Andrea Montanari Profile Banner
Andrea Montanari Profile
Andrea Montanari

@Andrea__M

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Professor, Statistics and Mathematics, Stanford University.

Joined April 2009
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@Andrea__M
Andrea Montanari
3 years
A reminder that: 1. Number of citations is not a measure of scientific validity; 2. Number of publications is not a measure of scientific value; 3. The argument from authority is not acceptable in science. (On a personal note, 3 is why I started loving science as a kid.)
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@Andrea__M
Andrea Montanari
3 years
The Nobel prize to Giorgio Parisi is such a joy and and multiply well deserved recognition: In random order: (1) Stochastic quantization; (2) The KPZ equation; (3) Matrix models; (4) Mean field spin glasses (!); (5) Random constraint satisfaction problems. 1/2
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@Andrea__M
Andrea Montanari
2 years
I can see the end-game in the current quest for the mathematical foundations of deep learning: A neural network will find a (very long) proof that deep learning works. No human will be able to understand it, but the proof will be formally verified.
@OpenAI
OpenAI
2 years
We trained a neural network that solved two problems from the International Math Olympiad.
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@Andrea__M
Andrea Montanari
3 years
With Peter Bartlett and Sasha Rakhlin, an overview of a slice of research in theoretical machine learning (and an effort to connect some dots):
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@Andrea__M
Andrea Montanari
10 months
A professional update. After 17 years in the wonderful Stanford Electrical Engineering Department, I am moving 50% of my appointment to Stanford Math. I am grateful to the University for making this change possible.
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@Andrea__M
Andrea Montanari
4 years
I devoted good part of the last 2 years trying to understand interpolators: models with vanishing training error. These behave well when the interpolator is selected via a `min norm' principle. Do I understand this phenomenon better now? A bit, summarized here in a cartoon. 1/n
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@Andrea__M
Andrea Montanari
2 years
The Shannon award to Rudiger Urbanke is both terrific news and a long overdue recognition to one of the most influential information and coding theorists of the last quarter century. 1/n
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@Andrea__M
Andrea Montanari
3 years
If you were ever curious about what theoretical physics has to do with high-dimensional statistics. (Next week, in Stanford, in person!)
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@Andrea__M
Andrea Montanari
2 years
I was asked several times for the lecture notes of this class (taught at Stanford in Winter 2017). At long last, with @subhabratasen90 , we managed to clean them up and make them presentable. Hopefully, they will be useful.
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@Andrea__M
Andrea Montanari
4 years
When do neural networks outperform kernel methods? High-dimensional noise orthogonal to the discriminative direction seem to hurt kernel methods more than nnets. (Experiment: tweeting a paper for the first time. Curious to see the feedback.)
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@Andrea__M
Andrea Montanari
2 years
Slides from a talk at MCQMC2022. Discussed (among other things) sampling via stochastic localization and its connection to diffusions.
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@Andrea__M
Andrea Montanari
2 years
I do not think that this is ok @Stanford
@minilek
Jelani Nelson
2 years
A @Stanford professor just threatened me with police. After BBQ Becky, Permit Patty, Golfcart Gail, and all the memes, we now have Retweet Rachel. Public advisory: don't call the cops on black people for no reason. Black people disagreeing with you on Twitter is not a crime.
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@Andrea__M
Andrea Montanari
3 years
Sorry, looks like I cannot finish. Giorgio also represent a unique way of doing science, which privileges creativity and wonder, while disregarding boundaries between disciplines and their mannerisms. 4/4
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@Andrea__M
Andrea Montanari
4 years
David Blackwell about how he was hired in Berkeley: via @YouTube
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@Andrea__M
Andrea Montanari
4 years
If you are interested doing research on the mathematics of machine learning, consider applying for a postdoc with our collaboration:
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@Andrea__M
Andrea Montanari
3 years
The dangers of statistical modeling with non-iid data. (And an excellent example of poor communication.)
