How the Bizarre Path of Mars Reshaped Astronomy [Keplers Laws Part 1] Welch Labs 2026-09-19 | How the Bizarre Path of Mars Reshaped Astronomy [Keplers Laws Part 1]
Can humans make AI any better? Welch Labs 2026-01-31 | Apply to work at Tufalabs: tufalabs.ai/joinWelch Labs Book: welchlabs.com/resources/ai-book-ezrzmWelch Labs eBook: welchlabs.com/resources/the-welch-labs-illustrated-guide-to-ai-digital-downloadPatreon: patreon.com/welchlabsSECTIONS0:00 - Harpy4:39 - The Bitter Lesson5:58 - Sutton Goes on a Podcast8:22 - LLMs are Not Bitter Lesson Pilled?9:19 - Supervised Learning10:04 - Reinforcement Learning10:32 - Work for Tufalabs!11:50 - How AlphaGo Surpassed Humans17:49 - RLHF and RLVR18:41 - The Era of Experience20:27 - My Take21:05 - Welch Labs Book!TECHNICAL NOTESwelchlabs.com/blog/2026/1/31/the-bitter-lesson-video-technical-notesCODEgithub.com/stephencwelch/manim_videosREFERENCESThe Bitter Lesson: http://www.incompleteideas.net/IncIdeas/BitterLesson.htmlDwarkesh Patel's interview with Richard Sutton: youtube.com/watch?v=21EYKqUsPfgAlphaGo vs Lee Sedol Match 4: youtube.com/watch?v=yCALyQRN3hwRepurposed some board setups and heatmaps from: lesswrong.com/posts/FF8i6SLfKb4g7C4EL/inside-the-mind-of-a-superhuman-go-model-how-does-leela-zero-2?utm_source=chatgpt.comGreat HARPY video: youtube.com/watch?v=NiiDe2n-GeQSutton, Richard S., and Andrew G. Barto. *Reinforcement learning: An introduction*. Vol. 1. No. 1. Cambridge: MIT press, 1998.Averbuch, Amir, et al. "An IBM PC based large-vocabulary isolated-utterance speech recognizer." ICASSP'86. IEEE International Conference on Acoustics, Speech, and Signal Processing. Vol. 11. IEEE, 1986.Radford, Alec, et al. "Language models are unsupervised multitask learners." OpenAI blog 1.8 (2019): 9.Silver, David, et al. "Mastering the game of Go with deep neural networks and tree search." *nature* 529.7587 (2016): 484-489.Silver, David, et al. "Mastering the game of go without human knowledge." *nature* 550.7676 (2017): 354-359.Lowerre, Bruce. "The HARPY speech understanding system." *Readings in speech recognition*. 1990. 576-586.Silver, David, and Richard S. Sutton. "Welcome to the era of experience." Google AI 1 (2025).PATRONSJuan Benet, Ross Hanson, Yan Babitski, AJ Englehardt, Alvin Khaled, Eduardo Barraza, Hitoshi Yamauchi, Jaewon Jung, Mrgoodlight, Shinichi Hayashi, Sid Sarasvati, Dominic Beaumont, Shannon Prater, Ubiquity Ventures, Matias Forti, Brian Henry, Tim Palade, Petar Vecutin, Nicolas baumann, Jason Singh, Robert Riley, vornska, Barry Silverman, Jake Ehrlich, Mitch Jacobs, Lauren Steely, Jeff Eastman, Rodolfo Ibarra, Clark Barrus, Rob Napier, Andrew White, Richard B Johnston, abhiteja mandava, Burt Humburg, Kevin Mitchell, Daniel Sanchez, Ferdie Wang, Tripp Hill, Richard Harbaugh Jr, Prasad Raje, Kalle Aaltonen, Midori Switch Hound, Zach Wilson, Chris Seltzer, Ven Popov, Hunter Nelson, Amit Bueno, Scott Olsen, Johan Rimez, Shehryar Saroya, Tyler Christensen, Beckett Madden-Woods, Darrell Thomas, Javier Soto, U007D, Caleb Begly, Rick Rubenstein, Brent Hunsaker, Dan Patterson, Tchsurvives, Alex Adai, Walter Reade, Zyansheep, Walter Reade, Duncan Stannett, Reginald Carey, Jean-Manuel Izaret, dh71633, Adrian Rodriguez, Dimitar Stojanovski, Michael Harder, Peter Maldonado, Emily Pesce, David Johnston, Insang Song, FaeTheWolf, Stephen Taylor, KittenKaboodle, EMatter, PATRICKMCCORMACK, John Beahan, Cameron, Cole Jones, Garrett Thornburg, Jeroen W, Rohit Sharma, GlennB, Emmanuel Cortes, Katie Quinn, Karina C, Cakra WW, Mike Ton, Eric Gometz, MacCallister Higgins, Niko Drossos, David Eraso, Tom Zehle, Steve, Brian Lineburg, rjbl, Michael Loh, Perry Vais, Bengal0, Farhad Manjoo, Sara Chipps, Ellis Driscoll, William Taysom, Will Harmon, CK, Abdullah, Peter Cho, Leo Nikora, Griffin Smith, Ash Katnoria, Alex, Markus Hays Nielsen, Catherine H., Vi, David Dobáš, Peter Wang, Sina Sohangir, Danny Thomas, Julian Francis, Hans Adler, Jiayu Peng, Weston M, Youssouf da Silva, John Thomas, Samuel Costello, Sam Adams, Bryan Liles, Malaya Zemlya, Karl, Vahe Andonians, Mike Doughty, Larry Novelo, Jonas Acres, Ludicrum Rex, Robert Blumofe, Anthony ZCreated by: Sam Baskin, Matthew Cohen, Pranav Gundu, and Stephen Welch Content ID: CFAQJOTYQHT7JYIT
The most complex model we actually understand Welch Labs 2025-12-20 | New AI Book! welchlabs.com/resources/ai-book-ezrzm Get a free ebook version today when you order a copy from our January 2026 print run! You’ll receive a discount code for 100% off the ebook in your purchase confirmation email. ebook: welchlabs.com/resources/the-welch-labs-illustrated-guide-to-ai-digital-downloadPatreon: patreon.com/welchlabsSections0:00 - Intro2:39 - Modular Addition3:54 - The Model’s Perspective6:52 - An Accidental Discovery at OpenAI7:56 - It Groks!8:49 - Some Clues13:17 - New Welch Labs Book!13:57 - Deeper into the model15:13 - Linear Probes16:59 - Clocks perform modular addition19:17 - How do x and y interact exactly?23:19 - It learns a trig identity?!26:38 - Putting the pieces together & excluded loss30:02 - Anthropic finds 6D manifolds32:24 - Final thoughts34:02 - Welch Labs updateSpecial thanks to Neel Nanda for discussing his work and Mech Interp with me, if you want to learn more about Mech Interp, check out Neel’s getting started post here: neelnanda.io/getting-startedThanks to Emmanuel Ameisen and Wes Gurnee for discussing their work on Claude Haiku. Their paper is incredibly in depth and interesting: https://transformer-circuits.pub/2025/linebreaks/index.htmlReally nice deeper breakdown on polynomial double descent from viewer Avaneesh Narla:avaneeshnarla.com/blog/double-descent.htmlOpenAI team’s grokking paper: arxiv.org/pdf/2201.02177. I wasn’t able to reach to team for comment on the origin story, but it is told here: youtube.com/watch?v=gYGWFjMf9JA&t=1236sNanda et al. arxiv.org/pdf/2301.05217v1More on Grokking: quantamagazine.org/how-do-machines-grok-data-20240412Code based on excellent these notebooks from Neel Nanda and collaborators:colab.research.google.com/drive/1F6_1_cWXE5M7WocUcpQWp3v8z4b1jL20Andrej Karapathy on “Summoning Ghosts”: