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1. Multi-Criteria Dimensionality Reduction with Applications to Fairness

   Tantipongpipat, Uthaipon and ( December 2019 , Advances in Neural Information Processing Systems 32 (NeurIPS 2019))

   Dimensionality reduction is a classical technique widely used for data analysis. One foundational instantiation is Principal Component Analysis (PCA), which minimizes the average reconstruction error. In this paper, we introduce the multi-criteria dimensionality reduction problem where we are given multiple objectives that need to be optimized simultaneously. As an application, our model captures several fairness criteria for dimensionality reduction such as the Fair-PCA problem introduced by Samadi et al. [NeurIPS18] and the Nash Social Welfare (NSW) problem. In the Fair-PCA problem, the input data is divided into k groups, and the goal is to find a single d-dimensional representation for all groups for which the maximum reconstruction error of any one group is minimized. In NSW the goal is to maximize the product of the individual variances of the groups achieved by the common low-dimensinal space. Our main result is an exact polynomial-time algorithm for the two-criteria dimensionality reduction problem when the two criteria are increasing concave functions. As an application of this result, we obtain a polynomial time algorithm for Fair-PCA for k=2 groups, resolving an open problem of Samadi et al.[NeurIPS18], and a polynomial time algorithm for NSW objective for k=2 groups. We also give approximation algorithms for k>2. Our technical contribution in the above results is to prove new low-rank properties of extreme point solutions to semi-definite programs. We conclude with the results of several experiments indicating improved performance and generalized application of our algorithm on real-world datasets. 

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2. Truncated Linear Regression in High Dimensions

   Daskalakis, C ; Rohatgi, D ; Zampetakis, M ( January 2020 , 34th Annual Conference on Neural Information Processing Systems (NeurIPS), NeurIPS 2020)

   null (Ed.)

   https://arxiv.org/abs/2007.14539 As in standard linear regression, in truncated linear regression, we are given access to observations (Ai,yi)i whose dependent variable equals yi=ATi⋅x∗+ηi, where x∗ is some fixed unknown vector of interest and ηi is independent noise; except we are only given an observation if its dependent variable yi lies in some "truncation set" S⊂ℝ. The goal is to recover x∗ under some favorable conditions on the Ai's and the noise distribution. We prove that there exists a computationally and statistically efficient method for recovering k-sparse n-dimensional vectors x∗ from m truncated samples, which attains an optimal ℓ2 reconstruction error of O((klogn)/m‾‾‾‾‾‾‾‾‾‾√). As a corollary, our guarantees imply a computationally efficient and information-theoretically optimal algorithm for compressed sensing with truncation, which may arise from measurement saturation effects. Our result follows from a statistical and computational analysis of the Stochastic Gradient Descent (SGD) algorithm for solving a natural adaptation of the LASSO optimization problem that accommodates truncation. This generalizes the works of both: (1) [Daskalakis et al. 2018], where no regularization is needed due to the low-dimensionality of the data, and (2) [Wainright 2009], where the objective function is simple due to the absence of truncation. In order to deal with both truncation and high-dimensionality at the same time, we develop new techniques that not only generalize the existing ones but we believe are of independent interest. 

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3. Approximation Algorithms for Socially Fair Clustering

   Makarychev, Yury ; Vakilian, Ali ( January 2021 , Proceedings of the Conference on Learning Theory, PMLR)

   Belkin, Mikhail ; Kpotufe, Samor (Ed.)

   We present an $e^{O(p)} (\log \ell) / (\log \log \ell)$-approximation algorithm for socially fair clustering with the $\ell_p$-objective. In this problem, we are given a set of points in a metric space. Each point belongs to one (or several) of $\ell$ groups. The goal is to find a $k$-medians, $k$-means, or, more generally, $\ell_p$-clustering that is simultaneously good for all of the groups. More precisely, we need to find a set of $k$ centers $C$ so as to minimize the maximum over all groups $j$ of $\sum_{u \text{ in group } j} d(u, C)^p$. The socially fair clustering problem was independently proposed by Abbasi, Bhaskara, and Venkatasubramanian (2021) and Ghadiri, Samadi, and Vempala (2021). Our algorithm improves and generalizes their $O(\ell)$-approximation algorithms for the problem. The natural LP relaxation for the problem has an integrality gap of $\Omega(\ell)$. In order to obtain our result, we introduce a strengthened LP relaxation and show that it has an integrality gap of $\Theta((\log \ell) / (\log \log \ell))$ for a fixed p. Additionally, we present a bicriteria approximation algorithm, which generalizes the bicriteria approximation of Abbasi et al. (2021). 

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4. Dimensionality Reduction for General {KDE} Mode Finding

   Luo, Xinyu ; Musco, Christopher ; Widdershoven, Cas ( June 2023 , Proceedings of the 40th International Conference on Machine Learning)

   Finding the mode of a high dimensional probability distribution $\mathcal{D}$ is a fundamental algorithmic problem in statistics and data analysis. There has been particular interest in efficient methods for solving the problem when $\mathcal{D}$ is represented as a mixture model or kernel density estimate, although few algorithmic results with worst-case approximation and runtime guarantees are known. In this work, we significantly generalize a result of (LeeLiMusco:2021) on mode approximation for Gaussian mixture models. We develop randomized dimensionality reduction methods for mixtures involving a broader class of kernels, including the popular logistic, sigmoid, and generalized Gaussian kernels. As in Lee et al.’s work, our dimensionality reduction results yield quasi-polynomial algorithms for mode finding with multiplicative accuracy $(1-\epsilon)$ for any $\epsilon > 0$. Moreover, when combined with gradient descent, they yield efficient practical heuristics for the problem. In addition to our positive results, we prove a hardness result for box kernels, showing that there is no polynomial time algorithm for finding the mode of a kernel density estimate, unless $\mathit{P} = \mathit{NP}$. Obtaining similar hardness results for kernels used in practice (like Gaussian or logistic kernels) is an interesting future direction. 

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5. Individually Fair Learning with One Sided Feedback

   Bechavod, Yahav ; Roth, Aaroh ( January 2023 , International Conference on Machine Learning)

   We consider an online learning problem with one-sided feedback, in which the learner is able to observe the true label only for positively predicted instances. On each round, k instances arrive and receive classification outcomes according to a randomized policy deployed by the learner, whose goal is to maximize accuracy while deploying individually fair policies. We first extend the framework of Bechavod et al. (2020), which relies on the existence of a human fairness auditor for detecting fairness violations, to instead incorporate feedback from dynamically-selected panels of multiple, possibly inconsistent, auditors. We then construct an efficient reduction from our problem of online learning with one-sided feedback and a panel reporting fairness violations to the contextual combinatorial semi-bandit problem (Cesa-Bianchi & Lugosi, 2009, György et al., 2007). Finally, we show how to leverage the guarantees of two algorithms in the contextual combinatorial semi-bandit setting: Exp2 (Bubeck et al., 2012) and the oracle-efficient Context-Semi-Bandit-FTPL (Syrgkanis et al., 2016), to provide multi-criteria no regret guarantees simultaneously for accuracy and fairness. Our results eliminate two potential sources of bias from prior work: the "hidden outcomes" that are not available to an algorithm operating in the full information setting, and human biases that might be present in any single human auditor, but can be mitigated by selecting a well chosen panel. 

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