Batch List-Decodable Linear Regression via Higher Moments

Ilias Diakonikolas, Daniel Kane, Sushrut Karmalkar, Sihan Liu, Thanasis Pittas
Proceedings of the 42nd International Conference on Machine Learning, PMLR 267:13622-13642, 2025.

Abstract

We study the task of list-decodable linear regression using batches, recently introduced by Das et al. 2023.. In this setting, we are given $m$ batches with each batch containing $n$ points in $\mathbb R^d$. A batch is called clean if the points it contains are i.i.d. samples from an unknown linear regression distribution. For a parameter $\alpha \in (0, 1/2)$, an unknown $\alpha$-fraction of the batches are clean and no assumptions are made on the remaining batches. The goal is to output a small list of vectors at least one of which is close to the true regressor vector in $\ell_2$-norm. Das et al. 2023 gave an efficient algorithm for this task, under natural distributional assumptions, with the following guarantee. Under the assumption that the batch size satisfies $n \geq \tilde{\Omega}(\alpha^{-1})$ and the total number of batches is $m = \text{poly}(d, n, 1/\alpha)$, their algorithm runs in polynomial time and outputs a list of $O(1/\alpha^2)$ vectors at least one of which is $\tilde{O}(\alpha^{-1/2}/\sqrt{n})$ close to the target regressor. Here we design a new polynomial-time algorithm for this task with significantly stronger guarantees under the assumption that the low-degree moments of the covariates distribution are Sum-of-Squares (SoS) certifiably bounded. Specifically, for any constant $\delta>0$, as long as the batch size is $n \geq \Omega_{\delta}(\alpha^{-\delta})$ and the degree-$\Theta(1/\delta)$ moments of the covariates are SoS certifiably bounded, our algorithm uses $m = \text{poly}((dn)^{1/\delta}, 1/\alpha)$ batches, runs in polynomial-time, and outputs an $O(1/\alpha)$-sized list of vectors one of which is $O(\alpha^{-\delta/2}/\sqrt{n})$ close to the target. That is, our algorithm substantially improves both the minimum batch size and the final error guarantee, while achieving the optimal list size. Our approach leverages higher-order moment information by carefully combining the SoS paradigm interleaved with an iterative method and a novel list pruning procedure for this setting. In the process, we give an SoS proof of the Marcinkiewicz-Zygmund inequality that may be of broader applicability.

