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						Hello, I'm <br>
						<span>Clayton.</span>					
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					<p class="mx-auto text-white  mt-20 mb-40">
						Welcome to my personal webpage.
						I am a Research Scientist at Google based in New York City.
						I recently finished a Ph.D. in Computer Science from Columbia University under the supervision of <a href="http://www.cs.columbia.edu/~rocco/">Rocco Servedio</a> and <a href="http://www.cs.columbia.edu/~djhsu/">Daniel Hsu</a>. 
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					<a href="doc/resume.pdf" class="text-uppercase header-btn" target="_blank">My Resume (short-form)</span></a><br><br>
					<a href="doc/cv.pdf" class="text-uppercase header-btn" target="_blank">My CV (long-form)</span></a><br>
					
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							<h1 class="mb-0 mt-0">About Me</h1>
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					<p>
						I'm a New York City-based researcher studying the overlap between machine learning and theoretical computer science.
						My research is broadly motivated by a desire to improve the interpretability, transparency, and accountability of neural networks by understanding their mathematical properties. 
						My Ph.D. research was funded by an NSF GRFP fellowship, which I received in March 2021.
					</p>
					<p>
						Before starting my PhD at Columbia, I studied applied math and computer science as an undergrad at Brown and worked as a data scientist at LinkedIn.
						I've completed internships at LinkedIn (data science), Lumi Labs (engineering at a 15-person startup), Allen Institute for AI (climate modeling), Microsoft Research (transformer theory), and Google Research (transformers and graph reasoning).
					</p>
					<h4>My work</h4>
					<p>
						Most of my research consists of mathematical results about the capabilities and limitations of neural networks and other machine learning algorithms. In particular, I study: 
						<ul style="margin-left: 30px">
							<li>the fundamental limitations of architectures like <a href=http://proceedings.mlr.press/v134/hsu21a.html>random feature models</a>, <a href=https://arxiv.org/abs/2110.10295>deep neural networks</a>, and <a href=https://arxiv.org/abs/2306.02896>self-attention units</a>;</li>
							<li>the generalization and optimization properties of machine learning models like <a href=https://arxiv.org/abs/2105.14084>maximum-margin linear classifiers</a>, <a href=https://arxiv.org/abs/2210.15651>neural nets trained with gradient descent on low-dimensional data</a>, <a href=https://arxiv.org/abs/2210.05212>randomly initialized recurrent neural networks</a>, and <a href=https://arxiv.org/abs/2206.05317>minimum-norm interpolating neural networks</a>;</li>
							<li>the empirical and theoretical abilities of transformer models trained to solve combinatorial planning tasks (work in progress from a Summer 2023 internship at Microsoft Research NYC).</li>
						</ul>
					</p>
					<p>
						I applied <a href=https://www.authorea.com/doi/full/10.22541/essoar.168500343.32924398>machine learning techniques to climate modeling research</a> as an internship at the Allen Institute for AI (AI2) during Summer 2022. My contributions led to my recognition as an <a href=https://allenai.org/outstanding-interns>Outstanding Intern</a>.
					</p>
					<p>
						As an undergraduate, I completed a variety of research projects on <a href="https://cs.brown.edu/research/pubs/theses/ugrad/2018/sanford.clayton.pdf">mathematical modeling</a> (as a winner of the COMAP Interdisciplinary Contest for Modeling), <a href="https://link.springer.com/epdf/10.1007/s10884-021-10127-w?sharing_token=gI1_HWixoEZjNTzbi3JRc_e4RwlQNchNByi7wbcMAY5nuWYOEuol8KoG4XdUVobokGy0WfKu_ziU5pCMOzp_Fewn7MynqACmRKOJYTxqSRFwZJfzZcyQXMmFFyj4RSxva0WDIiIyViyuLzI_-0Ox8QC0VqML2jDB69SW4Q740-U%3D">dynamical systems</a> (with <a href=https://bjornsandstede.com/>Bjorn Sandstede</a>), <a href="https://academic.oup.com/bioinformatics/article/33/18/2943/3887237">molecular biology</a> (with <a href=http://fairbrother.biomed.brown.edu/research.php>William Fairbrother</a>), and <a href="https://cs.brown.edu/research/pubs/theses/ugrad/2018/sanford.clayton.pdf">machine learning theory</a> (with <a href="http://cs.brown.edu/people/eupfal/">Eli Upfal</a>).
