BALTIMORE, MD. February 9, 2017 – Audacious Inquiry (Ai), a health information technology and policy company, today announced that former Oregon state government technology leader, Sarah Miller, has joined Ai effective February 1, 2017. Sarah has more than 12 years of experience in implementing large scale technology projects in Oregon state government, most recently leading the successful design and implementation of the MAGI Medicaid Eligibility & Enrollment Project.
Prior to coming to Ai, Sarah served as the Oregon Department of Human Services' Chief Operating Officer for Technology, the Oregon Integrated Eligibility Determination Project Director, and the Statewide Chief Operating Officer at Oregon's Department of Administrative Services.
A certified Project Management Professional, Sarah is a subject matter expert in project planning and execution, information technology, change management for system implementations, and vendor and contract management ($100M+ contracts) for large-scale Medicaid technology projects.
During her tenure at Oregon DHS, Sarah successfully negotiated a $162 Million expansion to the MAGI Medicaid System for the purposes of Integrated Eligibility Determination, including Non-MAGI Medicaid, Supplemental Nutrition Assistance Program, Temporary Assistance for Needy Families, and Employment Related Day Care. Sarah led Oregon's integrated Advanced Planning Document process for the project, including successfully negotiating an A-87 Cost Allocation Exception Waiver from the Center for Medicare & Medicaid Services, and US Department of Agriculture Food & Nutrition Services. Sarah holds a Master of Public Health degree from Idaho State University and a B.A. degree from Carroll College in Helena, Montana.


just a quick question - is a mosquito net necessary at all in CR?
Two friends are recently back from holidays (in Rhodes and Thailand) and both reported unprecedented mossie activity and got smothered in bites despite sprays etc... so it seems like the little blighters might be rather virulent this year!!
I'd say that most people do not take nets with them, but it's never a bad thing if you're particularly susceptible to bites from the little pests. Some of the hotels and lodges you'll be staying at will supply them (and have them hanging ready above your bed), but having your own, freshly soaked in DEET will provide better protection.
Ultimately, it's your call as to how much they bother you in the night. Personally, if it's not too hot, I find it very comforting to be sleeping under a mosquito net when in the forest and so tend to take them.


text:
Q: Zend Framework Router Getting /module/VALUE/controller/action I've been googling around and I can't seem to find anything which explains the use of ZF router well. I've read the documentation on the site, which seems to only talk about re-routing.
I am trying to make the format:
/module/value/controller/action give /module/controller/action passing on value as a parameter
e.g.
/store/johnsmithbigsale/home/newstuff would route to /store/home/newstuff passing on johnsmithbigsale as the value to a parameter with a hidden namespace e.g. storeName.
Some help would be greatful!

A: You can use Zend_Controller_Router_Route to map your url parts to modules, controllers, actions, and parameters that can be used in the controller by $this->_getParam('varName').  You can define these routes in the application.ini file or in the application bootstrap.
// custom city route
$route = new Zend_Controller_Router_Route(
    'cities/:city',
    array(
        'controller' => 'city',
        'action'     => 'view'
    )
);
$this->addRoute('city', $route);

// custom buy widgets route
$route = new Zend_Controller_Router_Route_Regex(
    'buy_(.+)_widgets/([0-9]+)(.*)',
    array(
        'controller' => 'widgets',
        'action'     => 'view'
    ),
    array(
        1 => 'nothing',
        2 => 'widget_id',
        3 => 'vars'
    )
);
$this->addRoute('widgets', $route);

The regex route is kind of specific to my app, but you can see that each match can get mapped to a parameter.

