arXiv subjects

Jacob R. Gardner

Publications and source records attributed to Jacob R. Gardner.

At least 37 records · Page 2Linked to original sources

arXiv2025

Demystifying SGD with Doubly Stochastic Gradients

Optimization objectives in the form of a sum of intractable expectations are rising in importance (e.g., diffusion models, variational autoencoders, and many more), a setting also known as "finite sum with infinite data." For these problems, a popular strategy is to employ SGD with doubly stochastic gradients (doubly SGD): the expectations are estimated using the gradient estimator of each component, while the sum is estimated by subsampling over these estimators. Despite its popularity, little is known about the convergence properties of doubly SGD, except under strong assumptions such as bounded variance. In this work, we establish the convergence of doubly SGD with independent minibatching and random reshuffling under general conditions, which encompasses dependent component gradient estimators. In particular, for dependent estimators, our analysis allows fined-grained analysis of the effect correlations. As a result, under a per-iteration computational budget of $b \times m$, where $b$ is the minibatch size and $m$ is the number of Monte Carlo samples, our analysis suggests where one should invest most of the budget in general. Furthermore, we prove that random reshuffling (RR) improves the complexity dependence on the subsampling noise.

stat.ML↗

arXiv2025

Learned Offline Query Planning via Bayesian Optimization

Analytics database workloads often contain queries that are executed repeatedly. Existing optimization techniques generally prioritize keeping optimization cost low, normally well below the time it takes to execute a single instance of a query. If a given query is going to be executed thousands of times, could it be worth investing significantly more optimization time? In contrast to traditional online query optimizers, we propose an offline query optimizer that searches a wide variety of plans and incorporates query execution as a primitive. Our offline query optimizer combines variational auto-encoders with Bayesian optimization to find optimized plans for a given query. We compare our technique to the optimal plans possible with PostgreSQL and recent RL-based systems over several datasets, and show that our technique finds faster query plans.

cs.DB↗

arXiv2024

A Fast, Robust Elliptical Slice Sampling Implementation for Linearly Truncated Multivariate Normal Distributions

Elliptical slice sampling, when adapted to linearly truncated multivariate normal distributions, is a rejection-free Markov chain Monte Carlo method. At its core, it requires analytically constructing an ellipse-polytope intersection. The main novelty of this paper is an algorithm that computes this intersection in $\mathcal{O}(m \log m)$ time, where $m$ is the number of linear inequality constraints representing the polytope. We show that an implementation based on this algorithm enhances numerical stability, speeds up running time, and is easy to parallelize for launching multiple Markov chains.

cs.LG↗

arXiv2024

Zeroth-Order Fine-Tuning of LLMs with Extreme Sparsity

Zeroth-order optimization (ZO) is a memory-efficient strategy for fine-tuning Large Language Models using only forward passes. However, the application of ZO fine-tuning in memory-constrained settings such as mobile phones and laptops is still challenging since full precision forward passes are infeasible. In this study, we address this limitation by integrating sparsity and quantization into ZO fine-tuning of LLMs. Specifically, we investigate the feasibility of fine-tuning an extremely small subset of LLM parameters using ZO. This approach allows the majority of un-tuned parameters to be quantized to accommodate the constraint of limited device memory. Our findings reveal that the pre-training process can identify a set of "sensitive parameters" that can guide the ZO fine-tuning of LLMs on downstream tasks. Our results demonstrate that fine-tuning 0.1% sensitive parameters in the LLM with ZO can outperform the full ZO fine-tuning performance, while offering wall-clock time speedup. Additionally, we show that ZO fine-tuning targeting these 0.1% sensitive parameters, combined with 4 bit quantization, enables efficient ZO fine-tuning of an Llama2-7B model on a GPU device with less than 8 GiB of memory and notably reduced latency.

cs.LG↗

arXiv2024

Understanding Stochastic Natural Gradient Variational Inference

Stochastic natural gradient variational inference (NGVI) is a popular posterior inference method with applications in various probabilistic models. Despite its wide usage, little is known about the non-asymptotic convergence rate in the \emph{stochastic} setting. We aim to lessen this gap and provide a better understanding. For conjugate likelihoods, we prove the first $\mathcal{O}(\frac{1}{T})$ non-asymptotic convergence rate of stochastic NGVI. The complexity is no worse than stochastic gradient descent (\aka black-box variational inference) and the rate likely has better constant dependency that leads to faster convergence in practice. For non-conjugate likelihoods, we show that stochastic NGVI with the canonical parameterization implicitly optimizes a non-convex objective. Thus, a global convergence rate of $\mathcal{O}(\frac{1}{T})$ is unlikely without some significant new understanding of optimizing the ELBO using natural gradients.

cs.LG↗

arXiv2024

The Behavior and Convergence of Local Bayesian Optimization

A recent development in Bayesian optimization is the use of local optimization strategies, which can deliver strong empirical performance on high-dimensional problems compared to traditional global strategies. The "folk wisdom" in the literature is that the focus on local optimization sidesteps the curse of dimensionality; however, little is known concretely about the expected behavior or convergence of Bayesian local optimization routines. We first study the behavior of the local approach, and find that the statistics of individual local solutions of Gaussian process sample paths are surprisingly good compared to what we would expect to recover from global methods. We then present the first rigorous analysis of such a Bayesian local optimization algorithm recently proposed by Müller et al. (2021), and derive convergence rates in both the noisy and noiseless settings.

