Alexander Kozachinskiy

Valentino Delle Rose, Alexander Kozachinskiy, Cristóbal Rojas et al.

The Weisfeiler--Lehman (WL) test is a fundamental iterative algorithm for checking isomorphism of graphs. It has also been observed that it underlies the design of several graph neural network architectures, whose capabilities and performance can be understood in terms of the expressive power of this test. Motivated by recent developments in machine learning applications to datasets involving three-dimensional objects, we study when the WL test is {\em complete} for clouds of euclidean points represented by complete distance graphs, i.e., when it can distinguish, up to isometry, any arbitrary such cloud. %arbitrary clouds of euclidean points represented by complete distance graphs. % How many dimensions of the Weisfeiler--Lehman test is enough to distinguish any two non-isometric point clouds in $d$-dimensional Euclidean space, assuming that these point clouds are given as complete graphs labeled by distances between the points? This question is important for understanding, which architectures of graph neural networks are capable of fully exploiting the spacial structure of a point cloud. Our main result states that the $(d-1)$-dimensional WL test is complete for point clouds in $d$-dimensional Euclidean space, for any $d\ge 2$, and that only three iterations of the test suffice. We also observe that the $d$-dimensional WL test only requires one iteration to achieve completeness. Our paper thus provides complete understanding of the 3-dimensional case: it was shown in previous works that 1-WL is not complete in $\mathbb{R}^3$, and we show that 2-WL is complete there. We also strengthen the lower bound for 1-WL by showing that it is unable to recognize planar point clouds in $\mathbb{R}^3$. Finally, we show that 2-WL is not complete in $\mathbb{R}^6$, leaving as an open question, whether it is complete in $\mathbb{R}^{d}$ for $d = 4,5$.

Valentino Delle Rose, Alexander Kozachinskiy, Cristobal Rojas et al.

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.

Pablo Barcelo, Alexander Kozachinskiy, Anthony Widjaja Lin et al.

We contribute to the study of formal languages that can be recognized by transformer encoders. We focus on two self-attention mechanisms: (1) UHAT (Unique Hard Attention Transformers) and (2) AHAT (Average Hard Attention Transformers). UHAT encoders are known to recognize only languages inside the circuit complexity class ${\sf AC}^0$, i.e., accepted by a family of poly-sized and depth-bounded boolean circuits with unbounded fan-ins. On the other hand, AHAT encoders can recognize languages outside ${\sf AC}^0$), but their expressive power still lies within the bigger circuit complexity class ${\sf TC}^0$, i.e., ${\sf AC}^0$-circuits extended by majority gates. We first show a negative result that there is an ${\sf AC}^0$-language that cannot be recognized by an UHAT encoder. On the positive side, we show that UHAT encoders can recognize a rich fragment of ${\sf AC}^0$-languages, namely, all languages definable in first-order logic with arbitrary unary numerical predicates. This logic, includes, for example, all regular languages from ${\sf AC}^0$. We then show that AHAT encoders can recognize all languages of our logic even when we enrich it with counting terms. We apply these results to derive new results on the expressive power of UHAT and AHAT up to permutation of letters (a.k.a. Parikh images).

Alexander Kozachinskiy, Tomasz Steifer

We consider online learning in the model where a learning algorithm can access the class only via the \emph{consistent 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 computationally intractable problem. 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 show that there exists no algorithm in this model that makes less than $3^d$ mistakes.

Alexander Kozachinskiy, Tomasz Steifer, Przemysław Wałȩga

The transformer architecture is almost a decade old. Despite that, we still have a limited understanding of what this architecture can or cannot compute. For instance, can a 1-layer transformer solve PARITY -- or more generally -- which kinds of transformers can do it? Known constructions for PARITY have at least 2 layers and employ impractical features: either a length-dependent positional encoding, or hardmax, or layernorm without the regularization parameter, or they are not implementable with causal masking. We give a new construction of a transformer for PARITY with softmax, length-independent and polynomially bounded positional encoding, no layernorm, working both with and without causal masking. We also give the first lower bound for transformers solving PARITY -- by showing that it cannot be done with only one layer and one head.

Marcelo Arenas, Pablo Barceló, Luis Cofré et al.