					</p>
					<p>
						I primarily code in Python and am experienced with core data science and deep learning packages (e.g. Pytorch, Tensorflow, Sklearn, and Pandas). I programmed in Java and Scala during my undergraduate years and when I worked at LinkedIn and Lumi Labs. The climate modeling repository I contributed to while interning at AI2 is <a href="https://github.com/ai2cm/fv3net">publicly available</a>, and my Microsoft Research code will be published once we upload our preprint.
					</p>
					<h4>Teaching experience</h4>
					<p>
						I TA'd five different courses at Brown: <a href="http://cs.brown.edu/courses/csci0190/">accelerated intro to CS</a>, <a href="http://cs.brown.edu/courses/csci0220/">discrete math</a>, <a href="http://cs.brown.edu/courses/csci1010/">CS theory</a>, intro to dynamical systems, and <a href="http://cs.brown.edu/courses/csci1570/">algorithms</a>, for which I was Head TA.
					</p>
					<p>
						I was a graduate TA for courses on <a href="http://www.cs.columbia.edu/~cs4252/">computational learning theory</a> with Rocco Servedio, natural and artificial neural networks with Christos Papadimitriou and John Morrison, and <a href=https://www.proditus.com/teaching/mlclimate2023/>ML and climate</a> with Alp Kucukelbir.
						Natural and artificial neural networks was a new course, for which I developed a lab component from scratch in collaboration with Sam Deng; all materials are <a href="https://samuel-deng.github.io/coms4995-nat-art-neural-nets/">available</a> online.
					</p>
					<p>
						As a graduate student, I coordinated <a href=https://theory.cs.columbia.edu/undergrad-tcs-seminar>undergraduate seminars</a> on deep learning theory during Summer 2021, Fall 2021, and <a href=https://docs.google.com/document/d/1S_8a3pAC7Hsh7WDG75Vu8rfarla6pzBNiXB0T5nJTNA/edit?usp=sharing>Spring 2023</a>. 
					</p>					
					<h4>Departmental and academic service</h4>
					<p>
						I served as a reviewer for ICLR 2024, NeurIPS 2023, JMLR, SODA 2023, and STOC 2022.
					</p>
					<p>
						I was currently a <a href="https://www.cs.columbia.edu/education/phd/phd-reps/">PhD representative</a> alongside the great <a href=https://twrand.github.io/>Tim Randolph</a>. We represented computer science PhD student interests and concerns to CS faculty and administrators. 
					</p>
					<p>
						In the Columbia CS theory group, I coordinated the student retreat for fall 2021 and 2022 and started (but no longer run) the <a href="http://theory.cs.columbia.edu/theory-student-seminar.html">CS Theory Student Seminar</a>.
					</p>
					<p>
						I was the leader of <a href="https://www.instagram.com/qstematcolumbia/">qSTEM</a>, an organization for LGBTQ+ students in the Columbia School of Engineering and Applied Sciences.
					</p>

					<h4>Find me online: </h4>
					<p>
						<a href=https://www.linkedin.com/in/claytonsanford/>[linkedin]</a>
						<a href=https://github.com/chsanford>[github]</a>
						<a href="https://arxiv.org/search/cs?searchtype=author&query=Sanford%2C+C">[arxiv]</a>
						<a href=https://scholar.google.com/citations?hl=en&user=Qo18yHAAAAAJ>[google scholar]</a>
						<a href=https://dblp.org/pid/232/1797.html>[dblp]</a>
					</p>
					<p>
						I am indebted to many fantastic mentors over many years among my teachers, professors, advisors, coworkers, and friends. <b>I am always happy to pass it forward and chat with anyone interested in learning more about PhD programs in ML and/or theory, Columbia CS, NYC living, or niche neighborhood concerns about Morningside Heights.</b>
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							<h1 class="mb-0">My Papers</h1>
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				<h4>Neural Networks</h4><br>

