arXiv subjects

Avrim Blum

Publications and source records attributed to Avrim Blum.

At least 37 records · Page 2Linked to original sources

arXiv2024

On the Vulnerability of Fairness Constrained Learning to Malicious Noise

We consider the vulnerability of fairness-constrained learning to small amounts of malicious noise in the training data. Konstantinov and Lampert (2021) initiated the study of this question and presented negative results showing there exist data distributions where for several fairness constraints, any proper learner will exhibit high vulnerability when group sizes are imbalanced. Here, we present a more optimistic view, showing that if we allow randomized classifiers, then the landscape is much more nuanced. For example, for Demographic Parity we show we can incur only a $Θ(α)$ loss in accuracy, where $α$ is the malicious noise rate, matching the best possible even without fairness constraints. For Equal Opportunity, we show we can incur an $O(\sqrtα)$ loss, and give a matching $Ω(\sqrtα)$lower bound. In contrast, Konstantinov and Lampert (2021) showed for proper learners the loss in accuracy for both notions is $Ω(1)$. The key technical novelty of our work is how randomization can bypass simple "tricks" an adversary can use to amplify his power. We also consider additional fairness notions including Equalized Odds and Calibration. For these fairness notions, the excess accuracy clusters into three natural regimes $O(α)$,$O(\sqrtα)$ and $O(1)$. These results provide a more fine-grained view of the sensitivity of fairness-constrained learning to adversarial noise in training data.

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arXiv2024

A Model for Combinatorial Dictionary Learning and Inference

We are often interested in decomposing complex, structured data into simple components that explain the data. The linear version of this problem is well-studied as dictionary learning and factor analysis. In this work, we propose a combinatorial model in which to study this question, motivated by the way objects occlude each other in a scene to form an image. First, we identify a property we call "well-structuredness" of a set of low-dimensional components which ensures that no two components in the set are too similar. We show how well-structuredness is sufficient for learning the set of latent components comprising a set of sample instances. We then consider the problem: given a set of components and an instance generated from some unknown subset of them, identify which parts of the instance arise from which components. We consider two variants: (1) determine the minimal number of components required to explain the instance; (2) determine the correct explanation for as many locations as possible. For the latter goal, we also devise a version that is robust to adversarial corruptions, with just a slightly stronger assumption on the components. Finally, we show that the learning problem is computationally infeasible in the absence of any assumptions.

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arXiv2024

Fundamental Bounds on Online Strategic Classification

We study the problem of online binary classification where strategic agents can manipulate their observable features in predefined ways, modeled by a manipulation graph, in order to receive a positive classification. We show this setting differs in fundamental ways from non-strategic online classification. For instance, whereas in the non-strategic case, a mistake bound of $\ln|H|$ is achievable via the halving algorithm when the target function belongs to a known class $H$, we show that no deterministic algorithm can achieve a mistake bound $o(Δ)$ in the strategic setting, where $Δ$ is the maximum degree of the manipulation graph (even when $|H|=O(Δ)$). We obtain an algorithm achieving mistake bound $O(Δ\ln|H|)$. We also extend this to the agnostic setting and obtain an algorithm with a $Δ$ multiplicative regret, and we show no deterministic algorithm can achieve $o(Δ)$ multiplicative regret. Next, we study two randomized models based on whether the random choices are made before or after agents respond, and show they exhibit fundamental differences. In the first model, at each round the learner deterministically chooses a probability distribution over classifiers inducing expected values on each vertex (probabilities of being classified as positive), which the strategic agents respond to. We show that any learner in this model has to suffer linear regret. On the other hand, in the second model, while the adversary who selects the next agent must respond to the learner's probability distribution over classifiers, the agent then responds to the actual hypothesis classifier drawn from this distribution. Surprisingly, we show this model is more advantageous to the learner, and we design randomized algorithms that achieve sublinear regret bounds against both oblivious and adaptive adversaries.