https://karpathy.bearblog.dev/animals-vs-ghosts/Code: github.com/stephencwelch/manim_videos/tree/master/_2025/grokkingTechnical Notes- It’s very natural for the attention layer to take the sum of it’s inputs (e.g. cos(kx)+cos(ky)), however we also find strong product terms. There’s a couple ways the network can compute products like cos(kx)cos(ky). One option is to approximate the product using ReLU activation functions (see Nanda’s notebooks for more). It’s also feasible for the attention block to do this, I found evidence of this is my own exploration - In the first 2D fourier decomposition, we’re leaving out one component, specifically a “negative frequency component” → 0.26 * np.cos(2*np.pi*((4*i)/113)) * np.cos(2*np.pi*((109*j)/113)). We left this out to avoid digging into a discussion of negative/aliased frequencies, and having this 4th component doesn’t add to our intuition about what the network is doing here.- at 28:00 we’re not showing removing the 8pi/113 frequency from the model’s final output surface.Patrons Juan Benet, Ross Hanson, Yan Babitski, AJ Englehardt, Alvin Khaled, Eduardo Barraza, Hitoshi Yamauchi, Jaewon Jung, Mrgoodlight, Shinichi Hayashi, Sid Sarasvati, Dominic Beaumont, Shannon Prater, Ubiquity Ventures, Matias Forti, Brian Henry, Tim Palade, Petar Vecutin, Nicolas baumann, Jason Singh, Robert Riley, vornska, Barry Silverman, Jake Ehrlich, Mitch Jacobs, Lauren Steely, Jeff Eastman, Rodolfo Ibarra, Clark Barrus, Rob Napier, Andrew White, Richard B Johnston, abhiteja mandava, Burt Humburg, Kevin Mitchell, Daniel Sanchez, Ferdie Wang, Tripp Hill, Richard Harbaugh Jr, Prasad Raje, Kalle Aaltonen, Midori Switch Hound, Zach Wilson, Chris Seltzer, Ven Popov, Hunter Nelson, Amit Bueno, Scott Olsen, Johan Rimez, Shehryar Saroya, Tyler Christensen, Beckett Madden-Woods, Darrell Thomas, Javier Soto, U007D, Caleb Begly, Rick Rubenstein, Brent Hunsaker, Dan Patterson, Tchsurvives, Alex Adai, Walter Reade, Zyansheep, Walter Reade, Duncan Stannett, Reginald Carey, Jean-Manuel Izaret, dh71633, Adrian Rodriguez, Dimitar Stojanovski, Michael Harder, Peter Maldonado, Emily Pesce, David Johnston, Insang Song, FaeTheWolf, Stephen Taylor, KittenKaboodle, EMatter, PATRICKMCCORMACK, John Beahan, Cameron, Cole Jones, Garrett Thornburg, Jeroen W, Rohit Sharma, GlennB, Emmanuel Cortes, Katie Quinn, Karina C, Cakra WW, Mike Ton, Eric Gometz, MacCallister Higgins, Niko Drossos, David Eraso, Tom Zehle, Steve, Brian Lineburg, rjbl, Michael Loh, Perry Vais, Bengal0, Farhad Manjoo, Sara Chipps, Ellis Driscoll, William Taysom, Will Harmon, CK, Abdullah, Peter Cho, Leo Nikora, Griffin Smith, Ash Katnoria, Alex, Markus Hays Nielsen, Catherine H., Vi, David Dobáš, Peter Wang, Sina Sohangir, Danny Thomas, Julian Francis, Hans Adler, Jiayu Peng Created by: Stephen Welch, Sam Baskin, and Pranav GunduCFAQJOTYQHT7JYIT
What the Books Get Wrong about AI [Double Descent] Welch Labs 2025-10-19 | Huge thanks to KiwiCo for sponsoring today’s video! Go to kiwico.com/welchlabs and use code WELCHLABS for 50% off your first monthly crate or for 20% off your first Panda Crate!New Book Available for Preorder Now! The Welch Labs Illustrated Guide to AI (30:47): welchlabs.com/resources/ai-book-ezrzmSections 0:00 - Intro3:43 - AlexNet & Overfitting5:19 - Overfitting6:45 - Rethinking Generalization11:05 - KiwiCo is Awesome12:28 - The Double Descent Hypothesis13:57 - Double Descent is Real!16:01 - Double Descent with Polynomial Curvefitting?!20:36 - But why?22:35 - Should I throw out my books?24:28 - The Bias-Variance Tradeoff28:30 - My take30:47 - I’ve written a new book on AI!Books with U-shaped test set error curves:Murphy, Kevin P. Probabilistic machine learning: an introduction. MIT press, 2022.Goodfellow, Ian, et al. *Deep learning*. Vol. 1. No. 2. Cambridge: MIT press, 2016.Russell, Stuart Jonathan, and Peter Norvig, eds. *Prentice Hall series in artificial intelligence*. Englewood Cliffs, NJ:: Prentice Hall, 1995.Bishop, Christopher M., and Nasser M. Nasrabadi. *Pattern recognition and machine learning*. Vol. 4. No. 4. New York: springer, 2006.Learning, Machine. "Tom mitchell." *Publisher: McGraw Hill* (1997): 31.Hastie, Trevor, Robert Tibshirani, and Jerome Friedman. "The elements of statistical learning." (2009).Hastie, Trevor, Robert Tibshirani, and Jerome Friedman. "An introduction to statistical learning." (2009).Abu-Mostafa, Yaser S., Malik Magdon-Ismail, and Hsuan-Tien Lin. *Learning from data*. Vol. 4. New York: AMLBook, 2012.MacKay, David JC. *Information theory, inference and learning algorithms*. Cambridge university press, 2003.Harvard Team’s code & results: gitlab.com/harvard-machine-learning/double-descentGreat repo showing polynomial double descent: github.com/RylanSchaeffer/Stanford-AI-Alignment-Double-Descent-TutorialTechnical Notes- 26:25 For these linear fits, we’re using N=15 instead of N=5 points. This increases the bias and reduces the variance of these fits, making the bias variance trade-off more clear, but also pushes out the interpolation threshold. Full results are here: github.com/stephencwelch/manim_videos/blob/master/_2025/generalization/Final Video Polynomial Examples.ipynb- 27:38 It’s tricky to show the full bias-variance results here, as the variance explodes ad Degree=4. Instead we’ve chosen to show qualitative breakdowns, showing which terms dominate the overall error at each degree. Full results can be seen here: github.com/stephencwelch/manim_videos/blob/master/_2025/generalization/Final%20Video%20Polynomial%20Examples.ipynbSpecial Thanks to Patrons patreon.com/welchlabsJuan Benet, Ross Hanson, Yan Babitski, AJ Englehardt, Alvin Khaled, Eduardo Barraza, Hitoshi Yamauchi, Jaewon Jung, Mrgoodlight, Shinichi Hayashi, Sid Sarasvati, Dominic Beaumont, Shannon Prater, Ubiquity Ventures, Matias Forti, Brian Henry, Tim Palade, Petar Vecutin, Nicolas baumann, Jason Singh, Robert Riley, vornska, Barry Silverman, Jake Ehrlich, Mitch Jacobs, Lauren Steely, Jeff Eastman, Rodolfo Ibarra, Clark Barrus, Rob Napier, Andrew White, Richard B Johnston, abhiteja mandava, Burt Humburg, Kevin Mitchell, Daniel Sanchez, Ferdie Wang, Tripp