redpajama_set_name: RedPajamaStackExchange


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text:
Patrick Byrne – DEAR CONGRESS, I CALLED FOR PEACEFUL DC RALLIES JANUARY 4-6. RESPECTFULLY, INVITE ME IN. I PROPOSE 10 AGENDA ITEMS:
30 Sep Patrick Byrne – DEAR CONGRESS, I CALLED FOR PEACEFUL DC RALLIES JANUARY 4-6. RESPECTFULLY, INVITE ME IN. I PROPOSE 10 AGENDA ITEMS:
Patrick Byrne has been the subject of many previous posts on this site (see here and here). He's never backed off from his detailed perspective on the election fraud in 2020, and he seems singularly familiar with yet unintimidated by what he calls the 'goons'. His latest is a useful survey of the election fraud landscape as of the end of September 2021.
There are not going to be enough leftist fingers to plug all the holes in the 'most secure election in history' fantasy; the dike will not hold.
An important reminder from Byrne's most recent post:
There is a word you have not yet read: "Trump". That is because this is not about Donald Trump. He has made clear publicly he is not particularly keen on me, and to me he made clear he knew I had said uncharitable things about him years ago. That said, in our four hours together he was gracious towards me, and as I have made clear elsewhere, I liked him more than I had expected. All of which has precisely 0 to do with the matter at hand. The matter at hand is, If an election can be as dirty as Maricopa's has turned out to be, given all the institutional vouching that has been done about it, what reason is there to believe in any election nationwide? Apparently the authorities are capable of being wrong or tricked…. or are they in on it? If this is rigged then are they who are behind the rig taking our country over permanently?
1_) I sincerely regret the emotional trauma some of you experienced when 1/1000th of the January 6 rally turned unruly. That was surely an ethical, tactical, and strategic error of those who involved. Please accept my condolences for the fear and trauma any of you experienced on January 6 in the US Capitol at the hands of any in the MAGA movement.
2) While I do not claim to have originated the idea of the January 4-6 rally, I welcomed it, and in the weeks leading up to it repeatedly invited citizens to rally in Washington DC January 4-6 to register objection to an election we saw as rigged. While some were distressed at the putative defeat of a president they loved, the distress of many originated in clear insult to the foundational principle of our republic, that principle being (as our Declaration's opening phrases it), "just government derives its power from the consent of the governed" (such consent being determined in elections that are free, fair, and transparent, we believe). November 3's election had been anything but free, fair, and transparent, and millions decided to assemble in DC on January 6 to petition our government for redress of grievance.
3) In all interviews before the rally I emphasized that it was to be peaceful, with language such as, "We don't raise hell. We're not the violent guys. We're better than the other guys. Several steps better than the other guys. The other guys are probably going to try to bring violence and intimidation. Let's show them with love and in peace that we are not going to be intimidated… Only we're going to do it peaceably. We're not going to be like the goons… We want to have the moral high ground… We want to be better than the other guys…" [i] In radio and podcast interviews I repeated such exhortations, and on January 5 published an essay drawing attention to the need for those in the rally to ignore or suppress anyone inciting violence, tweeting it to ≈ 300,000 followers, twice.[ii]
4) On the morning of January 6, the January 5 Georgia Senate Run-off results appeared. The data showed egregious artifacts of manipulation.[iii] Perhaps for this reason, temperatures on January 6 were running higher than they otherwise might.
5) Investigation into Maricopa's November 3 election has uncovered massive election fraud, vindicating the concerns (not the unruliness) of those who rallied on January 6.
A) CANVASS – A 4,500-interview door-to-door canvass was disclosed two weeks ago. The canvassing organization asked five benign questions (none of which were any form of, "For whom did you vote?") They report that citizens overwhelmingly welcomed their interviews, providing robust documentation and asking, What took you so long?
Maricopa has 2.6 million registered voters, of whom 2.1 million voted November 3.
Thus, 500,000 did not vote.
Yet of the 500,000 who did not vote, ≈170,000 (34%) think they did vote. Many are adamant, telling stories about driving to the polls or mailing their ballots, and providing witnesses. Many have given affidavits to this effect.
Thus, it seems that ≈170,000 Maricopans experienced voter suppression.
Of 2.1 million Maricopa voters who voted (per the records), ≈100,000 are ghosts. The humans do not exist (which explains such arcane mysteries as 52 ballots being mailed from a 2-bedroom home, or ballots being mailed from empty sandlots).
Another 30,000 ballots which people say they voted in-person do not show up, but those same 30,000 people show up as having voted-by-mail. What happened to their in-person votes, and from where came their vote-by-mail votes?
In sum, at the low end of the estimate the Maricopa canvass reveals at least ≈300,000 suppressed votes and stuffed ballots, and at the high-end the estimate is over 400,000. That is to say, 15% – 20% of the votes in Maricopa were either suppressed or stuffed. This swamps the outcome of 12 down-ballot races (two of which were Maricopa Board of Supervisors who won by tiny margins and who have since spent their time attacking the audit). The Maricopa rig also swamped the presidential race in Arizona, which was decided by 10,457 votes.
B) FORENSIC AUDIT A Maricopa County forensic audit delivered last week revealed:
Maricopa deleted 1 million election files the day before turning their equipment over;
What was left did not reconcile or "tick-and-tie" with anything or itself;
There are 255,326 Early Votes with no provenance (they just appeared);
The digital images of 284,412 ballots are corrupted so they cannot be checked;
There are at least 57,734 ballots with illegal provenance (e.g., voters who moved out of state decades ago still having ballots returned from their former address);
There are 34,448 (unambiguously illegal) photocopies from 17,126 voters;
There were 15 other felonies occurred whose effects cannot be quantified;
A heat map showing signature-legibility percentage flipped from 95% legible and 5% illegible four weeks before the election to an inverse relationship of 5% legible and 95% illegible four days after the election;
Thousands of images of ballot envelopes have an approval stamp behind the graphics of the envelope, suggesting tampering;
A 25% surge in duplicates from November 4 to November 9th;
There were 1,919 blank signatures stamped as approved;
There were 587 bad signatures;
There were 934 late returns that should not have been counted;
There were 2,580 "scribble" signatures.
There were 9,041 more mail ballots returned and recorded than official number sent.
There were 3,432 more ballots cast than people who cast a vote.
Another 23,344 people show as voting via mail-in-ballot even though they had moved and no one with their name shows as living at their old address.
There were 2,600 excess duplicate ballots.
There were 2,382 people who voted in person after having moved out of the county.
There were 5,047 who voted in more than one county for up to 5,295 votes.
At least one batch of 50 ballots was tabulated twice.
The VM55 holds 255,326 Early Votes that do not have a corresponding EV33 entry.
Deceased people voted 282 times.
There were 393 ballots with incomplete names.
There were 2,861 voters who shared an AFFSEQ number with another voter.
There were days when 92% of the ballots received were all duplicates.
Pre-Meditated Fraud: Votes were pre-printed, "certified and approved".
Ben Cotton described massive IT and Cybersecurity Issues and confirmed the Maricopa County system was breached during the 2020 election.
There were 1,064,746 Election files deleted. Many contained scanned ballots.
There were 284,412 Ballot images on the EMS that were corrupt or missing.
There were 27, 807 ballots cast from individuals who had moved before the election.
After election day, 95% of signatures were illegible (versus 5% before).
Hundreds of ballots had incomplete names.
Hundreds of people flagged as deceased prior to October 5, 2020, voted anyway.
Audit Interference: Runbeck Election Services is a private company that printed the ballots for Maricopa county. When asked questions by Cyber Ninjas, Runbeck said that Maricopa County instructed them to not speak with Cyber Ninjas.
IMPROPER PAPER UTILIZED: A large number of in-person ballots used paper not recommended by the manufacturer of the tabulators.
HIGH BLEED-THROUGH RATES ON BALLOTS: Many ballots from in-person voting experienced bleed-through (marks from one-side of the ballot were clearly visible on the other side of the ballot). This does not happen when the manufacturer recommended paper is utilized.
OUT OF CALIBRATION BALLOT PRINTERS: A large number of ballots appear to have been printed on printers not properly calibrated. This means that the front page of the ballot is not consistently aligned with the back page of the ballot. This is contrary to manufacturer guidelines and recommendations and could theoretically result in inconstant reading of votes across all the different tabulators.
Maricopa's systems violate CISA/industry standard cyber-security best practices.
The machines were connected to the internet.
No security patches were done in 2 years since purchase.
No anti-virus definitions were updated in 2 years since purchase.
.exe files were found all over the server hard drives with modifications done.
One Administrator name and password was used for ALL machines in the county.
Logs produced by Maricopa County did not contain the Windows Security Logs.
Security logs, set to maintain 20MB of data, erased the DAY before the audit began.
Intentional overwriting of the Security Logs by the EMSADMIN account.
2/11/2021 – 462 log entries overwritten
3/3/2021 – 37,686 log entries overwritten
Oldest date in Windows Security Log was 2/5/21 (didn't cover election).
Hardware Configuration Issues
Dual Boot configuration discovered on Adjudication 02 Endpoint.
Two hard drives internal to system
Both drives bootable to different configurations
Clearly not an approved configuration
Second hard drive contained non-Maricopa County Election Data (from South Carolina And Washington State).
Note that this is all that was found only because Maricopa has refused to supply important subpoenaed items:
Rejected provisional ballots;
Unsecured mail ballots;
Ballots returned to the County as undeliverable.
Our craven and reading-comprehension-challenged Mainstream Media laughably summarizes this pig's breakfast: "Maricopa audit shows Biden still won by 10,800 votes."
Good luck making that stick.
6) The Attorney General of Arizona, Mark Brnovich, is not so reading-comprehension-challenged as the MSM. We already knew that because of the fine brushback pitch he threw at the United States Department of Justice in June, reminding them about who created whom and where the Constitution assigns discretion to manage elections.[iv]
This Monday AG Brnovich's office continued to demonstrate its literacy by sending a muscular letter to the Senate asking for the entire production of the Maricopa Audit,[v] and sending a serious-as-a-heart-attack letter to the County telling them to preserve their records.[vi] Taken together, that means the efforts by the RINO Establishment of Arizona to throttle down what got included in the Senate report were for naught: it all goes to the AG.[vii]
Everyone's cards are getting laid on the table. Everyone is going-for-broke.
DeepCapture's 15 year arc reaches apogee and bursts: a man with a badge, a gun, and an ambition is investigating the corruption we are naming. It is the Götterdämmerung.[viii]
7) Since November 10, 2020, I have been calling for "opening the ballot boxes" in six counties (those holding Las Vegas, Phoenix, Milwaukee, Detroit, Philadelphia, and Atlanta), predicting that one would find between 300,000 and 600,000 votes of election fraud in each of them. In only one of the six did an investigation come to pass (Maricopa County, Arizona), and the number of fake and suppressed votes comes to 300,000 – 400,000 (per a canvass that was conducted), or 15%-20% of the ballots cast, and somewhere in the low hundreds of thousands (per a forensic audit).
To any rational person who is open to evidence, this a settled question.
"Of course you'd say that Byrne," I hear the choagies cry, "With Maricopa you're batting 1000 and we're batting .0000!" So out of good sportsmanship I offer double or nothing: choose from the other five cities, Choagies, I'll organize funding, and we will do it again.
But perhaps the Election Fraud Denial camp might practice a moment of sportsmanship themselves, and realize what their denialism has cost us as a nation. Perhaps it is time they let us move on to the next step. Now that we know the truth of Maricopa, citizens everywhere are reflecting on all the institutions of society that asked them to dismiss these claims as "baseless": DHS and CISA ("the most secure election in history"), the state courts, the DOJ, the FEC, Mainstream Media, and politicians of all stripes. They were wrong: the Maricopa ballot boxes were at least 15% and perhaps as much as 20% suppressed and stuffed, as an extensive canvass revealed and a Senate-organized audit has corroborated. It was high-tech ballot stuffing and suppression, no doubt, but no different in principle than the Tammany Hall of yesteryear. The presidential race in Arizona was decided by 10,457 votes, of which we now know that over 57,000 were illegal and several hundred thousand may have been fake. If we ignore this, then we should dispense with the whole charade and usher in an unelected uniparty of elites.
Let the Hunger Games Begin!
Otherwise, perhaps it is time for the Election Fraud Denial camp to pipe down a bit. Along with all the institutions of society that dismissed, canceled, and threatened tens of millions who saw the truth immediately and fought for it to be revealed.
8) Which brings us back to Congress.
The mystery of what happened at the Capitol on January 6 remains unresolved, and I welcome honest Congressional investigation into its origins. But since Congress is making such a production out of finding that origin, and since that origin is partially with me (in that I publicly echoed calls for peaceful assembly on January 4-6), and since I am also he who helped Arizona citizens nurture the conditions of the Maricopa audit into existence, and arranged the better part of its funding, and this audit vindicates the concerns of those January 6 rally-goers, I think it only right and proper that I offer to meet to address questions or concerns you may have.
Please be forewarned that I will take these two additional positions:
9) Those who suppressed and stuffed Maricopa ballot boxes 15-20%, thus flipping Arizona's 11 electoral votes and swinging 22 electoral votes, are the real "Insurrectionists". They betrayed our Constitution, not the citizens who saw through the scheme that has now been exposed in Arizona (and which I am confident would be exposed in five other states if other audits went forward), saw the import of a hijacked national election, rallied, and some minute fraction of which grew unruly (from their lack of discipline, or from agitation, as honest Congressional inquiry would discover).
10) That terrible feeling of fear that you experienced on January 6 at the hands of those unruly and misguided protesters (and whatever agents provocateurs existed amongst the crowd and police), that crippling insecurity that traumatized you, you are now visiting on hundreds of millions of other citizens. Because they are afraid. That is because they get the joke: only 31% of Americans will tell a pollster they believe Biden won fair and square, while 69% say that fraud was "Significant", "Very Significant", or they are "Unsure". And that was before this month's news about Maricopa's canvassing results and forensic audit.
So over 2/3 of Americans believe they have a government of uncertain legitimacy that, since the day it entered office, has seemed unusually keen to make three changes:
To change election law to disrupt Constitutional tenets (e.g., States manage their elections), but would forever enable Maricopa-style election fraud nationwide;
To change the electorate by jamming through the citizenship of tens of millions;
To change the Supreme Court by stacking it so it sprinkles Holy Water on this mess.
Naturally enough, 69% of the citizenry smells skunk. In addition, the aversion of MSM to treating any of this with a modicum of intellectual honesty has led anyone with an IQ above room temperature to suspect that we are living through an engineered reality.[ix]
That's why it feels weird, I assure all. It is supposed to be weird. The price to exit Weird is to accept a new engineered reality where 15-20% ballot stuffing is gaslit as "baseless". The price to exit Weird and regain Normal is: "Let our Constitutional Republic die."
Just tell me when, and I'll come. I don't need a lawyer. Let's skip a few steps and meet.
Most cordially and respectfully,
Patrick M. Byrne
PS There is a word you have not yet read: "Trump". That is because this is not about Donald Trump. He has made clear publicly he is not particularly keen on me, and to me he made clear he knew I had said uncharitable things about him years ago. That said, in our four hours together he was gracious towards me, and as I have made clear elsewhere, I liked him more than I had expected. All of which has precisely 0 to do with the matter at hand. The matter at hand is, If an election can be as dirty as Maricopa's has turned out to be, given all the institutional vouching that has been done about it, what reason is there to believe in any election nationwide? Apparently the authorities are capable of being wrong or tricked…. or are they in on it? If this is rigged then are they who are behind the rig taking our country over permanently?
PPS In September, 2006 (15 years ago this month), seven prominent US senators from both parties had me to come alone to Washington, DC so they could make an unusual request of me. In support of their request they showed me a letter and promised me that when I needed it years down the road, it would be available unconditionally, on a "no take-backs" basis. On that point, seven US Senators shook my hand and gave me their word as gentlemen. Six of those men are still alive and three are still Senators (still from both parties and even more prominent than before). Naturally I am not requesting any of them comment on my recent activities, but now, 15 years after they made it, I am triggering their "no take-back" promise: I respectfully request those three remaining Senators' make that letter public. Doing so now would only be… Gentlemanly.
[i] Interview with Dr. Jerome Corsi, December 28, 2020:
For numerous more examples of my statements about peace, see those gathered here:
https://patrickbyrne.locals.com/post/1040956/crowdsourcing-small-research-project
http://www.freedomsphoenix.com/News/297446-2021-01-05-patrick-byrne-we-are-on-the-verge-of-losing-our.htm
Jerry Garcia on Confrontation & "The Main Asshole"
A Message to Militias Across America Regarding the Goon-Left and Agents Provocateurs (Not the Lingerie)
And this lengthy tweet to ≈ 300,000 followers on January 4-5: http://web.archive.org/web/20210105233902/https://twitter.com/PatrickByrne
[ii] See, "Jerry Garcia on Confrontation & 'The Main Asshole'" at Deepcapture: https://www.deepcapture.com/2021/01/jerry-garcia-on-confrontation-the-main-asshole/
[iii] The Georgia run-off resembled and amplified a key artifact from November 3, 2020:
[iv]
[vii] As a side narrative: once all the truth is out, and voters find out all that is really there, and how RINO politicians cooperated with Democrats to keep it out of the Arizona Senate report…. That is going to be another scandal in its own right. On RINOs.
[viii] NEWSFLASH: Anyone in Arizona who has been involved with these shenanigans and who has any sense is buying a burner phone, calling the Arizona AG's office at 602-542-4266, and saying, "I was involved with the rigging of the Arizona election. I want to meet someone as soon as possible, even today. I may be under surveillance and would like to meet in a motel. I won't bring a lawyer." As far as I can tell, there may have been as many as 200 people who played a role in the Maricopa Mischief: if even 1 of them goes in and talks, the whole thing will unravel, reach you eventually, and you will go down for hard time. So if you stay silent now, you are betting that not 1 in 200 people will squeal.
Good luck with that one, too.

redpajama_set_name: RedPajamaCommonCrawl


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text:
\section{Introduction}
Motivated by the abundance of unlabeled data and the expensiveness of obtaining labels, the paradigm of active learning has been proposed and extensively studied in the literature~\cite[see e.g.][for  comprehensive surveys]{settles2009active,hanneke2014theory}.
In active learning, a learner starts with a set of unlabeled examples,
and can adaptively select subsets of them to query for their labels. Thanks to its adaptivity, an active learner can focus on obtaining informative labels, and can thus substantially reduce labeling effort compared to conventional supervised learning.

Halfspaces, also known as linear separators, are arguably one of the most fundamental concept classes studied in machine learning and data analysis.
Significant research efforts on halfspace learning from computational and statistical perspectives have resulted in rich theory~\cite[e.g.][]{vapnik1998statistical,blum1996polynomial,zhang2004statistical,bartlett2006convexity, kalai2008agnostically} and many practical algorithms~\citep[e.g.][]{cortes1995support, cristianini2000introduction}.

Label noise is ubiquitous in machine learning applications due to various factors, such as human error, sensor failure, etc~\citep{balcan2020noise}, and it is therefore important to design learning algorithms that are robust to label noise.
If computational efficiency is not of  concern, classical methods such as empirical risk minimization are known to achieve statistical consistency~\citep{vapnik1998statistical}.
However, in many practical applications, it is often necessary for learners to process its training examples in a computationally efficient manner.
Therefore, it is of importance to develop computationally efficient, noise-tolerant learning algorithms with statistical consistency guarantees.

