From External to Swap Regret 2.0: An Efficient Reduction and Oblivious Adversary for Large Action Spaces (2310.19786v3)

Abstract: We provide a novel reduction from swap-regret minimization to external-regret minimization, which improves upon the classical reductions of Blum-Mansour [BM07] and Stolz-Lugosi [SL05] in that it does not require finiteness of the space of actions. We show that, whenever there exists a no-external-regret algorithm for some hypothesis class, there must also exist a no-swap-regret algorithm for that same class. For the problem of learning with expert advice, our result implies that it is possible to guarantee that the swap regret is bounded by {\epsilon} after $\log(N)^{O(1/\epsilon)}$ rounds and with $O(N)$ per iteration complexity, where $N$ is the number of experts, while the classical reductions of Blum-Mansour and Stolz-Lugosi require $O(N/\epsilon^2)$ rounds and at least $\Omega(N^2)$ per iteration complexity. Our result comes with an associated lower bound, which -- in contrast to that in [BM07] -- holds for oblivious and $\ell_1$-constrained adversaries and learners that can employ distributions over experts, showing that the number of rounds must be $\tilde\Omega(N/\epsilon^2)$ or exponential in $1/\epsilon$. Our reduction implies that, if no-regret learning is possible in some game, then this game must have approximate correlated equilibria, of arbitrarily good approximation. This strengthens the folklore implication of no-regret learning that approximate coarse correlated equilibria exist. Importantly, it provides a sufficient condition for the existence of correlated equilibrium which vastly extends the requirement that the action set is finite, thus answering a question left open by [DG22; Ass+23]. Moreover, it answers several outstanding questions about equilibrium computation and learning in games.

• The paper introduces an efficient reduction from swap regret minimization to external regret, applicable even to large or infinite action spaces.
• The proposed TreeSwap algorithm achieves significant computational gains, requiring fewer rounds than classical methods for $\epsilon$ swap regret.
• This work enables efficient computation of approximate correlated equilibria in various games, including extensive-form games, with improved complexity.

An Analysis of Efficient Reduction from External to Swap Regret for Large Action Spaces

The paper "From External to Swap Regret 2.0: An Efficient Reduction for Large Action Spaces" by Yuval Dagan, Constantinos Daskalakis, Maxwell Fishelson, and Noah Golowich presents a novel reduction technique from swap-regret minimization to external-regret minimization. This approach extends classic reductions by Blum-Mansour and Stoltz-Lugosi and does not require finiteness of the action space, thereby broadening the applicability of the method to scenarios with large or infinite action sets.

Core Contribution and Results

The primary contribution of the paper is the development of a method where any no-external-regret algorithm can be transformed into a no-swap-regret algorithm. For the learning with expert advice problem, the presented methodology ensures that swap regret can be bounded by $\epsilon$ after $(\log N)^{\tilde O(1/\epsilon)}$ rounds, with $O(N)$ computational complexity per iteration. Classical approaches would require a minimum of $\Omega(N/\epsilon^2)$ rounds, demonstrating a substantial computational advantage.

Furthermore, the paper establishes theoretical lower bounds for the necessary rounds in achieving sublinear swap regret, which depend on the oblivious nature of the adversary and $\ell_1$ constraints. The results imply that the number of rounds must be $\tilde{\Omega}(N/\epsilon^2)$ or exponential in $1/\epsilon$ for achieving these bounds.

Theoretical and Practical Implications

One of the key theoretical implications is the extension of no-regret learning guarantees to large action spaces, having potential significance in decentralized equilibrium computation in game theory. The paper's results suggest that approximate correlated equilibria exist for expansive classes of games with finite Littlestone or finite sequential fat-shattering dimensions, significantly answering open questions in the theory of strategic learning environments.

Practically, this research enables efficient computation of $\epsilon$-approximate correlated equilibria in various types of games, such as extensive-form games, with improvements in computational efficiency. Specifically, the ability to compute $\epsilon$-approximate correlated equilibria in polynomial time in extensive-form games represents a significant advancement in solving these complex strategic scenarios.

Discussion of Methodology

The methodological innovation stems from the way the paper structures multiple instances of no-external regret algorithms in a tree-like manner to strategically converge on swap regret minimization. The proposed "TreeSwap" algorithm leverages a hierarchical interaction mechanism between different instances to efficiently balance the trade-offs in utilitarian gain, showcasing how complexity can be distributed effectively in large-scale learning tasks.

The authors provide detailed analysis supporting the validity of the proposed bounds and demonstrate how established principles like Rademacher complexity are incorporated to derive regret bounds applicable even to infinite action spaces.

Future Directions

The potential for future research includes extending these results to more complex adversarial models and exploring the implications in other areas, such as reinforcement learning in dynamic environments. Furthermore, examining the applicability of this reduction methodology to real-world scenarios, including multi-agent systems and large-scale digital marketplaces, can offer rich avenues for both theoretical exploration and practical implementation.

In conclusion, this paper significantly advances the understanding and application of swap regret, offering a robust framework for efficient computation in settings traditionally seen as computationally prohibitive. This work not only strengthens the theoretical underpinnings of regret minimization but also broadens its utilities across a range of strategic and computational complexities.