@business
Bloomberg
3 years
Goldman Sachs predicts that England will win its first major tournament in 55 years, based on a probability model using 6,000 matches
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@Andrea__M
Andrea Montanari
8 months
Slides on Optimization of Random Cost Functions (29th Solvay Conference of Physics). Tried to summarize what did we learn in the last 40 years about the fundamental complexity of random optimization, using Fu-Anderson 1986 as a starting point: …
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@Andrea__M
Andrea Montanari
2 years
I will try to explain (the motivation for) a very technical paper in a thread. Let's see if I manage. The paper is here and here is a talk [tldr: This is about the analysis of gradient algorithms for random cost functions.] 1/n
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@Andrea__M
Andrea Montanari
3 years
(6) Turbulence; (7) Stochastic resonance; (8) Deep inelastic scattering; (9) Lattice QCD and parallel computers. Each of these has inspired the work of entire communities of physicists and mathematicians. I have been influenced by Parisi's work since a student. 2/3
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@Andrea__M
Andrea Montanari
4 years
A workshop on computational (and statistical!) phase transitions at Simons Institute. A lot of exciting progress.
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@Andrea__M
Andrea Montanari
4 months
Fantastic choice
@abel_prize
The Abel Prize
4 months
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@Andrea__M
Andrea Montanari
6 months
I find theory helpful when I see something empirically, that looks weird. Is it a bug? One way to convince myself is to construct a model in which I can prove the weirdness happens, and I can prove it. Recent example: Weakly supervised subsampling data and reducing test error.
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@Andrea__M
Andrea Montanari
4 years
@HelenBranswell If applied to Lombardia, the same argument with 0.3% CFR and 1% infections would result in 300 deaths. We are at 1640 and growing with lockdown! This article is ignoring data that doesn’t fit its narrative.
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@Andrea__M
Andrea Montanari
4 years
If it was up to me, I would ban George E. P. Box famous quote ("all models are wrong.."). If you really need to quote your way out, please use this:
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@Andrea__M
Andrea Montanari
3 months
The latest chapter in the saga of using stochastic localization/denoising diffusions to sample from highly multi-modal Gibbs measures. With the amazing Brice Huang and Huy Tuan Pham. (1/4)
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@Andrea__M
Andrea Montanari
3 years
Just to remind something (minor with respect to the Nobel motivations): for a brief period in the 80s Italy had one of the most powerful parallel computers of the world due to the work of Cabibbo, Parisi and a group of scientists and engineers that built it for science. 3/4
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@Andrea__M
Andrea Montanari
2 years
Very new for me: I will be on partial leave and devote a good fraction of next year to work on a non-academic project. Contact me if you are interested in the opportunity.
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@Andrea__M
Andrea Montanari
4 months
Looks like LLMs put the last nail in the coffin of peer reviewing.
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@Andrea__M
Andrea Montanari
3 years
According to a well-known anecdote, Pablo Casals, one of the greatest cello players ever, while in is eighties was asked why he kept practicing five hours a day. The answer was: "Because I feel I am making progress." 1/4
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@Andrea__M
Andrea Montanari
17 days
I think it is instead important that scientists, technology experts and researchers stay engaged with political life. It is a small “price” we pay to live in a democracy.
@tengyuma
Tengyu Ma
18 days
To improve my productivity for H2 of 2024, I am trying to find an automatic political news blocker. Any recommendations?😂
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@Andrea__M
Andrea Montanari
3 years
I will add one personal comment about what is eloquently explained in this letter. We do not teach calculus or (for that matter) plane geometry only because of their intrinsic content. 1/x
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@Andrea__M
Andrea Montanari
4 years
Happy to have helped this group get together:
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@Andrea__M
Andrea Montanari
1 year
True, Large Language Models are impressively approaching humans, but what really scares me is humans increasingly behaving as Large Language Models.
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@Andrea__M
Andrea Montanari
2 years
I was always taught that asking the right questions is more important to being able to answer them. I never imagined this could become as obviously true as it is now.
