Can Mamba Always Enjoy the "Free Lunch"?

Transformers have been the cornerstone of current Large Language Models (LLMs); however, its linear growth in overhead during inference with respect to sequence length poses challenges for modeling long sequences. In this context, Mamba has gradually attracted attention due to its constant-level size during inference and existing empirical results have shown that it can perform comparably to Transformers in sequence modeling while offering significant savings. However, one may ask that, can Mamba always enjoy the ``free lunch"? In this paper, we focus on analyzing the expressive ability of Mamba from a theoretical standpoint. First, inspired by the connection between Mamba and linear attention, we investigate potential shortcomings of the Mamba when performing the COPY operation. Our results indicate that Mamba with constant size may encounter bottlenecks when handling COPY, while it can achieve perfect performance when the size scales linearly with sequence length. Based on this observation, we analyze Mamba's ability to tackle DP problems when equipped with Chain of Thought (CoT). Our findings suggest that to solve arbitrary DP problems, the total cost of Mamba is comparable to standard and efficient Transformers. However, similar to efficient Transformers, when facing DP problems with favorable properties such as locality, Mamba can provide savings in overhead. Our results contribute to a deeper understanding of Mamba.

Understanding Fairness Surrogate Functions in Algorithmic Fairness

It has been observed that machine learning algorithms exhibit biased predictions against certain population groups. To mitigate such bias while achieving comparable accuracy, a promising approach is to introduce surrogate functions of the concerned fairness definition and solve a constrained optimization problem. However, an intriguing issue in previous work is that such fairness surrogate functions may yield unfair results. In this work, in order to deeply understand this issue, taking a widely used fairness definition, demographic parity as an example, we both theoretically and empirically show that there is a surrogate-fairness gap between the fairness definition and the fairness surrogate function. The "gap" directly determines whether a surrogate function is an appropriate substitute for a fairness definition. Also, the theoretical analysis and experimental results about the "gap" motivate us that the unbounded surrogate functions will be affected by the points far from the decision boundary, which is the large margin points issue investigated in this paper. To address it, we propose the general sigmoid surrogate with a rigorous and reliable fairness guarantee. Interestingly, the theory also provides insights into two important issues that deal with the large margin points as well as obtaining a more balanced dataset are beneficial to fairness. Furthermore, we elaborate a novel and general algorithm called Balanced Surrogate, which iteratively reduces the "gap" to improve fairness. Finally, we provide empirical evidence showing that our methods achieve better fairness performance in three real-world datasets.