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**Published in:** LIPIcs, Volume 287, 15th Innovations in Theoretical Computer Science Conference (ITCS 2024)

Multicalibration is a notion of fairness for predictors that requires them to provide calibrated predictions across a large set of protected groups. Multicalibration is known to be a distinct goal than loss minimization, even for simple predictors such as linear functions.
In this work, we consider the setting where the protected groups can be represented by neural networks of size k, and the predictors are neural networks of size n > k. We show that minimizing the squared loss over all neural nets of size n implies multicalibration for all but a bounded number of unlucky values of n. We also give evidence that our bound on the number of unlucky values is tight, given our proof technique. Previously, results of the flavor that loss minimization yields multicalibration were known only for predictors that were near the ground truth, hence were rather limited in applicability. Unlike these, our results rely on the expressivity of neural nets and utilize the representation of the predictor.

Jarosław Błasiok, Parikshit Gopalan, Lunjia Hu, Adam Tauman Kalai, and Preetum Nakkiran. Loss Minimization Yields Multicalibration for Large Neural Networks. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 17:1-17:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{blasiok_et_al:LIPIcs.ITCS.2024.17, author = {B{\l}asiok, Jaros{\l}aw and Gopalan, Parikshit and Hu, Lunjia and Kalai, Adam Tauman and Nakkiran, Preetum}, title = {{Loss Minimization Yields Multicalibration for Large Neural Networks}}, booktitle = {15th Innovations in Theoretical Computer Science Conference (ITCS 2024)}, pages = {17:1--17:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-309-6}, ISSN = {1868-8969}, year = {2024}, volume = {287}, editor = {Guruswami, Venkatesan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.17}, URN = {urn:nbn:de:0030-drops-195452}, doi = {10.4230/LIPIcs.ITCS.2024.17}, annote = {Keywords: Multi-group fairness, loss minimization, neural networks} }

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APPROX

**Published in:** LIPIcs, Volume 145, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)

The l_2 tracking problem is the task of obtaining a streaming algorithm that, given access to a stream of items a_1,a_2,a_3,... from a universe [n], outputs at each time t an estimate to the l_2 norm of the frequency vector f^{(t)}in R^n (where f^{(t)}_i is the number of occurrences of item i in the stream up to time t). The previous work [Braverman-Chestnut-Ivkin-Nelson-Wang-Woodruff, PODS 2017] gave a streaming algorithm with (the optimal) space using O(epsilon^{-2}log(1/delta)) words and O(epsilon^{-2}log(1/delta)) update time to obtain an epsilon-accurate estimate with probability at least 1-delta. We give the first algorithm that achieves update time of O(log 1/delta) which is independent of the accuracy parameter epsilon, together with the nearly optimal space using O(epsilon^{-2}log(1/delta)) words. Our algorithm is obtained using the Count Sketch of [Charilkar-Chen-Farach-Colton, ICALP 2002].

Chi-Ning Chou, Zhixian Lei, and Preetum Nakkiran. Tracking the l_2 Norm with Constant Update Time. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 2:1-2:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{chou_et_al:LIPIcs.APPROX-RANDOM.2019.2, author = {Chou, Chi-Ning and Lei, Zhixian and Nakkiran, Preetum}, title = {{Tracking the l\underline2 Norm with Constant Update Time}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)}, pages = {2:1--2:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-125-2}, ISSN = {1868-8969}, year = {2019}, volume = {145}, editor = {Achlioptas, Dimitris and V\'{e}gh, L\'{a}szl\'{o} A.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2019.2}, URN = {urn:nbn:de:0030-drops-112175}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2019.2}, annote = {Keywords: Streaming algorithms, Sketching algorithms, Tracking, CountSketch} }

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**Published in:** LIPIcs, Volume 124, 10th Innovations in Theoretical Computer Science Conference (ITCS 2019)