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@Andrea__M
Andrea Montanari
3 years
I tried my best to explain this area of research to a broad audience of non-specialists (at the risk of being boring to the many world experts sitting in the room).
@SimonsInstitute
Simons Institute for the Theory of Computing
3 years
Watch the video of Andrea Montanari's recent Richard M. Karp Distinguished Lecture. "Computational Barriers in Statistical Estimation and Learning"
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@Andrea__M
Andrea Montanari
3 years
Took us several months, but we recently posted a revision of our work on the generalization error of interpolating NT models of 2 layer neural network. The new version removes the assumption that the network width is at most linear in the input dimension An instructive fig: 1/2
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@Andrea__M
Andrea Montanari
4 years
A small story about science and research, that perhaps is worth telling once more (in the painting, two of the main characters as portrayed by Raphael more than one millenium later). 1/n
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@Andrea__M
Andrea Montanari
4 years
For many years, Stanford Statistics Department has offered a free consulting service to Stanford's research community. For Summer 2020, we will open this service to anybody around the world for COVID-related data
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@Andrea__M
Andrea Montanari
1 year
As for many others, I was deeply impacted by Sanjoy Mitter. I met Sanjoy when I was finishing my PhD. He invited me to spend a quarter at MIT. My thesis was in lattice field theory, but I had started to work on the connections with information theory. 1/n
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@Andrea__M
Andrea Montanari
5 months
Here it is:
@Andrea__M
Andrea Montanari
8 months
Slides on Optimization of Random Cost Functions (29th Solvay Conference of Physics). Tried to summarize what did we learn in the last 40 years about the fundamental complexity of random optimization, using Fu-Anderson 1986 as a starting point: …
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@Andrea__M
Andrea Montanari
7 months
Back at it. (Oh, I love this material)
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@Andrea__M
Andrea Montanari
2 years
Congratulations to @KrzakalaF and @zdeborova for a well organized summer school. Videos:
@KrzakalaF
Krzakala Florent
2 years
Neural networks, a non parametric viewpoint by @Andrea__M Montanari in #leshouches2022
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@Andrea__M
Andrea Montanari
1 year
By the way, the second of two papers this talk was based on is now out. (With my fantastic collaborator Yuchen Wu!)
@Andrea__M
Andrea Montanari
2 years
Slides from a talk at MCQMC2022. Discussed (among other things) sampling via stochastic localization and its connection to diffusions.
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@Andrea__M
Andrea Montanari
2 years
Confession: I am not sure what is the function of so many "best xyz" awards in academia, nor what do they really measure. I know I am naive, but shouldn't good work and strong results speak for themselves?
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@Andrea__M
Andrea Montanari
4 years
Amid a major public health crisis, we are also going through a reproducibility crisis on steroids. Too many studies are popping out with poor or unfinished analysis. Tentative findings are picked up by the media and pushed onto the public without caution. 1/n
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@Andrea__M
Andrea Montanari
4 months
I am partial to the original version, but then again, if this is what it takes…
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@Andrea__M
Andrea Montanari
4 years
I learnt this story from Lucio Russo's beautiful book `The forgotten revolution'. What do I learn from it? Many things, but crucially that every breakthrough is actually the consequence of a laborious accumulation of knowledge, which involves many researchers. 5/5
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@Andrea__M
Andrea Montanari
2 years
For balance: a good fraction of probability theory used in high-dim statistics and ML has roots in physics. (Few of my papers inspired by statphys: , , , , )
@YiMaTweets
Yi Ma
2 years
What are the similarity and difference between statistical physics and data science? They all deal with high-dimensional distributions. But statistical physics often has (form of) the distribution but no samples; data science often has plenty of samples, but not the distribution.
