No-Regret M${}^{\natural}$-Concave Function Maximization: Stochastic Bandit Algorithms and NP-Hardness of Adversarial Full-Information Setting (2405.12439v1)

Abstract: M${}^{\natural}$-concave functions, a.k.a. gross substitute valuation functions, play a fundamental role in many fields, including discrete mathematics and economics. In practice, perfect knowledge of M${}^{\natural}$-concave functions is often unavailable a priori, and we can optimize them only interactively based on some feedback. Motivated by such situations, we study online M${}^{\natural}$-concave function maximization problems, which are interactive versions of the problem studied by Murota and Shioura (1999). For the stochastic bandit setting, we present $O(T^{-1/2})$-simple regret and $O(T^{2/3})$-regret algorithms under $T$ times access to unbiased noisy value oracles of M${}^{\natural}$-concave functions. A key to proving these results is the robustness of the greedy algorithm to local errors in M${}^{\natural}$-concave function maximization, which is one of our main technical results. While we obtain those positive results for the stochastic setting, another main result of our work is an impossibility in the adversarial setting. We prove that, even with full-information feedback, no algorithms that run in polynomial time per round can achieve $O(T^{1-c})$ regret for any constant $c > 0$ unless $\mathsf{P} = \mathsf{NP}$. Our proof is based on a reduction from the matroid intersection problem for three matroids, which would be a novel idea in the context of online learning.

• The paper introduces an algorithm achieving O(T^(-1/2)) simple regret in stochastic bandit settings using a greedy approach.
• It extends to cumulative regret with an explore-then-commit strategy reaching O(T^(2/3)) performance without approximation loss.
• The work proves NP-hardness in adversarial full-information settings, implying no polynomial-time algorithm can achieve sublinear regret unless P=NP.

Online Maximization of M${}^\natural$-Concave Functions

Let's dive into an interesting paper on a complex yet practically significant topic: the online maximization of M${}^\natural$-concave functions. M${}^\natural$-concave (or gross substitute) functions are crucial in fields like discrete mathematics and economics. This paper hones in on scenarios where we lack perfect knowledge of these functions and have to work with noisy or adversarial feedback.

What Are M${}^\natural$-Concave Functions?

M${}^\natural$-concave functions play a fundamental role in various domains:

• Economics: Known as gross substitute valuations, they model scenarios where increasing the price of some goods doesn't drastically change the demand for others.
• Operations Research: Useful in resource allocation problems, such as maximizing the flow in networks or demand in supply chains.

These functions are not only theoretically intriguing but also practically important. When working with M${}^\natural$-concave functions, one often needs to optimize them interactively due to imperfect information, which is where this paper steps in.

Key Contributions of the Paper

This paper explores two main scenarios for the online maximization of M${}^\natural$-concave functions: the stochastic bandit setting and the adversarial setting.

Stochastic Bandit Setting

In this scenario, the learner receives noisy evaluations of the function. Here are the key results:

1. Simple Regret Algorithm: The paper presents an algorithm that achieves $O(T^{-1/2})$ simple regret, meaning that as $T$ (the number of rounds) increases, the gap between the function value of the chosen action and the optimal action decreases. This is achieved via a greedy approach robust to local errors.
2. Cumulative Regret Algorithm: Additionally, using the explore-then-commit strategy, the authors extend their findings to achieve $O(T^{2/3})$ cumulative regret. In layman's terms, this strategy balances exploration (trying new actions) and exploitation (sticking with the best action found so far).

These results are significant because they provide strong performance guarantees without any approximation factors, unlike similar results for submodular functions.

Adversarial Full-Information Setting

The paper also tackles a tougher scenario where feedback can be strategically adversarial:

• Impossibility Result: A standout result of this paper is proving that no polynomial-time algorithm can achieve sub-linear regret ($O(T^{1-c})$ for any constant $c > 0$) in adversarial settings unless $\mathsf{P} = \mathsf{NP}$. This result hinges on a reduction from the 3-matroid intersection problem, which is known to be NP-hard.

Practical and Theoretical Implications

The practical implications of this work are clear for fields requiring real-time decision-making under uncertainty, such as online auctions and network routing. The ability to robustly optimize M${}^\natural$-concave functions interactively can lead to significant efficiency improvements.

Theoretically, these results highlight the limits of computability in adversarial environments and contribute to our understanding of the complexity of online learning problems.

Future Directions

This paper opens several avenues for future research:

1. Enhanced Algorithms: Exploring more sophisticated algorithms that can handle larger classes of functions or perform better in practice.
2. Broader Applications: Applying these concepts to other domains where similar types of optimization problems exist.
3. Complexity Insights: Further investigating the boundaries between tractable and intractable problems in online learning.

In summary, while this paper provides valuable algorithms for stochastic settings, it also underscores the inherent difficulty of the problem in adversarial settings. These insights and results can be pivotal for both theoretical advancement and practical applications in various fields.