Summary of Major Achievements:
In the two years since the inception of this STG project, me and my research group have made a substantial progress on the following four
problems mentioned in the CODY project proposal:
•Problem 2: How can dynamic data structures and sketching reduce the cost-per-iteration of IPMs and other path-following homotopy methods?
•Problem 3: Are Linear Programs and Linear systems computationally equivalent?
•Problem 4 : Can we prove non-black-box lower bounds on the iteration complexity of interior point methods (IPMs)?
•Problem 5 : How can we design practical algorithms for computing high-accuracy approximate matrix products in close to O(n^2) (or ideally matrix sparsity) time?
For Problems 2 and 4, my research group has developed novel upper and lower bounds on the iteration complexity and cost-per-iteration of Interior Point Methods
(IPMS) for Linear and Semidefinite Programming (items 5+6 and papers [2],[8] below), perhaps the most important question of this Grant project. Another set of
main achievements of this reporting period (addressing Problem 3 and 5) include novel dynamic data structures for estimating distances of general symmetric norms
(paper [2] below), for sparse recovery of Fourier-spare signals (paper [3]), and for dynamic least-squares regression (paper [6], resolving the complexity of a
fundamental problem in data analysis);
**UPDATE** : Papers [9] + [10] accepted to STOC'25 .
*UPCOMING Papers and work in progress:
*For Problem 5: In my upcoming (solo-author) paper on ”Approximate Matrix Multiplication via Spherical Convolutions” (under preparation), I present a cheaper
(quadratic instead of cubic-time) Bilinear operator as an alternative to MatMul in deep neural networks, which preserves the number of parameters of the model,
providing an end-to-end speedup of up to x3 in FLOPs over naiive MatMul.
*In an upcoming paper with an ERC Postdoc (Item [4] below, to be submitted to STOC'25), we have developed the first accelerated multiplicative-weights method
for approximate packing/covering LPs after nearly 20 years of research (Garg-Konneman,2007), providing quadratic speed up in the number of iterations (m --> m^{1/2})
using a different oracle. I was invited to give this preliminary track at the Simons Institute Data Structures & Optimization Semester (Berkeley, Nov'23).
* In another upcoming project with an ERC Postdoc (Item [7] below), we have developed a novel approach To the longstanding Dynamic Optimzality Conjecture
(Tarjan & Sleator, 1985), based on a continuous relaxation -- We have managed to encode the execution of any online binary search tree on a sequence x1...xm of lookups,
as an (implicit) online flow-based LP. Using This formulation provides an approach for attacking the Dynamic Optimality conjecture via online convex-body chasing,
and designing An entire new class of BST algorithms based on regularized gradient descent (online Bergman projections).
[1] [Y. Deng, Z.Song , O.WeinsteinC R.Zhang] Fast Distance Oracles for Any Symmetric Norm. (2022). NeurIPS23.
[2] [S.Jiang B.Peng O.Weinstein] Dynamic Least-Squares regression. 2023 IEEE Symposium on Foundations of Computer Science (FOCS23).
[3] [Z. Song, B. Sun, O. Weinstein, R. Zhang] Quartic Samples Suffice for Fourier Interpolation. 2023 IEEE Symposium on Foundations of Computer Science (FOCS23).
[4] [Z. Kuhn-Koh, O.Weinstein S.Yincharantawanchai]: An Accelerated Multiplicative-Weights Update Method for Implicit Packing-Covering LPs (to be submitted to STOC'25)
[5] [S.Jiang D. Ming, R.Kyng O.Weinstein]: How Many Linear Systems are Required to Solve a Linear Program? (To be submitted to ITCS'24)
[6] [S.Jiang A.Vladu O.Weinstein]: IPMs Beyond Self-Concordance via Energy-Based Perturbations (In preparation).
[7] [D.Dorfman P.Kalmansook O.Weinstein S.Yincharantawanchai]: Dynamic Optimality via Continuous Optimization (work in progress).
[8] [S.Jiang B. Natura , O.Weinstein]: “A Faster Interior-Point Method for Sum of Squares Optimization” (ICALP'22)
[9] [Z. Kuhn-Koh, O.Weinstein S.Yincharantawanchai]: Approximating the Held–Karp Bound of Metric TSP in Nearly-Linear Work and Polylogarithmic Depth (accepted to STOC'25)
[10] [A. Andoni, S. Jiang, O.Weinstein]: "A New Framework for Building Data Structures from Communication Protocols" (accepted to STOC'25)