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@Andrea__M
Andrea Montanari
8 months
First Latin translation of the Almagest (iirc 1541, Basel). It is mind boggling that the modern scientific revolution might have happened not thanks of a handful of geniuses but because of a technical innovation that expanded dramatically the access to 1000s years old manuscripts
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@Andrea__M
Andrea Montanari
4 years
To all students defending these days: I have been in a few online defenses by now, and I realize this can feel a bit of an anticlimactic experience. However: (1) A thesis defense is the anticlimactic event by definition. It felt that way for me. 1/n
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@Andrea__M
Andrea Montanari
3 years
Proud spouse moment (my wife, Ivana Maric , is a co-recipient the Rosenkranz prize)
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@Andrea__M
Andrea Montanari
2 months
Talk at FFT2022 (UMD). In the first few slides an example of the fact that overparametrizing and under-regularizing can be a bad idea in low dimension.
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@Andrea__M
Andrea Montanari
3 months
@ylecun Do not agree. This is regularized, so it is not really demonstrating double descent. (If you regularize it is obvious that you can have more parameters than samples)
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@Andrea__M
Andrea Montanari
29 days
A recent talk about solving overparametrized systems of nonlinear equations via Hessian descent. (Celebrating Andrew Barron) …Slides here
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@Andrea__M
Andrea Montanari
1 year
Shouldn’t open source be a fundamental ethical principle of scientific ML research? Am I missing something?
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@Andrea__M
Andrea Montanari
2 months
Another great blackboard
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@Andrea__M
Andrea Montanari
3 months
No.
@ylecun
Yann LeCun
3 months
Here ya go: double descent with fitting polynomials. Automatically demonstrated.
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@Andrea__M
Andrea Montanari
3 years
I feel that the distinction between pure and applied mathematics is no longer useful. I am in favor of replacing if with the distinction between pure and impure mathematics. 1/3
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@Andrea__M
Andrea Montanari
4 years
Remove Trump account from Twitter @jack
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@Andrea__M
Andrea Montanari
3 years
As a parent, I can witness every day that California public schools have already a weak math curriculum. These proposed changes go exactly in the wrong direction and risk to cause huge damage to children @GavinNewsom
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@Andrea__M
Andrea Montanari
2 months
Talk at ICERM, last week:
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@Andrea__M
Andrea Montanari
2 years
Please @InstMathStat , write a statement about what is actually happening, instead of an exercise in neutrality.
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@Andrea__M
Andrea Montanari
3 years
As much as I enjoy browsing the math genealogy project, I must say it risks to convey the wrong message that academic lineage matters. It shouldn't (except for historians).
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@Andrea__M
Andrea Montanari
3 months
Explanation: if you adapt the level of regularization to the overparametrization, there is nothing double (or particularly surprising) about double descent:
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@Andrea__M
Andrea Montanari
2 years
This tweet was a joke. Since a few people took it seriously (which is often the right thing to do with jokes), I will add a couple of comments. 1/n
@Andrea__M
Andrea Montanari
2 years
I can see the end-game in the current quest for the mathematical foundations of deep learning: A neural network will find a (very long) proof that deep learning works. No human will be able to understand it, but the proof will be formally verified.
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@Andrea__M
Andrea Montanari
4 years
A workshop on mathematical epidemiology at MSRI, Berkeley, from last week. Videos of several interesting talks
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@Andrea__M
Andrea Montanari
3 years
Attended several great talks from this workshop last week. The amount of progress towards a rigorous understanding of computational barriers in statistical estimation and learning is impressive.
@SimonsInstitute
Simons Institute for the Theory of Computing
3 years
Streaming live this week: the workshop on Rigorous Evidence for Information Computation Trade-Offs. This workshop is part of the Fall 2021 Simons Institute research program on Computational Complexity of Statistical Inference. Schedule and video:
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@Andrea__M
Andrea Montanari
3 months
Congratulations to Germain and Pulkit @tandon_pulkit ! (honorable mention @iclr_conf )
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@iclr_conf
ICLR 2025
3 months
Announcing the #ICLR2024 Outstanding Paper Awards: Shoutout to the awards committee: @eunsolc , @katjahofmann , @liu_mingyu , @nanjiang_cs , @guennemann , @optiML , @tkipf , @CevherLIONS
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@Andrea__M
Andrea Montanari
4 years
I was on leave from Stanford for a good part of 2020, and did not teach since the pandemic started. I look forward to resuming in January, and I am really curious about remote teaching.
