Hantao Yu

Robi Bhattacharjee, Max Hopkins, Akash Kumar et al.

Developing simple, sample-efficient learning algorithms for robust classification is a pressing issue in today's tech-dominated world, and current theoretical techniques requiring exponential sample complexity and complicated improper learning rules fall far from answering the need. In this work we study the fundamental paradigm of (robust) $\textit{empirical risk minimization}$ (RERM), a simple process in which the learner outputs any hypothesis minimizing its training error. RERM famously fails to robustly learn VC classes (Montasser et al., 2019a), a bound we show extends even to `nice' settings such as (bounded) halfspaces. As such, we study a recent relaxation of the robust model called $\textit{tolerant}$ robust learning (Ashtiani et al., 2022) where the output classifier is compared to the best achievable error over slightly larger perturbation sets. We show that under geometric niceness conditions, a natural tolerant variant of RERM is indeed sufficient for $γ$-tolerant robust learning VC classes over $\mathbb{R}^d$, and requires only $\tilde{O}\left( \frac{VC(H)d\log \frac{D}{γδ}}{ε^2}\right)$ samples for robustness regions of (maximum) diameter $D$.

Hantao Yu

We prove that the minimum edge-outerplanarity of a planar graph can be computed in polynomial time, resolving an open problem of Bentz (2009). The proof was initially produced by GPT~5.5 Pro and then verified and polished manually.

Barna Saha, Yinzhan Xu, Christopher Ye et al.

The transformer has revolutionized modern AI across language, vision, and beyond. It consists of $L$ layers, each running $H$ attention heads in parallel and feeding the combined output to the subsequent layer. In attention, the input consists of $N$ tokens, each a vector of dimension $m$. The attention mechanism involves multiplying three $N \times m$ matrices, applying softmax to an intermediate product. Several recent works have advanced our understanding of the complexity of attention. Known algorithms for transformers compute each attention head independently. This raises a fundamental question that has recurred throughout TCS under the guise of ``direct sum'' problems: can multiple instances of the same problem be solved more efficiently than solving each instance separately? Many answers to this question, both positive and negative, have arisen in fields spanning communication complexity and algorithm design. Thus, we ask whether transformers can be computed more efficiently than $LH$ independent evaluations of attention. In this paper, we resolve this question in the negative, and give the first non-trivial computational lower bounds for multi-head multi-layer transformers. In the small embedding regime ($m = N^{o(1)}$), computing $LH$ attention heads separately takes $LHN^{2 + o(1)}$ time. We establish that this is essentially optimal under SETH. In the large embedding regime ($m = N$), one can compute $LH$ attention heads separately using $LHN^{ω+ o(1)}$ arithmetic operations (plus exponents), where $ω$ is the matrix multiplication exponent. We establish that this is optimal, by showing that $LHN^{ω- o(1)}$ arithmetic operations are necessary when $ω> 2$. Our lower bound in the large embedding regime relies on a novel application of the Baur-Strassen theorem, a powerful algorithmic tool underpinning the famous backpropagation algorithm.

Omri Ben-Eliezer, Max Hopkins, Chutong Yang et al.

We initiate the study of active learning polynomial threshold functions (PTFs). While traditional lower bounds imply that even univariate quadratics cannot be non-trivially actively learned, we show that allowing the learner basic access to the derivatives of the underlying classifier circumvents this issue and leads to a computationally efficient algorithm for active learning degree-$d$ univariate PTFs in $\tilde{O}(d^3\log(1/\varepsilonδ))$ queries. We also provide near-optimal algorithms and analyses for active learning PTFs in several average case settings. Finally, we prove that access to derivatives is insufficient for active learning multivariate PTFs, even those of just two variables.