Feyza Duman Keles

Feyza Duman Keles, Pruthuvi Mahesakya Wijewardena, Chinmay Hegde

Transformer architectures have led to remarkable progress in many state-of-art applications. However, despite their successes, modern transformers rely on the self-attention mechanism, whose time- and space-complexity is quadratic in the length of the input. Several approaches have been proposed to speed up self-attention mechanisms to achieve sub-quadratic running time; however, the large majority of these works are not accompanied by rigorous error guarantees. In this work, we establish lower bounds on the computational complexity of self-attention in a number of scenarios. We prove that the time complexity of self-attention is necessarily quadratic in the input length, unless the Strong Exponential Time Hypothesis (SETH) is false. This argument holds even if the attention computation is performed only approximately, and for a variety of attention mechanisms. As a complement to our lower bounds, we show that it is indeed possible to approximate dot-product self-attention using finite Taylor series in linear-time, at the cost of having an exponential dependence on the polynomial order.

Feyza Duman Keles, Chinmay Hegde

The objective of generative model inversion is to identify a size-$n$ latent vector that produces a generative model output that closely matches a given target. This operation is a core computational primitive in numerous modern applications involving computer vision and NLP. However, the problem is known to be computationally challenging and NP-hard in the worst case. This paper aims to provide a fine-grained view of the landscape of computational hardness for this problem. We establish several new hardness lower bounds for both exact and approximate model inversion. In exact inversion, the goal is to determine whether a target is contained within the range of a given generative model. Under the strong exponential time hypothesis (SETH), we demonstrate that the computational complexity of exact inversion is lower bounded by $Ω(2^n)$ via a reduction from $k$-SAT; this is a strengthening of known results. For the more practically relevant problem of approximate inversion, the goal is to determine whether a point in the model range is close to a given target with respect to the $\ell_p$-norm. When $p$ is a positive odd integer, under SETH, we provide an $Ω(2^n)$ complexity lower bound via a reduction from the closest vectors problem (CVP). Finally, when $p$ is even, under the exponential time hypothesis (ETH), we provide a lower bound of $2^{Ω(n)}$ via a reduction from Half-Clique and Vertex-Cover.

Noah Amsel, Pratyush Avi, Tyler Chen et al.

We study the problem of learning a structured approximation (low-rank, sparse, banded, etc.) to an unknown matrix $A$ given access to matrix-vector product (matvec) queries of the form $x \rightarrow Ax$ and $x \rightarrow A^Tx$. This problem is of central importance to algorithms across scientific computing and machine learning, with applications to fast multiplication and inversion for structured matrices, building preconditioners for first-order optimization, and as a model for differential operator learning. Prior work focuses on obtaining query complexity upper and lower bounds for learning specific structured matrix families that commonly arise in applications. We initiate the study of the problem in greater generality, aiming to understand the query complexity of learning approximations from general matrix families. Our main result focuses on finding a near-optimal approximation to $A$ from any finite-sized family of matrices, $\mathcal{F}$. Standard results from matrix sketching show that $O(\log|\mathcal{F}|)$ matvec queries suffice in this setting. This bound can also be achieved, and is optimal, for vector-matrix-vector queries of the form $x,y\rightarrow x^TAy$, which have been widely studied in work on rank-$1$ matrix sensing. Surprisingly, we show that, in the matvec model, it is possible to obtain a nearly quadratic improvement in complexity, to $\tilde{O}(\sqrt{\log|\mathcal{F}|})$. Further, we prove that this bound is tight up to log-log factors.Via covering number arguments, our result extends to well-studied infinite families. As an example, we establish that a near-optimal approximation from any \emph{linear matrix family} of dimension $q$ can be learned with $\tilde{O}(\sqrt{q})$ matvec queries, improving on an $O(q)$ bound achievable via sketching techniques and vector-matrix-vector queries.