\mathbf{R},i}$, and $R_{ii}=1-\sum_{j\in \inneighbor{\mathbf{R},i}}R_{ij}$. The matrix $\mathbf{R}_{\eta}$ is constructed by taking $\eta=0.01$.

	\begin{figure}[htbp]
	\centering
	\includegraphics[width=3.2in]{comparison.eps}
		\caption{Comparison of the performance of R-Push-Pull, Push-Pull/AB and Push-DIGing/ADDOPT, measured by $\frac{1}{n}\mathbb{E}[\|x_{i,k}-x^*\|^2]$. The expected errors are approximated by averaging over $50$ simulation results. Dimension $p=10$, stepsize $\alpha=0.01$ and penalty parameter $\rho=0.01$.}
		\label{Figure_comparison}
	\end{figure}

Fig. \ref{Figure_comparison} compares the performance of different algorithms with respect to $\frac{1}{n}\mathbb{E}[\|x_{i,k}-x^*\|^2]$.
It can be seen that initially all the errors decrease exponentially fast at comparable rates. However, the errors for Push-Pull/AB and Push-DIGing/ADDOPT eventually increase over time while R-Push-Pull achieves linear convergence to a small neighborhood of $0$. This sharp contrast verifies the effectiveness of the proposed algorithm.

	\section{Conclusions}
	\label{sec: conclusions}
	In this paper, we introduce a robust gradient tracking method (R-Push-Pull) for distributed optimization over directed networks.

	R-Push-Pull inherits the advantage of Push-pull and achieves linear convergence to the optimal solution with exact information fusion.
	Under noisy information exchange, R-Push-Pull is more robust than the other gradient tracking based algorithms. We show the solutions obtained by each agent reach a neighborhood of the optimum in expectation exponentially fast under a constant stepsize policy. We also provide a numerical example that demonstrate the effectiveness of R-Push-Pull.

	\section*{ACKNOWLEDGMENT}
	We would like to thank Wei Shi from Princeton University and Jinming Xu from Zhejiang University for helpful discussions.

	\bibliographystyle{IEEEtran}

redpajama_set_name: RedPajamaArXiv


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Spend a day in Vientiane before an overnight train to Bangkok. Spend the morning visiting the sights, getting a massage or enjoying the local food scene. In the late afternoon, cross the Friendship Bridge back into Thailand to board the overnight train to Bangkok.


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Influential Women of the 1800s
Presented by History Center of Niagara at The Historic Palace Theatre, Lockport NY
In a collaboration of free programs with the Historic Palace Theatre, The History Center presents a portrayal of two important women, Belva Lockwood and Eliza Mossell, who took on both political and societal injustices in order to change the world.
The History Center's own "Step Back in Time Players" will guide us on a narrative journey through the life and times of two very significant figures during the 19th century. Marsha Zimmer will portray Belva Lockwood. She was an author, lawyer, and politician; gaining entry as the first woman admitted to the Bar of the Supreme Court. Her work is particularly important to several social reforms during that time, including equal pay for working women.
Brenda Reaves will play Eliza Mossell,
Brenda Reaves will play Eliza Mossell, the wife and mother of an outstanding African American family who were the first to achieve success in many social and professional areas. She and her husband Aaron Albert Mossell, Sr moved to Lockport in 1866 where they gave their children an education to succeed. Along with their various other achievements, her husband and son Charles are credited with desegregating Lockport, New York schools in 1876 – eight decades before segregated schools were ruled unconstitutional.
Together they will show us the world in which they lived along with their many struggles and accomplishments. The Palace concessions will also be made available for purchase.
(Donations Appreciated)
Email: info@niagarahistory.org
The Historic Palace Theatre
2 East Ave, Lockport, NY 14094
CONNECT WITH History Center of Niagara


Classic car show. Part of a First Fridays event in small town Indiana.
High dynamic range (HDR) photography and other modern photo styles are great for destination tourism marketing of small town / city events.


