Previous work in fair machine learning has characterised the Fair Bayes Optimal Classifier (BOC) on a given distribution for both deterministic and randomized classifiers. We study the robustness of the Fair BOC to adversarial noise in the data distribution. Kearns & Li (1988) implies that the accuracy of the deterministic BOC without any fairness constraints is robust (Lipschitz) to malicious noise in the data distribution. We demonstrate that their robustness guarantee breaks down when we add fairness constraints. Hence, we consider the randomized Fair BOC, and our central result is that its accuracy is robust to malicious noise in the data distribution. Our robustness result applies to various fairness constraints---Demographic Parity, Equal Opportunity, Predictive Equality. Beyond robustness, we demonstrate that randomization leads to better accuracy and efficiency. We show that the randomized Fair BOC is nearly-deterministic, and gives randomized predictions on at most one data point, hence availing numerous benefits of randomness, while using very little of it.

To fix the `bias in, bias out' problem in fair machine learning, it is important to steer feature distributions of data or internal representations of Large Language Models (LLMs) to \emph{ideal} ones that guarantee group-fair outcomes. Previous work on fair generative models and representation steering could greatly benefit from provable fairness guarantees on the model output. We define a distribution as \emph{ideal} if the minimizer of any cost-sensitive risk on it is guaranteed to have exact group-fair outcomes (e.g., demographic parity, equal opportunity)---in other words, it has no fairness-utility trade-off. We formulate an optimization program for optimal steering by finding the nearest \emph{ideal} distribution in KL-divergence, and provide efficient algorithms for it when the underlying distributions come from well-known parametric families (e.g., normal, log-normal). Empirically, our optimal steering techniques on both synthetic and real-world datasets improve fairness without diminishing utility (and sometimes even improve utility). We demonstrate affine steering of LLM representations to reduce bias in multi-class classification, e.g., occupation prediction from a short biography in Bios dataset (De-Arteaga et al.). Furthermore, we steer internal representations of LLMs towards desired outputs so that it works equally well across different groups.

For machine learning models to be reliable and trustworthy, their decisions must be interpretable. As these models find increasing use in safety-critical applications, it is important that not just the model predictions but also their explanations (as feature attributions) be robust to small human-imperceptible input perturbations. Recent works have shown that many attribution methods are fragile and have proposed improvements in either these methods or the model training. We observe two main causes for fragile attributions: first, the existing metrics of robustness (e.g., top-k intersection) overpenalize even reasonable local shifts in attribution, thereby making random perturbations to appear as a strong attack, and second, the attribution can be concentrated in a small region even when there are multiple important parts in an image. To rectify this, we propose simple ways to strengthen existing metrics and attribution methods that incorporate locality of pixels in robustness metrics and diversity of pixel locations in attributions. Towards the role of model training in attributional robustness, we empirically observe that adversarially trained models have more robust attributions on smaller datasets, however, this advantage disappears in larger datasets. Code is made available at https://github.com/ksandeshk/LENS.

A general belief in fair classification is that fairness constraints incur a trade-off with accuracy, which biased data may worsen. Contrary to this belief, Blum & Stangl (2019) show that fair classification with equal opportunity constraints even on extremely biased data can recover optimally accurate and fair classifiers on the original data distribution. Their result is interesting because it demonstrates that fairness constraints can implicitly rectify data bias and simultaneously overcome a perceived fairness-accuracy trade-off. Their data bias model simulates under-representation and label bias in underprivileged population, and they show the above result on a stylized data distribution with i.i.d. label noise, under simple conditions on the data distribution and bias parameters. We propose a general approach to extend the result of Blum & Stangl (2019) to different fairness constraints, data bias models, data distributions, and hypothesis classes. We strengthen their result, and extend it to the case when their stylized distribution has labels with Massart noise instead of i.i.d. noise. We prove a similar recovery result for arbitrary data distributions using fair reject option classifiers. We further generalize it to arbitrary data distributions and arbitrary hypothesis classes, i.e., we prove that for any data distribution, if the optimally accurate classifier in a given hypothesis class is fair and robust, then it can be recovered through fair classification with equal opportunity constraints on the biased distribution whenever the bias parameters satisfy certain simple conditions. Finally, we show applications of our technique to time-varying data bias in classification and fair machine learning pipelines.

