Online Optimization of Discrete Convex Functions with Bounded Domains

Core Concepts

This paper introduces the problem of online optimization of L♮-convex functions, which generalize online submodular minimization to a broader class of discrete convex functions. The authors propose computationally efficient algorithms with tight regret bounds for both the full information and bandit settings.

Abstract

The paper introduces the problem of online optimization of L♮-convex functions, which generalize online submodular minimization to a broader class of discrete convex functions. The authors make the following key contributions: In the full information setting, they propose a randomized algorithm that achieves a regret bound of O(ˆLN√dT), which is shown to be tight up to a constant factor. In the bandit setting, they propose a randomized algorithm that achieves a regret bound of O(MdNT^(2/3)). The authors demonstrate that the online L♮-convex minimization problem can capture a wider range of applications compared to the existing online submodular minimization framework. The paper provides the necessary background on L♮-convex functions, their convex extensions, and efficient algorithms for computing subgradients. This lays the foundation for the design and analysis of the proposed online optimization algorithms. The regret analysis leverages techniques from online convex optimization, while carefully handling the discrete nature of L♮-convex functions through the use of the Lovász extension and properties of submodular functions. The authors also establish a lower bound on the regret, showing that their full information algorithm is optimal up to a constant factor.

Stats

The decision space K is a bounded L♮-convex set in the d-dimensional integer lattice Zd. The cost functions ft : K → [−M, M] are L♮-convex with l∞-Lipschitz constant ˆL. The parameter N = maxi∈[d]{ˆγ(i) - ˇγ(i)} bounds the size of the decision space.

Quotes

"To overcome this limitation, we introduce online L♮-convex function minimization, where the minimizing objective function is an L♮-convex function." "We analyze the regrets of these algorithms and show in particular that our algorithm for the full information setting obtains a tight regret bound up to a constant factor."

Key Insights Distilled From

Online $\mathrm{L}^{\natural}$-Convex Minimization

by Ken Yokoyama... at arxiv.org 04-29-2024

https://arxiv.org/pdf/2404.17158.pdf

Deeper Inquiries

How can the proposed online L♮-convex minimization framework be extended to handle stochastic or adversarial noise in the cost functions

To handle stochastic or adversarial noise in the cost functions within the online L♮-convex minimization framework, we can incorporate techniques from stochastic optimization and robust optimization. For stochastic noise, we can modify the algorithm to include stochastic gradient descent or stochastic subgradient methods. By considering the noise in the cost functions as random variables, we can update the decision variables based on noisy estimates of the gradients. This approach allows the algorithm to adapt to the uncertainty in the cost functions and converge to a near-optimal solution in expectation. In the case of adversarial noise, where an adversary deliberately introduces perturbations to the cost functions to mislead the optimization process, we can employ robust optimization techniques. Robust optimization considers a set of possible scenarios or perturbations and aims to find a solution that performs well under the worst-case scenario. By formulating the optimization problem with robust constraints or objectives, the algorithm can provide solutions that are resilient to adversarial noise. By integrating these strategies into the online L♮-convex minimization framework, we can enhance the algorithm's robustness and adaptability in the presence of stochastic or adversarial noise in the cost functions.

Can the techniques developed in this paper be applied to other classes of discrete convex functions beyond L♮-convex functions

The techniques developed in this paper for online L♮-convex minimization can be extended to handle other classes of discrete convex functions beyond L♮-convex functions. Some possible extensions include: Multimodular Functions: Similar to L♮-convex functions, multimodular functions exhibit a certain type of discrete convexity. By adapting the algorithms and analysis to accommodate multimodular functions, we can address optimization problems with this class of functions. General Discrete Convex Functions: By generalizing the framework to encompass a broader class of discrete convex functions, we can tackle a wider range of optimization problems. This extension may involve developing new algorithms and regret bounds tailored to the specific properties of the chosen class of functions. Structured Discrete Convex Functions: For optimization problems with structured discrete convex functions, such as matroid rank functions or polymatroid functions, the techniques can be adapted to exploit the structural properties of these functions. This adaptation can lead to more efficient algorithms and improved regret bounds. By exploring these extensions, we can apply the developed techniques to a diverse set of discrete convex functions, expanding the applicability of the framework to various optimization scenarios.

What are some practical applications that can directly benefit from the online optimization of L♮-convex functions

The online optimization of L♮-convex functions has numerous practical applications across different domains. Some specific applications that can directly benefit from this framework include: Resource Allocation in Networks: Optimizing resource allocation in communication networks, transportation systems, or cloud computing environments can be formulated as an online L♮-convex minimization problem. By efficiently allocating resources based on changing demands and constraints, the system can operate optimally in real-time. Dynamic Pricing and Revenue Management: Online platforms and e-commerce websites can utilize L♮-convex minimization to dynamically adjust pricing strategies based on customer behavior and market conditions. By optimizing pricing decisions in real-time, businesses can maximize revenue and profitability. Supply Chain Optimization: Managing inventory levels, production schedules, and distribution networks in supply chains can benefit from online L♮-convex minimization. By continuously optimizing decisions to minimize costs and meet demand fluctuations, supply chain operations can be more efficient and responsive. Online Advertising and Recommendation Systems: Personalizing ad placements, content recommendations, and marketing campaigns can be enhanced through online L♮-convex minimization. By adapting strategies based on user interactions and feedback, advertisers and content providers can improve engagement and conversion rates. These practical applications demonstrate the versatility and effectiveness of the online optimization of L♮-convex functions in addressing complex decision-making problems in various industries.