Hill, Richard Harbaugh Jr, Prasad Raje, Kalle Aaltonen, Midori Switch Hound, Zach Wilson, Chris Seltzer, Ven Popov, Hunter Nelson, Amit Bueno, Scott Olsen, Johan Rimez, Shehryar Saroya, Tyler Christensen, Beckett Madden-Woods, Darrell Thomas, Javier Soto, U007D, Caleb Begly, Rick Rubenstein, Brent Hunsaker, Dan Patterson, Tchsurvives, Alex Adai, Walter Reade, Zyansheep, Walter Reade, Duncan Stannett, Reginald Carey, Jean-Manuel Izaret, dh71633, Adrian Rodriguez, Dimitar Stojanovski, Michael Harder, Peter Maldonado, Emily Pesce, David Johnston, Insang Song, FaeTheWolf, Stephen Taylor, KittenKaboodle, EMatter, PATRICKMCCORMACK, John Beahan, Cameron, Cole Jones, Garrett Thornburg, Jeroen W, Rohit Sharma, GlennB, Emmanuel Cortes, Katie Quinn, Karina C, Cakra WW, Mike Ton, Eric Gometz, MacCallister Higgins, Niko Drossos, David Eraso, Tom Zehle, Steve, Brian Lineburg, rjbl, Michael Loh, Perry Vais, Bengal0, Farhad Manjoo, Sara Chipps, Ellis Driscoll, William Taysom, Will Harmon, CK, Abdullah, Peter Cho, Leo Nikora, Griffin Smith, Ash Katnoria, Alex, Markus Hays NielsenSpecial thanks to: Mikhail Belkin, Preetum Nakkiran, Emily Zhang, Varun ReddyCode for Welch Labs Videos: github.com/stephencwelch/manim_videosWritten by: Stephen Welch Produced by: Stephen Welch, Sam Baskin, and Pranav GunduPremium Beat IDsEEDYZ3FP44YX8OWTCFAQJOTYQHT7JYIT
These Numbers Can Make AI Dangerous [Subliminal Learning] Welch Labs 2025-09-04 | Checkout RunPod’s AI infrastructure platform: get.runpod.io/welchlabsDiscount code at checkout: WELCH10Note that need to buy $15 or more in runpod credits for the discount code to apply, $10 will be deducted from your total. See screen recording at 3:31. Subliminal Learning Poster at 31:09: welchlabs.com/resources/subliminal-learning-poster-17x22Subliminal Learning Bundle: welchlabs.com/resources/subliminal-learning-poster-book-bundleSubliminal Learning Poster - Digital Download: welchlabs.com/resources/subliminal-learning-poster-digital-downloadSections0:00 - Intro1:47 - Why Welch Labs uses runpod for AI infrastructure - sponsored ad3:49 - The subliminal learning phenomenon5:44 - In context learning6:56 - Why can’t we just train a classifier?7:45 - Other clues9:28 - Small scale replication on MNIST12:47 - Mathematical proof23:01 - Proof Take-aways25:38 - Solving the GPT 4.1/4o mystery26:14 - My take on what’s going on27:55 - The token entanglement hypothesis29:11 - Final thoughts & take-aways31:09 - Subliminal Learning Poster!ReferencesSubliminal Learning Paper and code: alignment.anthropic.com/2025/subliminal-learningGenerate Your Own Numbers: https://subliminaldata.streamlit.app/Token Entanglement: lesswrong.com/posts/m5XzhbZjEuF9uRgGR/it-s-owl-in-the-numbers-token-entanglement-in-subliminal-1Hinton et. al. 2015. Distilling the Knowledge in a Neural Network. arxiv.org/pdf/1503.02531Full Video on Backpropagation: youtu.be/VkHfRKewkWw?si=PPONLc5j9Xwlv4JwSoftmax Basics: youtu.be/VkHfRKewkWw?si=WWPlqu7y1nozl1Fo&t=377Softmax Gradient: youtu.be/VkHfRKewkWw?si=hd63mRFFIlF3wT-A&t=836Softmax Visualized: youtu.be/VkHfRKewkWw?si=QZmFau5DjjrFvMso&t=1418Big thanks to Alex Cloud, Minh Le, Jacob Hilton, and Owain Evans for graciously answering my questions as I worked on the script. Special Thanks to Patrons patreon.com/welchlabsJuan Benet, Ross Hanson, Yan Babitski, AJ Englehardt, Alvin Khaled, Eduardo Barraza, Hitoshi Yamauchi, Jaewon Jung, Mrgoodlight, Shinichi Hayashi, Sid Sarasvati, Dominic Beaumont, Shannon Prater, Ubiquity Ventures, Matias Forti, Brian Henry, Tim Palade, Petar Vecutin, Nicolas baumann, Jason Singh, Robert Riley, vornska, Barry Silverman, Jake Ehrlich, Mitch Jacobs, Lauren Steely, Jeff Eastman, Rodolfo Ibarra, Clark Barrus, Rob Napier, Andrew White, Richard B Johnston, abhiteja mandava, Burt Humburg, Kevin Mitchell, Daniel Sanchez, Ferdie Wang, Tripp Hill, Richard Harbaugh Jr, Prasad Raje, Kalle Aaltonen, Midori Switch Hound, Zach Wilson, Chris Seltzer, Ven Popov, Hunter Nelson, Amit Bueno, Scott Olsen, Johan Rimez, Shehryar Saroya, Tyler Christensen, Beckett Madden-Woods, Darrell Thomas, Javier Soto, U007D, Caleb Begly, Rick Rubenstein, Brent Hunsaker, Dan Patterson, Tchsurvives, Alex Adai, Walter Reade, Zyansheep, Walter Reade, Duncan Stannett, Reginald Carey, Jean-Manuel Izaret, dh71633, Adrian Rodriguez, Dimitar Stojanovski, Michael Harder, Peter Maldonado, Emily Pesce, David Johnston, Insang Song, FaeTheWolf, Stephen Taylor, KittenKaboodle, EMatter, PATRICKMCCORMACK, John Beahan, Cameron, Cole Jones, Garrett Thornburg, Jeroen W, Rohit Sharma, GlennB, Emmanuel Cortes, Katie Quinn, Karina C, Cakra WW, Mike Ton, Eric Gometz, MacCallister Higgins, Niko Drossos, David Eraso, Tom Zehle, Steve, Brian Lineburg, rjbl, Michael Loh, Perry Vais, Bengal0, Farhad Manjoo, Sara ChippsSpecial thank you to these readers for helping improve the Imaginary Numbers Book!Marwan Daar, Matt Ellis, Nico Weber, Rafa Barroso, Jacob Sorensen, Bob Hall, Evan Van Peursem, Phillipe Loher, Attila Medl, Abdul Wahid Tanner, A friendly critic, NuttySwiss, Dean Burdick, Paul Du Bois, Włodzimierz BzylCode for Welch Labs Videos: github.com/stephencwelch/manim_videosWritten by: Stephen Welch Produced by: Stephen Welch, Sam Baskin, and Pranav GunduPremium Beat IDsEEDYZ3FP44YX8OWTMWROXNAY0SPXCMBSCFAQJOTYQHT7JYIT