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DEFxColorado: Innovation @ Elevation
by Defense Entrepreneurs Forum
Thank you for your registration and we look forward to seeing you at the inaugural DEFxColorado!
Sat, August 12, 2017
8:30 AM – 8:00 PM MDT
Eisenhower Golf Course
U.S. Air Force Academy, CO 80840
Refunds up to 7 days before event
Colorado is home to numerous military bases and startup communities; DEFxColorado will leverage the strengths of both groups to create innovation @ elevation within the defense community. DEFxColorado will be held in Colorado Springs – our Nation's national security capital of the West – home of the Air Force Academy, Fort Carson, Peterson AFB, and Schriever AFB, with Buckley AFB just down the road. Denver, Boulder, Fort Collins and Colorado Springs all have active ROTC programs and boast growing startup scenes with innovative companies.
Building on the success of the monthly Colorado Springs Drink & Think meetups, we're hosting a day long DEFxColorado to tackle issues at a deeper level. We will begin the day with TED talk-style speakers and a panel of successful innovators. We'll then learn Lean Startup principles and Influence Strategies to develop real solutions and strategies to drive change by applying them to real-world challenges.
DEFxColorado is an event organized by the Defense Entrepreneurs Forum a registered 501(c)(3) non-profit organization disrupting the status quo by inspiring, connecting, and empowering leaders to create a culture of innovation in the defense community. As with all DEF events, uniforms and ties can be left at home.
Friday, August 11th - Happy Hour at Cockpit Craft Distillery (Evening)
Cockpit Craft Distillery is a local, veteran-owned business started recently by Calder Curtis, our very first Drink & Think speaker. Cockpit has some of the best drinks in town and is generously providing a complimentary first drink, discounted food and a tour of the distillery to DEFxColorado attendees.
Saturday, August 12th - USAFA's Eisenhower's Golf Course (All Day)
Our Keynote and Closing Speakers
Colonel Jeffrey Collins is the Director of Air Force CyberWorx, a new venture comprising a public-private research and design center located at USAFA. CyberWorx melds military, academic and industry expertise with the latest technology, policy- thinking and user-centered design to solve wicked operational problems.
Sean Maday is a cloud engineer at Google where he specializes in leveraging machine learning to solve business challenges. Sean is an Air Force reservist who has held a variety of technology and entrepreneurship positions and was the first full time director of Patriot Boot Camp, a non profit helping Veterans build tech companies.
Our Panel of DOD Intrapreneurs
Major Joe Byerly is the founder of the widely read blog, From the Green Notebook. The blog features leadership lessons from NCOs to 4-star generals. Joe is an Army Armor Officer currently assigned to 1st Stryker Brigade, 4th Infantry Division.
Major Matt Cavanaugh is an Army Strategist currently assigned to Army Space & Missile Defense Command (a component of Strategic Command). He co-founded and remains a Fellow with, the Modern War Institute at West Point. His writings have been featured in the New York Times, Washington Post, and Wall Street Journal, among other outlets.
Lieutenant Colonel Aaron Roof witnessed and led innovation in elite Counter Terrorism units for over a decade while serving in US Army Special Operations Command. Aaron is an Infantry Officer and currently the Professor of Military Science at the University of Colorado, Boulder.
Sara Kinney is the CEO of Rim Technologies where her solutioneers empower Innovation to solve our nation's toughest challenges. Their tech supports SMC Satellite Control Network, GPS, 24 AF, AFRL, AFSPC and the Pentagon. A former pilot, Sara has supported STRATCOM, Air Force One and Harris & Odyssey Systems.
Our Intrapreneur's Toolkit Speakers
Darren Halford is the former Executive Director of Hacking for Defense Incorporated (H4Di). Under Darren's leadership, H4Di expanded the course started at Stanford University to schools nationwide, driving change in the public sector by applying Lean Startup principles to real-world challenges. Darren is a retired Air Force Colonel who was instrumental in standing up the MC-12 Project Liberty intelligence collection aircraft program which was the fastest weapons system delivered from concept to combat since the P-51 Mustang in World War II. He will also be on the panel.
Lieutenant Colonel Aram Donigian co-founded the West Point Negotiation Project in 2009, has taught negotiation at West Point and the Air Force Academy and has provided negotiation training for dozens of units, including Special Operations. He has co-authored articles in the Harvard Business Review, Military Review, and elsewhere. Aram is an Infantry officer and currently, serves in the Department of Behavioral Sciences and Leadership at the Air Force Academy.