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@Andrea__M
Andrea Montanari
3 years
Miguel Angel Virasoro passed away today. He was a truly great scientist. His 1985 paper on "The microstructure of ultrametricity" is one of my all time favorites. It is one of those papers that even 35 years later conveys the excitement and surprise of a beautiful discovery.
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@Andrea__M
Andrea Montanari
4 years
I agree: this is the best metaphor for the counterintuitive behavior of exponential growth phenomena I have read so far.
@Bob_Wachter
Bob Wachter
4 years
UK variant – ~50% more infectious & maybe more fatal too – is now surging in Denmark. In @washingtonpost article , terrific metaphor by Danish prime minister about why redoubling safe behavior is needed, even when things seem OK. That's exponential growth.
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@Andrea__M
Andrea Montanari
2 months
Still, I wouldn't have minded being in Göttingen in the 1920s. (If I could keep the science and discard the broader historical context)
@SebastienBubeck
Sebastien Bubeck
2 months
Every day I witness the AI revolution in action, and every day I see 1 or 2 questions that would deserve an entire PhD thesis to explore fully ... Honestly, how lucky we are to do research in that era!! *Even if* there is no more magical leap like from gpt2 to 3, or 3 to 4,
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@Andrea__M
Andrea Montanari
1 year
My two favorite science popularization books are Hilbert, Cohn-Vossen (1932) and Courant, Robbins (1941). Can anyone recommend anything of the same caliber? It seems that many popularization books take shortcuts: they explain things "around" the science, not the science itself.
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@Andrea__M
Andrea Montanari
2 years
I agree with everything that Jelani says here! It is truly stunning that anybody can think of addressing inequality by removing the content from public education.
@minilek
Jelani Nelson
2 years
"any attempt to make our education system fairer is met with fierce resistance from affluent liberals worried that Democratic reforms might threaten their carefully laid plans to help their children get ahead." @nytimes essay gets it terribly wrong
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@Andrea__M
Andrea Montanari
3 months
If you are in Vienna next week, consider attending Pulkit Tandon's @tandon_pulkit oral presentation at ICLR. Pulkit is a very clear speaker! Paper Calendar Github
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@Andrea__M
Andrea Montanari
4 years
Textbook example of false objectivity. There wasn't two candidates trading insults. There was one of them (Trump) openly supporting white supremacy and authoritarian ideology.
@WSJ
The Wall Street Journal
4 years
Take an early look at the front page of The Wall Street Journal
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@Andrea__M
Andrea Montanari
4 years
Overparametrization in modern machine learning comes with its blessings and its challenges. Nice thread about the latter (and a recent paper) by @alexdamour .
@alexdamour
Alexander D'Amour ([email protected])
4 years
NEW from a big collaboration at Google: Underspecification Presents Challenges for Credibility in Modern Machine Learning Explores a common failure mode when applying ML to real-world problems. 🧵 1/14
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@Andrea__M
Andrea Montanari
3 years
I thanked science (and every human who worked to make this possible), but of course this is a personal choice :)
@NPRKelly
Mary Louise Kelly
3 years
Round 1, done. Thank God.
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@Andrea__M
Andrea Montanari
2 years
@carlorovelli Trivial whataboutism. Thought you could come up with something smarter.
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@Andrea__M
Andrea Montanari
1 year
Not long ago, I read yet another fascinating book by Lucio Russo, this time about the history of the theory of tides. The book is not translated to english, so I thought I should write a short thread. Some of its content is presented in this paper 1/n
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@Andrea__M
Andrea Montanari
3 years
We also teach them because they are the best way to learn a procedure that is the foundation of scientific reasoning: 1) Define abstract objects. 2) Describe their relation to real ones. 3) Logically derive consequences of abstract definitions. 4) Compare with observations. 2/x
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@Andrea__M
Andrea Montanari
4 years
This is an example of how to use data to create the impression of evidence, and instead push your own political agenda. I really hope that Stanford takes data science more seriously than this! @Stanford
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@Andrea__M
Andrea Montanari
2 years
As others, I spend a fraction of my time preparing lectures. It is great that these days many of these presentations are recorded. Makes me feel the energy spent in preparing slides was more useful. These are slides of a talk that was not recorded:
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@Andrea__M
Andrea Montanari
2 years
I occasionally feel that GitHub is the programmer's version of Borges' Library of Babel. One does not need to code: it is sufficient to know where to find the code that implement the desired function.