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arXiv2024

Competitive strategies to use "warm start" algorithms with predictions

We consider the problem of learning and using predictions for warm start algorithms with predictions. In this setting, an algorithm is given an instance of a problem, and a prediction of the solution. The runtime of the algorithm is bounded by the distance from the predicted solution to the true solution of the instance. Previous work has shown that when instances are drawn iid from some distribution, it is possible to learn an approximately optimal fixed prediction (Dinitz et al, NeurIPS 2021), and in the adversarial online case, it is possible to compete with the best fixed prediction in hindsight (Khodak et al, NeurIPS 2022). In this work we give competitive guarantees against stronger benchmarks that consider a set of $k$ predictions $\mathbf{P}$. That is, the "optimal offline cost" to solve an instance with respect to $\mathbf{P}$ is the distance from the true solution to the closest member of $\mathbf{P}$. This is analogous to the $k$-medians objective function. In the distributional setting, we show a simple strategy that incurs cost that is at most an $O(k)$ factor worse than the optimal offline cost. We then show a way to leverage learnable coarse information, in the form of partitions of the instance space into groups of "similar" instances, that allows us to potentially avoid this $O(k)$ factor. Finally, we consider an online version of the problem, where we compete against offline strategies that are allowed to maintain a moving set of $k$ predictions or "trajectories," and are charged for how much the predictions move. We give an algorithm that does at most $O(k^4 \ln^2 k)$ times as much work as any offline strategy of $k$ trajectories. This algorithm is deterministic (robust to an adaptive adversary), and oblivious to the setting of $k$. Thus the guarantee holds for all $k$ simultaneously.

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arXiv2024

Nearly-tight Approximation Guarantees for the Improving Multi-Armed Bandits Problem

We give nearly-tight upper and lower bounds for the improving multi-armed bandits problem. An instance of this problem has $k$ arms, each of whose reward function is a concave and increasing function of the number of times that arm has been pulled so far. We show that for any randomized online algorithm, there exists an instance on which it must suffer at least an $Ω(\sqrt{k})$ approximation factor relative to the optimal reward. We then provide a randomized online algorithm that guarantees an $O(\sqrt{k})$ approximation factor, if it is told the maximum reward achievable by the optimal arm in advance. We then show how to remove this assumption at the cost of an extra $O(\log k)$ approximation factor, achieving an overall $O(\sqrt{k} \log k)$ approximation relative to optimal.

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arXiv2024

Winning Without Observing Payoffs: Exploiting Behavioral Biases to Win Nearly Every Round

Gameplay under various forms of uncertainty has been widely studied. Feldman et al. (2010) studied a particularly low-information setting in which one observes the opponent's actions but no payoffs, not even one's own, and introduced an algorithm which guarantees one's payoff nonetheless approaches the minimax optimal value (i.e., zero) in a symmetric zero-sum game. Against an opponent playing a minimax-optimal strategy, approaching the value of the game is the best one can hope to guarantee. However, a wealth of research in behavioral economics shows that people often do not make perfectly rational, optimal decisions. Here we consider whether it is possible to actually win in this setting if the opponent is behaviorally biased. We model several deterministic, biased opponents and show that even without knowing the game matrix in advance or observing any payoffs, it is possible to take advantage of each bias in order to win nearly every round (so long as the game has the property that each action beats and is beaten by at least one other action). We also provide a partial characterization of the kinds of biased strategies that can be exploited to win nearly every round, and provide algorithms for beating some kinds of biased strategies even when we don't know which strategy the opponent uses.

cs.GT↗

arXiv2024

Agnostic Multi-Robust Learning Using ERM

A fundamental problem in robust learning is asymmetry: a learner needs to correctly classify every one of exponentially-many perturbations that an adversary might make to a test-time natural example. In contrast, the attacker only needs to find one successful perturbation. Xiang et al.[2022] proposed an algorithm that in the context of patch attacks for image classification, reduces the effective number of perturbations from an exponential to a polynomial number of perturbations and learns using an ERM oracle. However, to achieve its guarantee, their algorithm requires the natural examples to be robustly realizable. This prompts the natural question; can we extend their approach to the non-robustly-realizable case where there is no classifier with zero robust error? Our first contribution is to answer this question affirmatively by reducing this problem to a setting in which an algorithm proposed by Feige et al.[2015] can be applied, and in the process extend their guarantees. Next, we extend our results to a multi-group setting and introduce a novel agnostic multi-robust learning problem where the goal is to learn a predictor that achieves low robust loss on a (potentially) rich collection of subgroups.