All of our sponsors are small businesses owned and operated by veterans in Colorado.
Lead Sponsor - BluCore Shooting Center is a full-service shooting facility based in Denver, owned and operated by former special operations personnel.
Happy Hour Sponsor - Cockpit Craft Distillery a new Colorado Springs-based distillery started by Calder Curtis.
Video & Photography Sponsor - Jay Billups Creative Media a new creative media business started by Jay Billups in Colorado Springs.
Our partners are other organizations with similar missions in Colorado Springs.
Peak Startup is growing a strong, vibrant startup ecosystem of companies, entrepreneurs, and talent in Colorado Springs.
Colorado Springs Rising Professionals is a non-profit, created by and for individuals between the ages of 21-39, focused on engaging young business professionals in our community.
Colorado Springs World Affairs Council's mission is to create and sustain study, discussion, and public participation in international relations—in essence, non-partisan, grassroots education on world affairs.
Colorado Springs Young Professionals has been around since 2004 and was founded to bring together young professionals in all industries and help them establish roots in Colorado Springs.
VISITING COLORADO SPRINGS
If you want to arrive early or stay late there are plenty of activities to fill your schedule. Book a tee time at USAFA's #1 ranked golf course in the DoD, visit Garden of the Gods, go fly fishing and hike the Manitou Incline with 2,000 feet of elevation change in less than a mile. You can also hike, drive or take a cog train up Pikes Peak "America's Mountain" and great Whitewater Rafting isn't far away either. Check out Visit Colorado Springs Website for more information on things to do in the area. Email us at DEFxCOS@defenseentrepreneurs.org if you have any questions or would like suggestions.
What does the ticket price include?
The price includes everything you will need to attend and participate in the event including food and drinks. There are no admissions fees to access USAFA or the golf course for the event.
Friday Happy Hour - Complimentary first drink at Cockpit distillery and discounted food.
Breakfast - Fresh fruit, pastries, and yogurt with coffee.
Lunch - Smoked turkey, ham and roast beef sandwiches with chips, cookies, iced tea, and lemonade.
Dinner - Barbeque, beans, chips, and drinks out on the back patio of the clubhouse.
How much are the tickets?
Early Bird tickets (through June 30th) are $35, Normal tickets are $40 (through July 31st) and Last Call tickets are $45 (through August 8th).
I'm coming from out of town, where should I stay?
We've reserved a block of rooms from 10-13 August at the Hampton Inn Colorado Springs located just minutes from the Air Force Academy at 7245 Commerce Center Dr. Rooms are $124.00 per night plus tax and include complimentary parking, breakfast, and wifi. Rooms can be booked online or by calling 719-593-9700 to book your room and let them know you're with the DEFxColorado event. Make your reservations soon as the rate is only guaranteed until July 10th.
Which gate should I use?
The North Gate off of I-25, Exit 156.
What are my transportation/parking options for getting to and from the event?
Parking is available on-site at the Eisenhower Golf Course.
What is the attire for the event?
As with all DEF events, uniforms and ties can be left at home, business casual is fine.
I'm a civilian and don't have a Department of Defense ID card, what do I do?
If you purchase a ticket before July 31st, you are good to go - just make sure to bring a form of government ID. We will be providing the gate guards with a roster beforehand.
If you purchased your ticket on or after August 1st, please contact us at DEFxCOS@defenseentrepreneurs.org to make special arrangements. USAFA requires a roster several days in advance, but we'll make it work.
For more information visit USAFA's Visitor Information Page.
Are there minimum age requirements to enter the event?
No, although alcohol will be served you can attend the events without being 21 or older.
How can I contact the organizer with any questions?
Send any questions to DEFxCOS@defenseentrepreneurs.org
Things to do in United States Air Force Academy, CO
United States Air Force Academy Classes
United States Air Force Academy Government Classes
Organizer Defense Entrepreneurs Forum
Organizer of DEFxColorado: Innovation @ Elevation
The Defense Entrepreneurs Forum is a 501(c)(3) non-profit. We inspire, connect and empower people to promote a culture of innovation in the U.S. national security community by convening events, forging partnerships and delivering tangible solutions.
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