However, it is now well-understood that without additional assumptions on the label noise, agnostically learning halfspaces is computationally hard~\citep{feldman2006new,guruswami2009hardness,daniely2016complexity}, even under well-behaved unlabeled data distributions such as standard Gaussian~\citep{klivans2014embedding,diakonikolas2020near}. This motivates the study of learning halfspaces under more benign label noise conditions. Massart noise~\citep{massart2006risk} and Tsybakov noise~\citep{tsybakov2004optimal} are two noise models widely studied in the literature; specialized to the halfspace learning setting, they are formally defined as:

\begin{definition}[Massart noise condition]
Given $\eta \in [0,\frac12)$,
a distribution $D$ over $\RR^d \times \cbr{-1,+1}$ is said to satisfy the {\em $\eta$-Massart noise condition} with respect to halfspace $w^\star \in \RR^d$, if for all examples $x$, $\eta(x) \leq \eta$, where $\eta(x) = \PP_D(y \neq \sign(\inner{w^\star}{x}) \mid x)$.
\label{def:mnc}
\end{definition}

\begin{definition}[Tsybakov noise condition]
Given $A > 0$ and $\alpha \in (0,1]$, a distribution $D$ over $\RR^d \times \cbr{-1,+1}$ is said to satisfy the {\em $(A, \alpha)$-Tsybakov noise condition} with respect to halfspace $w^\star \in \RR^d$, if for all $t \in [0,\frac12)$,  $\PP_D\rbr{ \frac12 - \eta(x) \leq t} \leq A t^{\frac{\alpha}{1-\alpha}}$,
where $\eta(x) = \PP_D(y \neq \sign(\inner{w^\star}{x}) \mid x)$.
\label{def:tnc}
\end{definition}

Although nearly-matching upper and lower bounds on sample and label complexities have been established in both supervised (passive) and active learning settings under these two noise conditions~\citep[e.g.][]{massart2006risk, tsybakov2004optimal, castro2008minimax, hanneke2011rates, balcan2013active, wang2016noise}, most of these results are only {\em statistical}: the algorithms that achieve the sample or label complexity upper bounds are computationally inefficient. Only recently have computationally efficient algorithms been proposed in the literature~\citep[e.g.][]{awasthi2015efficient, diakonikolas2019distribution, diakonikolas2020polynomial} under these noise conditions; see Section~\ref{sec:relwork} for detailed discussions. Still, these works leave out two important open questions:
\begin{enumerate}
    \item {\sloppy  Are there efficient active halfspace learning algorithms that tolerate Massart noise with near-optimal label complexity, under a broad range of unlabeled data distributions? Specifically, the algorithm of~\cite{yan2017revisiting}
    achieves an information-theoretically near-optimal label complexity of $O\rbr{\frac{d}{(1-2\eta)^2} \polylog(\frac1\epsilon)}$, but relies on the strong assumption that the unlabeled data distribution is uniform on the unit sphere; under broader distributional assumptions such as log-concave distributions,
    the state-of-the-art algorithm of~\cite{zhang2020efficient} only achieves a suboptimal label complexity of $O\rbr{\frac{d}{(1-2\eta)^4} \polylog(\frac1\epsilon)}$. Can we design efficient algorithms with $\tilde{O}\rbr{\frac{d}{(1-2\eta)^2} \polylog(\frac1\epsilon)}$ label complexity under broader unlabeled data distributions, for example, the family of isotropic log-concave distributions~\citep{lovasz2007geometry}? }

    \item Are there efficient active halfspace learning algorithms that tolerate Tsybakov noise with label complexities better than passive learning? The state-of-the-art work of~\cite{diakonikolas2020polynomial} propose efficient algorithms for passive learning halfspaces with Tsybakov noise, with sample complexities  $O\rbr{(\frac{d}{\epsilon})^{O(\frac{1}{\alpha})}}$ and $O\rbr{ \poly(d) (\frac 1 \epsilon)^{O(\frac1{\alpha^2})} }$; moreover, computationally inefficient algorithms such as empirical risk minimization achieve a sharper sample complexity of $O\rbr{  d(\frac{1}{\epsilon})^{2-\alpha}}$~\cite[e.g.][Chapter 3]{hanneke2014theory}.
    Can we design efficient active learning algorithms with label complexities of strictly lower order than these? A positive answer to this question can serve as a stepping stone towards developing efficient active algorithms with label complexity matching those of computationally inefficient active learning algorithms~\citep[e.g.][]{balcan2013active}, which is $O\rbr{ d(\frac 1 \epsilon)^{2-2\alpha}}$.
\end{enumerate}

\paragraph{Our results.} Our work answers the above two questions in the affirmative. Specifically, under a set of structural assumptions on the unlabeled data distribution~\citep{diakonikolas2020polynomial} (see also Definition~\ref{def:well-behaved} in Section~\ref{sec:setting}), we give an efficient PAC active halfspace learning algorithm, such that  with appropriate settings of its parameters:
\begin{enumerate}
    \item under the $\eta$-Massart noise condition, it has  an information-theoretically near-optimal label complexity of $O\rbr{\frac{d}{(1-2\eta)^2} \polylog(\frac1\epsilon)}$. This substantially weakens the distributional requirements to achieve such near-optimal label complexity results; before our work, such result is only known when the unlabeled data distribution is  uniform over the unit sphere~\citep{yan2017revisiting}, or when the noise parameter $\eta$ is $3 \times 10^{-6}$, a tiny constant~\citep{awasthi2015efficient}. Furthermore, when the unlabeled data distribution is isotropic log-concave, our result improves over the recent work of of~\cite{zhang2020efficient}, where an efficient algorithm with suboptimal label complexity $O\rbr{\frac{d}{(1-2\eta)^4} \polylog(\frac1\epsilon)}$ is proposed.

    \item under the $(A, \alpha)$-Tsybakov noise condition with $\alpha \in (\frac12, 1]$, it has a label complexity of $\tilde{O}(d (\frac{1}{\epsilon})^{\frac{2-2\alpha}{2\alpha-1}})$.
    Specifically, when $\alpha > \frac{7-\sqrt{17}}{4} \approx 0.719$, $\frac{2-2\alpha}{2\alpha-1} < 2-\alpha$, in which case our algorithm has a better label complexity than passive learning.

    Furthermore, in the special case of $(B,\alpha)$-geometric Tsybakov noise condition (see Definition~\ref{def:gtnc} in Section~\ref{sec:setting}), our algorithm achieves a lower label complexity of $O\rbr{ d (\frac{1}{\epsilon})^{\frac{2}{\alpha} - 2} }$ for all $\alpha \in (0,1]$; specifically, when $\alpha > 2 - \sqrt{2} \approx 0.585$, $\frac{2-2\alpha}{\alpha} < 2 - \alpha$, in which case our algorithm has a better label complexity than passive learning.

\end{enumerate}

\paragraph{Techniques.} Our algorithm and analysis bear similarities to the recent work of~\cite{zhang2020efficient}, who observe that online mirror descent-style updates, when composed with a margin-based active sample selection rule~\citep{balcan2007margin}, implicitly minimizes a nonstandard proximity measure against the Bayes-optimal halfspace $w^\star$.
Compared to~\citet{zhang2020efficient}, our results are novel in two aspects.
First, the algorithm of~\cite{zhang2020efficient} is specialized to Massart noise, as its update rule crucially relies on the knowledge of the Massart noise level $\eta$. In contrast, we propose a new and simpler update rule that can work under both Massart and Tsybakov noise conditions.
Second, under the $\eta$-Massart noise setting, ~\cite{zhang2020efficient} use an averaging-based initialization procedure, which leads to an algorithm and analysis requiring a suboptimal label complexity of $\tilde{O}\rbr{ \frac{d}{(1-2\eta)^4} \polylog(\frac1\epsilon)}$. In this paper, we design a new initialization procedure with improved label efficiency, leading to an algorithm with an information-theoretically near-optimal label complexity of $\tilde{O}\rbr{ \frac{d}{(1-2\eta)^2} \polylog(\frac1\epsilon) }$ under Massart noise.

\section{Related work}
\label{sec:relwork}

\paragraph{Learning halfspaces under Massart and Tsybakov noise: statistical rates.} For passive learning, it is well-known that empirical risk minimization achieves minimax-optimal sample complexities of $\tilde{O}\rbr{\frac{d}{(1-2\eta) \epsilon}}$ and $\tilde{O}\rbr{d (\frac{1}{\epsilon})^{2-\alpha}}$ under $\eta$-Massart noise and $(A,\alpha)$-Tsybakov noise conditions respectively~\citep[e.g.][Chapter 3]{hanneke2014theory}.
For active learning, many works have provided  distribution-specific label complexity upper bounds, including the general analyses of~\cite{hanneke2011rates,beygelzimer2010agnostic,zhang2014beyond} and more specialized analyses of~\cite{balcan2013active,wang2016noise}.
In the setting of $\eta$-Massart noise,  under the assumption that the unlabeled data distribution is isotropic log-concave, the state-of-the-art algorithms of~\cite{balcan2013active, zhang2014beyond} have a label complexity of $\tilde{O}\rbr{ \frac{d}{(1-2\eta)^2} \polylog( \frac1\epsilon) }$; in the setting of $(A,\alpha)$-Tsybakov noise under log-concave unlabeled distributions, the state-of-the-art algorithms of~\cite{balcan2013active,zhang2014beyond,wang2016noise} achieve a label complexity of $\tilde{O}\rbr{ d (\frac{1}{\epsilon})^{2-2\alpha} }$.
Although these algorithms provide sharp label complexity guarantees,
they all suffer from computational inefficiency: they need to perform empirical 0-1 loss minimization, which is known to be NP-hard in general~\citep{arora1997hardness}.

\paragraph{Efficient passive learning halfspaces with Massart noise.}
The study of computationally efficient halfspace learning  under Massart noise is initiated by the work of~\cite{awasthi2015efficient}, who provide a PAC learning algorithm that works under the assumptions that the unlabeled distribution is isotropic log-concave, and the Massart noise parameter $\eta$ is smaller than a tiny constant ($3 \times 10^{-6}$). Prior to this work, positive results mainly focus on the much weaker random classification noise~\citep[e.g.][]{blum1996polynomial,balcan2013statistical}. Under similar distributional assumptions,~\citet{awasthi2016learning} propose an algorithm with sample complexity $O\rbr{ d^{O\rbr{\frac{1}{(1-2\eta)^4}}} \frac{1}{\epsilon} }$ for any $\eta \in [0,\frac12)$. Recent works of~\cite{zhang2020efficient} and~\cite{diakonikolas2020learning} provide passive learning algorithms with fully-polynomial sample complexities in this setting, achieving sample complexities of $O\rbr{ \frac{d}{\epsilon(1-2\eta)^5} }$ and $O\rbr{ \frac{d^9}{\epsilon^4 (1-2\eta)^{10}} }$ respectively.

In the distribution-free PAC learning setting, that is, when no assumptions are imposed on the unlabeled data distribution, efficient halfspace learning is much more challenging.
Recent breakthrough of~\cite{diakonikolas2019distribution} provides an efficient improper learner that can guarantee to output a halfspace with error $\eta+\epsilon$ with sample complexity $\poly(d,\frac1\epsilon)$. ~\cite{chen2020classification} improves over this result by proposing a proper learner, along with a generic ``distillation'' procedure that converts any improper learner to a proper one. In the same paper, they also show that for any statistical query algorithm, obtaining a classifier that achieves an error rate of $\mathrm{opt} + o(1)$ requires a superpolynomial number of statistical queries, where $\mathrm{opt}$ denotes the error rate of the Bayes-optimal halfspace $w^\star$. This lower bound is recently strengthened by~\cite{diakonikolas2020hardness}, showing that even achieving a weaker $\poly(\mathrm{opt})$ error rate requires a superpolynomial number of statistical queries.

\paragraph{Efficient passive learning halfspaces with Tsybakov noise.} Recently,~\cite{diakonikolas2020learningb} obtains an algorithm with a quasi-polynomial time and sample complexity of $O\rbr{d^{O(\frac{1}{\alpha^2} \ln\frac{1}{\epsilon})}}$ for PAC learning halfspaces under $(A,\alpha)$-Tsybakov noise condition, under distributions with certain structural properties.
This result is further improved by~\cite{diakonikolas2020polynomial}, who obtain two algorithms with time and sample complexities of $O\rbr{\poly(d) \cdot (\frac1\epsilon)^{O(\frac{1}{\alpha^2})}}$ and $O\rbr{(\frac d \epsilon)^{O(\frac 1 \alpha)}}$ respectively.