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@Andrea__M
Andrea Montanari
2 months
Don’t agree entirely with this. Not sure we understand mathematically how a jet flies in the same way that, say, we understand kernel ridge regression. Should expect to understand a deep net better than a plane? Still there are lots of important aspects to study/understand.
@jasondeanlee
Jason Lee
2 months
@SebastienBubeck @Andrea__M No strong reason but more a personal belief that math has been useful in most eng fields. And almost all fields move to more and more quantitative.
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Andrea Montanari
3 months
@hippopedoid @ylecun I must say this hurts more than the 120% probability meme.
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@Andrea__M
Andrea Montanari
3 years
This is a topic I have been thinking about for over 10 years. Why do I keep thinking about it? Well, because I feel I am making progress. 4/4
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@Andrea__M
Andrea Montanari
2 years
Bravo Roman
@UkrainianMath
Ukrainian Mathematicians
2 years
Prof. Roman Vershynin (UCI) will be teaching on zoom Probability for Data Science course for National University of Kyiv students this upcoming semester.
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@Andrea__M
Andrea Montanari
2 years
Is Italy the only country in the world towards which negative stereotypes are not only accepted, but mainstream? I lived and worked abroad for a quarter century. I have seen dozens of these. Please, try to be more creative.
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@Andrea__M
Andrea Montanari
4 years
Third, the underlying simple model is regularized (even if the complex initial model isn't). More details in this talk tomorrow (including recent work with Yiqiao Zhong ) or this talk of a few weeks ago 5/n
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@Andrea__M
Andrea Montanari
3 years
Very honored of giving this talk. Merci a toutes et a tous les presents.
@ValRobert974
𝕍alérie ℝ. 🔆🌋🪄🐿️
3 years
Conférence Lucien Le Cam d'Andrea Montanari #JdS2021 #statistics @StatFr
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@Andrea__M
Andrea Montanari
2 years
Rudiger had also outsized influence on so many colleagues and students. Surely he had a huge influence on me. I came to coding and information theory from theoretical physics, with a slightly snobbish attitude that physicists sometimes have towards..., well, everybody else. 4/n
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@Andrea__M
Andrea Montanari
2 years
Sorry, it is important to add one last thing. Rudiger has always been exceptionally open and and genuinely interested in understanding and applying ideas from other fields (physics, cs, math, etc). This had remarkable effects on the whole community. n+1/n
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@Andrea__M
Andrea Montanari
7 months
Thought: We might not be the only thinking thing but we seem to be the only thing to whom thinking matters (so far).
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@Andrea__M
Andrea Montanari
2 years
Sounds familiar
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@Andrea__M
Andrea Montanari
1 year
I believe part of this is due to a point of view in the computational sciences, according to which assumptions are "good" or not because of some fundamental reason. In the natural sciences, assumptions are typically made so as their implications match observed phenomena.
@ylecun
Yann LeCun
1 year
When empirical evidence clashes with a theory, it is the theory that is wrong (or misinterpreted), not the universe. The well-documented initial resistance of many ML theorists to deep learning was due to complete theory-fueled disbelief: this can't possibly work!
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@Andrea__M
Andrea Montanari
4 years
Thanks, I'll take CS-y as a compliment!
@boazbaraktcs
Boaz Barak
4 years
The Online Open Probability school looks fantastic! CS-y lecturers include @Andrea__M , Tselil Schramm, Amin Cojah Oghlan, and Elchanan Mossel
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