Kleinberg and Mullainathan showed that, in principle, language generation is always possible: with sufficiently many positive examples, a learner can eventually produce sentences indistinguishable from those of a target language. However, the existence of such a guarantee does not speak to its practical feasibility. In this work, we show that even for simple and well-studied language families -- such as regular and context-free languages -- the number of examples required for successful generation can be extraordinarily large, and in some cases not bounded by any computable function. These results reveal a substantial gap between theoretical possibility and efficient learnability. They suggest that explaining the empirical success of modern language models requires a refined perspective -- one that takes into account structural properties of natural language that make effective generation possible in practice.

Alexander Kozachinskiy

In this note, we establish the hardness of approximation of the problem of computing the minimal size of a $δ$-sufficient reason for decision trees.

Pablo Barceló, Alexander Kozachinskiy, Miguel Romero Orth et al.

Despite the wide use of $k$-Nearest Neighbors as classification models, their explainability properties remain poorly understood from a theoretical perspective. While nearest neighbors classifiers offer interpretability from a "data perspective", in which the classification of an input vector $\bar{x}$ is explained by identifying the vectors $\bar{v}_1, \ldots, \bar{v}_k$ in the training set that determine the classification of $\bar{x}$, we argue that such explanations can be impractical in high-dimensional applications, where each vector has hundreds or thousands of features and it is not clear what their relative importance is. Hence, we focus on understanding nearest neighbor classifications through a "feature perspective", in which the goal is to identify how the values of the features in $\bar{x}$ affect its classification. Concretely, we study abductive explanations such as "minimum sufficient reasons", which correspond to sets of features in $\bar{x}$ that are enough to guarantee its classification, and "counterfactual explanations" based on the minimum distance feature changes one would have to perform in $\bar{x}$ to change its classification. We present a detailed landscape of positive and negative complexity results for counterfactual and abductive explanations, distinguishing between discrete and continuous feature spaces, and considering the impact of the choice of distance function involved. Finally, we show that despite some negative complexity results, Integer Quadratic Programming and SAT solving allow for computing explanations in practice.

César Bravo, Alexander Kozachinskiy, Cristóbal Rojas

We revisit the classical result of Morris et al.~(AAAI'19) that message-passing graphs neural networks (MPNNs) are equal in their distinguishing power to the Weisfeiler--Leman (WL) isomorphism test. Morris et al.~show their simulation result with ReLU activation function and $O(n)$-dimensional feature vectors, where $n$ is the number of nodes of the graph. By introducing randomness into the architecture, Aamand et al.~(NeurIPS'22) were able to improve this bound to $O(\log n)$-dimensional feature vectors, again for ReLU activation, although at the expense of guaranteeing perfect simulation only with high probability. Recently, Amir et al.~(NeurIPS'23) have shown that for any non-polynomial analytic activation function, it is enough to use just 1-dimensional feature vectors. In this paper, we give a simple proof of the result of Amit et al.~and provide an independent experimental validation of it.

Alexander Kozachinskiy, Alexander Shen, Tomasz Steifer

In perpetual voting, multiple decisions are made at different moments in time. Taking the history of previous decisions into account allows us to satisfy properties such as proportionality over periods of time. In this paper, we consider the following question: is there a perpetual approval voting method that guarantees that no voter is dissatisfied too many times? We identify a sufficient condition on voter behavior -- which we call 'bounded conflicts' condition -- under which a sublinear growth of dissatisfaction is possible. We provide a tight upper bound on the growth of dissatisfaction under bounded conflicts, using techniques from Kolmogorov complexity. We also observe that the approval voting with binary choices mimics the machine learning setting of prediction with expert advice. This allows us to present a voting method with sublinear guarantees on dissatisfaction under bounded conflicts, based on the standard techniques from prediction with expert advice.

Alexander Kozachinskiy, Felipe Urrutia, Hector Jimenez et al.

We propose the first method to show theoretical limitations for one-layer softmax transformers with arbitrarily many precision bits (even infinite). We establish those limitations for three tasks that require advanced reasoning. The first task, Match 3 (Sanford et al., 2023), requires looking at all possible token triplets in an input sequence. The second and third tasks address compositionality-based reasoning: function composition (Peng et al., 2024) and binary relations composition, respectively. We formally prove the inability of one-layer softmax Transformers to solve any of these tasks. To overcome these limitations, we introduce Strassen attention and prove that, equipped with this mechanism, a one-layer transformer can in principle solve all these tasks. Importantly, we 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.

