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Learnability is a Compact Property
Feb 15, 2024
Recent work on learning has yielded a striking result: the learnability of various problems can be undecidable, or independent of the standard ZFC axioms of set theory. Furthermore, the learnability of such problems can fail to be a property of finite character: informally, it cannot be detected by examining finite projections of the problem. On the other hand, learning theory abounds with notions of dimension that characterize learning and consider only finite restrictions of the problem, i.e., are properties of finite character. How can these results be reconciled? More precisely, which classes of learning problems are vulnerable to logical undecidability, and which are within the grasp of finite characterizations? We demonstrate that the difficulty of supervised learning with metric losses admits a tight finite characterization. In particular, we prove that the sample complexity of learning a hypothesis class can be detected by examining its finite projections. For realizable and agnostic learning with respect to a wide class of proper loss functions, we demonstrate an exact compactness result: a class is learnable with a given sample complexity precisely when the same is true of all its finite projections. For realizable learning with improper loss functions, we show that exact compactness of sample complexity can fail, and provide matching upper and lower bounds of a factor of 2 on the extent to which such sample complexities can differ. We conjecture that larger gaps are possible for the agnostic case. At the heart of our technical work is a compactness result concerning assignments of variables that maintain a class of functions below a target value, which generalizes Hall's classic matching theorem and may be of independent interest.
Stability and Multigroup Fairness in Ranking with Uncertain Predictions
Feb 14, 2024Rankings are ubiquitous across many applications, from search engines to hiring committees. In practice, many rankings are derived from the output of predictors. However, when predictors trained for classification tasks have intrinsic uncertainty, it is not obvious how this uncertainty should be represented in the derived rankings. Our work considers ranking functions: maps from individual predictions for a classification task to distributions over rankings. We focus on two aspects of ranking functions: stability to perturbations in predictions and fairness towards both individuals and subgroups. Not only is stability an important requirement for its own sake, but -- as we show -- it composes harmoniously with individual fairness in the sense of Dwork et al. (2012). While deterministic ranking functions cannot be stable aside from trivial scenarios, we show that the recently proposed uncertainty aware (UA) ranking functions of Singh et al. (2021) are stable. Our main result is that UA rankings also achieve multigroup fairness through successful composition with multiaccurate or multicalibrated predictors. Our work demonstrates that UA rankings naturally interpolate between group and individual level fairness guarantees, while simultaneously satisfying stability guarantees important whenever machine-learned predictions are used.
Regularization and Optimal Multiclass Learning
Sep 24, 2023The quintessential learning algorithm of empirical risk minimization (ERM) is known to fail in various settings for which uniform convergence does not characterize learning. It is therefore unsurprising that the practice of machine learning is rife with considerably richer algorithmic techniques for successfully controlling model capacity. Nevertheless, no such technique or principle has broken away from the pack to characterize optimal learning in these more general settings. The purpose of this work is to characterize the role of regularization in perhaps the simplest setting for which ERM fails: multiclass learning with arbitrary label sets. Using one-inclusion graphs (OIGs), we exhibit optimal learning algorithms that dovetail with tried-and-true algorithmic principles: Occam's Razor as embodied by structural risk minimization (SRM), the principle of maximum entropy, and Bayesian reasoning. Most notably, we introduce an optimal learner which relaxes structural risk minimization on two dimensions: it allows the regularization function to be "local" to datapoints, and uses an unsupervised learning stage to learn this regularizer at the outset. We justify these relaxations by showing that they are necessary: removing either dimension fails to yield a near-optimal learner. We also extract from OIGs a combinatorial sequence we term the Hall complexity, which is the first to characterize a problem's transductive error rate exactly. Lastly, we introduce a generalization of OIGs and the transductive learning setting to the agnostic case, where we show that optimal orientations of Hamming graphs -- judged using nodes' outdegrees minus a system of node-dependent credits -- characterize optimal learners exactly. We demonstrate that an agnostic version of the Hall complexity again characterizes error rates exactly, and exhibit an optimal learner using maximum entropy programs.
Fairness in Matching under Uncertainty
Feb 08, 2023
The prevalence and importance of algorithmic two-sided marketplaces has drawn attention to the issue of fairness in such settings. Algorithmic decisions are used in assigning students to schools, users to advertisers, and applicants to job interviews. These decisions should heed the preferences of individuals, and simultaneously be fair with respect to their merits (synonymous with fit, future performance, or need). Merits conditioned on observable features are always uncertain, a fact that is exacerbated by the widespread use of machine learning algorithms to infer merit from the observables. As our key contribution, we carefully axiomatize a notion of individual fairness in the two-sided marketplace setting which respects the uncertainty in the merits; indeed, it simultaneously recognizes uncertainty as the primary potential cause of unfairness and an approach to address it. We design a linear programming framework to find fair utility-maximizing distributions over allocations, and we show that the linear program is robust to perturbations in the estimated parameters of the uncertain merit distributions, a key property in combining the approach with machine learning techniques.
Polynomial Time Reinforcement Learning in Correlated FMDPs with Linear Value Functions
Jul 12, 2021Many reinforcement learning (RL) environments in practice feature enormous state spaces that may be described compactly by a "factored" structure, that may be modeled by Factored Markov Decision Processes (FMDPs). We present the first polynomial-time algorithm for RL with FMDPs that does not rely on an oracle planner, and instead of requiring a linear transition model, only requires a linear value function with a suitable local basis with respect to the factorization. With this assumption, we can solve FMDPs in polynomial time by constructing an efficient separation oracle for convex optimization. Importantly, and in contrast to prior work, we do not assume that the transitions on various factors are independent.
Failout: Achieving Failure-Resilient Inference in Distributed Neural Networks
Feb 18, 2020
When a neural network is partitioned and distributed across physical nodes, failure of physical nodes causes the failure of the neural units that are placed on those nodes, which results in a significant performance drop. Current approaches focus on resiliency of training in distributed neural networks. However, resiliency of inference in distributed neural networks is less explored. We introduce ResiliNet, a scheme for making inference in distributed neural networks resilient to physical node failures. ResiliNet combines two concepts to provide resiliency: skip connection in residual neural networks, and a novel technique called failout, which is introduced in this paper. Failout simulates physical node failure conditions during training using dropout, and is specifically designed to improve the resiliency of distributed neural networks. The results of the experiments and ablation studies using three datasets confirm the ability of ResiliNet to provide inference resiliency for distributed neural networks.
Guardians of the Deep Fog: Failure-Resilient DNN Inference from Edge to Cloud
Sep 21, 2019
Partitioning and distributing deep neural networks (DNNs) over physical nodes such as edge, fog, or cloud nodes, could enhance sensor fusion, and reduce bandwidth and inference latency. However, when a DNN is distributed over physical nodes, failure of the physical nodes causes the failure of the DNN units that are placed on these nodes. The performance of the inference task will be unpredictable, and most likely, poor, if the distributed DNN is not specifically designed and properly trained for failures. Motivated by this, we introduce deepFogGuard, a DNN architecture augmentation scheme for making the distributed DNN inference task failure-resilient. To articulate deepFogGuard, we introduce the elements and a model for the resiliency of distributed DNN inference. Inspired by the concept of residual connections in DNNs, we introduce skip hyperconnections in distributed DNNs, which are the basis of deepFogGuard's design to provide resiliency. Next, our extensive experiments using two existing datasets for the sensing and vision applications confirm the ability of deepFogGuard to provide resiliency for distributed DNNs in edge-cloud networks.