Strassen Attention: Unlocking Compositional Abilities in Transformers Based on a New Lower Bound Method

We propose a novel method to evaluate the theoretical limits of Transformers, allowing us to prove the first lower bounds against one-layer softmax Transformers with infinite precision. We establish those bounds for three tasks that require advanced reasoning. The first task, Match3 (Sanford et al., 2023), requires looking at all triples of positions. The second and third tasks address compositionality-based reasoning: one is composition of functions (Peng et al., 2024) and the other is composition of binary relations. We formally prove the inability of one-layer softmax Transformers to solve any of these tasks. In an attempt to overcome these limitations, we introduce Strassen attention and prove that with this mechanism a one-layer Transformer can in principle solve all these tasks. We also show that it enjoys sub-cubic running-time complexity, making it more scalable than similar previously proposed mechanisms, such as higher-order attention (Sanford et al., 2023). To complement our theoretical findings, we experimentally studied Strassen attention and compared it against standard (Vaswani et al, 2017), higher-order attention (Sanford et al., 2023) and triangular attention (Bergen et al. 2021). Our results help to disentangle all these attention mechanisms, highlighting their strengths and limitations. In particular, Strassen attention outperforms standard attention significantly on all the tasks. Altogether, understanding the theoretical limitations can guide research towards scalable attention mechanisms that improve the reasoning abilities of Transformers.

A completely uniform transformer for parity

We construct a 3-layer constant-dimension transformer, recognizing the parity language, where neither parameter matrices nor the positional encoding depend on the input length. This improves upon a construction of Chiang and Cholak who use a positional encoding, depending on the input length (but their construction has 2 layers).

Effective Littlestone Dimension

Delle Rose et al.~(COLT'23) introduced an effective version of the Vapnik-Chervonenkis dimension, and showed that it characterizes improper PAC learning with total computable learners. In this paper, we introduce and study a similar effectivization of the notion of Littlestone dimension. Finite effective Littlestone dimension is a necessary condition for computable online learning but is not a sufficient one -- which we already establish for classes of the effective Littlestone dimension 2. However, the effective Littlestone dimension equals the optimal mistake bound for computable learners in two special cases: a) for classes of Littlestone dimension 1 and b) when the learner receives as additional information an upper bound on the numbers to be guessed. Interestingly, finite effective Littlestone dimension also guarantees that the class consists only of computable functions.

Simple online learning with consistency oracle

We consider online learning in the model where a learning algorithm can access the class only via the consistency oracle -- an oracle, that, at any moment, can give a function from the class that agrees with all examples seen so far. This model was recently considered by Assos et al. (COLT'23). It is motivated by the fact that standard methods of online learning rely on computing the Littlestone dimension of subclasses, a problem that is computationally intractable. Assos et al. gave an online learning algorithm in this model that makes at most $C^d$ mistakes on classes of Littlestone dimension $d$, for some absolute unspecified constant $C > 0$. We give a novel algorithm that makes at most $O(256^d)$ mistakes. Our proof is significantly simpler and uses only very basic properties of the Littlestone dimension. We also observe that there exists no algorithm in this model that makes at most $2^{d+1}-2$ mistakes. We also observe that our algorithm (as well as the algorithm of Assos et al.) solves an open problem by Hasrati and Ben-David (ALT'23). Namely, it demonstrates that every class of finite Littlestone dimension with recursively enumerable representation admits a computable online learner (that may be undefined on unrealizable samples).

Find a witness or shatter: the landscape of computable PAC learning

This paper contributes to the study of CPAC learnability -- a computable version of PAC learning -- by solving three open questions from recent papers. Firstly, we prove that every improperly CPAC learnable class is contained in a class which is properly CPAC learnable with polynomial sample complexity. This confirms a conjecture by Agarwal et al (COLT 2021). Secondly, we show that there exists a decidable class of hypothesis which is properly CPAC learnable, but only with uncomputably fast growing sample complexity. This solves a question from Sterkenburg (COLT 2022). Finally, we construct a decidable class of finite Littlestone dimension which is not improperly CPAC learnable, strengthening a recent result of Sterkenburg (2022) and answering a question posed by Hasrati and Ben-David (ALT 2023). Together with previous work, our results provide a complete landscape for the learnability problem in the CPAC setting.

No Agreement Without Loss: Learning and Social Choice in Peer Review

In peer review systems, reviewers are often asked to evaluate various features of submissions, such as technical quality or novelty. A score is given to each of the predefined features and based on these the reviewer has to provide an overall quantitative recommendation. However, reviewers differ in how much they value different features. It may be assumed that each reviewer has her own mapping from a set of criteria scores (score vectors) to a recommendation, and that different reviewers have different mappings in mind. Recently, Noothigattu, Shah and Procaccia introduced a novel framework for obtaining an aggregated mapping by means of Empirical Risk Minimization based on $L(p,q)$ loss functions, and studied its axiomatic properties in the sense of social choice theory. We provide a body of new results about this framework. On the one hand we study a trade-off between strategy-proofness and the ability of the method to properly capture agreements of the majority of reviewers. On the other hand, we show that dropping a certain unrealistic assumption makes the previously reported results to be no longer valid. Moreover, in the general case, strategy-proofness fails dramatically in the sense that a reviewer is able to make significant changes to the solution in her favor by arbitrarily small changes to their true beliefs. In particular, no approximate version of strategy-proofness is possible in this general setting since the method is not even continuous w.r.t. the data. Finally we propose a modified aggregation algorithm which is continuous and show that it has good axiomatic properties.