Using a mild variant of polar codes we design linear compression schemes compressing Hidden Markov sources (where the source is a Markov chain, but whose state is not necessarily observable from its output), and to decode from Hidden Markov channels (where the channel has a state and the error introduced depends on the state). We give the first polynomial time algorithms that manage to compress and decompress (or encode and decode) at input lengths that are polynomial both in the gap to capacity and the mixing time of the Markov chain. Prior work achieved capacity only asymptotically in the limit of large lengths, and polynomial bounds were not available with respect to either the gap to capacity or mixing time. Our results operate in the setting where the source (or the channel) is known. If the source is unknown then compression at such short lengths would lead to effective algorithms for learning parity with noise - thus our results are the first to suggest a separation between the complexity of the problem when the source is known versus when it is unknown.

Venkatesan Guruswami, Preetum Nakkiran, and Madhu Sudan. Algorithmic Polarization for Hidden Markov Models. In 10th Innovations in Theoretical Computer Science Conference (ITCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 124, pp. 39:1-39:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{guruswami_et_al:LIPIcs.ITCS.2019.39, author = {Guruswami, Venkatesan and Nakkiran, Preetum and Sudan, Madhu}, title = {{Algorithmic Polarization for Hidden Markov Models}}, booktitle = {10th Innovations in Theoretical Computer Science Conference (ITCS 2019)}, pages = {39:1--39:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-095-8}, ISSN = {1868-8969}, year = {2019}, volume = {124}, editor = {Blum, Avrim}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2019.39}, URN = {urn:nbn:de:0030-drops-101326}, doi = {10.4230/LIPIcs.ITCS.2019.39}, annote = {Keywords: polar codes, error-correcting codes, compression, hidden markov model} }

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**Published in:** LIPIcs, Volume 60, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016)

In this paper, we prove an almost-optimal hardness for Max k-CSP_R based on Khot's Unique Games Conjecture (UGC). In Max k-CSP_R, we are given a set of predicates each of which depends on exactly k variables. Each variable can take any value from 1, 2, ..., R. The goal is to find an assignment to variables that maximizes the number of satisfied predicates.
Assuming the Unique Games Conjecture, we show that it is NP-hard to approximate Max k-CSP_R to within factor 2^{O(k log k)}(log R)^{k/2}/R^{k - 1} for any k, R.
To the best of our knowledge, this result improves on all the known hardness of approximation results when 3 <= k = o(log R/log log R). In this case, the previous best hardness result was NP-hardness of approximating within a factor O(k/R^{k-2}) by Chan. When k = 2, our result matches the best known UGC-hardness result of Khot, Kindler, Mossel and O'Donnell.
In addition, by extending an algorithm for Max 2-CSP_R by Kindler, Kolla and Trevisan, we provide an Omega(log R/R^{k - 1})-approximation algorithm for Max k-CSP_R. This algorithm implies that our inapproximability result is tight up to a factor of 2^{O(k \log k)}(\log R)^{k/2 - 1}. In comparison, when 3 <= k is a constant, the previously known gap was $O(R)$, which is significantly larger than our gap of O(polylog R).
Finally, we show that we can replace the Unique Games Conjecture assumption with Khot's d-to-1 Conjecture and still get asymptotically the same hardness of approximation.

Pasin Manurangsi, Preetum Nakkiran, and Luca Trevisan. Near-Optimal UGC-hardness of Approximating Max k-CSP_R. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 60, pp. 15:1-15:28, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{manurangsi_et_al:LIPIcs.APPROX-RANDOM.2016.15, author = {Manurangsi, Pasin and Nakkiran, Preetum and Trevisan, Luca}, title = {{Near-Optimal UGC-hardness of Approximating Max k-CSP\underlineR}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016)}, pages = {15:1--15:28}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-018-7}, ISSN = {1868-8969}, year = {2016}, volume = {60}, editor = {Jansen, Klaus and Mathieu, Claire and Rolim, Jos\'{e} D. P. and Umans, Chris}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2016.15}, URN = {urn:nbn:de:0030-drops-66388}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2016.15}, annote = {Keywords: inapproximability, unique games conjecture, constraint satisfaction problem, invariance principle} }