\paragraph{Efficient active learning halfspaces with Massart and Tsybakov noise.} By combining the agnostic halfspace learning algorithm of~\cite{kalai2008agnostically}, and margin-based sampling~\citep{balcan2007margin, balcan2013active},  \cite{awasthi2016learning} obtains an active halfspace learning algorithm that tolerates $\eta$-Massart noise with a sample complexity of $O\rbr{ d^{O(\frac{1}{(1-2\eta)^4})} \ln\frac1\epsilon }$ under the isotropic log-concavity assumption on the unlabeled data distribution. This result is recently substantially improved by~\cite{zhang2020efficient}, who obtain a label complexity of $O\rbr{ \frac{d}{(1-2\eta)^4} \polylog(\frac1\epsilon) }$ in the same setting; however this label complexity bound still does not match the information-theoretic lower bound of $\Omega\rbr{ \frac{d}{(1-2\eta)^2} \ln\frac1\epsilon }$. The only label-optimal result on PAC active halfspace learning under Massart noise we are aware of  is~\cite{yan2017revisiting}, however it relies on the strong assumption that the unlabeled data distribution is uniform over the unit sphere. Under Tsybakov noise condition, to the best of our knowledge,
all prior active learning works require well-specified model assumptions on the conditional distribution of label given feature \citep{cesa2009robust,dekel2012selective,agarwal2013selective,krishnamurthy2017active}, e.g. assuming $\EE[ y \mid x ] = \sigma(w^\star \cdot x)$ for some known function $\sigma: \RR \to \RR$.
This is relatively strong, as it requires all examples $x$ that have the same projection on $w^\star$ to have the same value of $\eta(x)$. In contrast, our work does not require such assumptions.

\section{Preliminaries}
\label{sec:setting}

We consider the standard PAC active learning for binary classfication setup~\citep{valiant1985learning,balcan2009agnostic}. Specifically, the instance space $\Xcal$ is $\RR^d$, the label space $\Ycal$ is $\cbr{-1,+1}$, and there is a data distribution $D$ supported on $\Xcal \times \Ycal$. The hypothesis class of interest is the set of linear classifiers, also known as halfspaces, defined as $\Hcal = \cbr{h_w: w \in \RR^d}$, where for every $w \in \RR^d$, $h_w$ denotes the corresponding linear classifier that maps $x$ to $\sign(\inner{w}{x})$. We use {\em error rate} to measure the performance of a classifier $h:  \Xcal \to \Ycal$, defined as $\err(h, D) = \PP_{(x,y) \sim D}(h(x) \neq y)$.
Given classifier $h$, and a set of labeled examples $S$, denote $\err(h, S) = \frac1{|S|} \sum_{(x,y) \in S} \ind\rbr{h(x) \neq y}$ as the empirical error rate of $h$ on $S$.
Throughout this paper, we assume that the Bayes-optimal classifier is a halfspace $h_{w^\star}$, where $w^\star \in \RR^d$ is a unit vector. It can be verified that $h_{w^\star}$ is indeed Bayes-optimal under Massart or Tsybakov noise conditions (recall Definitions \ref{def:mnc} and \ref{def:tnc}).

An active learning algorithm has access to two labeling oracles: first, an unlabeled example oracle $\mathrm{EX}$, which, upon query, returns an unlabeled example $x$ drawn from $D_X$, the marginal distribution of $D$ over $\Xcal$; second, a labeling oracle $\Ocal$, which, upon query with input example $x$, returns a label $y$ drawn from $D_{Y \mid X=x}$, the conditional distribution of $Y$ given $X=x$. A learner is said to achieve $(\epsilon,\delta)$-PAC active learning guarantee, if, by interactively querying the unlabeled example oracle $\mathrm{EX}$ and the labeling oracle $\Ocal$, it outputs a classifier $\tilde{h}$, such that with probability $1-\delta$, $\err(\tilde{h}, D) - \err(h_{w^\star}, D) \leq \epsilon$. Its label complexity is the total number of queries to $\Ocal$ throughout the learning process.

For a natural number $N$, denote by $[N] := \cbr{1,2,\ldots,N}$.
For a vector $w$ in $\RR^d$, denote by its $\ell_2$-normalization $\hat{w} := \frac{w}{\|w\|_2}$ if $w \neq \overrightarrow{0}$, and $\hat{w} = (1,0,\ldots,0)$ if $w = \overrightarrow{0}$. {\em Throughout this paper, we reserve the ``hat'' symbol and notations such as $\hat{w}$, $\hat{v}$ for $\ell_2$-normalization and $\ell_2$-normalized vectors.} Unless explicitly stated, we use $\| \cdot \|$ to denote the vector $\ell_2$ norm.
For two vectors $u$ and $v$, denote by $\theta(u,v) = \arccos \rbr{\frac{\inner{u}{v}}{\|u\|\|v\|}} \in [0,\pi]$ the angle between them; also, denote by $\tilde{\theta}(u,v) = \min(\theta(u,v), \pi - \theta(u,v)) \in [0, \frac \pi 2]$.
In our algorithm and analysis below, we will be frequently using the following definition of the distribution $D$ conditioned on a band: given a unit vector $\hat{w}$ and a threshold $b > 0$, denote by $B_{\hat{w},b} = \cbr{x \in \RR^d: \abr{\inner{\hat{w}}{x}} \leq b}$; in addition, denote by $D_{\hat{w}, b}$ the conditional distribution of $D$ on the set $\cbr{(x,y) \in \RR^d \times \cbr{-1,+1}: x \in B_{\hat{w},b}}$; similarly, denote by $D_{X \mid \hat{w}, b}$ the conditional distribution of $D_X$ on the set $B_{\hat{w},b}$.

Computational hardness results~\citep{chen2020classification,diakonikolas2020hardness} strongly suggest that efficient learning halfspaces with noise may be computationally intractable if no assumptions on the unlabeled data distribution are made, even under benign noise conditions such as Massart noise. Therefore, throughout this paper, following~\cite{diakonikolas2020polynomial}, we assume the unlabeled distribution to lie in a family of structured, or well-behaved distributions, defined as follows:

\begin{definition}[Well-behaved distributions~\citep{diakonikolas2020polynomial}]
\label{def:well-behaved}
Fix  $L, R, U, \beta > 0$. A distribution $D_X$ over $\RR^d$ is said to be $(2,L,R, U, \beta)$ well-behaved, if for any $2$-dimensional linear subspace $V$ of $\RR^d$, we have:
given an $x$ randomly drawn from $D_X$,
$x_V$, the projected coordinates of $x$ onto $V$\footnote{Formally, pick $(v_1,v_2)$ as an orthonormal basis of $V$; define $x_V := (\inner{v_1}{x}, \inner{v_2}{x}) \in \RR^2$.}, has a probability density function $p_V$ on $\RR^2$, such that:
\begin{enumerate}
    \item $p_V(z) \geq L$, for all $z$ such that $\| z \|_2 \leq R$;
    \item $p_V(z) \leq U$, for all $z \in \RR^2$;

\end{enumerate}
in addition, for any unit vector $w$ in $\RR^d$ and any $t > 0$, $\PP_{D_X}(\abr{\inner{w}{x}} \geq t) \leq \exp(1-\frac{t}{\beta})$.
\end{definition}

The well-behavedness assumption captures the well-studied family of isotropic log-concave distributions~\citep{lovasz2007geometry,balcan2013active}, and can potentially be more general.\footnote{However, the well-behavedness assumption here does not capture the family of $s$-concave distributions ($s \geq -\frac{1}{2d+3}$) studied in recent works~\citep{balcan2017sample}, as it requires any 1-d projection of distribution to have sub-exponential tail. Whether our analysis can be extended to $s$-concave distributions is an interesting open question.}

In addition to Massart and Tsybakov noise conditions, we also study a subfamily of Tsybakov noise, namely geometric Tsybakov noise. Such noise assumption was first considered in nonparametric active learning literature~\citep{castro2008minimax}. It generalizes the ``strong Massart noise'' condition considered in~\cite{diakonikolas2020learning,zhang2017hitting}, in that it allows $\frac12 - \eta(x)$ to grow  polynomially with $\abr{\inner{w^\star}{x}}$, the distance between $x$ and the Bayes-optimal decision boundary $\cbr{x \in \RR^d: \inner{w^\star}{x} = 0}$.

\begin{definition}[Geometric Tsybakov noise condition]
\label{def:gtnc}
Given $B > 0$ and $\alpha \in (0,1]$, a distribution $D$ over $\RR^d \times \cbr{-1,+1}$ is said to satisfy the {\em $(B,\alpha)$-geometric Tsybakov noise condition} with respect to halfspace $w^\star$, if for all $x$ in $\RR^d$, $\frac12 - \eta(x) \geq \min\rbr{\frac12, B \abr{\inner{w^\star}{x}}^{\frac{1-\alpha}{\alpha}} }$, where $\eta(x) = \PP(y \neq \sign(\inner{w^\star}{x}) \mid x)$.
\end{definition}

It can be shown that, if the unlabeled distribution is well-behaved (Definition~\ref{def:well-behaved}),  modulo a logarithmic factor, $(B,\alpha)$-geometric Tsybakov noise condition implies $(A,\alpha)$-Tsybakov noise condition with $A = \tilde{O}\rbr{(\frac 1 B)^{\frac \alpha {1-\alpha}}}$; see Lemma~\ref{lem:geomtnc-tnc} in Appendix~\ref{sec:auxiliary} for a formal statement.

\section{Algorithm}

We now describe our noise-tolerant active halfspace learning algorithm in detail. The main algorithm, Algorithm 1, has a simple structure: it first calls subprocedure $\ensuremath{\textsc{Initialize}}\xspace$ (line~\ref{step:init}) to generate a vector $v_1$ with $\ell_2$ distance at most $\frac{1}{4}$ to $w^\star$ with high probability. After obtaining $v_1$, it repeatedly calls $\ensuremath{\textsc{Optimize}}\xspace$ to refine its iterates $v_j$; as we will see, $\ensuremath{\textsc{Optimize}}\xspace$ guarantees that, with high probability, after iteration $j$, iterate $v_{j+1}$ is such that $\| v_{j+1} - w^\star \| \leq 4^{-(j+1)}$, i.e. after each iteration, an upper bound on $\|v_j - w^\star \|$ shrinks by a constant factor.
The algorithm returns after its final iteration $k_\epsilon$ is finished; after this iteration, we have $\theta(v_{k_\epsilon+1}, w^\star) \leq r_{\epsilon} = \tilde{O}\rbr{ \epsilon }$, which implies that
$\err(h_{\tilde{v}}, D) - \err(h_{w^\star}, D) \leq \PP_{D}(h_{\tilde{v}}(x) \neq h_{w^\star}(x)) \leq  \epsilon$ (see Lemma~\ref{lem:prob-angle} in Appendix \ref{sec:auxiliary}).

Depending on different noise conditions on $D$, we use different schedules of sampling region bandwidths $\cbr{b_j}$ and numbers of label queries $\cbr{T_j}$:

{
\sloppy
\begin{enumerate}
    \item Under the $\eta$-Massart noise condition, $b_j = b_{\mathsf{MNC}}(\eta, 4^{-(j+1)})$ and $T_j = T_{\mathsf{MNC}}(\eta, 4^{-(j+1)})$ for all $j \in \NN$, where $b_{\mathsf{MNC}}(\eta, r) = \tilde{\Theta}\rbr{ (1-2\eta) r}$ and $T_{\mathsf{MNC}}(\eta, r) = \tilde{O}\rbr{ \frac{d}{(1-2\eta)^2} (\ln\frac 1 {\delta r})^3 }$.