Cite this Paper

BibTeX
@InProceedings{pmlr-v267-diakonikolas25c, title = {Batch List-Decodable Linear Regression via Higher Moments}, author = {Diakonikolas, Ilias and Kane, Daniel and Karmalkar, Sushrut and Liu, Sihan and Pittas, Thanasis}, booktitle = {Proceedings of the 42nd International Conference on Machine Learning}, pages = {13622--13642}, year = {2025}, editor = {Singh, Aarti and Fazel, Maryam and Hsu, Daniel and Lacoste-Julien, Simon and Berkenkamp, Felix and Maharaj, Tegan and Wagstaff, Kiri and Zhu, Jerry}, volume = {267}, series = {Proceedings of Machine Learning Research}, month = {13--19 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v267/main/assets/diakonikolas25c/diakonikolas25c.pdf}, url = {https://proceedings.mlr.press/v267/diakonikolas25c.html}, abstract = {We study the task of list-decodable linear regression using batches, recently introduced by Das et al. 2023.. In this setting, we are given $m$ batches with each batch containing $n$ points in $\mathbb R^d$. A batch is called clean if the points it contains are i.i.d. samples from an unknown linear regression distribution. For a parameter $\alpha \in (0, 1/2)$, an unknown $\alpha$-fraction of the batches are clean and no assumptions are made on the remaining batches. The goal is to output a small list of vectors at least one of which is close to the true regressor vector in $\ell_2$-norm. Das et al. 2023 gave an efficient algorithm for this task, under natural distributional assumptions, with the following guarantee. Under the assumption that the batch size satisfies $n \geq \tilde{\Omega}(\alpha^{-1})$ and the total number of batches is $m = \text{poly}(d, n, 1/\alpha)$, their algorithm runs in polynomial time and outputs a list of $O(1/\alpha^2)$ vectors at least one of which is $\tilde{O}(\alpha^{-1/2}/\sqrt{n})$ close to the target regressor. Here we design a new polynomial-time algorithm for this task with significantly stronger guarantees under the assumption that the low-degree moments of the covariates distribution are Sum-of-Squares (SoS) certifiably bounded. Specifically, for any constant $\delta>0$, as long as the batch size is $n \geq \Omega_{\delta}(\alpha^{-\delta})$ and the degree-$\Theta(1/\delta)$ moments of the covariates are SoS certifiably bounded, our algorithm uses $m = \text{poly}((dn)^{1/\delta}, 1/\alpha)$ batches, runs in polynomial-time, and outputs an $O(1/\alpha)$-sized list of vectors one of which is $O(\alpha^{-\delta/2}/\sqrt{n})$ close to the target. That is, our algorithm substantially improves both the minimum batch size and the final error guarantee, while achieving the optimal list size. Our approach leverages higher-order moment information by carefully combining the SoS paradigm interleaved with an iterative method and a novel list pruning procedure for this setting. In the process, we give an SoS proof of the Marcinkiewicz-Zygmund inequality that may be of broader applicability.} }
Endnote
%0 Conference Paper %T Batch List-Decodable Linear Regression via Higher Moments %A Ilias Diakonikolas %A Daniel Kane %A Sushrut Karmalkar %A Sihan Liu %A Thanasis Pittas %B Proceedings of the 42nd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2025 %E Aarti Singh %E Maryam Fazel %E Daniel Hsu %E Simon Lacoste-Julien %E Felix Berkenkamp %E Tegan Maharaj %E Kiri Wagstaff %E Jerry Zhu %F pmlr-v267-diakonikolas25c %I PMLR %P 13622--13642 %U https://proceedings.mlr.press/v267/diakonikolas25c.html %V 267 %X We study the task of list-decodable linear regression using batches, recently introduced by Das et al. 2023.. In this setting, we are given $m$ batches with each batch containing $n$ points in $\mathbb R^d$. A batch is called clean if the points it contains are i.i.d. samples from an unknown linear regression distribution. For a parameter $\alpha \in (0, 1/2)$, an unknown $\alpha$-fraction of the batches are clean and no assumptions are made on the remaining batches. The goal is to output a small list of vectors at least one of which is close to the true regressor vector in $\ell_2$-norm. Das et al. 2023 gave an efficient algorithm for this task, under natural distributional assumptions, with the following guarantee. Under the assumption that the batch size satisfies $n \geq \tilde{\Omega}(\alpha^{-1})$ and the total number of batches is $m = \text{poly}(d, n, 1/\alpha)$, their algorithm runs in polynomial time and outputs a list of $O(1/\alpha^2)$ vectors at least one of which is $\tilde{O}(\alpha^{-1/2}/\sqrt{n})$ close to the target regressor. Here we design a new polynomial-time algorithm for this task with significantly stronger guarantees under the assumption that the low-degree moments of the covariates distribution are Sum-of-Squares (SoS) certifiably bounded. Specifically, for any constant $\delta>0$, as long as the batch size is $n \geq \Omega_{\delta}(\alpha^{-\delta})$ and the degree-$\Theta(1/\delta)$ moments of the covariates are SoS certifiably bounded, our algorithm uses $m = \text{poly}((dn)^{1/\delta}, 1/\alpha)$ batches, runs in polynomial-time, and outputs an $O(1/\alpha)$-sized list of vectors one of which is $O(\alpha^{-\delta/2}/\sqrt{n})$ close to the target. That is, our algorithm substantially improves both the minimum batch size and the final error guarantee, while achieving the optimal list size. Our approach leverages higher-order moment information by carefully combining the SoS paradigm interleaved with an iterative method and a novel list pruning procedure for this setting. In the process, we give an SoS proof of the Marcinkiewicz-Zygmund inequality that may be of broader applicability.
APA
Diakonikolas, I., Kane, D., Karmalkar, S., Liu, S. & Pittas, T.. (2025). Batch List-Decodable Linear Regression via Higher Moments. Proceedings of the 42nd International Conference on Machine Learning, in Proceedings of Machine Learning Research 267:13622-13642 Available from https://proceedings.mlr.press/v267/diakonikolas25c.html.

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