Learning Randomized Algorithms with Transformers (2408.10818v1)

Abstract: Randomization is a powerful tool that endows algorithms with remarkable properties. For instance, randomized algorithms excel in adversarial settings, often surpassing the worst-case performance of deterministic algorithms with large margins. Furthermore, their success probability can be amplified by simple strategies such as repetition and majority voting. In this paper, we enhance deep neural networks, in particular transformer models, with randomization. We demonstrate for the first time that randomized algorithms can be instilled in transformers through learning, in a purely data- and objective-driven manner. First, we analyze known adversarial objectives for which randomized algorithms offer a distinct advantage over deterministic ones. We then show that common optimization techniques, such as gradient descent or evolutionary strategies, can effectively learn transformer parameters that make use of the randomness provided to the model. To illustrate the broad applicability of randomization in empowering neural networks, we study three conceptual tasks: associative recall, graph coloring, and agents that explore grid worlds. In addition to demonstrating increased robustness against oblivious adversaries through learned randomization, our experiments reveal remarkable performance improvements due to the inherently random nature of the neural networks' computation and predictions.

• The paper proposes a novel adversarial objective that trains transformers to learn robust randomized strategies.
• It establishes a theoretical link between model capacity and randomness, showing that standard ERM tends to produce deterministic models.
• Empirical results demonstrate significant performance gains over deterministic models in associative recall, graph coloring, and grid world exploration.

Learning Randomized Algorithms with Transformers

This paper addresses an ambitious objective: integrating the robust properties of randomized algorithms into transformer models. The introduction of randomization in transformers is undertaken with a comprehensive theoretical framework and validated through empirical studies. The authors focus on the robustness advantages that randomized algorithms traditionally offer over deterministic ones, particularly in adversarial settings. They propose an intuitive yet potent hypothesis that deep neural networks, especially transformers, can learn robust randomized strategies through data- and objective-driven methods.

Theoretical Foundations

The paper establishes a firm theoretical groundwork by discussing classical results from the literature on randomized algorithms. One of the vital theoretical insights derived is the principle that excessive model capacity—or the lack of it—determines whether a model can benefit from randomization. If a model can fit the training data perfectly, the need for randomization diminishes. However, the real strength of the paper lies in demonstrating that common optimization techniques like empirical risk minimization (ERM) tend to create deterministic models, even when randomness is explicitly provided. The authors argue that ERM does not maximize the utility of randomization by referencing Yao’s Minimax Principle, illustrating that expected risk minimization inherently biases models towards determinism.

They propose an alternative optimization objective centered on minimizing a relaxed adversarial loss, driven by an overarching goal to perform well under adversarial, worst-case scenarios. This approach hinges on the insight that randomization can provide significant robustness against worst-case inputs, ultimately leading to lower loss values in adversarial contexts. By defining a min-max loss function and demonstrating how to approximate it using a multi-seed strategy, the authors set the stage for learning powerful randomized algorithms in transformers.

Empirical Validation

To substantiate their theoretical claims, the authors conducted several experiments that span different conceptual tasks: associative recall, graph coloring, and grid world exploration.

Associative Recall

In associative recall tasks, transformers with linear self-attention layers were trained to memorize and recall arbitrary value vectors associated with unique keys. The empirical results illustrate that models trained on the relaxed adversarial loss (with $q=100$) exhibit robust randomization, especially evident in improved performance on worst-case inputs via majority voting. This enhancement is contrasted against transformers trained with single fixed seeds, which failed to deliver comparable results. The experiments confirm that randomization significantly reduces recall errors, shielding the model from adversarial failures.

Graph Coloring

The paper leverages the classical problem of 3-coloring cycles to examine how transformers handle distributed graph coloring tasks. Here, the hallmark of success for randomized algorithms—simplicity and robustness—was distinctly observed. Transformers trained on the relaxed adversarial loss exhibited markedly better performance when compared to their deterministic counterparts. The most striking improvements were observed using majority voting over different seeds, demonstrating nearly optimal performance. This task vividly showcases the advantage of learned randomization over traditional deterministic strategies.

Grid World Exploration

Finally, the experiments encompassing grid world exploration elegantly bridge the gap between theoretical insights and practical non-differentiable environments. Utilizing evolutionary strategies for optimization, the authors demonstrated that transformers could learn randomized exploration strategies effectively. The randomized models outperformed deterministic ones in environments where adversarial settings were simulated by varying treasure locations. These results underscore the potential of learned randomization to enhance exploration efficiency in reinforcement learning contexts.

Implications and Future Directions

The integration of randomized algorithms into transformers has far-reaching theoretical and practical implications:

1. Theoretical Advancements: The paper bridges a critical gap between theoretical computer science and deep learning. It brings the robust properties of randomized algorithms into neural architecture design, marking a significant step in the evolution of AI algorithm design.
2. Practical Impact: The demonstrated improvements in robustness and performance suggest practical applications in adversarial environments, such as security, autonomous navigation, and game theory.
3. Future Research: The concept of learning from data to develop randomized algorithms opens numerous avenues for future research. It invites further exploration into scaling these methods, integrating them with existing adversarial training techniques, and potentially even drawing parallels with human and biological cognition.

Conclusion

"Learning Randomized Algorithms with Transformers" provides a compelling narrative supported by robust theoretical constructs and empirical validation. The proposed approach, through its optimization of a novel adversarial objective, effectively learns and leverages randomization in transformers. This work lays the foundation for future investigations into the interplay between deterministic and randomized strategies in neural networks, especially as we push the boundaries of what deep learning models can achieve in complex, adversarial environments.