Lower Complexity Bounds of Finite-Sum Optimization Problems: The Results and Construction

The contribution of this paper includes two aspects. First, we study the lower bound complexity for the minimax optimization problem whose objective function is the average of $n$ individual smooth component functions. We consider Proximal Incremental First-order (PIFO) algorithms which have access to gradient and proximal oracle for each individual component. We develop a novel approach for constructing adversarial problems, which partitions the tridiagonal matrix of classical examples into $n$ groups. This construction is friendly to the analysis of incremental gradient and proximal oracle. With this approach, we demonstrate the lower bounds of first-order algorithms for finding an $\varepsilon$-suboptimal point and an $\varepsilon$-stationary point in different settings. Second, we also derive the lower bounds of minimization optimization with PIFO algorithms from our approach, which can cover the results in \citep{woodworth2016tight} and improve the results in \citep{zhou2019lower}

Revisiting Co-Occurring Directions: Sharper Analysis and Efficient Algorithm for Sparse Matrices

We study the streaming model for approximate matrix multiplication (AMM). We are interested in the scenario that the algorithm can only take one pass over the data with limited memory. The state-of-the-art deterministic sketching algorithm for streaming AMM is the co-occurring directions (COD), which has much smaller approximation errors than randomized algorithms and outperforms other deterministic sketching methods empirically. In this paper, we provide a tighter error bound for COD whose leading term considers the potential approximate low-rank structure and the correlation of input matrices. We prove COD is space optimal with respect to our improved error bound. We also propose a variant of COD for sparse matrices with theoretical guarantees. The experiments on real-world sparse datasets show that the proposed algorithm is more efficient than baseline methods