Why Deep Learning Works Unreasonably Well [How Models Learn Part 3] Welch Labs 2025-08-10 | Take your personal data back with Incogni! Use code WELCHLABS and get 60% off an annual plan: http://incogni.com/welchlabsNew Patreon Rewards 33:31- own a piece of Welch Labs history!patreon.com/welchlabsBooks & Posterswelchlabs.com/resourcesSections0:00 - Intro 4:49 - How Incogni Saves Me Time6:32 - Part 2 Recap8:10 - Moving to Two Layers9:15 - How Activation Functions Fold Space11:45 - Numerical Walkthrough13:42 - Universal Approximation Theorem15:45 - The Geometry of Backpropagation19:52 - The Geometry of Depth24:27 - Exponentially Better?30:23 - Neural Networks Demystifed31:50 - The Time I Quit YouTube33:31 - New Patreon Rewards!Special Thanks to Patrons patreon.com/welchlabsJuan Benet, Ross Hanson, Yan Babitski, AJ Englehardt, Alvin Khaled, Eduardo Barraza, Hitoshi Yamauchi, Jaewon Jung, Mrgoodlight, Shinichi Hayashi, Sid Sarasvati, Dominic Beaumont, Shannon Prater, Ubiquity Ventures, Matias Forti, Brian Henry, Tim Palade, Petar Vecutin, Nicolas baumann, Jason Singh, Robert Riley, vornska, Barry Silverman, Jake Ehrlich, Mitch Jacobs, Lauren Steely, Jeff Eastman, Rodolfo Ibarra, Clark Barrus, Rob Napier, Andrew White, Richard B Johnston, abhiteja mandava, Burt Humburg, Kevin Mitchell, Daniel Sanchez, Ferdie Wang, Tripp Hill, Richard Harbaugh Jr, Prasad Raje, Kalle Aaltonen, Midori Switch Hound, Zach Wilson, Chris Seltzer, Ven Popov, Hunter Nelson, Amit Bueno, Scott Olsen, Johan Rimez, Shehryar Saroya, Tyler Christensen, Beckett Madden-Woods, Darrell Thomas, Javier SotoReferencesSimon Prince, Understanding Deep Learning. udlbook.github.io/udlbookLiang, Shiyu, and Rayadurgam Srikant. "Why deep neural networks for function approximation?." arXiv preprint arXiv:1610.04161 (2016).Hanin, Boris, and David Rolnick. "Deep relu networks have surprisingly few activation patterns." *Advances in neural information processing systems* 32 (2019).Hanin, Boris, and David Rolnick. "Complexity of linear regions in deep networks." *International Conference on Machine Learning*. PMLR, 2019.Fan, Feng-Lei, et al. "Deep relu networks have surprisingly simple polytopes." *arXiv preprint arXiv:2305.09145* (2023).All Code: github.com/stephencwelch/manim_videos100k neuron wide example training code: github.com/stephencwelch/manim_videos/blob/master/_2025/backprop_3/notebooks/Wide%20Training%20Example.ipynbWritten by: Stephen Welch Produced by: Stephen Welch, Sam Baskin, and Pranav GunduPremium Beat IDsEEDYZ3FP44YX8OWTeMWROXNAY0SPXCMBSCFAQJOTYQHT7JYIT
The F=ma of Artificial Intelligence [Backpropagation, How Models Learn Part 2] Welch Labs 2025-06-11 | Take your personal data back with Incogni! Use code WELCHLABS and get 60% off an annual plan: http://incogni.com/welchlabsNew Patreon Rewards 29:48 - own a piece of Welch Labs history! patreon.com/welchlabsBooks & Posterswelchlabs.com/resourcesSections0:00 - Intro2:08 - No more spam calls w/ Incogni3:45 - Toy Model5:20 - y=mx+b6:17 - Softmax7:48 - Cross Entropy Loss9:08 - Computing Gradients12:31 - Backpropagation18:23 - Gradient Descent20:17 - Watching our Model Learn23:53 - Scaling Up25:45 - The Map of Language28:13 - The time I quit YouTube29:48 - New Patreon Rewards!Nice Implementation for a viewer in C++:kirit.com/Tiny%20Classifiers/tiny-classifier.cppSpecial Thanks to Patrons patreon.com/welchlabsJuan Benet, Ross Hanson, Yan Babitski, AJ Englehardt, Alvin Khaled, Eduardo Barraza, Hitoshi Yamauchi, Jaewon Jung, Mrgoodlight, Shinichi Hayashi, Sid Sarasvati, Dominic Beaumont, Shannon Prater, Ubiquity Ventures, Matias Forti, Brian Henry, Tim Palade, Petar Vecutin, Nicolas baumann, Jason Singh, Robert Riley, vornska, Barry Silverman, Jake Ehrlich, Mitch Jacobs, Lauren SteelyReferencesWerbos, P. J. (1994). The roots of backpropagation : from ordered derivatives to neural networks and political forecasting. United Kingdom: Wiley. Newton quote is on p4, Werbos expands on the analogy on p4. Olazaran, Mikel. "A sociological study of the official history of the perceptrons controversy." *Social Studies of Science* 26.3 (1996): 611-659. Minsky quote is on p 393.Widrow, Bernard. "Generalization and information storage in networks of adaline neurons.” Self-organizing systems (1962): 435-461.Historical Videoshttp://youtube.com/watch?v=FwFduRA_L6Qyoutube.com/watch?v=ntIczNQKfjQCode: github.com/stephencwelch/manim_videosTechnical NotesLarge Llama training animation shows 8/16 layers. Specifically layers 1, 2, 7, 8, 9, 10, 15, and 16. Every third attention pattern is shown, and special tokens are ignored. MLP neurons are downsampled using max pooling. Only the weights and gradients above a specific percentile based threshold are shown. Only query weights are shown going into each attention layer. The coordinates of Paris are subtracted from all training examples in the 4 city example as a simple normalization - this helps with convergence. In some scenes, math is happening at higher precision behind the scenes, and results are rounded, which may create apparent inconsistencies. Written by: Stephen Welch Produced by: Stephen Welch, Sam Baskin, and Pranav GunduSpecial thanks to: Emily ZhangPremium Beat IDsEEDYZ3FP44YX8OWTMWROXNAY0SPXCMBSCFAQJOTYQHT7JYIT
The Misconception that Almost Stopped AI [How Models Learn Part 1] Welch Labs 2025-05-09 | Take your personal data back with Incogni! Use code WELCHLABS and get 60% off an annual plan: http://incogni.com/welchlabsLoss Landscape Posters! 21:23welchlabs.com/resources/loss-landscape-poster-17x19welchlabs.com/resources/loss-landscape-poster-digital-downloadPoster and Book Bundlewelchlabs.com/resources/loss-landscape-bundle-w-imaginary-numbers-bookSpecial Matte Black Edition Posterwelchlabs.com/resources/loss-landscape-poster-17x22-matte-black-special-editionWelch Labs Bookwelchlabs.com/resources/imaginary-numbers-bookSections0:00 - Intro1:18 - How Incogni gets me more focus time3:01 - What are we measuring again? 6:18 - How to make our loss go down? 7:32 - Tuning one parameter9:11 - Tuning two parameters together11:01 - Gradient descent13:18 - Visualizing high dimensional surfaces15:10 - Loss Landscapes16:55 - Wormholes!17:55 - Wikitext18:55 - But where do the wormholes come from?20:00 - Why local minima are not a problem21:23 - PostersSpecial Thanks to Patrons patreon.com/welchlabsJuan Benet, Ross Hanson, Yan Babitski, AJ Englehardt, Alvin Khaled, Eduardo Barraza, Hitoshi Yamauchi, Jaewon Jung, Mrgoodlight, Shinichi Hayashi, Sid Sarasvati, Dominic Beaumont, Shannon Prater, Ubiquity Ventures, Matias Forti, Brian Henry, Tim Palade, Petar Vecutin, Nicolas baumann, Jason Singh, Robert Riley, vornska, Barry Silverman, Jake Ehrlich, Mitch JacobsReferencesLi et al: Visualizing the Loss Landscape of Neural Nets. arxiv.org/abs/1712.09913Talking Nets: An Oral History of Neural Networks. (2000). United Kingdom: MIT Press. Hinton quote is on p376. Goodfellow, I., Bengio, Y., Courville, A. (2016). Deep Learning. United Kingdom: MIT Press.Prince, S. J. (2023). Understanding Deep Learning. United Kingdom: MIT Press.Manim Animations: github.com/stephencwelch/manim_videosPremium Beat IDsMWROXNAY0SPXCMBSCFAQJOTYQHT7JYIT