				<p><b>[SHT24b]</b> Clayton Sanford, Daniel Hsu, Matus Telgarsky. "One-layer transformers fail to solve the induction heads task." <i>Note.</i> <a href="https://arxiv.org/abs/2408.14332">[arxiv]</a></p>

				<p><b>[SFH+24]</b> Clayton Sanford, Bahare Fatemi, Ethan Hall, Anton Tsitsulin, Mehran Kazemi, Jonathan Halcrow, Bryan Perozzi, Vahab Mirrokni. "Understanding Transformer Reasoning Capabilities via Graph Algorithms." <i>Preprint.</i> <a href="https://arxiv.org/abs/2405.18512">[arxiv]</a> <a href="doc/sfh24-flatiron.pdf">[Flatiron slides]</a></p>

				<p><b>[Sanford24]</b> Clayton Sanford. "Representational Capabilities of Feed-forward and Sequential Neural Architectures." <i>Doctoral Thesis.</i> 2024. <a href="https://www.proquest.com/docview/3046052302">[thesis]</a> <a href="doc/defense-slides.pdf">[defense slides]</a></p>

				<p><b>[SHT24a]</b> Clayton Sanford, Daniel Hsu, Matus Telgarsky. "Transformers, Parallel Computation, and Logarithmic Depth." <i>ICML 2024.</i> <a href="https://arxiv.org/abs/2402.09268">[arxiv]</a> <a href="doc/sht24-google-slides.pdf">[Google slides]</a> <a href="doc/sht24-cu-slides.pdf">[Columbia slides]</a></p>

				<p><b>[SHT23]</b> Clayton Sanford, Daniel Hsu, Matus Telgarsky. "Representational Strengths and Limitations of Transformers." <i>NeurIPS 2023.</i> <a href="https://proceedings.neurips.cc/paper_files/paper/2023/hash/73bf692447f174984f30499ec9b20e04-Abstract-Conference.html">[paper]</a> <a href="https://arxiv.org/abs/2306.02896">[arxiv]</a> <a href="doc/sht23-ucsd.pdf">[UCSD seminar slides]</a> <a href="doc/sht23-google.pdf">[Google NYC algorithms seminar slides]</a> <a href="doc/sht23-cu-poster.pdf">[Columbia StatML workshop poster]</a> <a href="sht23-cu-slides">[Columbia StatML workshop slides]</a></p>

				<p><b>[AHS23]</b> Navid Ardeshir*, Daniel Hsu*, Clayton Sanford*. "Intrinsic dimensionality and generalization properties of the R-norm inductive bias." <i>COLT 2023.</i> <a href="https://proceedings.mlr.press/v195/ardeshir23a.html">[paper]</a> <a href="https://arxiv.org/abs/2206.05317">[arxiv]</a> <a href="https://blog.claytonsanford.com/2023/07/12/ahs23.html">[blog post]</a> <a href="doc/ahs23-poster.pdf">[COLT poster]</a></p>

				<p><b>[BBSS22]</b> Alberto Bietti*, Joan Bruna*, Clayton Sanford*, Min Jae Song*. "Learning single-index models with shallow neural networks." <i>NeurIPS 2022.</i> <a href="https://proceedings.neurips.cc/paper_files/paper/2022/hash/3fb6c52aeb11e09053c16eabee74dd7b-Abstract-Conference.html">[paper]</a> <a href="doc/bbss22-poster.pdf">[NeurIPS poster]</a> <a href="https://arxiv.org/abs/2210.15651">[arxiv]</a></p>

				<p><b>[CPSS22]</b> Vaggos Chatziafratis*, Ioannis Panageas*, Clayton Sanford*, Stelios Stavroulakis*. "On Scrambling Phenomena for Randomly Initialized Recurrent Networks." <i>NeurIPS 2022.</i> <a href="https://proceedings.neurips.cc/paper_files/paper/2022/hash/755acd0c7c07180d78959b6d89768207-Abstract-Conference.html">[paper]</a> <a href="https://arxiv.org/abs/2210.05212">[arxiv]</a></p>