In supervised learning, it is known that overparameterized neural networks with one hidden layer provably and efficiently learn and generalize, when trained using stochastic gradient descent with a sufficiently small learning rate and suitable initialization. In contrast, the benefit of overparameterization in unsupervised learning is not well understood. Normalizing flows (NFs) constitute an important class of models in unsupervised learning for sampling and density estimation. In this paper, we theoretically and empirically analyze these models when the underlying neural network is a one-hidden-layer overparametrized network. Our main contributions are two-fold: (1) On the one hand, we provide theoretical and empirical evidence that for constrained NFs (this class of NFs underlies most NF constructions) with the one-hidden-layer network, overparametrization hurts training. (2) On the other hand, we prove that unconstrained NFs, a recently introduced model, can efficiently learn any reasonable data distribution under minimal assumptions when the underlying network is overparametrized and has one hidden-layer.

In optimization or machine learning problems we are given a set of items, usually points in some metric space, and the goal is to minimize or maximize an objective function over some space of candidate solutions. For example, in clustering problems, the input is a set of points in some metric space, and a common goal is to compute a set of centers in some other space (points, lines) that will minimize the sum of distances to these points. In database queries, we may need to compute such a some for a specific query set of $k$ centers. However, traditional algorithms cannot handle modern systems that require parallel real-time computations of infinite distributed streams from sensors such as GPS, audio or video that arrive to a cloud, or networks of weaker devices such as smartphones or robots. Core-set is a "small data" summarization of the input "big data", where every possible query has approximately the same answer on both data sets. Generic techniques enable efficient coreset \changed{maintenance} of streaming, distributed and dynamic data. Traditional algorithms can then be applied on these coresets to maintain the approximated optimal solutions. The challenge is to design coresets with provable tradeoff between their size and approximation error. This survey summarizes such constructions in a retrospective way, that aims to unified and simplify the state-of-the-art.

Group-fairness in classification aims for equality of a predictive utility across different sensitive sub-populations, e.g., race or gender. Equality or near-equality constraints in group-fairness often worsen not only the aggregate utility but also the utility for the least advantaged sub-population. In this paper, we apply the principles of Pareto-efficiency and least-difference to the utility being accuracy, as an illustrative example, and arrive at the Rawls classifier that minimizes the error rate on the worst-off sensitive sub-population. Our mathematical characterization shows that the Rawls classifier uniformly applies a threshold to an ideal score of features, in the spirit of fair equality of opportunity. In practice, such a score or a feature representation is often computed by a black-box model that has been useful but unfair. Our second contribution is practical Rawlsian fair adaptation of any given black-box deep learning model, without changing the score or feature representation it computes. Given any score function or feature representation and only its second-order statistics on the sensitive sub-populations, we seek a threshold classifier on the given score or a linear threshold classifier on the given feature representation that achieves the Rawls error rate restricted to this hypothesis class. Our technical contribution is to formulate the above problems using ambiguous chance constraints, and to provide efficient algorithms for Rawlsian fair adaptation, along with provable upper bounds on the Rawls error rate. Our empirical results show significant improvement over state-of-the-art group-fair algorithms, even without retraining for fairness.

(Non-)robustness of neural networks to small, adversarial pixel-wise perturbations, and as more recently shown, to even random spatial transformations (e.g., translations, rotations) entreats both theoretical and empirical understanding. Spatial robustness to random translations and rotations is commonly attained via equivariant models (e.g., StdCNNs, GCNNs) and training augmentation, whereas adversarial robustness is typically achieved by adversarial training. In this paper, we prove a quantitative trade-off between spatial and adversarial robustness in a simple statistical setting. We complement this empirically by showing that: (a) as the spatial robustness of equivariant models improves by training augmentation with progressively larger transformations, their adversarial robustness worsens progressively, and (b) as the state-of-the-art robust models are adversarially trained with progressively larger pixel-wise perturbations, their spatial robustness drops progressively. Towards achieving Pareto-optimality in this trade-off, we propose a method based on curriculum learning that trains gradually on more difficult perturbations (both spatial and adversarial) to improve spatial and adversarial robustness simultaneously.