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arXiv2024

Strategic Classification under Unknown Personalized Manipulation

We study the fundamental mistake bound and sample complexity in the strategic classification, where agents can strategically manipulate their feature vector up to an extent in order to be predicted as positive. For example, given a classifier determining college admission, student candidates may try to take easier classes to improve their GPA, retake SAT and change schools in an effort to fool the classifier. Ball manipulations are a widely studied class of manipulations in the literature, where agents can modify their feature vector within a bounded radius ball. Unlike most prior work, our work considers manipulations to be personalized, meaning that agents can have different levels of manipulation abilities (e.g., varying radii for ball manipulations), and unknown to the learner. We formalize the learning problem in an interaction model where the learner first deploys a classifier and the agent manipulates the feature vector within their manipulation set to game the deployed classifier. We investigate various scenarios in terms of the information available to the learner during the interaction, such as observing the original feature vector before or after deployment, observing the manipulated feature vector, or not seeing either the original or the manipulated feature vector. We begin by providing online mistake bounds and PAC sample complexity in these scenarios for ball manipulations. We also explore non-ball manipulations and show that, even in the simplest scenario where both the original and the manipulated feature vectors are revealed, the mistake bounds and sample complexity are lower bounded by $Ω(|H|)$ when the target function belongs to a known class $H$.

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arXiv2023

Dueling Optimization with a Monotone Adversary

We introduce and study the problem of dueling optimization with a monotone adversary, which is a generalization of (noiseless) dueling convex optimization. The goal is to design an online algorithm to find a minimizer $\mathbf{x}^{*}$ for a function $f\colon X \to \mathbb{R}$, where $X \subseteq \mathbb{R}^d$. In each round, the algorithm submits a pair of guesses, i.e., $\mathbf{x}^{(1)}$ and $\mathbf{x}^{(2)}$, and the adversary responds with any point in the space that is at least as good as both guesses. The cost of each query is the suboptimality of the worse of the two guesses; i.e., ${\max} \left( f(\mathbf{x}^{(1)}), f(\mathbf{x}^{(2)}) \right) - f(\mathbf{x}^{*})$. The goal is to minimize the number of iterations required to find an $\varepsilon$-optimal point and to minimize the total cost (regret) of the guesses over many rounds. Our main result is an efficient randomized algorithm for several natural choices of the function $f$ and set $X$ that incurs cost $O(d)$ and iteration complexity $O(d\log(1/\varepsilon)^2)$. Moreover, our dependence on $d$ is asymptotically optimal, as we show examples in which any randomized algorithm for this problem must incur $Ω(d)$ cost and iteration complexity.

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arXiv2023

Eliciting User Preferences for Personalized Multi-Objective Decision Making through Comparative Feedback

In classic reinforcement learning (RL) and decision making problems, policies are evaluated with respect to a scalar reward function, and all optimal policies are the same with regards to their expected return. However, many real-world problems involve balancing multiple, sometimes conflicting, objectives whose relative priority will vary according to the preferences of each user. Consequently, a policy that is optimal for one user might be sub-optimal for another. In this work, we propose a multi-objective decision making framework that accommodates different user preferences over objectives, where preferences are learned via policy comparisons. Our model consists of a Markov decision process with a vector-valued reward function, with each user having an unknown preference vector that expresses the relative importance of each objective. The goal is to efficiently compute a near-optimal policy for a given user. We consider two user feedback models. We first address the case where a user is provided with two policies and returns their preferred policy as feedback. We then move to a different user feedback model, where a user is instead provided with two small weighted sets of representative trajectories and selects the preferred one. In both cases, we suggest an algorithm that finds a nearly optimal policy for the user using a small number of comparison queries.

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arXiv2023

An Analysis of Robustness of Non-Lipschitz Networks

Despite significant advances, deep networks remain highly susceptible to adversarial attack. One fundamental challenge is that small input perturbations can often produce large movements in the network's final-layer feature space. In this paper, we define an attack model that abstracts this challenge, to help understand its intrinsic properties. In our model, the adversary may move data an arbitrary distance in feature space but only in random low-dimensional subspaces. We prove such adversaries can be quite powerful: defeating any algorithm that must classify any input it is given. However, by allowing the algorithm to abstain on unusual inputs, we show such adversaries can be overcome when classes are reasonably well-separated in feature space. We further provide strong theoretical guarantees for setting algorithm parameters to optimize over accuracy-abstention trade-offs using data-driven methods. Our results provide new robustness guarantees for nearest-neighbor style algorithms, and also have application to contrastive learning, where we empirically demonstrate the ability of such algorithms to obtain high robust accuracy with low abstention rates. Our model is also motivated by strategic classification, where entities being classified aim to manipulate their observable features to produce a preferred classification, and we provide new insights into that area as well.