    \item Under the $(A,\alpha)$-Tsybakov noise condition for $\alpha \in (\frac12, 1]$, $b_j = b_{\mathsf{TNC}}(A, \alpha, 4^{-(j+1)})$ and $T_j = T_{\mathsf{TNC}}(A, \alpha, 4^{-(j+1)})$ for all $j \in \NN$, where  $b_{\mathsf{TNC}}(A, \alpha, r) = \tilde{\Theta}\rbr{ \min(r, (\frac{1}{A})^{\frac{1-\alpha}{2\alpha-1}} r^{\frac \alpha {2\alpha-1}}) }$ and $T_{\mathsf{TNC}}(A, \alpha, r) = \tilde{O}\rbr{ d  (\ln\frac 1 {\delta r})^3 \cdot (1 + (\frac{A}{r})^{\frac{2-2\alpha}{2\alpha-1}})}$.

    \item Under the $(B, \alpha)$-geometric Tsybakov noise condition, $b_j = b_{\mathsf{GTNC}}(B,\alpha, 4^{-(j+1)})$ and $T_j = T_{\mathsf{GTNC}}(B, \alpha, 4^{-(j+1)})$ for all $j \in \NN$, where $b_{\mathsf{GTNC}}(B, \alpha, r) = \tilde{\Theta}\rbr{ \min(r, B  r^{\frac1\alpha}) }$, and
    $T_{\mathsf{GTNC}}(B, \alpha, r) = \tilde{O}\left( d (\ln\frac 1 {\delta r})^3 (1  + \frac{1}{B^2 } (\frac{1}{r})^{\frac{2-2\alpha}{\alpha}} )  \right)$.
\end{enumerate}
}

For brevity, in the above definitions, the dependence on the unlabeled distribution parameters $(L, R, U, \beta)$ is ignored; we refer the readers to Appendix~\ref{sec:params} for more precise definitions of these functions.
The choices of $\cbr{b_j}$ and $\cbr{T_j}$ are to ensure that the algorithm's iterates $v_j$ are brought progressively closer to $w^\star$ with increasing $j$; this will be discussed in greater detail in Section~\ref{sec:theory}.

\begin{algorithm}[t]
\caption{Main algorithm}
\label{alg:main}
\begin{algorithmic}[1]
\REQUIRE{Bandwidth schedule $\cbr{b_j}$, iteration schedule $\cbr{T_j}$, target excess error $\epsilon \in (0,1)$, failure probability $\delta \in (0,1)$.}
\ENSURE{A halfspace $\tilde{v}$ such that $\err(h_{\tilde{v}}, D) - \err(h_{w^\star}, D) \leq \epsilon$.}
\STATE Define $r_\epsilon := \frac{\epsilon}{32 U \beta^2  (\ln\frac{12}{\epsilon})^2}$, and set $k_\epsilon := \lceil \log_4 \frac{1}{r_\epsilon} \rceil$ be the total number of iterations.
\STATE $v_1 \gets \ensuremath{\textsc{Initialize}}\xspace(\cbr{b_j}, \cbr{T_j})$.
\label{step:init}
\FOR{$j=1,\ldots,k_\epsilon$}
\STATE $v_{j+1} \gets \ensuremath{\textsc{Optimize}}\xspace(v_{j}, 4^{-(j+1)}, b_j, T_j, \mathsf{average})$.
\label{step:refine}
\ENDFOR
\RETURN $v_{k_\epsilon+1}$.
\end{algorithmic}
\end{algorithm}

We now discuss the two subprocedures employed by the main algorithm, \ensuremath{\textsc{Optimize}}\xspace and \ensuremath{\textsc{Initialize}}\xspace, in detail.

\subsection{Procedure \ensuremath{\textsc{Optimize}}\xspace and its guarantees}
\label{subsec:optimize}

Procedure \ensuremath{\textsc{Optimize}}\xspace (Algorithm~\ref{alg:optimize}) aims at  refining its input halfspace $w_1$, so that it outputs a halfspace $\tilde{w}$ whose $\ell_2$ distance to $w^\star$ has an upper bound ($r$) at most a factor of $\frac14$ times the original $\ell_2$ distance upper bound between $w_1$ and  $w^\star$ ($4r$), with good probability.

To this end, it maintains an iterate $w_t$; at each iteration, it performs adaptive sampling to obtain a labeled example $(x_t, y_t)$ drawn from $D_{\hat{w}_t, b}$, and updates with this new example using the well-known online gradient descent algorithm~\citep[e.g.][Chapter 11]{cesa2006prediction}.
Following standard active learning sampling strategies~\cite[e.g.][]{balcan2009agnostic}, every draw from distributions $D_{X \mid \hat{w}_t, b}$ is done by rejection sampling, i.e., keep querying $\mathrm{EX}$ until it returns an example in $B_{\hat{w}_t,b}$.
After generating the iterates $\cbr{w_t}_{t=1}^T$, it aggregates them using either a normalize-and-average step (lines~\ref{line:avg-start} to~\ref{line:avg-end}), or draws a vector uniformly at random from this set, multiplied by a random sign (lines~\ref{line:rnd-start} to~\ref{line:rnd-end}), depending on its aggregation mode $\mathrm{agg}$.

\ensuremath{\textsc{Optimize}}\xspace is similar to \ensuremath{\textsc{Refine}}\xspace  in~\cite{zhang2020efficient}, but has two key differences. First, the update vector $g_t$ used in our \ensuremath{\textsc{Optimize}}\xspace algorithm is $-y_t x_t$, whereas the update vector $g_t$ in \ensuremath{\textsc{Refine}}\xspace depends on the Massart noise parameter $\eta$; this undesirable dependence on $\eta$ implies that \ensuremath{\textsc{Refine}}\xspace cannot be used for handling broader noise conditions such as Tsybakov noise. Second, it allows two aggregation modes to be used, and when to use which aggregation mode depends on the precision of the input (i.e. the $\ell_2$ closeness of input $w_1$ and $w^\star$): as we will see, in early learning stages, we will call \ensuremath{\textsc{Optimize}}\xspace with  mode $\mathsf{random}$; in later stages, we will call \ensuremath{\textsc{Optimize}}\xspace with  mode $\mathsf{average}$. \ensuremath{\textsc{Optimize}}\xspace is algorithmically similar to the nonconvex SGD algorithm of~\cite{diakonikolas2020learning} (see also earlier algorithmic insights of~\citet{guillory2009active}), who carefully construct a nonconvex learning objective such that under Massart noise, any stationary point of the objective corresponds to a vector close to $w^\star$; however, as we will see next, our analysis techniques are fairly different from theirs.

As its update rule suggest, \ensuremath{\textsc{Optimize}}\xspace performs online linear optimization with adaptively-chosen linear functions. Generalizing insights from prior work~\citep{zhang2020efficient}, our key observation is that, somewhat intriguingly, \ensuremath{\textsc{Optimize}}\xspace can be alternatively viewed as minimizing the following ``proximity function'' to $w^\star$:
\begin{definition}
Given distribution $D$ with its Bayes optimal classifier being a halfspace $h_{w^\star}$, and a positive number $b > 0$, define $\psi_{D,b}(w) := \EE_{D_{\hat{w},b}} \sbr{ (1-2\eta(x)) \abr{\inner{w^\star}{x}} }$.
\end{definition}

\begin{algorithm}[t]
\caption{$\ensuremath{\textsc{Optimize}}\xspace$}
\label{alg:optimize}
\begin{algorithmic}[1]
\REQUIRE vector $w_1$, target proximity $r$ (such that $\| w_1 - w^\star \| \leq 4 r$), bandwidth $b$, number of iterations $T$, aggregation mode $\mathrm{agg}$.
\ENSURE Optimized halfspace $\tilde{w}$ such that $\| \tilde{w} - w^\star \| \leq r$ with good probability.

\STATE Define $\Kcal = \cbr{w: \| w - w_1 \| \leq  4 r}$, step size $\alpha = \frac{r}{\beta} \sqrt{\frac{1}{dT}} / \rbr{\ln\frac{Td}{\delta r b R L}}$.
\FOR{$t=1, 2, \dots, T$}
\label{line:omd-loop-start}
\STATE Sample $x_t$ from $D_{X \mid \hat{w}_t, b}$ using rejection sampling, and query $\Ocal$ for its label $y_t$.
\STATE Update $w_{t+1} \gets\arg\min_{w \in \Kcal} \| w_t - \alpha g_t \|$,
where $g_t = - y_t x_t$.
\label{line:omd}
\ENDFOR
\label{line:omd-loop-end}
\IF{$\mathrm{agg} = \mathsf{average}$}
\label{line:avg-start}
\RETURN $\tilde{w} \gets \frac{1}{T} \sum_{t=1}^T \hat{w}_t$.
\label{line:avg-end}
\ELSIF{$\mathrm{agg} = \mathsf{random}$}
\label{line:rnd-start}
\RETURN $\tilde{w} \gets \sigma \cdot \hat{w}_\tau$, where $\tau$ is chosen uniformly at random from $[T]$, and $\sigma$ is chosen uniformly at random from $\cbr{-1,+1}$.
\label{line:rnd-end}
\ENDIF
\end{algorithmic}
\end{algorithm}

Recall that the three noise conditions considered in our paper all assume that $h_{w^\star}$ is the Bayes optimal classifier. Therefore, $\eta(x) \leq \frac12$ for all $x$, and consequently $\psi_{D,b}$ always takes nonnegative values.
In addition, $\psi_{D,b}$ is scale-invariant: as $B_{\widehat{\alpha w}, b} = B_{\hat{w}, b}$, $\psi_{D,b}(\alpha w) = \psi_{D,b}(w)$ for any $w \in \RR$ and $\alpha \neq 0$.
Informally, $\psi_{D,b}$ is a distance proxy function that measures the closeness of input $w$ and the optimal $w^\star$, although the closeness here is defined by a nonstandard measure $\tilde{\theta}(\cdot, \cdot)$ (recall its definition in Section~\ref{sec:setting}). This is formalized in the following lemma (proof in Appendix~\ref{sec:pot}):
\begin{lemma}
Suppose $D_X$ is $(2, L, R, U, \beta)$-well-behaved. In addition, $w \in \RR^d$ and $b > 0$ is such that $\tilde{\theta} = \tilde{\theta}(w, w^\star) \geq \Omega(b)$. Then:
\begin{enumerate}
    \item if $D$ satisfies $\eta$-Massart noise condition, $\psi_{D,b}(w) \geq \tilde{\Omega}\rbr{ (1-2\eta) \tilde{\theta} }$.
    \item If $D$ satisfies $(A, \alpha)$-Tsybakov noise condition, $\psi_{D,b}(w) \geq \Omega\rbr{ (\frac b A)^{\frac{1-\alpha}{\alpha}} \tilde{\theta} }$.

    \item If $D$ satisfies $(B, \alpha)$-geometric Tsybakov noise condition, then $\psi_{D,b}(w) \geq \tilde{\Omega}\rbr{ \min( \tilde{\theta}, B\tilde{\theta}^{\frac{1}{\alpha}}) }$.

    \end{enumerate}
    \label{lem:pot-lb-main-text}
\end{lemma}

The following key lemma formalizes the aforementioned claim that \ensuremath{\textsc{Optimize}}\xspace produces iterates $w_t$'s that approximately minimize $\psi_{D,b}$; specifically, the average value of $\psi_{D,b}(w_t)$'s is well-controlled, if the sampling bandwidth $b$ is small and the number of iterations $T$ is large.

\begin{lemma}
Suppose $D_X$ is $(2, L, R, U, \beta)$-well-behaved. There exists a numerical constant $c > 0$ such that the following holds. \ensuremath{\textsc{Optimize}}\xspace, with input initial vector $w_1$, target proximity $r \in (0,\frac 14]$ such that $\| w_1 - w^\star \| \leq 4r$, bandwidth $b \leq \frac R 2$, number of iterations $T$, produces iterates $\cbr{w_t}_{t=1}^T$, such that
with probability $1-\delta r$,
\[
\frac1T \sum_{t=1}^T \psi_{D,b}(w_t)
\leq
c \rbr{ b
+
(b + \beta r) \cdot \rbr{\ln\frac{Td}{\delta r b R L}} (\sqrt{\frac{d + \ln\frac1{\delta r}}{T}} + \frac{\ln\frac1{\delta r}}{T}) }.
\]
\label{lem:optimize-g}
\end{lemma}

The proof of Lemma~\ref{lem:optimize-g} can be found at Appendix~\ref{sec:optimize-g}.
Its key insight is similar to the ideas in~\cite{zhang2020efficient}: we derive a regret guarantee of the online linear optimization problem induced by the adaptively-chosen gradient vectors $\cbr{g_t}_{t=1}^T$, which implies an upper bound on the {\em negative benchmark} term. Thanks to the adaptive sampling scheme, the negative benchmark concentrates to $\sum_{t=1}^T \psi_{D,b}(w_t)$. Although the proof of Lemma~\ref{lem:optimize-g}  uses standard regret results on online linear optimization, it is not a direct consequence of the standard reduction from online convex optimization to online linear optimization -- $\psi_{D,b}(w)$ is not necessarily convex in $w$.

