Learning Randomized Algorithms with Transformers

Authors: Johannes von Oswald, Seijin Kobayashi, Yassir Akram, Angelika Steger
Primary area: other topics in machine learning (i.e., none of the above)
Keywords: Randomized algorithms, Learning under adversarial losses, Adversarial robustness, In-context learning algorithms

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.

Decision: accept (oral)
Reviewer scores: 8, 6, 6, 6

Reviews and discussion:
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creation date:
1730479626198

thread_type:
official_review

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creator: reviewer_bfrq

<content>
summary:
This paper explores the integration of randomization into transformer models, traditionally used in deterministic settings, to enhance their performance, especially in adversarial contexts. The authors demonstrate that randomized algorithms can be instilled in transformers through learning, in a purely data- and objective-driven manner. Though an analysis of known adversarial objectives for which randomized algorithms offer a distinct advantage over deterministic ones, they 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.

strengths:
- The paper is very well written, clear, with a good motivation. They first introduce the setting by presenting an example with associative recall, which is a variant of the classical paging problem in computer science for which randomized solutions exist. This problem is also a well-studied problem for transformers with the goal of evaluating recall capabilities. The paper continues with a theoretical analysis in which the authors study when randomization can be beneficial and when it is not. From there, the authors propose a training objective. 
Finally, the authors experiment with 3 use cases: associative recall, graph coloring, and agents exploring lattice worlds.
- The contribution is, to the best of my knowledge, novel and could lead to improvements in how transformers can solve certain tasks.

weaknesses:
- The experiments are performed on three different small tasks, the proposed approach should increase the computational complexity, which is already high with transformers, can the authors comment on this in terms of training and inference? 
- I am a bit puzzled by the paragraph on adversarial robustness (small input perturbations) in the related work, first, the cited paper (Rakin et al. (2018)) shows an approach to increase robustness with randomization, but now such approaches have been mostly disproved (see Gnecco et al. 2023). Furthermore, the randomized smoothing approach uses randomization _only in the training phase_, the approach is inherently a training procedure for a smooth function. The inference _is_ deterministic. Overall, I find this discussion could be made clearer. 

Gnecco et al. On the Role of Randomization in Adversarially Robust Classification. NeurIPS 2023.

questions:
See first Weakness
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creation date:
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creator: reviewer_mqsr

<content>
summary:
The paper explores how transformers can learn and implement randomized algorithms, which are advantageous in certain adversarial and game-theoretical contexts. Randomized algorithms typically perform well in adversarial environments and exhibit robustness against worst-case scenarios. The authors propose a novel objective function for training transformers that optimizes their performance in these settings by introducing randomization via an input "seed."

strengths:
- The authors provide a strong theoretical foundation for why randomization is advantageous in certain adversarial contexts, referencing game theory and established concepts like Yao’s Minimax Principle.
- The study highlights that randomization can increase resilience against adversarial attacks.

weaknesses:
- While the paper presents conceptual tasks to validate the approach, it does not provide empirical results on large-scale or real-world datasets. This limitation raises questions about how well the method would scale and perform in more complex, realistic environments.
- The approach requires sampling multiple seeds during training (controlled by the hyperparameter mm), which can increase computational overhead significantly. The authors note that this limitation affects memory, training time, and inference time, making the method potentially impractical for large-scale applications.
- The paper’s theoretical sections are rigorous but may be dense for readers without a background in adversarial training or randomized algorithms.
- The study assumes relatively static environments for tasks like grid world exploration and graph coloring. However, it does not address how the transformer’s randomized strategies would perform in dynamic, non-stationary settings where environmental conditions change over time.

questions:
- What are the computational and memory challenges of scaling the proposed randomization method to larger, more complex datasets or real-world tasks? Are there specific optimizations that could make the approach more efficient?
- The study introduces hyperparameters qq and mm in the training objective. How sensitive are the final model’s performance and robustness to variations in these hyperparameters?
- Could the proposed randomization approach be adapted for more complex reinforcement learning environments, such as those with continuous action spaces or requiring long-term strategy?
- How does this approach compare with other randomization techniques, such as dropout (at inference)?
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<thread>
creation date:
1730571137093

thread_type:
official_review

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creator: reviewer_pyup

<content>
summary:
The paper explores the incorporation of randomized algorithms into transformer models through learning, in order to enhance robustness and performance in adversarial environments. By training transformers on expected and adversarial losso, single and multiple seeds, the study demonstrates that their proposed approach can outperform their deterministic counterparts, particularly when faced with adversarial challenges. The authors focus on three main tasks: associative recall, graph coloring and exploration in grid worlds, to evaluate the impact of randomized strategies on model performance. The results indicate that, by employing randomness, transformers achieve higher robustness and adaptability (implementing simple strategies repetition or majority voting) over predictions with different random seeds.

strengths:
1) Innovative integration of randomization: Altough randomization was a concept that has been used with Transformers during the years and at different levels (e.g. positional encodings, attention weights) the paper introduces an original concept of definining randomized algorithms within neural networks through learning: by simple epmloying repetitions and strategies such as majority voting this approach outperforms deterministic approach

2) Comprehensive experimental design: The paper provides a solid experimental framework across varied tasks, such as associative recall, graph coloring, and grid world exploration. Each task is carefully selected to illustrate different advantages of randomization, such as memory management, combinatorial problem-solving, and exploration.

