Zhiyuan Li

Zhiyuan Li, Hong Liu, Denny Zhou et al.

Instructing the model to generate a sequence of intermediate steps, a.k.a., a chain of thought (CoT), is a highly effective method to improve the accuracy of large language models (LLMs) on arithmetics and symbolic reasoning tasks. However, the mechanism behind CoT remains unclear. This work provides a theoretical understanding of the power of CoT for decoder-only transformers through the lens of expressiveness. Conceptually, CoT empowers the model with the ability to perform inherently serial computation, which is otherwise lacking in transformers, especially when depth is low. Given input length $n$, previous works have shown that constant-depth transformers with finite precision $\mathsf{poly}(n)$ embedding size can only solve problems in $\mathsf{TC}^0$ without CoT. We first show an even tighter expressiveness upper bound for constant-depth transformers with constant-bit precision, which can only solve problems in $\mathsf{AC}^0$, a proper subset of $ \mathsf{TC}^0$. However, with $T$ steps of CoT, constant-depth transformers using constant-bit precision and $O(\log n)$ embedding size can solve any problem solvable by boolean circuits of size $T$. Empirically, enabling CoT dramatically improves the accuracy for tasks that are hard for parallel computation, including the composition of permutation groups, iterated squaring, and circuit value problems, especially for low-depth transformers.

Shizhong Han, Zibo Meng, Zhiyuan Li et al.

Recognizing facial action units (AUs) during spontaneous facial displays is a challenging problem. Most recently, Convolutional Neural Networks (CNNs) have shown promise for facial AU recognition, where predefined and fixed convolution filter sizes are employed. In order to achieve the best performance, the optimal filter size is often empirically found by conducting extensive experimental validation. Such a training process suffers from expensive training cost, especially as the network becomes deeper. This paper proposes a novel Optimized Filter Size CNN (OFS-CNN), where the filter sizes and weights of all convolutional layers are learned simultaneously from the training data along with learning convolution filters. Specifically, the filter size is defined as a continuous variable, which is optimized by minimizing the training loss. Experimental results on two AU-coded spontaneous databases have shown that the proposed OFS-CNN is capable of estimating optimal filter size for varying image resolution and outperforms traditional CNNs with the best filter size obtained by exhaustive search. The OFS-CNN also beats the CNN using multiple filter sizes and more importantly, is much more efficient during testing with the proposed forward-backward propagation algorithm.