We now discuss the aggregation mode $\mathrm{agg}$ in more detail. As \ensuremath{\textsc{Optimize}}\xspace is called by the main algorithm (Algorithm~\ref{alg:main}) repeatedly, we discuss its different settings in earlier and later stages of the main algorithm respectively.

In later stages of calling $\ensuremath{\textsc{Optimize}}\xspace$ (specifically, when its input target proximity $r \leq \frac{1}{16}$), the constraint set $\Kcal$ ensures that all $w_t$'s have acute angles with $w^\star$; in this case, for all $t$, $\tilde{\theta}(w_t, w^\star) = \theta(w_t, w^\star)$, and therefore  upper bounds on $\psi_{D,b}(w_t)$ imply upper bounds on $\theta(w_t, w^\star)$.
In this ``local convergence'' regime, applying the guarantees provided by Lemma~\ref{lem:optimize-g}, a deterministic average over the normalized iterates $\hat{w}_t$'s achieves the target $\ell_2$ proximity to $w^\star$; see item~\ref{item:avg} of Lemma~\ref{lem:optimize-main} or Lemma \ref{lem:optimize-main-avg-only} for a precise statement. This corresponds to the $\mathsf{average}$ mode (lines~\ref{line:avg-start} to~\ref{line:avg-end}).

In contrast, in early stages of calling \ensuremath{\textsc{Optimize}}\xspace (specifically, when $r > \frac1{16}$), its iterates $\cbr{w_t}_{t=1}^T$ may not yet be in acute angle with $w^\star$; in this case, it is hard to guarantee that the average normalized iterate $\frac{1}{T} \sum_{t=1}^T \hat{w}_t$ is close to $w^\star$. This motivates the second mode $\mathsf{random}$ (lines~\ref{line:rnd-start} to~\ref{line:rnd-end}), which returns a vector uniformly at random from $\hat{w}_t$'s, times a random sign $\sigma$. By the guarantees of Lemma~\ref{lem:optimize-g}, it can be shown that with appropriate settings of parameters $\alpha, T, b$, \ensuremath{\textsc{Optimize}}\xspace guarantees to return $\tilde{w}$ with a constant $\ell_2$ distance to $w^\star$ with a constant probability; see item~\ref{item:rnd} of Lemma~\ref{lem:optimize-main} for a precise statement.

\subsection{Procedure \ensuremath{\textsc{Initialize}}\xspace and its guarantees}
Procedure \ensuremath{\textsc{Initialize}}\xspace (Algorithm~\ref{alg:initialize}) aims at label-efficiently learning a halfspace $\hat{u}_0$ amenable to local refinement: it guarantees that with high probability, the output halfspace $\hat{u}_0$ satisfies that $\| \hat{u}_0 - w^\star \| \leq \frac14$.

At a high level, \ensuremath{\textsc{Initialize}}\xspace uses \ensuremath{\textsc{Optimize}}\xspace as a black box in a label efficient manner.
Recall from the previous subsection that, with appropriate settings of input parameters, running \ensuremath{\textsc{Optimize}}\xspace with aggregation mode $\mathsf{random}$ guarantees to output a halfspace $\frac14$-close to $w^\star$ with  constant probability. \ensuremath{\textsc{Initialize}}\xspace   ``boosts'' the above guarantee, in that it increases the probability of outputting a vector $\frac14$-close to $w^\star$ from a small constant to $1-O(\delta)$.

\ensuremath{\textsc{Initialize}}\xspace consists of two stages. In the first stage (lines~\ref{line:stage-1-start} to~\ref{line:stage-1-end}), it generates $U$, a set of halfspaces of size $N = \lceil 10 \ln\frac 4 \delta \rceil$, such that at least one  element in $U$ has a small excess error rate, specifically $O(\epsilon_0)$; here the choice of $\epsilon_0$ depends on the noise condition of $D$, as will be discussed next. To achieve this target excess error rate, it runs \ensuremath{\textsc{Optimize}}\xspace for $k_0 = \tilde{O}(\log\frac{1}{\epsilon_0})$ iterations; Claim~\ref{claim:stage1} in Appendix~\ref{sec:initialize} shows that each trial $i$ generates $v_{i,k_0+1}$ with excess error $O(\epsilon_0)$ with constant probability. Because of the independence of the $N$ trials, with high probability, one of the $v_{i,k_0+1}$'s will have excess error $O(\epsilon_0)$.

\begin{algorithm}[t]
\caption{\ensuremath{\textsc{Initialize}}\xspace: a label efficient acute initialization procedure}
\label{alg:initialize}
\begin{algorithmic}[1]
\REQUIRE{Bandwidth schedule $\cbr{b_j}$,  iterations schedule $\cbr{T_j}$.}
\ENSURE{Unit vector $\hat{u}_0$ such that $\| \hat{u}_0 - w^\star \| \leq \frac14$.}

\STATE Define $r_0 = \frac{\epsilon_0}{64 U \beta^2  (\ln\frac{24}{\epsilon_0})^2}$, where $\epsilon_0$ is defined according to the noise condition of $D$; $k_0 = \lceil \log_4 (\frac{1}{r_0}) \rceil$ is the number of iterations per trial; $N = \lceil 10 \ln \frac {4}{\delta} \rceil$ the number of trials in the first stage.

\FOR{$i=1,2,\ldots,N$}
\label{line:stage-1-start}

\STATE Initialize $v_{i,0} \gets \overrightarrow{0}$.

\FOR{$j=0,1,2,\ldots,k_0$}

\STATE Set $\mathrm{agg} \gets \mathsf{random}$ if $j = 0$; $\mathrm{agg} \gets \mathsf{average}$ otherwise.

\STATE $v_{i,j+1} \gets \ensuremath{\textsc{Optimize}}\xspace(v_{i,j}, 4^{-(j+1)}, b_j, T_j, \mathrm{agg})$.
\ENDFOR
\ENDFOR

\STATE $U \gets \cbr{v_{i,k_0+1}: i \in [N]}$.
\label{line:stage-1-end}

\STATE $S \leftarrow$ draw $O(\frac{1}{\epsilon_0^2}\ln \frac {N}{\delta} )$ random unlabeled examples from $D_X$, and query $\Ocal$ for their labels.
\label{line:stage-2-start}

\RETURN $\hat{u}_0 = \frac{u_0}{\| u_0\|}$, where $u_0 = \argmin_{u \in U} \err(h_u, S)$.
\label{line:stage-2-end}
\end{algorithmic}
\end{algorithm}

In the second stage (lines~\ref{line:stage-2-start} to~\ref{line:stage-2-end}), it draws $S$, a set of labeled examples from $D$, and selects the halfspace $\hat{u}$ in $U$ with the smallest empirical error on $S$.
Combining the guarantees of $U$ in the first stage, the choice of $|S| = \tilde{O}(\epsilon_0^{-2})$, and standard guarantees of empirical risk minimization, it is guaranteed that $\hat{u}$ has excess error  $\tilde{O}(\epsilon_0)$ with high probability.
Parameter $\epsilon_0$ is set to ensure that the above excess error guarantee can be translated to a geometric $\ell_2$-proximity guarantee, and therefore depends on different noise conditions on $D$. Specifically, we set $\epsilon_0$ as:
\begin{enumerate}
    \item $\epsilon_{\mathsf{MNC}}(\eta) = \tilde{O}\rbr{ 1-2\eta }$, under the $\eta$-Massart noise condition;

    \item $\epsilon_{\mathsf{TNC}}(A, \alpha) = \tilde{O}\rbr{  (\frac{1}{A})^{\frac{1-\alpha} {\alpha}} }$, under the $(A, \alpha)$-Tsybakov noise condition with $\alpha \in (\frac12, 1]$;

    \item $\epsilon_{\mathsf{GTNC}}(B, \alpha) = \tilde{O}\rbr{ B }$, under the $(B,\alpha)$-geometric Tsybakov noise condition.
\end{enumerate}
We again refer the readers to Appendix~\ref{sec:params} for more precise definitions of these functions, with their dependence on $(L, R, U, \beta)$ explicit.

We show the following guarantee of \ensuremath{\textsc{Initialize}}\xspace, under any one of the three noise conditions considered.
\begin{lemma}
Suppose $D$ is $(2, L, R, U, \beta)$-well behaved and satisfies one of the three noise conditions. With the respective settings of $\cbr{b_j}, \cbr{T_j}, \epsilon_0$, \ensuremath{\textsc{Initialize}}\xspace  outputs a unit vector $\hat{u}_0$, such that with probability $1-\delta/2$, $\| \hat{u}_0 - w^\star \| \leq \frac{1}{4}$. The total number of label queries by \ensuremath{\textsc{Initialize}}\xspace is at most:
\begin{enumerate}
    \item $\tilde{O} \rbr{ \frac{d}{(1-2\eta)^2} }$, if $D$ satisfies $\eta$-Massart noise;
    \item $\tilde{O} \rbr{ d \cdot \rbr{ 1 + A^{\frac{2-2\alpha}{\alpha(2\alpha-1)}} } }$, if $D$ satisfies $(A, \alpha)$-Tsybakov noise with $\alpha \in (\frac12, 1]$;
    \item $\tilde{O} \rbr{ d \cdot \rbr{ 1 + (\frac{1}{B})^{\frac 2 \alpha} } }$, if $D$ satisfies $(B,\alpha)$-geometric Tsybakov noise.
\end{enumerate}
\label{lem:initialize}
\end{lemma}

Specifically, under the $\eta$-Massart noise condition, this yields a procedure that can  output a vector $\hat{u}$ with constant proximity to the optimal halfspace $w^\star$ with high probability, using $\tilde{O}(\frac{d}{(1-2\eta)^2})$ label queries.
This label complexity matches the $\Omega(\frac{d}{(1-2\eta)^2})$ information-theoretic lower bound~\cite[e.g.][Theorem 1]{yan2017revisiting}. When specialized to isotropic log-concave unlabeled distribution settings, this resolves an open problem by~\cite{zhang2020efficient} on whether there is an efficient and label-optimal initialization procedure that  reliably computes a vector with small constant angle with $w^\star$.