cs.LG↗

arXiv2022

A Theory of PAC Learnability under Transformation Invariances

Transformation invariances are present in many real-world problems. For example, image classification is usually invariant to rotation and color transformation: a rotated car in a different color is still identified as a car. Data augmentation, which adds the transformed data into the training set and trains a model on the augmented data, is one commonly used technique to build these invariances into the learning process. However, it is unclear how data augmentation performs theoretically and what the optimal algorithm is in presence of transformation invariances. In this paper, we study PAC learnability under transformation invariances in three settings according to different levels of realizability: (i) A hypothesis fits the augmented data; (ii) A hypothesis fits only the original data and the transformed data lying in the support of the data distribution; (iii) Agnostic case. One interesting observation is that distinguishing between the original data and the transformed data is necessary to achieve optimal accuracy in setting (ii) and (iii), which implies that any algorithm not differentiating between the original and transformed data (including data augmentation) is not optimal. Furthermore, this type of algorithms can even "harm" the accuracy. In setting (i), although it is unnecessary to distinguish between the two data sets, data augmentation still does not perform optimally. Due to such a difference, we propose two combinatorial measures characterizing the optimal sample complexity in setting (i) and (ii)(iii) and provide the optimal algorithms.

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arXiv2022

Multi Stage Screening: Enforcing Fairness and Maximizing Efficiency in a Pre-Existing Pipeline

Consider an actor making selection decisions using a series of classifiers, which we term a sequential screening process. The early stages filter out some applicants, and in the final stage an expensive but accurate test is applied to the individuals that make it to the final stage. Since the final stage is expensive, if there are multiple groups with different fractions of positives at the penultimate stage (even if a slight gap), then the firm may naturally only choose to the apply the final (interview) stage solely to the highest precision group which would be clearly unfair to the other groups. Even if the firm is required to interview all of those who pass the final round, the tests themselves could have the property that qualified individuals from some groups pass more easily than qualified individuals from others. Thus, we consider requiring Equality of Opportunity (qualified individuals from each each group have the same chance of reaching the final stage and being interviewed). We then examine the goal of maximizing quantities of interest to the decision maker subject to this constraint, via modification of the probabilities of promotion through the screening process at each stage based on performance at the previous stage. We exhibit algorithms for satisfying Equal Opportunity over the selection process and maximizing precision (the fraction of interview that yield qualified candidates) as well as linear combinations of precision and recall (recall determines the number of applicants needed per hire) at the end of the final stage. We also present examples showing that the solution space is non-convex, which motivate our exact and (FPTAS) approximation algorithms for maximizing the linear combination of precision and recall. Finally, we discuss the `price of' adding additional restrictions, such as not allowing the decision maker to use group membership in its decision process.

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arXiv2022

Robustly-reliable learners under poisoning attacks

Data poisoning attacks, in which an adversary corrupts a training set with the goal of inducing specific desired mistakes, have raised substantial concern: even just the possibility of such an attack can make a user no longer trust the results of a learning system. In this work, we show how to achieve strong robustness guarantees in the face of such attacks across multiple axes. We provide robustly-reliable predictions, in which the predicted label is guaranteed to be correct so long as the adversary has not exceeded a given corruption budget, even in the presence of instance targeted attacks, where the adversary knows the test example in advance and aims to cause a specific failure on that example. Our guarantees are substantially stronger than those in prior approaches, which were only able to provide certificates that the prediction of the learning algorithm does not change, as opposed to certifying that the prediction is correct, as we are able to achieve in our work. Remarkably, we provide a complete characterization of learnability in this setting, in particular, nearly-tight matching upper and lower bounds on the region that can be certified, as well as efficient algorithms for computing this region given an ERM oracle. Moreover, for the case of linear separators over logconcave distributions, we provide efficient truly polynomial time algorithms (i.e., non-oracle algorithms) for such robustly-reliable predictions. We also extend these results to the active setting where the algorithm adaptively asks for labels of specific informative examples, and the difficulty is that the adversary might even be adaptive to this interaction, as well as to the agnostic learning setting where there is no perfect classifier even over the uncorrupted data.