A good seeding or initialization of cluster centers for the $k$-means method is important from both theoretical and practical standpoints. The $k$-means objective is inherently non-robust and sensitive to outliers. A popular seeding such as the $k$-means++ [3] that is more likely to pick outliers in the worst case may compound this drawback, thereby affecting the quality of clustering on noisy data.For any $0 < \delta \leq 1$, we show that using a mixture of $D^{2}$ [3] and uniform sampling, we can pick $O(k/\delta)$ candidate centers with the following guarantee: they contain some $k$ centers that give $O(1)$-approximation to the optimal robust $k$-means solution while discarding at most $\delta n$ more points than the outliers discarded by the optimal solution. That is, if the optimal solution discards its farthest $\beta n$ points as outliers, our solution discards its $(\beta + \delta) n$ points as outliers. The constant factor in our $O(1)$-approximation does not depend on $\delta$. This is an improvement over previous results for $k$-means with outliers based on LP relaxation and rounding [7] and local search [17]. The $O(k/\delta)$ sized subset can be found in time $O(ndk)$. Our \emph{robust} $k$-means++ is also easily amenable to scalable, faster, parallel implementations of $k$-means++ [5]. Our empirical results show a comparison of the above \emph{robust} variant of $k$-means++ with the usual $k$-means++, uniform random seeding, threshold $k$-means++ [6] and local search on real world and synthetic data.

Sampling methods that choose a subset of the data proportional to its diversity in the feature space are popular for data summarization. However, recent studies have noted the occurrence of bias {–} e.g., under or over representation of a particular gender or ethnicity {–} in such data summarization methods. In this paper we initiate a study of the problem of outputting a diverse and fair summary of a given dataset. We work with a well-studied determinantal measure of diversity and corresponding distributions (DPPs) and present a framework that allows us to incorporate a general class of fairness constraints into such distributions. Designing efficient algorithms to sample from these constrained determinantal distributions, however, suffers from a complexity barrier; we present a fast sampler that is provably good when the input vectors satisfy a natural property. Our empirical results on both real-world and synthetic datasets show that the diversity of the samples produced by adding fairness constraints is not too far from the unconstrained case.

Determinantal Point Processes (DPPs) are probabilistic models that arise in quantum physics and random matrix theory and have recently found numerous applications in theoretical computer science and machine learning. DPPs define probability distributions over subsets of a given ground set, they exhibit interesting properties such as negative correlation, and, unlike other models of negative correlation such as Markov random fields, have efficient algorithms for sampling. When applied to kernel methods in machine learning, DPPs favor subsets of the given data with more diverse features. However, many real-world applications require efficient algorithms to sample from DPPs with additional constraints on the sampled subset, e.g., partition or matroid constraints that are important from the viewpoint of ensuring priors, resource or fairness constraints on the sampled subset. Whether one can efficiently sample from DPPs in such constrained settings is an important problem that was first raised in a survey of DPPs for machine learning by Kulesza and Taskar and studied in some recent works. The main contribution of this paper is the first resolution of the complexity of sampling from DPPs with constraints. On the one hand, we give exact efficient algorithms for sampling from constrained DPPs when the description of the constraints is in unary; this includes special cases of practical importance such as a small number of partition, knapsack or budget constraints. On the other hand, we prove that when the constraints are specified in binary, this problem is #P-hard via a reduction from the problem of computing mixed discriminants; implying that it may be unlikely that there is an FPRAS. Technically, our algorithmic result benefits from viewing the constrained sampling problem via the lens of polynomials and we obtain our complexity results by providing an equivalence between computing mixed discriminants and sampling from partition constrained DPPs. As a consequence, we obtain a few corollaries of independent interest: 1) An algorithm to count, sample (and, hence, optimize) over the base polytope of regular matroids when there are additional (succinct) budget constraints and, 2) An algorithm to evaluate and compute mixed characteristic polynomials, that played a central role in the resolution of the Kadison-Singer problem, for certain special cases.