How DeepSeek Rewrote the Transformer [MLA] Welch Labs 2025-03-05 | Thanks to KiwiCo for sponsoring today’s video! Go to kiwico.com/welchlabs and use code WELCHLABS for 50% off your first monthly club crate or for 20% off your first Panda Crate!MLA/DeepSeek Poster at 17:12 (Free shipping for a limited time with code DEEPSEEK):welchlabs.com/resources/mladeepseek-attention-poster-13x19Limited edition MLA Poster and Signed Book:welchlabs.com/resources/deepseek-bundle-mla-poster-and-signed-book-limited-runImaginary Numbers book is back in stock!welchlabs.com/resources/imaginary-numbers-bookSpecial Thanks to Patrons patreon.com/c/welchlabsJuan Benet, Ross Hanson, Yan Babitski, AJ Englehardt, Alvin Khaled, Eduardo Barraza, Hitoshi Yamauchi, Jaewon Jung, Mrgoodlight, Shinichi Hayashi, Sid Sarasvati, Dominic Beaumont, Shannon Prater, Ubiquity Ventures, Matias Forti, Brian Henry, Tim Palade, Petar Vecutin, Nicolas baumann, Jason Singh, Robert Riley, vornska, Barry Silverman, Jake EhrlichReferencesDeepSeek-V2 paper: arxiv.org/pdf/2405.04434DeepSeek-R1 paper: arxiv.org/abs/2501.12948Great Article by Ege Erdil: epoch.ai/gradient-updates/how-has-deepseek-improved-the-transformer-architectureGPT-2 Visualizaiton: github.com/TransformerLensOrg/TransformerLensManim Animations: github.com/stephencwelch/manim_videosTechnical Notes1. Note that DeepSeek-V2 paper claims a KV cache size reduction of 93.3%. They don’t exactly publish their methodology, but as far as I can tell it’s something likes this: start with Deepseek-v2 hyperparameters here: huggingface.co/deepseek-ai/DeepSeek-V2/blob/main/configuration_deepseek.py. num_hidden_layers=30, num_attention_heads=32, v_head_dim = 128. If DeepSeek-v2 was implemented with traditional MHA, then KV cache size would be 2*32*128*30*2=491,520 B/token. With MLA with a KV cache size of 576, we get a total cache size of 576*30=34,560 B/token. The percent reduction in KV cache size is then equal to (491,520-34,560)/492,520=92.8%. The numbers I present in this video follow the same approach but are for DeepSeek-v3/R1 architecture: huggingface.co/deepseek-ai/DeepSeek-V3/blob/main/config.json. num_hidden_layers=61, num_attention_heads=128, v_head_dim = 128. So traditional MHA cache would be 2*128*128*61*2 = 3,997,696 B/token. MLA reduces this to 576*61*2=70,272 B/token. Tor the DeepSeek-V3/R1 architecture, MLA reduces the KV cache size by a factor of 3,997,696/70,272 =56.9X. 2. I claim a couple times that MLA allows DeepSeek to generate tokens more than 6x faster than a vanilla transformer. The DeepSeek-V2 paper claims a slightly less than 6x throughput improvement with MLA, but since the V3/R1 architecture is heavier, we expect a larger lift, which is why i claim “more than 6x faster than a vanilla transformer” - in reality it’s probably significantly more than 6x for the V3/R1 architecture.3. In all attention patterns and walkthroughs, we’re ignoring the |beginning of sentence| token. “The American flag is red, white, and” actually maps to 10 tokens if we include this starting token, and may attention patterns do assign high values to this token. 4. We’re ignoring bias terms matrix equations. 5. We’re ignoring positional embeddings. These are fascinating. See DeepSeek papers and ROPE.
ChatGPT is made from 100 million of these [The Perceptron] Welch Labs 2025-02-01 | Go to drinkag1.com/welchlabs to subscribe and save $20 off your first subscription of AG1! Thanks to AG1 for sponsoring today's video.Imaginary Numbers book is back in stock! Update at 23:11welchlabs.com/resources/imaginary-numbers-bookWelch Labs Posters:welchlabs.com/resourcesCool Interactive Perceptron Simulator made by viewer Priyangsu Banerjee!priyangsubanerjee.github.io/perceptron-simulatorSpecial Thanks to Patrons patreon.com/welchlabsJuan Benet, Ross Hanson, Yan Babitski, AJ Englehardt, Alvin Khaled, Eduardo Barraza, Hitoshi Yamauchi, Jaewon Jung, Mrgoodlight, Shinichi Hayashi, Sid Sarasvati, Dominic Beaumont, Shannon Prater, Ubiquity Ventures, Matias Forti, Brian Henry, Tim Palade, Petar Vecutin, Nicolas baumann, Jason Singh, Robert Riley, vornska, Barry SilvermanReferencesRumelhart, D. E., Mcclelland, J. L. (1987). Parallel Distributed Processing, Volume 1: Explorations in the Microstructure of Cognition: Foundations. United Kingdom: Penguin Random House LLC.Talking Nets: An Oral History of Neural Networks. (2000). United Kingdom: MIT Press.Prince, S. J. (2023). Understanding Deep Learning. United Kingdom: MIT Press.Crevier, D. (1993). AI : the tumultuous history of the search for artificial intelligence. New York: Basic Books.Cat and dog face dataset: kaggle.com/datasets/andrewmvd/animal-faces?resource=downloadMinsky, M., Papert, S. (2017). Perceptrons: An Introduction to Computational Geometry. United Kingdom: MIT Press.Widrow, Bernard, and Michael A. Lehr. "30 years of adaptive neural networks: perceptron, madaline, and backpropagation." *Proceedings of the IEEE* 78.9 (1990): 1415-1442.Olazaran, Mikel. "A sociological history of the neural network controversy." *Advances in computers*. Vol. 37. Elsevier, 1993. 335-425.Widrow, Bernard. "Generalization and information storage in networks of adaline neurons." *Self-organizing systems* (1962): 435-461.Widrow, Bernard. "Thinking about thinking: the discovery of the LMS algorithm." *IEEE Signal Processing Magazine* 22.1 (2005): 100-106.Technical NotesMethod for counting neurons in ChatGPT: Starting with GPT-2 implementation here: github.com/karpathy/build-nanogpt/blob/master/train_gpt2.py - keys, queries, and values are implemented in Linear layers with n_embd inputs and 3*n_embd outputs, where n_embd is the embedding dimension. Output projection layer has n_embd and n_embd outputs. So a single attention layer will have ~4*n_embd neurons. GPT-3 has an embedding dimension of 12,288, so each attention layer has ~49,152 neurons. Each MLP block has n_embd inputs, 4*n_embd hidden units, and n_embd outputs, so ~5*n_embd total neurons, or ~61,440. Total neuron count for GPT-3 is then 96*(49,152+61,440)=10,616,832, ignoring initial embedding and final unembedding. Finally, GPT-4 reportedly has ~1.8 Trillion parameters (semianalysis.com/2023/07/10/gpt-4-architecture-infrastructure), making it ~10x larger than GPT-3. Note that GPT-4 is reportedly a mixture of experts, and not all experts are used for each inference, so it appears that not all 1.8 trillion parameters are used for a given inference call. Assuming that ~10x the parameters means 10x the neurons, then GPT-4 should have ~100M neurons.CFAQJOTYQHT7JYIT