3) Theoretical justifications and adversarially driven objective : The authors define a description of theoretical considerations about randomized algorithms that bring to the definition of an adversarially driven objective: it is an interesting addition that supports the study's claims on robustness. The use of a relaxed adversarial loss and exploration of different adversarial strengths adds depth and completeness to the analysis of transformer behavior in challenging environments.

4) Empirical results on robustness: The empirical results are thorough and show randomized transformers advantages, eespecially in scenarios with worst-case inputs. The paper's analysis highlights how majority voting and the number of seeds influence performance, enhancing the robustness of the findings. Moreover tha authors discuss about their results and the major limitations of their approach.

5) Appendix: Further information on the training modality and parameters is appreciated.

weaknesses:
1) Scalability challenges: The most important corcern about the proposed methodology is the computationally cost, particularly with the reliance on multiple seeds and adversarial loss training. The authors acknowledge this in the "Summary and Limitations" paragraph, noting that scaling the approach to larger settings may require significant computational resources, which limits the practicality and broader applicability of the approach. I consider that such a problem should have been addressed in a more in-depth manner and not relegated to a second analysis, as it is a very important measure for judging the entire methodology.

2) Limited practical applications discussed: While the theoretical foundation is strong, the paper could benefit from discussing more real-world applications where randomized transformers might be beneficial. 

3) Dependency on the choice of q and m hyperparameters: The model's performance and randomization effectiveness heavily depend on hyperparameters, particularly q and m. The authors could delve further into automated ways to tune these values or provide guidelines for choosing optimal parameters for various tasks. High-computation hardware needed to perfom analysis about the hyperparameters can be prohibitive.

questions:
What were the reasons that pushed the authors not to present analyses (even theoretical) regarding the computational cost preferring to insert more general information such as section E of the appendix?
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<thread>
creation date:
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creator: reviewer_y3s8

<content>
summary:
Randomized algorithms can offer better worst-case performance than deterministic ones.  
Can transformers learn randomized algorithms?   
The authors study conditions for randomized algorithms to yield possible advantages and they offer a training methodology to emerge them in transformer networks. Namely design choices to emerge randomized algorithms involve around: a. the model; it needs to have limited capacity and receive random signal at its input, b. the loss function; it needs to be a surrogate for worst-input risk instead of expected risk.
They demonstrate application in three conceptual tasks: associative recall, graph coloring, agent exploring a grid-world
And they claim that randomized algorithms achieve increased robustness against adversaries.

strengths:
Very well-written paper, easy to follow, with extensive examples and related work material both in the main paper and in the appendix.
Well-motivated and executed study for the possibility to learn randomized algorithms.

1. Convincing results of superior worst-case performance in the considered tasks when compared to training alternatives, ablating some aspect of the loss, the multi-seed version trained on the relaxed adversarial loss is the best.
2. Experiments cover well important hyperparameters for the training setting. namely the relaxation hyperparameter which interpolates between ERM and worst-case risk, $q$, and the number of seeds per input, $m$.
3. Experiments cover variety of architectural constraints, such as linear attention (sec 3.2), structured local attention masks that are GNN-like (sec 3.3), and autoregressive settings (sec 3.4). As well as a variety of tasks and optimization techniques.

The paper finds the reader with a clear impression about the possibility of randomized algorithms for improving worst-case performance, as evidence by 95th percentile results in Figs 2,3,5 and 7. As well as inference-time strategies on improving them with majority voting.

weaknesses:
1.a. I would personally have enjoyed seeing experiments with actual adversarial robustness training, using a threat model on human preference alignment data, applied at generative LLMs.  

1.b. Lines 216-218 mention that adversarial robustness training is difficult, however recent literature has provided with framework for that [ 1]. While it does not decrease the contributions of this paper, comparing with the methodology mentioned in the paper would have been great!

[1] Xhonneux et al., “Efficient Adversarial Training in LLMs with Continuous Attacks”, May 2024 to appear at NeurIPS 2024.

2. Section 2.1 “Excessive Model Capacity Will not Enforce Randomness” the argument is not very convincing to me. In the face of a multi-seed worst-case objective like one described later, does the argument still hold? Proposition 1 also seems to go against the impossibility eluded by this section. Experiments on finetuning medium-sized (~1B) models on toy tasks could have provided some extra evidence for that.

