TL;DR#
Black-box optimization (BBO) is crucial for handling complex problems where only input-output data is available, with adversarial BBO dealing with scenarios involving multiple interacting agents. The ‘No Free Lunch’ theorem highlights limitations of traditional BBO algorithms. However, applying NFL to adversarial settings (maximin optimization), especially when focusing on Nash Equilibrium, remains a challenge; defining optimality and the impact of solution concepts are critical here. Existing work shows ‘free lunches’ for some adversarial scenarios. This paper addresses these open challenges.
This research rigorously proves that no universally superior algorithm exists for finding Nash Equilibria in two-player zero-sum games. It introduces a novel notion of black-box complexity to analyze adversarial optimization, providing general lower bounds for query complexity. The findings are illustrated using simple two-player zero-sum games, showing limitations for finding unique Nash Equilibria. The results emphasize the critical role of solution concepts in adversarial optimization and highlight the challenges in creating universally efficient algorithms.
Key Takeaways#
Why does it matter?#
This paper is crucial because it rigorously proves the impossibility of a universally effective algorithm for black-box adversarial optimization, challenging the conventional wisdom. It introduces a novel black-box complexity framework for analyzing adversarial optimization, providing valuable insights into the inherent difficulty of these problems and guiding future algorithm design. This is particularly relevant given the increasing importance of adversarial settings in various domains like machine learning and game theory.
Visual Insights#
🔼 This figure compares traditional black-box optimization and maximin black-box optimization. The traditional approach involves querying a black box with an input ‘x’ to get an output ‘f(x)’. In contrast, maximin optimization queries the black box with both a strategy ‘x’ and an opponent’s best response ‘y’, yielding an output ‘g(x,y)’. This illustrates the key difference: maximin considers the interaction between strategies.
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Figure 1: Comparison between traditional black-box optimisation and maximin black-box optimisation. Instead of querying at x in traditional optimisation, maximin optimisation queries at (x, y) include both strategy x and the best response y from the opponent, i.e. maxxex miny∈y g(x, y). Their interaction is converted to the payoff g(x, y) in the given black-box model.
🔼 This table visually compares traditional black-box optimization and maximin (or adversarial) black-box optimization. It highlights the key difference: in traditional optimization, a single point x is queried to evaluate the objective function f(x). In contrast, maximin optimization queries a pair (x, y), where y represents the best response from an opponent to strategy x in a game-theoretic setting. The payoff g(x,y) then reflects the outcome of this interaction. This illustrates the shift from single-objective optimization to a game-theoretic setting.
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Table 1: Comparison between traditional black-box optimisation and maximin black-box optimisation. Instead of querying at x in traditional optimisation, maximin optimisation queries at (x, y) include both strategy x and the best response y from the opponent, i.e., maxx∈X miny∈Y g(x, y). Their interaction is converted to the payoff g(x, y) in the given black-box model.
Full paper#
