Yikai Wu

Oren Mangoubi, Yikai Wu, Satyen Kale et al.

Consider the following optimization problem: Given $n \times n$ matrices $A$ and $Λ$, maximize $\langle A, UΛU^*\rangle$ where $U$ varies over the unitary group $\mathrm{U}(n)$. This problem seeks to approximate $A$ by a matrix whose spectrum is the same as $Λ$ and, by setting $Λ$ to be appropriate diagonal matrices, one can recover matrix approximation problems such as PCA and rank-$k$ approximation. We study the problem of designing differentially private algorithms for this optimization problem in settings where the matrix $A$ is constructed using users' private data. We give efficient and private algorithms that come with upper and lower bounds on the approximation error. Our results unify and improve upon several prior works on private matrix approximation problems. They rely on extensions of packing/covering number bounds for Grassmannians to unitary orbits which should be of independent interest.

Ori Press, Brandon Amos, Haoyu Zhao et al. · princeton, uw

Despite progress in language model (LM) capabilities, evaluations have thus far focused on models' performance on tasks that humans have previously solved, including in programming (Jimenez et al., 2024) and mathematics (Glazer et al., 2024). We therefore propose testing models' ability to design and implement algorithms in an open-ended benchmark: We task LMs with writing code that efficiently solves computationally challenging problems in computer science, physics, and mathematics. Our AlgoTune benchmark consists of 154 coding tasks collected from domain experts and a framework for validating and timing LM-synthesized solution code, which is compared to reference implementations from popular open-source packages. In addition, we develop a baseline LM agent, AlgoTuner, and evaluate its performance across a suite of frontier models. AlgoTuner uses a simple, budgeted loop that edits code, compiles and runs it, profiles performance, verifies correctness on tests, and selects the fastest valid version. AlgoTuner achieves an average 1.72x speedup against our reference solvers, which use libraries such as SciPy, sk-learn and CVXPY. However, we find that current models fail to discover algorithmic innovations, instead preferring surface-level optimizations. We hope that AlgoTune catalyzes the development of LM agents exhibiting creative problem solving beyond state-of-the-art human performance.

Yikai Wu, Haoyu Zhao, Sanjeev Arora

This position paper argues that the machine learning community should fundamentally rethink how AI-inspired methods are developed and evaluated for combinatorial optimization (CO). We present comprehensive empirical benchmarks comparing various recent AI-inspired GPU-based methods with several classical CPU-based solvers on the Maximum Independent Set (MIS) problem. Strikingly, even on in-distribution random graphs, leading AI-inspired methods are consistently outperformed by the state-of-the-art classical solver KaMIS, and some AI-inspired methods frequently fail to surpass even the simplest degree-based greedy heuristic. To better understand the source of these failures, we introduce a novel analysis, serialization, which reveals that non-backtracking AI methods, such as LTFT (based on GFlowNets), end up reasoning similarly to the simplest degree-based greedy heuristic, and thus worse than KaMIS. Our findings reveal three core issues: (1) Limited benchmarks and evaluation - AI-inspired methods are often tested only on small instances with very limited inference time, which covers up issues with scalability and resource usage; (2) Intrinsic hardness and learning limits - even under ideal, in-distribution conditions, learning-based approaches lag behind classical heuristics, highlighting inherent barriers that receive little attention; and (3) Insufficient use and understanding of classical heuristics - current learning frameworks often neglect to incorporate effective classical techniques. Although we use MIS as a testbed, similar gaps and challenges have been reported in other combinatorial optimization problems, suggesting broader relevance for our recommendations. We propose that future research must address these issues by rigorous benchmarking, deepening understanding of learning limitations, and integrating classical heuristics into AI-inspired methods.

David Pujol, Yikai Wu, Brandon Fain et al.

Large organizations that collect data about populations (like the US Census Bureau) release summary statistics that are used by multiple stakeholders for resource allocation and policy making problems. These organizations are also legally required to protect the privacy of individuals from whom they collect data. Differential Privacy (DP) provides a solution to release useful summary data while preserving privacy. Most DP mechanisms are designed to answer a single set of queries. In reality, there are often multiple stakeholders that use a given data release and have overlapping but not-identical queries. This introduces a novel joint optimization problem in DP where the privacy budget must be shared among different analysts. We initiate study into the problem of DP query answering across multiple analysts. To capture the competing goals and priorities of multiple analysts, we formulate three desiderata that any mechanism should satisfy in this setting -- The Sharing Incentive, Non-Interference, and Adaptivity -- while still optimizing for overall error. We demonstrate how existing DP query answering mechanisms in the multi-analyst settings fail to satisfy at least one of the desiderata. We present novel DP algorithms that provably satisfy all our desiderata and empirically show that they incur low error on realistic tasks.

Yikai Wu, Xingyu Zhu, Chenwei Wu et al.

Hessian captures important properties of the deep neural network loss landscape. Previous works have observed low rank structure in the Hessians of neural networks. In this paper, we propose a decoupling conjecture that decomposes the layer-wise Hessians of a network as the Kronecker product of two smaller matrices. We can analyze the properties of these smaller matrices and prove the structure of top eigenspace random 2-layer networks. The decoupling conjecture has several other interesting implications - top eigenspaces for different models have surprisingly high overlap, and top eigenvectors form low rank matrices when they are reshaped into the same shape as the corresponding weight matrix. All of these can be verified empirically for deeper networks. Finally, we use the structure of layer-wise Hessian to get better explicit generalization bounds for neural networks.

Yikai Wu, David Pujol, Ios Kotsogiannis et al.

In this work we explore the problem of answering a set of sum queries under Differential Privacy. This is a little understood, non-trivial problem especially in the case of numerical domains. We show that traditional techniques from the literature are not always the best choice and a more rigorous approach is necessary to develop low error algorithms.