Revenue Maximization and Learning in Products Ranking

We consider the revenue maximization problem for an online retailer who plans to display a set of products differing in their prices and qualities and rank them in order. The consumers have random attention spans and view the products sequentially before purchasing a ``satisficing'' product or leaving the platform empty-handed when the attention span gets exhausted. Our framework extends the cascade model in two directions: the consumers have random attention spans instead of fixed ones and the firm maximizes revenues instead of clicking probabilities. We show a nested structure of the optimal product ranking as a function of the attention span when the attention span is fixed and design a $1/e$-approximation algorithm accordingly for the random attention spans. When the conditional purchase probabilities are not known and may depend on consumer and product features, we devise an online learning algorithm that achieves $\tilde{\mathcal{O}}(\sqrt{T})$ regret relative to the approximation algorithm, despite of the censoring of information: the attention span of a customer who purchases an item is not observable. Numerical experiments demonstrate the outstanding performance of the approximation and online learning algorithms

Federated Multi-Level Optimization over Decentralized Networks

Multi-level optimization has gained increasing attention in recent years, as it provides a powerful framework for solving complex optimization problems that arise in many fields, such as meta-learning, multi-player games, reinforcement learning, and nested composition optimization. In this paper, we study the problem of distributed multi-level optimization over a network, where agents can only communicate with their immediate neighbors. This setting is motivated by the need for distributed optimization in large-scale systems, where centralized optimization may not be practical or feasible. To address this problem, we propose a novel gossip-based distributed multi-level optimization algorithm that enables networked agents to solve optimization problems at different levels in a single timescale and share information through network propagation. Our algorithm achieves optimal sample complexity, scaling linearly with the network size, and demonstrates state-of-the-art performance on various applications, including hyper-parameter tuning, decentralized reinforcement learning, and risk-averse optimization.Comment: arXiv admin note: substantial text overlap with arXiv:2206.1087

Optimization and revenue management in complex networks

This thesis consists of three papers in optimization and revenue management over complex networks: Robust Linear Control in Transmission Systems, Online Learning and Optimization Under a New Linear-Threshold Model with Negative Influence, and Revenue Management with Complementarity Products. This thesis contributes to analytical methods for optimization problems in complex networks, namely, power network, social network and product network. In Chapter 2, we describe a robust multiperiod transmission planning model including renewables and batteries, where battery output is used to partly offset renewable output deviations from forecast. A central element is a nonconvex battery operation model plus a robust model of forecast errors and a linear control scheme. Even though the problem is nonconvex we provide an efficient and theoretically valid algorithm that effectively solves cases on large transmission systems. In Chapter 3, we propose a new class of Linear Threshold Model-based information-diffusion model that incorporates the formation and spread of negative attitude. We call such models negativity-aware. We show that in these models, the expected positive influence is a monotone sub-modular function of the seed set. Thus we can use a greedy algorithm to construct a solution with constant approximation guarantee when the objective is to select a seed set of fixed size to maximize positive influence. Our models are flexible enough to account for both the features of local users and the features of the information being propagated in the diffusion. We analyze an online-learning setting for a multi-round influence-maximization problem, where an agent is actively learning the diffusion parameters over time while trying to maximize total cumulative positive influence. We develop a class of online learning algorithms and provide the theoretical upper bound on the regret. In Chapter 4, we propose a tractable information-diffusion-based framework to capture complementary relationships among products. Using this framework, we investigate how various revenue-management decisions can be optimized. In particular, we prove that several fundamental problems involving complementary products, such as promotional pricing, product recommendation, and category planning, can be formulated as sub-modular maximization problems, and can be solved by tractable greedy algorithms with guarantees on the quality of the solutions. We validate our model using a dataset that contains product reviews and metadata from Amazon from May 1996 to July 2014. We also analyze an online-learning setting for revenue-maximization with complementary products. In this setting, we assume that the retailer has access only to sales observations. That is, she can only observe whether a product is purchased from her. This assumption leads to diffusion models with novel node-level feedback, in contrast to classical models that have edge-level feedback. We conduct confidence region analysis on the maximum likelihood estimator for our models, develop online-learning algorithms, and analyze their performance in both theoretical and practical perspectives

Bridging Adversarial and Nonstationary Multi-armed Bandit

In the multi-armed bandit framework, there are two formulations that are commonly employed to handle time-varying reward distributions: adversarial bandit and nonstationary bandit. Although their oracles, algorithms, and regret analysis differ significantly, we provide a unified formulation in this paper that smoothly bridges the two as special cases. The formulation uses an oracle that takes the best-fixed arm within time windows. Depending on the window size, it turns into the oracle in hindsight in the adversarial bandit and dynamic oracle in the nonstationary bandit. We provide algorithms that attain the optimal regret with the matching lower bound

Multi-Level Stochastic Gradient Methods for Nested Composition Optimization

Stochastic gradient methods are scalable for solving large-scale optimization problems that involve empirical expectations of loss functions. Existing results mainly apply to optimization problems where the objectives are one- or two-level expectations. In this paper, we consider the multi-level compositional optimization problem that involves compositions of multi-level component functions and nested expectations over a random path. It finds applications in risk-averse optimization and sequential planning. We propose a class of multi-level stochastic gradient methods that are motivated from the method of multi-timescale stochastic approximation. First we propose a basic $T$-level stochastic compositional gradient algorithm, establish its almost sure convergence and obtain an $n$-iteration error bound $O (n^{-1/2^T})$. Then we develop accelerated multi-level stochastic gradient methods by using an extrapolation-interpolation scheme to take advantage of the smoothness of individual component functions. When all component functions are smooth, we show that the convergence rate improves to $O(n^{-4/(7+T)})$ for general objectives and $O (n^{-4/(3+T)})$ for strongly convex objectives. We also provide almost sure convergence and rate of convergence results for nonconvex problems. The proposed methods and theoretical results are validated using numerical experiments

AUCTION THEORY: COMMON VALUE AUCTIONS

Bachelor'sBACHELOR OF SCIENCE (HONOURS

Optimality Conditions and Algorithms for Top-K Arm Identification

We consider the top-k arm identification problem for multi-armed bandits with rewards belonging to a one-parameter canonical exponential family. The objective is to select the set of k arms with the highest mean rewards by sequential allocation of sampling efforts. We propose a unified optimal allocation problem that identifies the complexity measures of this problem under the fixed-confidence, fixed-budget settings, and the posterior convergence rate from the Bayesian perspective. We provide the first characterization of its optimality. We provide the first provably optimal algorithm in the fixed-confidence setting for k>1. We also propose an efficient heuristic algorithm for the top-k arm identification problem. Extensive numerical experiments demonstrate superior performance compare to existing methods in all three settings

Data-Driven Minimax Optimization with Expectation Constraints

Attention to data-driven optimization approaches, including the well-known stochastic gradient descent method, has grown significantly over recent decades, but data-driven constraints have rarely been studied, because of the computational challenges of projections onto the feasible set defined by these hard constraints. In this paper, we focus on the non-smooth convex-concave stochastic minimax regime and formulate the data-driven constraints as expectation constraints. The minimax expectation constrained problem subsumes a broad class of real-world applications, including two-player zero-sum game and data-driven robust optimization. We propose a class of efficient primal-dual algorithms to tackle the minimax expectation-constrained problem, and show that our algorithms converge at the optimal rate of $\mathcal{O}(\frac{1}{\sqrt{N}})$. We demonstrate the practical efficiency of our algorithms by conducting numerical experiments on large-scale real-world applications