3. On grid-world experiments: While the first episode demonstrates randomness in ways to explore for the target, why doesn’t the randomness follow as well in the second “exploitation” episode? There are multiple equivalent in length ways to reach the target. I would expect that in Figure 6.C. those would vary across different seeds.

4. Instead of using input randomness, it would have been very interesting to understand the ability or limitations on using output sample randomness which is fed back to the input in an autoreregressive case, as a drive to learn randomized algorithms. The authors avoid this explicitly in Section 3.4, however I do not understand the reason.


## Typos
L141: “for any distribution **over** $\mathcal{X}$ and $\mathcal{R}$.  
L233: “hyperparameters $m$ and $\mathbf{q}$.  
L965, L895, L853: change $p$ for $q$.  


## Missing citations
On linear attention (L291): Katharopoulos et al., “Transformers are RNNs: Fast Autoregressive Transformers with Linear Attention”, ICML 2020.

questions:
1. The transition to use the q-norm relaxed loss in L195-200 is not motivated sufficiently at that point in text. Can you elaborate on it a bit more?

2. L207-208: Are the same random input seeds reused for all inputs? Why? What would happen if fresh seeds are sampled for every input $x_i$?  

3. Batching strategy when finetuning for the emergence of a randomized algorithm is not clear at the text. I assume that $B \times N$ samples are processed independently at once. $B$ is the number of input data, $N$ is the number of random seeds. Is this correct?
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creator: authors

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title:
General Response about real-world applications and scalability

comment:
We want to thank all reviewers for their time and effort in evaluating this work and the generally positive reviews. We address here in a general answer the concerns about the scalability of the approach. 

1) We want to highlight that the goal of this paper is to build a theoretical understanding, backed by empirical evidence, providing insights under which assumptions neural networks, through learning, become randomized algorithms. We therefore focused on studying well known problems from the randomized algorithms literature, for which powerful randomized algorithms are known and well studied. 

2) Although the proposed objective does introduce an overhead compared to common empirical risk minimization (ERM), other related adversarial training objectives can also be considerably more expensive than ERM. For example, common gradient based adversarial training techniques require up to 10-100x more compute, see for example Table 1 of Madry et al., 2018. Furthermore, we stress that in all of our experiments rather small $m=10$ (leading to 10x more compute) led to near optimal performance, when compared to larger $m>10$. 

3) We want to put forward more comments and ideas on how to address scalability issues with the current approach:
  - Fine-tuning: We find an interesting future research avenue to investigate if pre-trained models can be randomized by fine-tuning on our objective. 
  - A relaxation of this abrupt 1) pre-training and 2) fine-tuning stages, would be to gradually increase $m$ and $q$ of our objective during training. In the beginning of training, the model could be trained as usually done via ERM (q=1, m=1) and then gradually increased to enforce randomization e.g. q=10, m=10. We argue that this is a scalable and intriguing future direction.	

4) We agree with the reviewers, and will highlight this in the revised manuscript, that other objectives, leveraging gradient based attacks, could exchange the arguably expensive  $m$ forward passes, which can easily be parallelized, with the costs of finding attacks by gradients. We agree that this is a promising future direction in order to scale our approach to real world settings. We will add, at least in the appendix if space does not permit, a discussion on real-world applications and possible future directions.  
Here, we chose our training objective, the relaxation of the min-max objective through the q-norm, to allow differentiability with respect to the parameters while tackling problems where differentiating with respect to the inputs, and therefore allowing for gradient based attacks, is not possible. Note that this situation is common for problems in field of randomized algorithms, reflected by the tasks that we study in our work.  

We again want to express our sincere gratitude for this set of high quality and insightful comments and suggestions which benefited the quality of the work greatly. We welcome any further suggestions and questions.
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creation date:
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thread_type:
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creator: area_chair_bguj

<content>
comment:
Dear reviewers,

A reminder that **November, 26** is the last day to interact with the authors, before the private discussion with the area chairs. At the very least, please acknowledge having read the rebuttal (if present). If the rebuttal was satisfying, please improve your score accordingly. Finally, if you have concerns that might be solved in time, this is the last chance before moving on to the next phase.

Thanks,
The AC
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Metareview:
metareview:
The paper considers transformers endowed with a further random seed as input, which are used to learn randomized algorithms. They show (both theoretically and empirically) that this is beneficial in adversarial situations, where it is known from standard computer science that the added randomness "breaks" the adversary attack.

The paper had 4 reviews, all of which were positive on the contributions and recommended acceptance. The authors provided a very strong rebuttal which clarified several points, and the reviewers maintained their positive score.

In general, all reviewers praised the paper's writing, its contributions, and the novelty. Their concerns were generally aligned (e.g., computational complexity, as described below) but well addressed in the rebuttal. No concerns remained after the rebuttal.

In summary, the paper is a very strong contribution describing an innovative framework for exploiting randomness in learning algorithms with transformers, which may have an impact also outside the scope of algorithmic reasoning. I see no reason to override the reviewer's consensus and I strongly recommend acceptance.