The Dark Matter of AI [Mechanistic Interpretability] Welch Labs 2024-12-23 | Take your personal data back with Incogni! Use code WELCHLABS at the link below and get 60% off an annual plan: http://incogni.com/welchlabsWelch Labs Imaginary Numbers Book!welchlabs.com/resources/imaginary-numbers-bookWelch Labs Posters:welchlabs.com/resourcesSpecial Thanks to Patrons patreon.com/welchlabsJuan Benet, Ross Hanson, Yan Babitski, AJ Englehardt, Alvin Khaled, Eduardo Barraza, Hitoshi Yamauchi, Jaewon Jung, Mrgoodlight, Shinichi Hayashi, Sid Sarasvati, Dominic Beaumont, Shannon Prater, Ubiquity Ventures, Matias Forti, Brian Henry, Tim Palade, Petar Vecutin, Nicolas baumann, Jason Singh, Robert Riley, vornska, Barry SilvermanMy Gemma walkthrough notebook: colab.research.google.com/drive/1Y68yNr5TcHr4G5RJ0QHZhKkDe55AUkVj?usp=sharingMost animations made with Manim: github.com/3b1b/manimReferences and Further ReadingChris Olah’s original “Dark Matter of Neural Networks” post: https://transformer-circuits.pub/2024/july-update/index.html#dark-matterGreat recent interview with Chris Olah: youtube.com/watch?v=ugvHCXCOmm4Gemma Scope: arxiv.org/pdf/2408.05147Experiment with SAEs yourself here! neuronpedia.orgRelevant work from the Anthropic team:https://transformer-circuits.pub/2022/toy_model/index.htmlhttps://transformer-circuits.pub/2023/monosemantic-featureshttps://transformer-circuits.pub/2024/scaling-monosemanticity/Excellent intro Mechanistic Interpretability: https://arena3-chapter1-transformer-interp.streamlit.app/%5B1.2%5D_Intro_to_Mech_InterpNeel Nanda’s Mechanistic Interpretability Explainer: dynalist.io/d/n2ZWtnoYHrU1s4vnFSAQ519JTransformer Lens: github.com/TransformerLensOrg/TransformerLensSAE Lens: jbloomaus.github.io/SAELensTechnical Notes1. There are more advanced and more meaningful ways to map mid layer vectors to outputs, see: arxiv.org/pdf/2303.08112, neuralblog.github.io/logit-prisms, lesswrong.com/posts/AcKRB8wDpdaN6v6ru/interpreting-gpt-the-logit-lens2. The 6x2304 matrix is actually 7x2304, we’re ignoring the /bos token.3. Gemma also includes positional embeddings and lots and lots of normalization layers, which we didn’t really cover4. I’m conflating tokens and words sometimes, in this example each word is a token, so we don’t have to worry about it too much5. The “_” characters represent spaces in the token stringsCFAQJOTYQHT7JYIT
What is the i really doing in Schrödingers equation? Welch Labs 2024-11-23 | Go to piavpn.com/WelchLabs to get 83% off Private Internet Access with 4 months free!Book Update at 23:28!Welch Labs Imaginary Numbers Book!welchlabs.com/resources/imaginary-numbers-bookWelch Labs Posters: welchlabs.com/resourcesHuge thanks to Grant Sanderson for a quick manim crash course - I used manim for some of the more complex animations. Schrodinger equation numerical simulations used for animations were computed with: github.com/quantum-visualizations/qmsolveSpecial thanks to Patrons: Juan Benet, Ross Hanson, Yan Babitski, AJ Englehardt, Alvin Khaled, Eduardo Barraza, Hitoshi Yamauchi, Jaewon Jung, Mrgoodlight, Shinichi Hayashi, Sid Sarasvati, Dominic Beaumont, Shannon Prater, Ubiquity Ventures, Matias Forti, Brian Henry, Tim Palade, Petar Vecutin, Nicolas baumann, Jason Singh, Robert RileyOrbitals Graphics Credits: Geek3, commons.wikimedia.org/wiki/File:Atomic_orbitals_spdf_m-eigenstates_and_superpositions.png, license: creativecommons.org/licenses/by-sa/4.0/deed.enPoorLeno, commons.wikimedia.org/wiki/File:Hydrogen_Density_Plots.pngReferencesDe Broglie, L. (1924). *Recherches sur la théorie des quanta* (Doctoral dissertation, Migration-université en cours d'affectation).Callender, C. (2023). Quantum mechanics: Keeping it real?. *The British Journal for the Philosophy of Science*, *74*(4), 837-851.Dyson, F. (2009). Birds and frogs. *Notices of the AMS*, *56*(2), 212-223.Einstein, A. (2011). Letters on Wave Mechanics: Correspondence with H. A. Lorentz, Max Planck, and Erwin Schrödinger. United States: Philosophical Library/Open Road.Fleisch, D. A. (2020). A Student's Guide to the Schrödinger Equation. India: Cambridge University Press.Fleisch, D., Kinnaman, L. (2015). A Student's Guide to Waves. United Kingdom: Cambridge University Press.Gamow, G. (2012). Thirty Years that Shook Physics: The Story of Quantum Theory. United States: Dover Publications.Jammer, M. (1989). The Conceptual Development of Quantum Mechanics. United Kingdom: Tomash Publishers.Karam, R. (2020). Schrödinger's original struggles with a complex wave function. *American Journal of Physics*, *88*(6), 433-438.Moore, W. (2015). Schrödinger: Life and Thought. United Kingdom: Cambridge University Press.Schrödinger: Centenary Celebration of a Polymath. (1987). United Kingdom: Cambridge University Press.Schrödinger, E. (2003). Collected Papers on Wave Mechanics: Together with His Four Lectures on Wave Mechanics. United States: AMS Chelsea Pub..CFAQJOTYQHT7JYIT