Pablo Barceló, Alexander Kozachinskiy, Tomasz Steifer

The notion of \emph{rank} of a Boolean function has been a cornerstone in PAC learning theory, enabling quasipolynomial-time learning algorithms for polynomial-size decision trees. We present a novel characterization of rank, grounded in the well-known Transformer architecture. We show that the rank of a function $f$ corresponds to the minimum number of \emph{Chain of Thought} (CoT) steps required by a single-layer Transformer with hard attention to compute $f$. Based on this characterization we establish tight bounds on the number of CoT steps required for specific problems, showing that \(\ell\)-fold function composition necessitates exactly \(\ell\) CoT steps. Furthermore, we analyze the problem of identifying the position of the \(k\)-th occurrence of 1 in a Boolean sequence, proving that it requires \(k\) CoT steps. Finally, we introduce the notion of the multi-head rank that captures multi-head single-layer transformers, and perform the analysis of PAC-learnability of the classes of functions with bounded multi-head rank.

Alexander Kozachinskiy

In this note, we use the VC dimension technique to prove the first lower bound against one-layer softmax transformers with infinite precision. We do so for two tasks: function composition, considered by Peng, Narayanan, and Papadimitriou, and the SUM$_2$ task, considered by Sanford, Hsu, and Telgarsky.

Valentino Delle Rose, Alexander Kozachinskiy, Tomasz Steifer

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.

Alexander Kozachinskiy, Tomasz Steifer

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).

Pablo Barceló, Fabian Jogl, Alexander Kozachinskiy et al.

We propose EB-1WL, an edge-based color-refinement test, and a corresponding GNN architecture, EB-GNN. Our architecture is inspired by a classic triangle counting algorithm by Chiba and Nishizeki, and explicitly uses triangles during message passing. We achieve the following results: (1)~EB-1WL is significantly more expressive than 1-WL. Further, we provide a complete logical characterization of EB-1WL based on first-order logic, and matching distinguishability results based on homomorphism counting. (2)~In an important distinction from previous proposals for more expressive GNN architectures, EB-1WL and EB-GNN require near-linear time and memory on practical graph learning tasks. (3)~Empirically, we show that EB-GNN is a highly-efficient general-purpose architecture: It substantially outperforms simple MPNNs, and remains competitive with task-specialized GNNs while being significantly more computationally efficient.

Felipe Urrutia, Jorge Salas, Alexander Kozachinskiy et al.

An important aspect subtending language understanding and production is the ability to independently encode positional and symbolic information of the words within a sentence. In Transformers, positional information is typically encoded using Positional Encodings (PEs). One such popular PE, namely Rotary PE (RoPE), has been widely used due to its empirical success. Recently, it has been argued that part of RoPE's success emerges from its ability to encode robust positional and semantic information using large and small frequencies, respectively. In this work, we perform a deeper dive into the positional versus symbolic dichotomy of attention heads behavior, both at the theoretical and empirical level. We provide general definitions of what it means for a head to behave positionally or symbolically, prove that these are two mutually exclusive behaviors and develop a metric to quantify them. We apply our framework to analyze Transformer-based LLMs using RoPE and find that all heads exhibit a strong correspondence between behavior and frequency use. Finally, we introduce canonical tasks designed to be either purely positional or symbolic, and demonstrate that the Transformer performance causally relates to the ability of attention heads to leverage the appropriate frequencies. In particular, we show that we can control the Transformer performance by controlling which frequencies the attention heads can access. Altogether, our work provides a detailed understanding of RoPE, and how its properties relate to model behavior.

Hector Pasten, Felipe Urrutia, Hector Jimenez et al.

Understanding how Transformers work and how they process information is key to the theoretical and empirical advancement of these machines. In this work, we demonstrate the existence of two phenomena in Transformers, namely isolation and continuity. Both of these phenomena hinder Transformers to learn even simple pattern sequences. Isolation expresses that any learnable sequence must be isolated from another learnable sequence, and hence some sequences cannot be learned by a single Transformer at the same time. Continuity entails that an attractor basin forms around a learned sequence, such that any sequence falling in that basin will collapse towards the learned sequence. Here, we mathematically prove these phenomena emerge in all Transformers that use compact positional encoding, and design rigorous experiments, demonstrating that the theoretical limitations we shed light on occur on the practical scale.