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Optimization/Sampling and Optimization

Vivek Bharadwaj, Osman Asif Malik, Riley Murray, Laura Grigori, Aydin Buluc, James Demmel

We present a data structure to randomly sample rows from the Khatri-Rao product of several matrices according to the exact distribution of its leverage scores. Our proposed sampler draws each row in time logarithmic in the height of the Khatri-Rao product and quadratic in its column count, with persistent space overhead at most the size of the input matrices. As a result, it tractably draws samples even when the matrices forming the Khatri-Rao product have tens of millions of rows each. When used to sketch the linear least-squares problems arising in Candecomp / PARAFAC decomposition, our method achieves lower asymptotic complexity per solve than recent state-of-the-art methods. Experiments on billion-scale sparse tensors and synthetic data validate our theoretical claims, with our algorithm achieving higher accuracy than competing methods as the decomposition rank grows.

Optimization/Sampling and Optimization

Katayoon Goshvadi, Haoran Sun, Xingchao Liu, Azade Nova, Ruqi Zhang, Will Grathwohl, Dale Schuurmans, Hanjun Dai

Sampling in discrete spaces, with critical applications in simulation and optimization, has recently been boosted by significant advances in gradient-based approaches that exploit modern accelerators like GPUs. However, two key challenges are hindering further advancement in research on discrete sampling. First, since there is no consensus on experimental settings and evaluation setups, the empirical results in different research papers are often not comparable. Second, implementing samplers and target distributions often requires a nontrivial amount of effort in terms of calibration and parallelism. To tackle these challenges, we propose DISCS (DISCrete Sampling), a tailored package and benchmark that supports unified and efficient experiment implementation and evaluations for discrete sampling in three types of tasks: sampling from classical graphical models and energy based generative models, and sampling for solving combinatorial optimization. Throughout the comprehensive evaluations in DISCS, we gained new insights into scalability, design principles for proposal distributions, and lessons for adaptive sampling design. DISCS efficiently implements representative discrete samplers in existing research works as baselines and offers a simple interface that researchers can conveniently add new discrete samplers and directly compare their performance with the benchmark result in a calibrated setup.

Optimization/Sampling and Optimization

Lin Wang, Yongxin Guo, Tao Lin, Xiaoying Tang

Partial client participation has been widely adopted in Federated Learning (FL) to reduce the communication burden efficiently. However, an inadequate client sampling scheme can lead to the selection of unrepresentative subsets, resulting in significant variance in model updates and slowed convergence. Existing sampling methods are either biased or can be further optimized for faster convergence.In this paper, we present DELTA, an unbiased sampling scheme designed to alleviate these issues. DELTA characterizes the effects of client diversity and local variance, and samples representative clients with valuable information for global model updates. In addition, DELTA is a proven optimal unbiased sampling scheme that minimizes variance caused by partial client participation and outperforms other unbiased sampling schemes in terms of convergence. Furthermore, to address full-client gradient dependence, we provide a practical version of DELTA depending on the available clients' information, and also analyze its convergence. Our results are validated through experiments on both synthetic and real-world datasets.

Optimization/Sampling and Optimization

Aniket Das, Dheeraj Nagaraj

Stein Variational Gradient Descent (SVGD) is a popular particle-based variational inference algorithm with impressive empirical performance across various domains. Although the population (i.e, infinite-particle) limit dynamics of SVGD is well characterized, its behavior in the finite-particle regime is far less understood. To this end, our work introduces the notion of *virtual particles* to develop novel stochastic approximations of population-limit SVGD dynamics in the space of probability measures, that are exactly realizable using finite particles. As a result, we design two computationally efficient variants of SVGD, namely VP-SVGD and GB-SVGD, with provably fast finite-particle convergence rates. Our algorithms can be viewed as specific random-batch approximations of SVGD, which are computationally more efficient than ordinary SVGD. We show that the $n$ particles output by VP-SVGD and GB-SVGD, run for $T$ steps with batch-size $K$, are at-least as good as i.i.d samples from a distribution whose Kernel Stein Discrepancy to the target is at most $O(\tfrac{d^{1/3}}{(KT)^{1/6}})$ under standard assumptions. Our results also hold under a mild growth condition on the potential function, which is much weaker than the isoperimetric (e.g. Poincare Inequality) or information-transport conditions (e.g. Talagrand's Inequality $\mathsf{T}_1$) generally considered in prior works. As a corollary, we analyze the convergence of the empirical measure (of the particles output by VP-SVGD and GB-SVGD) to the target distribution and demonstrate a **double exponential improvement** over the best known finite-particle analysis of SVGD. Beyond this, our results present the **first known oracle complexities for this setting with polynomial dimension dependence**, thereby completely eliminating the curse of dimensionality exhibited by previously known finite-particle rates.