				<p><b>[HSSV22]</b> Daniel Hsu*, Clayton Sanford*, Rocco Servedio*, Emmanouil-Vasileios Vlatakis-Gkaragkounis*. "Near-Optimal Statistical Query Lower Bounds for Agnostically Learning Intersections of Halfspaces with Gaussian Marginals." <i>COLT 2022.</i> <a href="https://proceedings.mlr.press/v178/hsu22a.html">[paper]</a> <a href="https://arxiv.org/abs/2202.05096">[arxiv]</a> <a href="doc/sq-nyu-seminar.pdf">[NYU seminar slides]</a> <a href="https://slideslive.com/38985576/nearoptimal-statistical-query-lower-bounds-for-agnostically-learning-intersections-of-halfspaces-with-gaussian-marginals?ref=search-presentations-clayton+sanford">[conference talk]</a></p>

				<p><b>[SC22]</b> Clayton Sanford, Vaggos Chatziafratis. "Expressivity of Neural Networks via Chaotic Itineraries beyond Sharkovsky's Theorem." <i>AISTATS 2022.</i> <a href="https://proceedings.mlr.press/v151/sanford22a.html">[paper]</a> <a href="https://arxiv.org/abs/2110.10295">[arxiv]</a> <a href="https://slideslive.com/38980743/expressivity-of-neural-networks-via-chaotic-itineraries-beyond-sharkovskys-theorem?ref=search-presentations-aistats+sanford">[conference talk]</a></p>


				<p><b>[ASH21]</b> Navid Ardeshir*, Clayton Sanford*, Daniel Hsu. "Support vector machines and linear regression coincide with very high-dimensional features." <i>NeurIPS 2021.</i> <a href="https://proceedings.neurips.cc/paper/2021/hash/26d4b4313a7e5828856bc0791fca39a2-Abstract.html">[paper]</a> <a href="https://arxiv.org/abs/2105.14084">[arxiv]</a> <a href="https://blog.claytonsanford.com/2021/12/07/ash21.html">[blog post]</a> <a href="https://openreview.net/forum?id=9bqxRuRwBlu">[reviews]</a> <a href="https://slideslive.com/38968209/support-vector-machines-coincide-with-regression-only-in-very-high-dimensions?ref=speaker-88261">[conference talk]</a> <a href="doc/ash21-brown.pdf">[Brown seminar slides]</a> <a href="doc/ash21-ucsc.pdf">[UCSC seminar slides]</a></p>

				<p><b>[HSSV21]</b> Daniel Hsu*, Clayton Sanford*, Rocco Servedio*, Emmanouil-Vasileios Vlatakis-Gkaragkounis*. "On the Approximation Power of Two-Layer Networks of Random ReLUs." <i>COLT 2021.</i> <a href="http://proceedings.mlr.press/v134/hsu21a.html">[paper]</a> <a href="https://arxiv.org/abs/2102.02336">[arxiv]</a> <a href="https://blog.claytonsanford.com/2021/08/15/hssv21.html">[blog post]</a> <a href="http://www.learningtheory.org/colt2021/virtual/poster_1178.html">[conference talks]</a> <a href="https://www.dsiposters.com/posters-21-22/computing-systems-team-5">[Columbia DSI poster session]</a> <a href="doc/hssv21-mit.pdf">[MIT and BU seminar slides]</a> <a href="doc/hssv21-uw.pdf">[UW seminar slides]</a>
				</p>
				<br>