Kepler’s Impossible Equation Welch Labs 2024-10-15 | Why is such a simple equation so difficult to solve? Head to kiwico.com/welchlabs and use code WELCHLABS for 50% off your first monthly club crate or 20% off your first Panda Crate.Welch Labs Imaginary Numbers Book!welchlabs.com/resources/imaginary-numbers-bookWelch Labs Posters: welchlabs.com/resourcesHow the Bizarre Path of Mars Reshaped Astronomy: youtu.be/Phscjl0u6TISupport Welch Labs on Patreon! patreon.com/welchlabsSpecial thanks to Patrons: Juan Benet, Ross Hanson, Yan Babitski, AJ Englehardt, Alvin Khaled, Eduardo Barraza, Hitoshi Yamauchi, Jaewon Jung, Mrgoodlight, Shinichi Hayashi, Sid Sarasvati, Dominic Beaumont, Shannon Prater, Ubiquity Ventures, Matias Forti, Brian Henry, Tim Palade, Petar Vecutin, Nicolas baumannLearn more about WelchLabs!welchlabs.comTikTok: tiktok.com/@welchlabsInstagram: instagram.com/welchlabsREFERENCESColwell, P. (1993). Solving Kepler's Equation Over Three Centuries. United Kingdom: Willmann-Bell.Needham, T. (1997). Visual Complex Analysis. United Kingdom: Clarendon Press. Bate, R. R., Mueller, D. D., White, J. E. (1971). Fundamentals of Astrodynamics. Egypt: Dover Publications.Vallado, D. (2001). Fundamentals of Astrodynamics and Applications. Netherlands: Springer Netherlands.Borghi R. On the Bessel Solution of Kepler’s Equation. *Mathematics*. 2024; 12(1):154. doi.org/10.3390/math12010154Tom Archibald, Craig Fraser, Ivor Grattan-Guinness, The History of Differential Equations, 1670–1950. Oberwolfach Rep. 1 (2004), no. 4, pp. 2729–2794Francisco G. M. Orlando, C. Farina, Carlos A. D. Zarro, P. Terra; Kepler's equation and some of its pearls. Am. J. Phys. 1 November 2018; 86 (11): 849–858.Arthur A. Rambaut, M.A. A Simple Method of obtaining an Approximate Solution of Kepler's Equation. *Monthly Notices of the Royal Astronomical Society*, Volume 50, Issue 5, March 1890, Pages 301–302. Ben Coleman. How to Find the Taylor Series of an Inverse Function. randorithms.com/2021/08/31/Taylor-Series-Inverse.html17 Recent papers on Kepler’s equation can be found in references 2-16 here: mdpi.com/2227-7390/12/1/154CFAQJOTYQHT7JYIT
AI cant cross this line and we dont know why. Welch Labs 2024-09-13 | Have we discovered an ideal gas law for AI? Head to brilliant.org/WelchLabs to try Brilliant for free for 30 days and get 20% off an annual premium subscription.Welch Labs Imaginary Numbers Book!welchlabs.com/resources/imaginary-numbers-bookWelch Labs Posters: welchlabs.com/resourcesSupport Welch Labs on Patreon! patreon.com/welchlabsSpecial thanks to Patrons: Juan Benet, Ross Hanson, Yan Babitski, AJ Englehardt, Alvin Khaled, Eduardo Barraza, Hitoshi Yamauchi, Jaewon Jung, Mrgoodlight, Shinichi Hayashi, Sid Sarasvati, Dominic Beaumont, Shannon Prater, Ubiquity Ventures, Matias Forti, Brian Henry, Tim Palade, Petar VecutinLearn more about WelchLabs! welchlabs.comTikTok: tiktok.com/@welchlabsInstagram: instagram.com/welchlabsREFERENCESA Neural Scaling Law from the Dimension of the Data Manifold: arxiv.org/pdf/2004.10802First 2020 OpenAI Scaling Paper: arxiv.org/pdf/2001.08361GPT-3 Paper: arxiv.org/pdf/2005.14165Second 202 OpenAI Scaling Paper: arxiv.org/pdf/2010.14701Google Deepmind “Chinchilla Scaling” Paper: arxiv.org/abs/2203.15556Nice summary of Chinchilla Scaling: lesswrong.com/posts/6Fpvch8RR29qLEWNH/chinchilla-s-wild-implicationsGPT-4 Technical Report: arxiv.org/pdf/2303.08774Nice Neural Scaling Laws Summary: lesswrong.com/posts/Yt5wAXMc7D2zLpQqx/an-140-theoretical-models-that-predict-scaling-lawsExplaining Neural Scaling Laws: arxiv.org/pdf/2102.06701High Cost of Training GPT-4: wired.com/story/openai-ceo-sam-altman-the-age-of-giant-ai-models-is-already-overNvidia V100 FLOPs: lambdalabs.com/blog/demystifying-gpt-3Nvidia V100 Original Price: [microway.com/hpc-tech-tips/nvidia-tesla-v100-price-analysis/#:~:text=Tesla GPU model,Key Points](microway.com/hpc-tech-tips/nvidia-tesla-v100-price-analysis/#:~:text=Tesla%20GPU%20model,Key%20Points)Great paper on scaling up training infrastructure: arxiv.org/pdf/2104.04473Eight Things to Know about LLMs: arxiv.org/abs/2304.00612Emergent Properties of LLMs: arxiv.org/abs/2206.07682Theoretical Motivation for Cross Entropy (Section 6.2): deeplearningbook.orgSome papers that appear to pass the compute efficient frontierarxiv.org/pdf/2206.14486arxiv.org/abs/2210.11399CFAQJOTYQHT7JYITLeaked GPT-4 training infopatmcguinness.substack.com/p/gpt-4-details-revealedsemianalysis.com/p/gpt-4-architecture-infrastructureepochai.org/blog/tracking-large-scale-ai-models
The most beautiful equation in math, explained visually [Euler’s Formula] Welch Labs 2024-08-12 | Welch Labs Imaginary Numbers Book!welchlabs.com/resources/imaginary-numbers-bookBook Digital Versionwelchlabs.com/resources/imaginary-numbers-are-real-book-digital-downloadEuler’s Formula Poster!welchlabs.com/resources/eulers-formula-poster-13x19Poster Digital Versionwelchlabs.com/resources/eulers-formula-dark-mode-poster-digital-downloadSpecial thanks to the Patrons:Juan Benet, Ross Hanson, Yan Babitski, AJ Englehardt, Alvin Khaled, Eduardo Barraza, Hitoshi Yamauchi, Jaewon Jung, Mrgoodlight, Shinichi Hayashi, Sid Sarasvati, Dominic Beaumont, Shannon Prater, Ubiquity Ventures, Matias FortiTattoo by @themutemaker - thank you! Check out her awesome work here:instagram.com/themutemakerWelch LabsAd free videos and exclusive perks: patreon.com/welchlabsWatch on TikTok: tiktok.com/@welchlabsLearn More or Contact: welchlabs.comInstagram: instagram.com/welchlabsX: twitter.com/welchlabsReferences & NotesWelch Labs Imaginary Numbers Series: youtube.com/watch?v=T647CGsuOVUExcellent History of Logarithms by Florian CajoriCajori, Florian. “History of the Exponential and Logarithmic Concepts.” *The American Mathematical Monthly*, vol. 20, no. 1, 1913, pp. 5–14. *JSTOR*, doi.org/10.2307/2973509. Accessed 22 July 2024.Cajori, Florian. “History of the Exponential and Logarithmic Concepts.” *The American Mathematical Monthly*, vol. 20, no. 2, 1913, pp. 35–47. *JSTOR*, doi.org/10.2307/2974078. Accessed 22 July 2024.Cajori, Florian. “History of the Exponential and Logarithmic Concepts:” *The American Mathematical Monthly*, vol. 20, no. 3, 1913, pp. 75–84. *JSTOR*, doi.org/10.2307/2973441. Accessed 22 July 2024.Cajori, Florian. “History of the Exponential and Logarithmic Concepts.” *The American Mathematical Monthly*, vol. 20, no. 4, 1913, pp. 107–17. *JSTOR*, doi.org/10.2307/2972960. Accessed 22 July 2024.Nice History of Euler’s FormulaSandifer, Ed. *e, pi and i: Why is “Euler” in the Euler identity?