Goemans showed that any n points x_1,..., x_n in d-dimensions satisfying l_2^2 triangle inequalities can be embedded into l_{1}, with worst-case distortion at most sqrt{d}. We consider an extension of this theorem to the case when the points are approximately low-dimensional as opposed to exactly low-dimensional, and prove the following analogous theorem, albeit with average distortion guarantees: There exists an l_{2}^{2}-to-l_{1} embedding with average distortion at most the stable rank, sr(M), of the matrix M consisting of columns {x_i-x_j}_{i

Gaussian Process bandit optimization has emerged as a powerful tool for optimizing noisy black box functions. One example in machine learning is hyper-parameter optimization where each evaluation of the target function may require training a model which may involve days or even weeks of computation. Most methods for this so-called “Bayesian optimization” only allow sequential exploration of the parameter space. However, it is often desirable to propose batches or sets of parameter values to explore simultaneously, especially when there are large parallel processing facilities at our disposal. Batch methods require modeling the interaction between the different evaluations in the batch, which can be expensive in complex scenarios. In this paper, we propose a new approach for parallelizing Bayesian optimization by modeling the diversity of a batch via Determinantal point processes (DPPs) whose kernels are learned automatically. This allows us to generalize a previous result as well as prove better regret bounds based on DPP sampling. Our experiments on a variety of synthetic and real-world robotics and hyper-parameter optimization tasks indicate that our DPP-based methods, especially those based on DPP sampling, outperform state-of-the-art methods.

Submodular function maximization is one of the key problems that arise in many machine learning tasks. Greedy selection algorithms are the proven choice to solve such problems, where prior theoretical work guarantees (1 - 1/e) approximation ratio. However, it has been empirically observed that greedy selection provides almost optimal solutions in practice. The main goal of this paper is to explore and answer why the greedy selection does significantly better than the theoretical guarantee of (1 - 1/e). Applications include, but are not limited to, sensor selection tasks which use both entropy and mutual information as a maximization criteria. We give a theoretical justification for the nearly optimal approximation ratio via detailed analysis of the curvature of these objective functions for Gaussian RBF kernels.

Guruswami and Sinop give a $O(1/δ)$ approximation guarantee for the non-uniform Sparsest Cut problem by solving $O(r)$-level Lasserre semidefinite constraints, provided that the generalized eigenvalues of the Laplacians of the cost and demand graphs satisfy a certain spectral condition, namely, $λ_{r+1} \geq Φ^{*}/(1-δ)$. Their key idea is a rounding technique that first maps a vector-valued solution to $[0, 1]$ using appropriately scaled projections onto Lasserre vectors. In this paper, we show that similar projections and analysis can be obtained using only $\ell_{2}^{2}$ triangle inequality constraints. This results in a $O(r/δ^{2})$ approximation guarantee for the non-uniform Sparsest Cut problem by adding only $\ell_{2}^{2}$ triangle inequality constraints to the usual semidefinite program, provided that the same spectral condition, $λ_{r+1} \geq Φ^{*}/(1-δ)$, holds.

The subspace approximation problem Subspace(k, p) asks for a k dimensional linear subspace that fits a given set of m points in ℝn optimally. The error for fitting is a generalization of the least squares fit and uses the ℓp norm of the distances (ℓ2 distances) of the points from the subspace, e.g., p = ∞ means minimizing the ℓ2 distance of the farthest point from the subspace. Previous work on subspace approximation considers either the case of small or constant k and p [27, 11, 14] or the case of p = ∞ [16, 8, 17, 7, 24, 23, 29]. In this paper, we study the algorithms and hardness for Subspace(k, p) in the natural range 1 ≤ k ≤ n and 2 ≤ p ≤ ∞. Our results are as follows. Extending the convex relaxation and rounding techniques of Varadarajan, Venkatesh, Ye and Zhang [29], we give a polynomial time approximation algorithm for Subspace(k, p), for any k and any p ≥ 2, with an approximation guarantee of roughly , where is the pth moment of a standard normal variable. This improves to γp for k = n − 1. We exhibit a simple integrality gap (or “rank gap”) instance for our convex relaxation giving a gap of γp(1 − ε), for any constant ε > 0. We show that, assuming the Unique Games Conjecture, the subspace approximation problem is hard to approximate within a factor better than γp(1 − ε), for any constant ε > 0. Our hardness reduction involves a dictatorship test which is somewhat different from “long code” based tests used in reductions from Unique Games, and seems better suited for problems of a continuous nature.

