Using this estimator, we propose a perturbation-based exploration strategy and develop a generalized regret analysis scheme that provides upper and lower regret bounds by revealing the relationship between the regret and the cumulative density function of the perturbation. From the proposed analysis scheme, we obtain gap-dependent and gap-independent upper and lower regret bounds of various perturbations. We also find the optimal hyperparameters for each perturbation, which can achieve the minimax optimal regret bound with respect to total rounds. In simulation, the proposed estimator shows favorable performance compared to existing robust estimators for various p values and, for MAB problems, the proposed perturbation strategy outperforms existing exploration methods.

We analyze the dynamics of finite width effects in wide but finite feature learning neural networks. Starting from a dynamical mean field theory description of infinite width deep neural network kernel and prediction dynamics, we provide a characterization of the O(1/ width) fluctuations of the DMFT order parameters over random initializations of the network weights. Our results, while perturbative in width, unlike prior analyses, are non-perturbative in the strength of feature learning. In the lazy limit of network training, all kernels are random but static in time and the prediction variance has a universal form. However, in the rich, feature learning regime, the fluctuations of the kernels and predictions are dynamically coupled with a variance that can be computed self-consistently.

We study the relationship between the notions of differentially private learning and online learning in games. Several recent works have shown that differentially private learning implies online learning, but an open problem of Neel, Roth, and Wu [27] asks whether this implication is efficient. Specifically, does an efficient differentially private learner imply an efficient online learner? In this paper we resolve this open question in the context of pure differential privacy. We derive an efficient black-box reduction from differentially private learning to online learning from expert advice.

B.1 Proof of Theorem 3 In Algorithm 3, we present the online randomized rounding scheme that combined with Projected Gradient Descent (Algorithm 1) produces a polynomial-time randomized online learning algorithm for GMSSC with (roughly) 28 regret. The randomized rounding scheme described in Algorithm 3 was introduced by [36] to provide a 28-approximation algorithm for the (offline) GMSSC. We remark that this LP relaxation cannot be translated to an equivalent relaxed online learning problem as the one we formulated using the fractional access cost of Definition 2. The goal of the section is to prove Theorem 3 which extends the result of [36] to the fractional access cost of Definition 2. Randomly pick α (0, 1) with probability density function f(α) = 2α. We first prove Claim 1 that is crucial for the subsequent analysis. B.2 Proof of Theorem 4 We first present the online sampling scheme, described in Algorithm 4, that produces the 11.713 guarantee of Theorem 4. Randomly pick α (0, 1) with probability density function f(α) = 2α.

The widely studied Generalized Min-Sum-Set-Cover (GMSSC) problem serves as a formal model for the setting above.

We refer to this simple illustrative example several times in the rest of the paper. Consider 4 types of resources and 4 types of products: Product No.1: (s, 1, 3, t), i.e., SQL Database EC2

We consider a price-based network revenue management problem with multiple products and multiple reusable resources. Each randomly arriving customer requests a product (service) that needs to occupy a sequence of reusable resources (servers). We adopt an incomplete information setting where the firm does not know the price-demand function for each product and the goal is to dynamically set prices of all products to maximize the total expected revenue of serving customers. We propose novel batched bandit learning algorithms for finding near-optimal pricing policies, and show that they admit a near-optimal cumulative regret bound of Õ(J XT), where J, X, and T are the numbers of products, candidate prices, and service periods, respectively. As part of our regret analysis, we develop the first finite-time mixing time analysis of an open network queueing system (i.e., the celebrated Jackson Network), which could be of independent interest. Our numerical studies show that the proposed approaches perform consistently well.

We study the problem of learning a real-valued function that satisfies the Demographic Parity constraint. It demands the distribution of the predicted output to be independent of the sensitive attribute. We consider the case that the sensitive attribute is available for prediction. We establish a connection between fair regression and optimal transport theory, based on which we derive a close form expression for the optimal fair predictor. Specifically, we show that the distribution of this optimum is the Wasserstein barycenter of the distributions induced by the standard regression function on the sensitive groups. This result offers an intuitive interpretation of the optimal fair prediction and suggests a simple post-processing algorithm to achieve fairness. We establish risk and distribution-free fairness guarantees for this procedure. Numerical experiments indicate that our method is very effective in learning fair models, with a relative increase in error rate that is inferior to the relative gain in fairness.

We study the problem of learning a real-valued function that satisfies the Demographic Parity constraint. It demands the distribution of the predicted output to be independent of the sensitive attribute. We consider the case that the sensitive attribute is available for prediction. We establish a connection between fair regression and optimal transport theory, based on which we derive a close form expression for the optimal fair predictor. Specifically, we show that the distribution of this optimum is the Wasserstein barycenter of the distributions induced by the standard regression function on the sensitive groups. This result offers an intuitive interpretation of the optimal fair prediction and suggests a simple post-processing algorithm to achieve fairness. We establish risk and distribution-free fairness guarantees for this procedure. Numerical experiments indicate that our method is very effective in learning fair models, with a relative increase in error rate that is inferior to the relative gain in fairness.

We consider the problem of learning to play a repeated multi-agent game with an unknown reward function. Single player online learning algorithms attain strong regret bounds when provided with full information feedback, which unfortunately is unavailable in many real-world scenarios. Bandit feedback alone, i.e., observing outcomes only for the selected action, yields substantially worse performance. In this paper, we consider a natural model where, besides a noisy measurement of the obtained reward, the player can also observe the opponents' actions. This feedback model, together with a regularity assumption on the reward function, allows us to exploit the correlations among different game outcomes by means of Gaussian processes (GPs). We propose a novel confidence-bound based bandit algorithm GP-MW, which utilizes the GP model for the reward function and runs a multiplicative weight (MW) method. We obtain novel kernel-dependent regret bounds that are comparable to the known bounds in the full information setting, while substantially improving upon the existing bandit results. We experimentally demonstrate the effectiveness of GP-MW in random matrix games, as well as realworld problems of traffic routing and movie recommendation. In our experiments, GP-MW consistently outperforms several baselines, while its performance is often comparable to methods that have access to full information feedback.