* http://eulerarchive.maa.org/hedi/HEDI-2007-08.pdfMuch of the visual approach presented here comes from Needham’s incredible book: Needham, T. (1997). Visual Complex Analysis. United Kingdom: Clarendon Press.Other books referencedMaor, E. (2011). E: The Story of a Number. Ukraine: Princeton University Press.Penrose, R. (2021). The Road to Reality: A Complete Guide to the Laws of the Universe. United Kingdom: Knopf Doubleday Publishing Group.Dunham, W. (2022). Euler: The Master of Us All. United States: AMM Press.Wilson, R. (2019). Euler's Pioneering Equation: The Most Beautiful Theorem in Mathematics. United Kingdom: Oxford University Press.Nahin, P. J. (2010). An Imaginary Tale: The Story of √-1. Ukraine: Princeton University Press.Stillwell, J. (2013). Mathematics and Its History. United Kingdom: Springer New York.Euler’s Amazing 1747 PaperEuler, Leonard. *"Sur les logarithmes des nombres négatifs et imaginaires”* Written in 1747, but not published until 1862. Euler did publish a similar paper in 1749. See Cajori #3 above. English Translation: https://scholarlycommons.pacific.edu/cgi/viewcontent.cgi?filename=0&article=1806&context=euler-works&type=additionalNote on Benroulli’s area of sectorsEuler’s counterexample using Bernoulli’s sector area comes in a couple flavors. The one presented in his 1747 paper "Sur les logarithmes des nombres négatifs et imaginaires” is a bit different than an earlier example in a letter to Bernoulli. I chose the earlier example for clarity. See Cajori vol 2 and Sandifer. Feynman Lectures - Algebrahttps://www.feynmanlectures.caltech.edu/I_22.htmlCFAQJOTYQHT7JYIT
The moment we stopped understanding AI [AlexNet] Welch Labs 2024-07-01 | Thanks to KiwiCo for sponsoring today's video! Go to kiwico.com/welchlabs and use code WELCHLABS for 50% off your first month of monthly lines and/or for 20% off your first Panda Crate.Activation Atlas Posters!welchlabs.com/resources/5gtnaauv6nb9lrhoz9cp604padxp5owelchlabs.com/resources/activation-atlas-poster-mixed5b-13x19welchlabs.com/resources/large-activation-atlas-poster-mixed4c-24x36welchlabs.com/resources/activation-atlas-poster-mixed4c-13x19Special thanks to the Patrons:Juan Benet, Ross Hanson, Yan Babitski, AJ Englehardt, Alvin Khaled, Eduardo Barraza, Hitoshi Yamauchi, Jaewon Jung, Mrgoodlight, Shinichi Hayashi, Sid Sarasvati, Dominic Beaumont, Shannon Prater, Ubiquity Ventures, Matias FortiWelch LabsAd free videos and exclusive perks: patreon.com/welchlabsWatch on TikTok: tiktok.com/@welchlabsLearn More or Contact: welchlabs.comInstagram: instagram.com/welchlabsX: twitter.com/welchlabsReferencesAlexNet Paperproceedings.neurips.cc/paper_files/paper/2012/file/c399862d3b9d6b76c8436e924a68c45b-Paper.pdfOriginal Activation Atlas Article- explore here - Great interactive Atlas! https://distill.pub/2019/activation-atlas/Carter, et al., "Activation Atlas", Distill, 2019.Feature Visualization Article: https://distill.pub/2017/feature-visualization/`Olah, et al., "Feature Visualization", Distill, 2017.`Great LLM Explainability work: https://transformer-circuits.pub/2024/scaling-monosemanticity/index.htmlTempleton, et al., "Scaling Monosemanticity: Extracting Interpretable Features from Claude 3 Sonnet", Transformer Circuits Thread, 2024.“Deep Visualization Toolbox" by Jason Yosinski video inspired many visuals: youtube.com/watch?v=AgkfIQ4IGaMGreat LLM/GPT Intro paperarxiv.org/pdf/2304.105573B1Bs GPT Videos are excellent, as always:youtube.com/watch?v=eMlx5fFNoYcyoutube.com/watch?v=wjZofJX0v4MAndrej Kerpathy's walkthrough is amazing:youtube.com/watch?v=kCc8FmEb1nYGoodfellow’s Deep Learning Bookdeeplearningbook.orgOpenAI’s 10,000 V100 GPU cluster (1+ exaflop) news.microsoft.com/source/features/innovation/openai-azure-supercomputerGPT-3 size, etc: Language Models are Few-Shot Learners, Brown et al, 2020. Unique token count for ChatGPT: cookbook.openai.com/examples/how_to_count_tokens_with_tiktoken GPT-4 training size etc, speculative:patmcguinness.substack.com/p/gpt-4-details-revealedsemianalysis.com/p/gpt-4-architecture-infrastructureHistorical Neural Network Videosyoutube.com/watch?v=FwFduRA_L6Qyoutube.com/watch?v=cNxadbrN_aIErrata1:40 should be: "word fragment is appended to the end of the original input". Thanks for Chris A for finding this one.CFAQJOTYQHT7JYIT
How the Bizarre Path of Mars Reshaped Astronomy [Keplers Laws Part 2] Welch Labs 2024-05-30 | Tycho Brahe Doppelmayr Poster: welchlabs.com/resources/tycho-brahe-doppelmayr-poster-13x15Tycho Brahe Cellarius Poster: welchlabs.com/resources/tycho-brahe-cellarius-poster-13x16Eclipse Posters! welchlabs.com/resourcesThis is the second of a two part series - part one here: youtube.com/watch?v=Phscjl0u6TI&feature=youtu.beSpecial thanks to the Patrons:Juan Benet, Ross Hanson, Yan Babitski, AJ Englehardt, Alvin Khaled, Eduardo Barraza, Hitoshi Yamauchi, Jaewon Jung, Mrgoodlight, Shinichi Hayashi, Sid Sarasvati, Dominic Beaumont, Shannon Prater, Ubiquity Ventures, Matias FortiWelch LabsAd free videos and exclusive perks: patreon.com/welchlabsWatch on TikTok: tiktok.com/@welchlabsLearn More or Contact: welchlabs.comInstagram: instagram.com/welchlabsX: twitter.com/welchlabsReferencesOn the Shoulders of Giants: The Great Works of Physics and Astronomy. (2003). Kiribati: Penguin.Koestler, A. (2017). The Sleepwalkers: A History of Man's Changing Vision of the Universe. United Kingdom: Penguin Books Limited.en.wikipedia.org/wiki/History_of_Mars_observationkeplersdiscovery.com/index.htmlThe Cambridge Concise History of Astronomy. (1999). United Kingdom: Cambridge University Press.Mazer, A. (2011). Shifting the Earth: The Mathematical Quest to Understand the Motion of the Universe. Germany: Wiley.Voelkel, J. R. (2021). The Composition of Kepler's Astronomia Nova. United Kingdom: Princeton University Press.stellarium.orgKepler, J. (2015). Astronomia Nova. United States: Green Lion Press.Stephenson, B. (2012). Kepler’s Physical Astronomy. Switzerland: Springer New York.Brahe, T., Dreyer, J. L. E. (1972). Tychonis Brahe Dani Opera omnia. Netherlands: Swets & Zeitlinger.dn790003.ca.archive.org/0/items/Astronomianovaa00Kepl/Astronomianovaa00Kepl.pdfFeynman Lecture on Gravity: https://www.feynmanlectures.caltech.edu/I_07.htmlCFAQJOTYQHT7JYIT
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