				<!-- <h4>Journal Publications</h4><br> -->

				<h4>ML + Climate</h4><br>
				<p><b>[SKW+23]</b> Clayton Sanford, Anna Kwa, Oliver Watt-Meyer, Spencer Clark, Noah Brenowitz, Jeremy McGibbon, Christopher Bretherton. "Improving the reliability of ML-corrected climate models with novelty detection." <i>Appearing in Journal of Advances in Modeling Earth Systems (JAMES).</i> <a href="https://www.authorea.com/doi/full/10.22541/essoar.168500343.32924398">[journal submission]</a> <a href="https://arxiv.org/abs/2211.13354">[NeurIPS workshop paper arxiv]</a> <a href="doc/skw22-neurips.pdf">[NeurIPS workshop slides]</a> <a href="https://ams.confex.com/ams/103ANNUAL/meetingapp.cgi/Paper/418032">[AMS workshop abstract]</a> <a href="doc/skw22-ams.pdf">[AMS workshop slides]</a></p><br> 

				<h4>Undergraduate Research</h4><br>

				<p><b>[CRSSCS22]</b> Tracy Chin*, Jacob Ruth*, Clayton Sanford*, Rebecca Santorella*, Paul Carter*, Bjorn Sandstede*. "Enabling equation-free modeling via diffusion maps." <i>Journal of Dynamics and Differential Equations, 2022.</i> <a href="https://link.springer.com/epdf/10.1007/s10884-021-10127-w?sharing_token=gI1_HWixoEZjNTzbi3JRc_e4RwlQNchNByi7wbcMAY5nuWYOEuol8KoG4XdUVobokGy0WfKu_ziU5pCMOzp_Fewn7MynqACmRKOJYTxqSRFwZJfzZcyQXMmFFyj4RSxva0WDIiIyViyuLzI_-0Ox8QC0VqML2jDB69SW4Q740-U%3D">[journal]</a> <a href="https://arxiv.org/abs/2112.15159">[arxiv]</a> 
				<br>

				<p>
					<b>[S18]</b> Clayton Sanford. "Applying Rademacher-Like Bounds to Combinatorial Samples and Function Selection." Honors Thesis, Brown Department of Computer Science, 2018. <a href="http://cs.brown.edu/research/pubs/theses/ugrad/2018/sanford.clayton.pdf">[thesis]</a>
				</p>
				<p>
					<b>[CSF17]</b> Kamil Cygan*, Clayton Sanford*, William Fairbrother. "Spliceman2 - A Computational Web Server That Predicts Sequence Variations in Pre-mRNA Splicing." <i>Bioinformatics</i> 33 (18), 2017. <a href="https://academic.oup.com/bioinformatics/article/33/18/2943/3887237">[paper]</a> 
				</p>
				<p>
					<b>[GSK16]</b> Julia Gross*, Clayton Sanford*, Geoff Kocks*. "Projected Water Needs and Intervention Strategies in India." <i>Undergraduate Mathematics and its Applications</i> 37 (2), 2016. <a href="http://www.dam.brown.edu/siam/2016/gsk.pdf">[paper]</a> <a href="https://www.browndailyherald.com/2016/04/19/undergraduates-win-top-honor-at-international-math-contest/">[article]</a>
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							<li>I grew up in Santa Cruz, California.</li>
								<ul style="margin-left: 30px"><li>Technically, I lived in Aptos, which is why this page uses a font with the <a href=https://en.wikipedia.org/wiki/Aptos_(typeface)>same name</a>.</li></ul>
							<li>I like running, backpacking, and all things nature.</li>
							<li>In spring 2023, I was appointed as a member of <a href="https://www.cb9m.org/officers_members">Manhattan Community Board 9</a>. Send me your thoughts and questions about the district, especially around housing and transit policy!</li>
							<li>I am an Eagle Scout.</li><!-- , and I have been involved with <a href="http://whitestagsierra.org/">White Stag leadership camp</a> for over ten years.</li> -->
							<li>The blond hair was a phase that started when I lived in San Francisco. The phase has since ended.</li>
							<li>I have strange exploration goals in NYC: <ul style="margin-left: 30px">
								<li>In 2021, I ran from the end of every subway line to Morningside Heights.</li>
								<li>In 2022 and 2023, I walked every block in Manhattan.</li>
								<li>If you have ideas what the next goal should be, let me know!</li>
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