We give efficient algorithms for volume sampling, i.e., for picking k-subsets of the rows of any given matrix with probabilities proportional to the squared volumes of the simplices defined by them and the origin (or the squared volumes of the parallelepipeds defined by these subsets of rows). This solves an open problem from the monograph on spectral algorithms by Kannan and Vempala (see Section 7.4 of [15], also implicit in [1], [5]). Our first algorithm for volume sampling k-subsets of rows from an m-by-n matrix runs in O(kmnωlog n) arithmetic operations (where ω is the exponent of matrix multiplication) and a second variant of it for (1 + ϵ)-approximate volume sampling runs in O(mn log m · k2/ϵ2+m logωm · k2ω+1/ϵ2ω· log(kϵ-1log m)) arithmetic operations, which is almost linear in the size of the input (i.e., the number of entries) for small k. Our efficient volume sampling algorithms imply the following results for low-rank matrix approximation: 1) Given A ∈ Rm×n, in O(kmnωlog n) arithmetic operations we can find k of its rows such that projecting onto their span gives a √k + 1-approximation to the matrix of rank fc closest to A under the Frobenius norm. This improves the O(k√log k)-approximation of Boutsidis, Drineas and Mahoney [1] and matches the lower bound shown in [5]. The method of conditional expectations gives a deterministic algorithm with the same complexity. The running time can be improved to O(mn log m · k2/e2+ m logωm·k2ω+1ϵ2ω-log(kϵ-1log m)) at the cost of losing an extra (1 + ϵ) in the approximation factor. 2) The same rows and projection as in the previous point give a √(k + 1)(n -k)-approximation to the matrix of rank k closest to A under the spectral norm. In this paper, we show an almost matching lower bound of √n, even for k = 1.

We give a randomized bi-criteria algorithm for the problem of finding a k-dimensional subspace that minimizesthe Lp-error for given points, i.e., p-th root of the sum of p-th powers of distances to given points,for any p ≥ 1. Our algorithm runs in time Õ (mn · pk3 (k/ε)2p) andproduces a subset of size Õ (pk2 (k/ε)2p) from the given points such that, withhigh probability, the span of these points gives a (1+ε)-approximation to the optimal k-dimensionalsubspace. We also show a dimension reduction type of result for this problem where we can efficiently find asubset of size Õ (pk2(p+1) + (k/ε)p+2) such that, with high probability, theirspan contains a k-dimensional subspace that gives (1+ε)-approximation to the optimum. We prove similarresults for the corresponding projective clustering problem where we need to find multiple k-dimensional subspaces.

Spielman and Teng (JACM '04), proved that the smoothed complexity of a two-phase shadow-vertex method for linear programming is polynomial in the number of constraints n, the number of variables d, and the parameter of perturbation 1//spl sigma/. The key geometric result in their proof was an upper bound of O(nd/sup 3//min (/spl sigma/, (9d ln n)/sup 1/2 /)/sup 6/) on the expected size of the shadow of the polytope defined by the perturbed linear program. In this paper, we give a much simpler proof of a better bound: O(n/sup 2/ d ln n/min (/spl sigma/, (4d ln n)/sup 1/2 /)/sup 2/). When evaluated at /spl sigma/ = (9d ln n)/sup 1/2 /, this improves the size estimate from O(nd/sup 6/ ln/sup 3/ n) to O(n/sup 2/d/sup 2/ ln n). The improvement only becomes better as /spl sigma/ decreases. The bound on the running time of the two-phase shadow vertex proved by Spielman and Teng is dominated by the exponent of /spl sigma/ in the shadow-size bound. By reducing this exponent from 6 to 2, we decrease the exponent in the smoothed complexity of the two-phase shadow vertex method by a multiplicative factor of 3.

An error-correcting code is said to be locally decodable if a randomized algorithm can recover any single bit of a message by reading only a small number of symbols of a possibly corrupted encoding of the message. Katz and Trevisan (2000) showed that any such code C: {0, 1} /spl rarr/ /spl Sigma//sup m/ with a decoding algorithm that makes at most q probes must satisfy m = /spl Omega/((n/log |/spl Sigma/|)/sup q/(q-1)/). They assumed that the decoding algorithm is non-adaptive, and left open the question of proving similar bounds for adaptive decoders. We improve the results of Katz and Trevisan (2000) in two ways. First, we give a more direct proof of their result. Second, and this is our main result, we prove that m = /spl Omega/((n/log|/spl Sigma/|)/sup q/(q-1)/) even if the decoding algorithm is adaptive. An important ingredient of our proof is a randomized method for smoothing an adaptive decoding algorithm. The main technical tool we employ is the Second Moment Method.