\section{Performance guarantees}
\label{sec:theory}

We now present Theorem~\ref{thm:main}, the main result of this paper.

\begin{theorem}
Fix $\epsilon \in (0,1)$ and $\delta \in (0, \frac{1}{10})$. Suppose $D$ is $(2, L, R, U, \beta)$-well behaved and satisfies one of the three noise conditions. With the settings of $\cbr{b_j}, \cbr{T_j}$, and $\epsilon_0$ under the respective noise conditions, with probability $1-\delta$, Algorithm~\ref{alg:main} outputs a halfspace $\tilde{v}$, such that $\err(h_{\tilde{v}}, D) - \err(h_{w^\star}, D) \leq \epsilon$. In addition, its total number of label queries is at most:
\begin{enumerate}
    \item $\tilde{O}\rbr{ \frac{d}{(1-2\eta)^2} \polylog(\frac1\epsilon)}$, if $D$ satisfies $\eta$-Massart noise;
    \item $\tilde{O}\rbr{ d \cdot \rbr{1 + A^{\frac{2-2\alpha}{\alpha(2\alpha-1)}} + (\frac{A}{\epsilon})^{\frac{2-2\alpha}{2\alpha-1}}} }$, if $D$ satisfies $(A, \alpha)$-Tsybakov noise with $\alpha \in (\frac12, 1]$;
    \item $\tilde{O}\rbr{ d \cdot \rbr{ 1 + (\frac{1}{B})^{\frac 2 \alpha} + \frac{1}{B^2} (\frac 1 \epsilon)^{\frac{2-2\alpha}{\alpha}}} }$, if $D$ satisfies $(B,\alpha)$-geometric Tsybakov noise.
\end{enumerate}
\label{thm:main}
\end{theorem}
Specialized to the $\eta$-Massart noise condition, our label complexity bound matches information-theoretic lower bounds~\cite[e.g.][Theorem 1]{yan2017revisiting} up to polylogarithmic factors, and improves over the state-of-the-art active halfspace learning algorithm of~\cite{zhang2020efficient} in two aspects. First, our label complexity is a factor of $O\rbr{ \frac{1}{(1-2\eta)^2} }$ lower than~\cite{zhang2020efficient}, thanks to the new initialization procedure; second, our algorithm and analysis allows for dealing with a broader set of unlabeled data distributions beyond isotropic log-concave, matching the assumptions employed in recent works~\citep[e.g.][]{diakonikolas2020polynomial}.

Under the $(A, \alpha)$-Tsybakov noise condition, our theorem provides nontrivial label complexity results when $\alpha \in (\frac12, 1]$. In the extreme case when $\alpha = 1$, our algorithm has a label complexity of $\tilde{O}\rbr{ d \polylog(\frac1\epsilon) }$. The label complexity bound becomes higher when $\alpha$ is further away from $1$. Compared to the recent passive learning algorithm of~\cite{diakonikolas2020polynomial} that can tolerate $(A,\alpha)$-Tsybakov noise for any $\alpha \in (0,1]$, our results cannot allow $\alpha$ to be in $(0,\frac12]$, but our algorithm has better label efficiency when $\alpha$ is close to 1.

Under the $(B, \alpha)$-geometric Tsybakov noise condition, our label complexity bound $\tilde{O}\rbr{ d (\frac{1}{\epsilon})^{\frac{2-2\alpha}{\alpha}} }$ is higher than the computationally inefficient algorithm of~\cite{balcan2013active}, which has a label complexity of $\tilde{O}\rbr{ d (\frac1\epsilon)^{2-2\alpha} }$.
This is due to a limitation of our current proof technique: we reduce the goal of achieving excess error guarantee to achieving geometric proximity.
Our proof in fact yields a stronger result: with $\tilde{O}\rbr{ d (\frac{1}{\epsilon})^{\frac{2-2\alpha}{\alpha}} }$ label queries, our algorithm outputs a halfspace that has angle $O(\epsilon)$ with $w^\star$ with high probability; this result matches the information-theoretic lower bound of~\cite{wang2016noise} in achieving closeness-in-angle guarantees, in the dependence on $\epsilon$. We leave whether it is possible to develop  efficient active learning algorithms with label complexity guarantees matching those of computationally inefficient algorithms in this setting as an important open question.

\begin{remark}[Unlabeled sample complexity of Algorithm~\ref{alg:main}]
Our active learning algorithm consumes a total of $O(\poly(d, \frac1\epsilon))$ unlabeled examples with high probability.
To see this, note that our sampling regions' bandwidths all satisfy $b = \Omega(\poly(\epsilon))$ under each of the three noise conditions and therefore have probability masses $\Omega(\poly(\epsilon))$ by Lemma~\ref{lem:1d-prob-ub}. This implies that with high probability, each active sampling invokes at most $O(\poly(\frac1\epsilon))$ calls to the unlabeled example oracle $\mathrm{EX}$; this implies that the total number of calls to $\mathrm{EX}$ is also $O(\poly(d, \frac1\epsilon))$.
\end{remark}

\begin{remark}[Attribute efficiency]
Our algorithm and analysis can be straightforwardly modified to achieve attribute efficiency \citep[e.g.][]{littlestone1987learning,blum1990learning,awasthi2016learning,zhang2018efficient}, i.e. achieving label complexities that exploit the sparsity of the Bayes-optimal linear classifier $w^\star$. Specifically, under the extra assumption that $w^\star$ is $s$-sparse ($s \ll d$), a variant of our algorithm achieves a guarantee similar to  Theorem~\ref{thm:main}, with the dimension $d$ in the label complexity bounds replaced with $s \polylog(d)$. We provide the details in Appendix~\ref{sec:sparsity}.
\end{remark}

\subsection{Proof sketch of Theorem~\ref{thm:main}}
We now outline the proof of Theorem~\ref{thm:main}.
Recall that from Lemma~\ref{lem:initialize}, line~\ref{step:init} of the main algorithm calls \ensuremath{\textsc{Initialize}}\xspace to generate vector $v_1$ such that $\| v_1 - w^\star \| \leq \frac14$ with probability $1-\delta/2$. This step uses $\tilde{O}\rbr{\frac{d}{(1-2\eta)^2}}$, $\tilde{O}\rbr{d \cdot (1 + A^{\frac{2-2\alpha}{\alpha(2\alpha-1)}}) }$, and $\tilde{O}\rbr{ d \cdot (1 + (\frac{1}{B})^{\frac 2 \alpha})}$ label queries to $\Ocal$, under the three noise conditions respectively.

For the guarantees in subsequent rounds, we rely on the following lemma, which shows that repeatedly applying \ensuremath{\textsc{Optimize}}\xspace yields local convergence guarantees.
Specifically, this lemma implies that, given an input halfspace $v_j \in \RR^d$ such that $\| v_j - w^\star \| \leq 4^{-j}$ at the beginning of the $j$-th iteration of the main algorithm (Algorithm~\ref{alg:main}), \ensuremath{\textsc{Optimize}}\xspace, with settings of bandwidth parameter $b_j$ and number of iterations $T_j$, outputs a refined halfspace $v_{j+1}$ such that its $\ell_2$ distance with $w^\star$ is at most $4^{-(j+1)}$ with high probability.

\begin{lemma}
Fix $r \in (0,\frac{1}{16}]$, and $\delta \in (0,\frac1{10})$.
Suppose $D$ is $(2, L, R, U, \beta)$-well behaved and satisfies one of the three noise conditions; in addition,
$b$ and $T$ are such that:
\begin{enumerate}
    \item $b = b_{\mathsf{MNC}}(\eta, r), T = T_{\mathsf{MNC}}(\eta, r)$, if $D$ satisfies $\eta$-Massart noise;
    \item $b = b_{\mathsf{TNC}}(A, \alpha, r), T = T_{\mathsf{TNC}}(A, \alpha, r)$, if $D$ satisfies $(A, \alpha)$-Tsybakov noise with $\alpha \in (\frac12, 1]$;
    \item $b = b_{\mathsf{GTNC}}(B, \alpha, r), T = T_{\mathsf{GTNC}}(B, \alpha, r)$, if $D$ satisfies $(B, \alpha)$-geometric Tsybakov noise.
\end{enumerate}
Then \ensuremath{\textsc{Optimize}}\xspace, with input initial $w_1$ satisfying $\| w_1 - w^\star \| \leq 4 r$, target proximity $r$, bandwidth $b$, number of iterations $T$, aggregation method $\mathrm{agg} = \mathsf{average}$, outputs $\tilde{w}$ such that probability $1-\delta r$,
    $\| \tilde{w} - w^\star \| \leq r$.
    \label{lem:optimize-main-avg-only}
\end{lemma}

The proof of Lemma~\ref{lem:optimize-main-avg-only} can be found at Appendix~\ref{sec:optimize}.
Some intuitions on this lemma have been given in Section~\ref{subsec:optimize}, and we elaborate on its key ideas in greater detail here. Recall that
Lemma~\ref{lem:optimize-g} shows that running \ensuremath{\textsc{Optimize}}\xspace gives an upper bound on $\frac 1 T \sum_{t=1}^T \psi_{D,b}(w_t)$, the average value of $\psi_{D,b}(w_t)$'s, in terms of $b$ and $T$.
We set $b$ and $T$ differently under different noise conditions, so that $\frac 1 T \sum_{t=1}^T \psi_{D,b}(w_t)$ can be controlled at an appropriate level.
Markov's Inequality implies that there is an overwhelming fraction ($\geq \frac{31}{32}$) of $w_t$'s with small $\psi_{D,b}(w_t)$ -  denote by $S$ the set of such $t$'s. Now, we conduct a case analysis:
\begin{enumerate}
\item For every $t$ in $S$, $\psi_{D,b}(w_t)$ is small. Recall that  Lemma~\ref{lem:pot-lb-main-text} shows that a small value of $\psi_{D,b}(w_t)$ implies a small value of $\tilde{\theta}(w_t, w^\star)$.
In addition, the diameter of the constraint set $\Kcal$ is at most $8 r \leq \frac12$, and both $w_t$ and $w^\star$ are in $\Kcal$, so $\theta(w_t, w^\star)$ is acute (see Lemma~\ref{lem:angle-l2}) and is equal to $\tilde{\theta}(w_t, w^\star)$.
Hence, for all $t$ in $S$, $\theta(w_t, w^\star) \leq \frac{r}{2}$ and consequently, $\| \hat{w}_t - w^\star \| \leq \frac r 2$.
\item On the other hand, for every $t$ in $[T] \setminus S$, we still have $w_t \in \Kcal$, so $\| \hat{w}_t - w^\star \|$ is  at most $16 r$.
\end{enumerate}

By averaging over the upper bounds on $\| \hat{w}_t - w^\star \|$, and using the convexity of $\ell_2$ norm, we conclude that $\|\tilde{w} - w^\star \| = \| \frac1T \sum_{t=1}^T \hat{w}_t - w^\star \| \leq \frac1T \sum_{t=1}^T \| \hat{w}_t - w^\star \| \leq r$.

Equipped with the above initialization and local convergence guarantees, the proof of Theorem~\ref{thm:main} is now straightforward; its details can be found at Appendix~\ref{sec:proof-main}.

\section{Conclusions and open problems}
We provide an efficient active halfspace learning algorithm that can achieve new label complexity guarantees under Massart and Tsybakov noise conditions, under certain structural assumptions on the unlabeled data distribution. Specifically, our algorithm achieves a near-optimal label complexity under the Massart noise condition, and achieves new label complexity guarantees under two subfamilies of Tsybakov noise conditions. A key open problem is to develop efficient algorithms with label complexities matching those of computationally inefficient approaches under  $(A,\alpha)$-Tsybakov noise, for all $\alpha \in (0,1]$.
Another interesting open question is to design efficient active learning algorithms that can adapt to unknown noise conditions.

