The theory of discrete-time online learning has been successfully applied in many problems that involve sequential decision-making under uncertainty. However, in many applications including contractual hiring in online freelancing platforms and server allocation in cloud computing systems, the outcome of each action is observed only after a random and action-dependent time. Furthermore, as a consequence of certain ethical and economic concerns, the controller may impose deadlines on the completion of each task, and require fairness across different groups in the allocation of total time budget $B$. In order to address these applications, we consider continuous-time online learning problem with fairness considerations, and present a novel framework based on continuous-time utility maximization. We show that this formulation recovers reward-maximizing, max-min fair and proportionally fair allocation rules across different groups as special cases. We characterize the optimal offline policy, which allocates the total time between different actions in an optimally fair way (as defined by the utility function), and impose deadlines to maximize time-efficiency. In the absence of any statistical knowledge, we propose a novel online learning algorithm based on dual ascent optimization for time averages, and prove that it achieves $\tilde{O}(B^{-1/2})$ regret bound.

Existing deep neural networks (DNNs), say for image classification, have been shown to be vulnerable to adversarial images that can cause a DNN misclassification, without any perceptible change to an image. In this work, we propose "shock-absorbing" robust features such as binarization (e.g., rounding) and group extraction (e.g., color or shape) to augment the classification pipeline, resulting in more robust classifiers. Experimentally, we show that augmenting ML models with these techniques leads to improved overall robustness on adversarial inputs as well as significant improvements in training time. On the MNIST dataset, we achieved 14x speedup in training time to obtain 90% adversarial accuracy compared to the state-of-the-art adversarial training method of Madry et al. [12], as well as retained higher adversarial accuracy over a broader range of attacks. We also find robustness improvements on traffic sign classification using robust feature augmentation. Finally, we give theoretical insights for why one can expect robust feature augmentation to reduce adversarial input space.

The original simplicial method (OSM), a variant of the classic Kelley’s cutting plane method, has been shown to converge to the minimizer of a composite convex and submodular objective, though no rate of convergence for this method was known. Moreover, OSM is required to solve subproblems in each iteration whose size grows linearly in the number of iterations. We propose a limited memory version of Kelley’s method (L-KM) and of OSM that requires limited memory (at most n+ 1 constraints for an n-dimensional problem) independent of the iteration. We prove convergence for L-KM when the convex part of the objective g is strongly convex and show it converges linearly when g is also smooth. Our analysis relies on duality between minimization of the composite convex and submodular objective and minimization of a convex function over the submodular base polytope. We introduce a limited memory version, L-FCFW, of the Fully-Corrective Frank-Wolfe (FCFW) method with approximate correction, to solve the dual problem. We show that L-FCFW and L-KM are dual algorithms that produce the same sequence of iterates; hence both converge linearly (when g is smooth and strongly convex) and with limited memory. We propose L-KM to minimize composite convex and submodular objectives; however, our results on L-FCFW hold for general polytopes and may be of independent interest.

The original simplicial method (OSM), a variant of the classic Kelley’s cutting plane method, has been shown to converge to the minimizer of a composite convex and submodular objective, though no rate of convergence for this method was known. Moreover, OSM is required to solve subproblems in each iteration whose size grows linearly in the number of iterations. We propose a limited memory version of Kelley’s method (L-KM) and of OSM that requires limited memory (at most n+ 1 constraints for an n-dimensional problem) independent of the iteration. We prove convergence for L-KM when the convex part of the objective g is strongly convex and show it converges linearly when g is also smooth. Our analysis relies on duality between minimization of the composite convex and submodular objective and minimization of a convex function over the submodular base polytope. We introduce a limited memory version, L-FCFW, of the Fully-Corrective Frank-Wolfe (FCFW) method with approximate correction, to solve the dual problem. We show that L-FCFW and L-KM are dual algorithms that produce the same sequence of iterates; hence both converge linearly (when g is smooth and strongly convex) and with limited memory. We propose L-KM to minimize composite convex and submodular objectives; however, our results on L-FCFW hold for general polytopes and may be of independent interest.

The concept of individual fairness advocates similar treatment of similar individuals to ensure equality in treatment [Dwork et al. 2012]. In this paper, we extend this notion to account for the time at which a decision is made, in settings where there exists a notion of "conduciveness" of decisions as perceived by individuals. We introduce two definitions: (i) fairness-across-time and (ii) fairness-in-hindsight. In the former, treatments of individuals are required to be individually fair relative to the past as well as future, while in the latter we only require individual fairness relative to the past. We show that these two definitions can have drastically different implications in the setting where the principal needs to learn the utility model: one can achieve a vanishing asymptotic loss in long-run average utility relative to the full-information optimum under the fairness-in-hindsight constraint, whereas this asymptotic loss can be bounded away from zero under the fairness-across-time constraint.