Related papers: Sums and differences of sets (improvement over Alp…
The rapid advancement of Large Language Models (LLMs) poses a significant challenge to existing mathematical reasoning benchmarks. However, these benchmarks tend to become easier over time as LLMs can learn from the published benchmarks.…
We establish new exponential in dimension lower bounds for the Maximum Halfspace Discrepancy problem, which models linear classification. Both are fundamental problems in computational geometry and machine learning in their exact and…
This paper demonstrates that by fine-tuning an autoregressive language model (GPT-Neo) on appropriately structured step-by-step demonstrations, it is possible to teach it to execute a mathematical task that has previously proved difficult…
Recently, Arjevani et al. [1] established a lower bound of iteration complexity for the first-order optimization under an $L$-smooth condition and a bounded noise variance assumption. However, a thorough review of existing literature on…
The DifSets package for GAP implements an algorithm for enumerating all difference sets in a group up to equivalence and provides access to a library of results. The algorithm functions by finding difference sums, which are potential images…
Ensembling multiple predictions is a widely used technique for improving the accuracy of various machine learning tasks. One obvious drawback of ensembling is its higher execution cost during inference. In this paper, we first describe our…
Merging together a result of Nathanson from the early 70s and a recent result of Granville and Walker, we show that for any finite set $A$ of integers with $\min(A)=0$ and $\gcd(A)=1$ there exist two sets, the "head" and the "tail", such…
We provide optimal upper bounds on the growth of iterated sumsets $hA=A+\dots+A$ for finite subsets $A$ of abelian semigroups. More precisely, we show that the new upper bounds recently derived from Macaulay's theorem in commutative algebra…
The main result in this paper concerns a new five-variable expander. It is proven that for any finite set of real numbers $A$, $$|\{(a_1+a_2+a_3+a_4)^2+\log a_5 :a_1,a_2,a_3,a_4,a_5 \in A \}| \gg \frac{|A|^2}{\log |A|}.$$ This bound is…
The sum of square roots is as follows: Given $x_1,\dots,x_n \in \mathbb{Z}$ and $a_1,\dots,a_n \in \mathbb{N}$ decide whether $ E=\sum_{i=1}^n x_i \sqrt{a_i} \geq 0$. It is a prominent open problem (Problem 33 of the Open Problems Project),…
Deep learning models have been criticized for their lack of easy interpretation, which undermines confidence in their use for important applications. Nevertheless, they are consistently utilized in many applications, consequential to…
Large language models (LLMs) exhibit complementary strengths arising from differences in pretraining data, model architectures, and decoding behaviors. Inference-time ensembling provides a practical way to combine these capabilities without…
A well-known discovery of Feige's is the following: Let $X_1, \ldots, X_n$ be nonnegative independent random variables, with $\mathbb{E}[X_i] \leq 1 \;\forall i$, and let $X = \sum_{i=1}^n X_i$. Then for any $n$, \[\Pr[X < \mathbb{E}[X] +…
Recent advances in optimization theory have shown that smooth strongly convex finite sums can be minimized faster than by treating them as a black box "batch" problem. In this work we introduce a new method in this class with a theoretical…
We obtain stabilization conditions and large time estimates for weak solutions of the inequality $$ \sum_{|\alpha| = m} \partial^\alpha a_\alpha (x, t, u) - u_t \ge f (x, t) g (u) \quad \mbox{in } \Omega \times (0, \infty), $$ where…
Modern approaches for semantic segmentation usually employ dilated convolutions in the backbone to extract high-resolution feature maps, which brings heavy computation complexity and memory footprint. To replace the time and memory…
Given the prevalence of crowd sourced labor in creating Natural Language processing datasets, these aforementioned sets have become increasingly large. For instance, the SQUAD dataset currently sits at over 80,000 records. However, because…
We explore an evolutionary search strategy for scaling inference time compute in Large Language Models. The proposed approach, Mind Evolution, uses a language model to generate, recombine and refine candidate responses. The proposed…
New lower bounds involving sum, difference, product, and ratio sets for a set $A\subset \C$ are given. The estimates involving the sum set match, up to constants, the one obtained by Solymosi for the reals and are obtained by generalising…
Deep Ensembles, as a type of Bayesian Neural Networks, can be used to estimate uncertainty on the prediction of multiple neural networks by collecting votes from each network and computing the difference in those predictions. In this paper,…