\paragraph{Acknowledgments.} We thank Yining Wang for helpful discussions on label complexity lower bounds in~\citep{wang2016noise} for active learning halfspaces under Tsybakov noise under the uniform distribution. We also thank the anonymous reviewers for their constructive feedback.

redpajama_set_name: RedPajamaArXiv


text:
Washington Watch: Donald, melech Yisrael
Trump regularly retweets conspiracy theories and many other cockamamie ideas, which he claims he's just innocently passing on. That won't wash. You retweet it, you own it.
By DOUGLAS BLOOMFIELD
Published: SEPTEMBER 1, 2019 10:52
U.S. President Donald Trump visiting the Western Wall in 2017
(photo credit: JONATHAN ERNST / REUTERS)
When US President Donald Trump questions the loyalty of American Jews, he's not just talking about whether they are more loyal to Israel than the United States; he's upset that they fail to show him the appreciation and loyalty he feels he personally deserves for all he has done for Israel.
And to show that fealty, he expects them to make him their Moses and follow him in a mass exodus to the Republican Party from the Democratic Party.
More than Moses, actually. The messiah, judging by his tweets and talk.
He recently retweeted comments by Wayne Allen Root, a right-wing writer, conspiracy theorist and Messianic Jew, who called Trump "the greatest president for the Jews and for Israel in the history of the world" and said Israelis "love him like he is the King of Israel… [and the] second coming of God."
When Trump gets heavy flak from some tweets or boasts, like when he referred to himself as "the Chosen One," he tries to retreat by saying he was just joking. Nothing doing. He's not one known for having a sense of humor.
The tweet about being the "King of Israel" should upset his wing-man, Prime Minister Benjamin Netanyahu, whose loyalists call him "King Bibi." Israel is a strong country, but can it handle such enormous and fragile royal egos?
But Trump's dual loyalty accusations are about more than showing him devotion. Such charges have incited hatred and violence against Jews throughout history. Trump's words are fueling the rise of white nationalism and antisemitic attacks across the United States, often by professed Trump followers.
The ADL reports that antisemitic hate crimes have been on the rise since Trump came to office.
If anyone ever doubted Trump was an antisemite, just go back over all those comments, tweets, tropes, references to "your prime minister," campaign postings and his history in real estate. That should clear up any doubts. And having Jewish grandchildren is no more redeeming than Hitler having Jewish ancestors.
Trump has "helped unleash a new wave of antisemitism," wrote New York Times columnist Michelle Goldberg. She quoted Sophie Ellman-Golan, an activist working to mobilize Jews against white nationalism, saying, "Only one political party is quite literally inciting white nationalists to shoot up our synagogues."
Trump may think he can draw Jews away from the Democratic Party by making two Muslim freshman congresswomen the face of the party, but it may be quite the opposite. Interestingly, what they said about Jewish dual loyalty is not much different from what Trump said, just the tone. He may think it is a compliment, and they know it isn't.
Trump does not comprehend that Jews are not single-issue voters and that Israel is not at the top of their agenda – and it may, in fact, be shrinking as a factor in the political choices they make. Moreover, when it comes to Israel, we Jews have as many diverse views of that country and its policies as we have of our own.
THE BIG mystery is why Republican Jews continue to support a man who praises them for their dual loyalty while denigrating Democrats for not being equally obsequious to an antisemite, racist xenophobe. To be a good Jewish Republican, he is saying, you must be loyal to another country and the bigot in the White House.
In 2018, Jews voted nearly 80% Democrat, five points higher than the previous election. Democratic pollster Jim Gerstein has said evidence suggests Trump is driving more Jews toward the Democratic Party than away.
Goldberg contends Trump's antisemitism and racism "has spurred a renaissance" in the Jewish Left in America.
And he is helping drive growing numbers of Jews away from Israel, accelerating a trend that keeps Jewish leaders up at night. Most Jewish voters are less enthusiastic about moving the US Embassy to Jerusalem or encouraging settlements and annexation of the Golan Heights and parts of the West Bank than they are about pursuing an Israeli-Palestinian peace process and the two-state solution. They are also repulsed by Trump's embrace of religious and nationalist extremists in both countries.
Trump's Israel policies are designed to appeal to his broad Evangelical base (and, to a lesser degree, a small group of wealthy Jewish donors). Christians United for Israel (CUFI) – the evangelical lobby with some seven million members – has eclipsed AIPAC in the eyes of the White House because it speaks for the largest portion of Trump's base. Evangelicals profess to love Israel and love the Jews so much they want to see us disappear in the Second Coming (we Jews are still waiting for the First). The most outrageous part of this ugly episode of dual loyalty charges may be Netanyahu's silence. His acquiescence only serves to further widen the rift between Israel and American Jews and strikes a damaging blow to the longstanding bipartisan consensus that is the foundation of US-Israel relations. Republicans, for purely partisan reasons, have tried to use Israel as a wedge issue to claim their party loves Israel more than the Democrats – and Bibi has been their most valuable and willing tool.
If members of Congress see that Israel is no longer a high priority for their Jewish constituents, they may begin having second thoughts about sending $3.8 billion a year to shore up a Republican stronghold.
Like no other Israeli leader, Netanyahu has plunged deeply into American partisan politics. Shallow dives by Yitzhak Rabin endorsing Richard Nixon and Bill Clinton endorsing Shimon Peres were nods, not passionate embraces.
Netanyahu has been quick to take offense when he thinks Israel is being criticized unfairly – but Trump, a swaggering bully, has apparently scared him silent. Instead the prime minister grovels before a bigoted president most American Jews consider unfit to lead their country.
Netanyahu is so intensely focused on his own September 17 Israeli election, and his desire to keep his job and stay out of prison, that he doesn't seem to care about the damage he is doing to American Jewish support. He keeps widening the chasm and tightening his embrace of a president most American Jews vigorously oppose for a multitude of reasons, starting with his antisemitism. If Bibi wins a record fifth term, don't look for much to change.


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Dash decked out in his hat and sunglasses for a ride on the tricycle.
A bowl of cherries -- and little red, white, and blue preview.
We had kind of a blah Fourth of July; Mr. E had to work. I took the boys to the park in their holiday outfits.
Larabars. A new discovery I've been eating a lot of lately.
Fun day on the farm with grandparents and cousins. And a very small pool.
Bedtime. Mr. E and my Kindle.


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Ecotourism Investment Begins to Flourish
Ecotourism is beginning to catch the eye of private investors and other funders who are interested in financing sustainable development, land rehabilitation, wildlife reintroduction, and/or conservation research. Examples from Africa show how ecotourism can synergistically build support for these goals.
The tourism sector is one of the fastest-growing parts of the global economy. According to the World Travel and Tourism Council, the tourism sector made up 9.8 percent of world GDP in 2015. This will rise to 10.8 percent by 2026. Tourism has grown into such a key part of economies and development that the United Nations has made 2017 the Year of Sustainable Tourism for Development.
According to the Center for Responsible Travel (CREST), a research organization focused on sustainable travel, ecotourism has begun to stand out as one area that is quickly growing in popularity among different responsible tourism markets. Examples of ecotourism include education and community projects, wildlife research, and energy-efficiency activities. Destinations vary widely in how much and how they support these projects.
Ecotourism advances conservation using the lens of sustainable development. It supports local communities by creating financial incentives to protect their natural spaces. It also provides alternative jobs that make local citizens proud of their environments and motivate them to reduce degrading practices.
According to Martha Honey, a leading expert in sustainable tourism and the executive director of CREST, one of the missing legs that would provide more support for ecotourism is financing. She said many of the other pieces are in place, such as local interest, certification tools, and government buy-in, but funding has remained a stumbling block that makes it harder for this industry to continue to grow.
Financing for ecotourism mainly comes from three sources: wealthy individuals, philanthropic foundations, and multilateral development banks. While other players such as nonprofits also invest in ecotourism, the bulk of the capital comes from these three sources.
Singita, a group of luxury safari lodges in Africa, is an example of a project with philanthropic financing. Their website describes how Luke Bailes, the founder, began a lodge on a piece of family-owned land in the Sabi Sand Reserve in South Africa to begin to raise funds for conservation management. Through its early success, Singita gained the support of a wealthy philanthropist from the United States and other individual investors.
Singita expanded to 12 lodges and properties in five different ecosystems across Africa, with more lodges coming. Neil Midlane, manager of Conservation Projects at Singita, said that through the ecotourism provided on these properties, Singita has been able to control land management to support land rehabilitation, wildlife monitoring, anti-poaching security, scientific research, and wildlife reintroduction.
According to Midlane, by using a high-cost, low-impact model, Singita uses revenue and donations from wealthy clients to fund land management. It has built upon its reputation to gain the support of new impact investors interested in financial and conservation returns of long-term investments. Another source of income that Singita has recently begun allows individuals to lease portions of land on which they can build private lodges or villas within the reserves. These properties provide returns to investors.
At the other end of the spectrum, multilateral development banks, such as the Inter-American Development Bank, have recognized the importance of ecotourism in providing sustainable jobs for local communities. As can be seen on their websites, many of the regional development banks have direct investment projects with ecolodges and other operators. Some others have worked with third parties such as nonprofit organizations or impact funds to funnel investments and ensure sustainable practices.
EcoEnterprises Fund (EcoE) is an impact fund that receives financing from both multilateral development banks and individual investors. According to its website, EcoE uses funds to help grow companies – including ecotourism operators. EcoE specializes in mezzanine financing and equity for growth. The company is currently working on the second of three rounds of funds. It chooses and works with environmentally oriented companies to scale up their practices and collect returns.
From both a conservation perspective and a financial standpoint, ecotourism is an industry that can have intersecting benefits. But it also needs a close amount of monitoring and evaluation.
When done effectively, ecotourism can have good results for wildlife and habitat. According to Midlane, Singita has seen major population growth of local buffalo, elephants, and other megafauna. It has even had game-changing growth in its black and white rhino populations, reintroducing the species into areas where they had disappeared completely.
From a financial standpoint, ecotourism investment is as much impact-focused as return-oriented, according to Nathalie Prado, managing director at EcoE. Prado said that it is a steady, long-term investment and a necessary part of any impact investing portfolio.
These investments provide human-interest and conservation results that investors want as well as long-term returns that balance portfolio averages. EcoE has found its ecotourism investments have higher returns than comparable ones that are in similar tourism businesses. They also produce more straightforward returns than investments in food, agriculture or beverages.
Obstacles and Potential
Many businesses remain too small for investment. Often, local and indigenous communities with interest in tourism have trouble finding the support they need to start businesses. Investors and development banks also have lower interest and a limited ability to be able to support many small operations.
While EcoE was previously financing a number of ecotourism ventures with the first of its three funds, it has reduced this as it has shifted its focus toward growing established companies, Prado said.
Due to the many parts of sustainability of ecotourism businesses, investments need more in-depth monitoring and analysis. EcoE staff have found that they have had to work more closely with their ecotourism investments in order to make sure proper practices are being used.
According to Honey, working closely with operators to expand energy and water efficiency is one of the main requirements for growing and ensuring sustainability. Investments in this infrastructure pay for themselves through the energy-cost savings of greening operations.
Ecotourism is also not a cure-all for funding conservation. There are few areas where lodging revenue covers the full cost of running a park, Midlane said. Since ecotourism is dependent on weather, politics, and other issues that impact appeal, it needs a wider net of investment and financial flows outside of operating revenue. This is especially true for the demonstrated conservation results at Singita.
While it comes with a high level of thoughtful monitoring, ecotourism, with its sustainable development overlap and long-term returns, has begun to and can continue to grow to play an important role for conservation in the future.
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My client is one of the largest suppliers of specialist laboratory and life sciences systems in the UK. This instantly recognisable market leader is renowned for their forward-thinking products, support services and complete solutions that range from systems used within the healthcare, life sciences, pharmaceutical and industrial market sectors. With over 15,000 employees operating in over 50 countries worldwide, they are a market leader in a number of equipment technologies and enjoy 'best in class' status with their innovative systems.
Following expansion they seek to recruit a customer focused Field Service Engineer responsible for the service, installation, repair and general maintenance of their range of highly advanced gas chromatography systems at customer sites.
To be considered for this post you should have previous experience either as a Field Service Engineer, Applications Engineer or Technical Support Engineer working on a range of GC / GCMS / Gas Chromatography systems – or as a laboratory technician or scientist with extensive experience of using GC / GCMS / Gas Chromatograpy systems. Ideally you will be educated with an HNC, HND or BSc in a life science or engineering discipline and have a full UK driving license and a minimum of 2 years experience with life sciences instrumentation. Of equal importance is your ability to communicate effectively at all levels with external customers and internal departments.
In return the successful applicant will benefit from first class product training, recognised by the industry, which will run on an ongoing basis as new products are introduced. A competitive remuneration package is also guaranteed, including a competitive basic salary of up to £40K, 10% bonus, car, private medical cover, life assurance and pension scheme.
This vacancy is being advertised by TRS Consulting. The services advertised by TRS Consulting are those of an employment agency and / or an employment business.