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arXiv2022

On classification of strategic agents who can both game and improve

In this work, we consider classification of agents who can both game and improve. For example, people wishing to get a loan may be able to take some actions that increase their perceived credit-worthiness and others that also increase their true credit-worthiness. A decision-maker would like to define a classification rule with few false-positives (does not give out many bad loans) while yielding many true positives (giving out many good loans), which includes encouraging agents to improve to become true positives if possible. We consider two models for this problem, a general discrete model and a linear model, and prove algorithmic, learning, and hardness results for each. For the general discrete model, we give an efficient algorithm for the problem of maximizing the number of true positives subject to no false positives, and show how to extend this to a partial-information learning setting. We also show hardness for the problem of maximizing the number of true positives subject to a nonzero bound on the number of false positives, and that this hardness holds even for a finite-point version of our linear model. We also show that maximizing the number of true positives subject to no false positive is NP-hard in our full linear model. We additionally provide an algorithm that determines whether there exists a linear classifier that classifies all agents accurately and causes all improvable agents to become qualified, and give additional results for low-dimensional data.

cs.GT↗

arXiv2022

Setting Fair Incentives to Maximize Improvement

We consider the problem of helping agents improve by setting short-term goals. Given a set of target skill levels, we assume each agent will try to improve from their initial skill level to the closest target level within reach or do nothing if no target level is within reach. We consider two models: the common improvement capacity model, where agents have the same limit on how much they can improve, and the individualized improvement capacity model, where agents have individualized limits. Our goal is to optimize the target levels for social welfare and fairness objectives, where social welfare is defined as the total amount of improvement, and fairness objectives are considered where the agents belong to different underlying populations. A key technical challenge of this problem is the non-monotonicity of social welfare in the set of target levels, i.e., adding a new target level may decrease the total amount of improvement as it may get easier for some agents to improve. This is especially challenging when considering multiple groups because optimizing target levels in isolation for each group and outputting the union may result in arbitrarily low improvement for a group, failing the fairness objective. Considering these properties, we provide algorithms for optimal and near-optimal improvement for both social welfare and fairness objectives. These algorithmic results work for both the common and individualized improvement capacity models. Furthermore, we show a placement of target levels exists that is approximately optimal for the social welfare of each group. Unlike the algorithmic results, this structural statement only holds in the common improvement capacity model, and we show counterexamples in the individualized improvement capacity model. Finally, we extend our algorithms to learning settings where we have only sample access to the initial skill levels of agents.

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arXiv2022

Boosting Barely Robust Learners: A New Perspective on Adversarial Robustness

We present an oracle-efficient algorithm for boosting the adversarial robustness of barely robust learners. Barely robust learning algorithms learn predictors that are adversarially robust only on a small fraction $β\ll 1$ of the data distribution. Our proposed notion of barely robust learning requires robustness with respect to a "larger" perturbation set; which we show is necessary for strongly robust learning, and that weaker relaxations are not sufficient for strongly robust learning. Our results reveal a qualitative and quantitative equivalence between two seemingly unrelated problems: strongly robust learning and barely robust learning.

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arXiv2021

Stochastic Vertex Cover with Few Queries

We study the minimum vertex cover problem in the following stochastic setting. Let $G$ be an arbitrary given graph, $p \in (0, 1]$ a parameter of the problem, and let $G_p$ be a random subgraph that includes each edge of $G$ independently with probability $p$. We are unaware of the realization $G_p$, but can learn if an edge $e$ exists in $G_p$ by querying it. The goal is to find an approximate minimum vertex cover (MVC) of $G_p$ by querying few edges of $G$ non-adaptively. This stochastic setting has been studied extensively for various problems such as minimum spanning trees, matroids, shortest paths, and matchings. To our knowledge, however, no non-trivial bound was known for MVC prior to our work. In this work, we present a: * $(2+ε)$-approximation for general graphs which queries $O(\frac{1}{ε^3 p})$ edges per vertex, and a * $1.367$-approximation for bipartite graphs which queries $poly(1/p)$ edges per vertex. Additionally, we show that at the expense of a triple-exponential dependence on $p^{-1}$ in the number of queries, the approximation ratio can be improved down to $(1+ε)$ for bipartite graphs. Our techniques also lead to improved bounds for bipartite stochastic matching. We obtain a $0.731$-approximation with nearly-linear in $1/p$ per-vertex queries. This is the first result to break the prevalent $(2/3 \sim 0.66)$-approximation barrier in the $poly(1/p)$ query regime, improving algorithms of [Behnezhad et al; SODA'19] and [Assadi and Bernstein; SOSA'19].

cs.DS↗