Optimization/Sampling and Optimization

Jiaxin Shi, Lester Mackey

We provide the first finite-particle convergence rate for Stein variational gradient descent (SVGD), a popular algorithm for approximating a probability distribution with a collection of particles. Specifically, whenever the target distribution is sub-Gaussian with a Lipschitz score, SVGD with $n$ particles and an appropriate step size sequence drives the kernel Stein discrepancy to zero at an order ${1/}{\sqrt{\log\log n}}$ rate. We suspect that the dependence on $n$ can be improved, and we hope that our explicit, non-asymptotic proof strategy will serve as a template for future refinements.

Optimization/Sampling and Optimization

Mohammad Reza Karimi Jaghargh, Ya-Ping Hsieh, Andreas Krause

Non-convex sampling is a key challenge in machine learning, central to non-convex optimization in deep learning as well as to approximate probabilistic inference. Despite its significance, theoretically there remain some important challenges: Existing guarantees suffer from the drawback of lacking guarantees for the last-iterates, and little is known beyond the elementary schemes of stochastic gradient Langevin dynamics. To address these issues, we develop a novel framework that lifts the above issues by harnessing several tools from the theory of dynamical systems. Our key result is that, for a large class of state-of-the-art sampling schemes, their last-iterate convergence in Wasserstein distances can be reduced to the study of their continuous-time counterparts, which is much better understood. Coupled with standard assumptions of MCMC sampling, our theory immediately yields the last-iterate Wasserstein convergence of many advanced sampling schemes such as mirror Langevin, proximal, randomized mid-point, and Runge-Kutta methods.

Optimization/Sampling and Optimization

Zhengmian Hu, Heng Huang

This paper proposes a novel algorithm, Transformative Bayesian Learning (TansBL), which bridges the gap between empirical risk minimization (ERM) and Bayesian learning for neural networks. We compare ERM, which uses gradient descent to optimize, and Bayesian learning with importance sampling for their generalization and computational complexity. We derive the first algorithm-dependent PAC-Bayesian generalization bound for infinitely wide networks based on an exact KL divergence between the trained posterior distribution obtained by infinitesimal step size gradient descent and a Gaussian prior. Moreover, we show how to transform gradient-based optimization into importance sampling by incorporating a weight. While Bayesian learning has better generalization, it suffers from low sampling efficiency. Optimization methods, on the other hand, have good sampling efficiency but poor generalization. Our proposed algorithm TansBL enables a trade-off between generalization and sampling efficiency.

Optimization/Sampling and Optimization

Yuanshi Liu, Cong Fang, Tong Zhang

This paper focuses on the high-dimensional sampling of log-concave distributions with composite structures: $p^*(\mathrm{d}x)\propto \exp(-g(x)-f(x))\mathrm{d}x$. We develop a double randomization technique, which leads to a fast underdamped Langevin algorithm with a dimension-independent convergence guarantee. We prove that the algorithm enjoys an overall $\tilde{\mathcal{O}}\left(\frac{\left(\mathrm{tr}(H)\right)^{1/3}}{\epsilon^{2/3}}\right)$ iteration complexity to reach an $\epsilon$-tolerated sample whose distribution $p$ admits $W_2(p,p^*)\leq \epsilon$. Here, $H$ is an upper bound of the Hessian matrices for $f$ and does not explicitly depend on dimension $d$. For the posterior sampling over linear models with normalized data, we show a clear superiority of convergence rate which is dimension-free and outperforms the previous best-known results by a $d^{1/3}$ factor. The analysis to achieve a faster convergence rate brings new insights into high-dimensional sampling.