cs.LG↗

arXiv2024

Large-Scale Gaussian Processes via Alternating Projection

Training and inference in Gaussian processes (GPs) require solving linear systems with $n\times n$ kernel matrices. To address the prohibitive $\mathcal{O}(n^3)$ time complexity, recent work has employed fast iterative methods, like conjugate gradients (CG). However, as datasets increase in magnitude, the kernel matrices become increasingly ill-conditioned and still require $\mathcal{O}(n^2)$ space without partitioning. Thus, while CG increases the size of datasets GPs can be trained on, modern datasets reach scales beyond its applicability. In this work, we propose an iterative method which only accesses subblocks of the kernel matrix, effectively enabling mini-batching. Our algorithm, based on alternating projection, has $\mathcal{O}(n)$ per-iteration time and space complexity, solving many of the practical challenges of scaling GPs to very large datasets. Theoretically, we prove the method enjoys linear convergence. Empirically, we demonstrate its fast convergence in practice and robustness to ill-conditioning. On large-scale benchmark datasets with up to four million data points, our approach accelerates GP training and inference by speed-up factors up to $27\times$ and $72 \times$, respectively, compared to CG.

cs.LG↗

arXiv2024

Stochastic Approximation with Biased MCMC for Expectation Maximization

The expectation maximization (EM) algorithm is a widespread method for empirical Bayesian inference, but its expectation step (E-step) is often intractable. Employing a stochastic approximation scheme with Markov chain Monte Carlo (MCMC) can circumvent this issue, resulting in an algorithm known as MCMC-SAEM. While theoretical guarantees for MCMC-SAEM have previously been established, these results are restricted to the case where asymptotically unbiased MCMC algorithms are used. In practice, MCMC-SAEM is often run with asymptotically biased MCMC, for which the consequences are theoretically less understood. In this work, we fill this gap by analyzing the asymptotics and non-asymptotics of SAEM with biased MCMC steps, particularly the effect of bias. We also provide numerical experiments comparing the Metropolis-adjusted Langevin algorithm (MALA), which is asymptotically unbiased, and the unadjusted Langevin algorithm (ULA), which is asymptotically biased, on synthetic and real datasets. Experimental results show that ULA is more stable with respect to the choice of Langevin stepsize and can sometimes result in faster convergence.

stat.CO↗

arXiv2024

On the Convergence of Black-Box Variational Inference

We provide the first convergence guarantee for full black-box variational inference (BBVI), also known as Monte Carlo variational inference. While preliminary investigations worked on simplified versions of BBVI (e.g., bounded domain, bounded support, only optimizing for the scale, and such), our setup does not need any such algorithmic modifications. Our results hold for log-smooth posterior densities with and without strong log-concavity and the location-scale variational family. Also, our analysis reveals that certain algorithm design choices commonly employed in practice, particularly, nonlinear parameterizations of the scale of the variational approximation, can result in suboptimal convergence rates. Fortunately, running BBVI with proximal stochastic gradient descent fixes these limitations, and thus achieves the strongest known convergence rate guarantees. We evaluate this theoretical insight by comparing proximal SGD against other standard implementations of BBVI on large-scale Bayesian inference problems.

cs.LG↗

arXiv2023

Practical and Matching Gradient Variance Bounds for Black-Box Variational Bayesian Inference

Understanding the gradient variance of black-box variational inference (BBVI) is a crucial step for establishing its convergence and developing algorithmic improvements. However, existing studies have yet to show that the gradient variance of BBVI satisfies the conditions used to study the convergence of stochastic gradient descent (SGD), the workhorse of BBVI. In this work, we show that BBVI satisfies a matching bound corresponding to the $ABC$ condition used in the SGD literature when applied to smooth and quadratically-growing log-likelihoods. Our results generalize to nonlinear covariance parameterizations widely used in the practice of BBVI. Furthermore, we show that the variance of the mean-field parameterization has provably superior dimensional dependence.

cs.LG↗

arXiv2023

Inverse Protein Folding Using Deep Bayesian Optimization

Inverse protein folding -- the task of predicting a protein sequence from its backbone atom coordinates -- has surfaced as an important problem in the "top down", de novo design of proteins. Contemporary approaches have cast this problem as a conditional generative modelling problem, where a large generative model over protein sequences is conditioned on the backbone. While these generative models very rapidly produce promising sequences, independent draws from generative models may fail to produce sequences that reliably fold to the correct backbone. Furthermore, it is challenging to adapt pure generative approaches to other settings, e.g., when constraints exist. In this paper, we cast the problem of improving generated inverse folds as an optimization problem that we solve using recent advances in "deep" or "latent space" Bayesian optimization. Our approach consistently produces protein sequences with greatly reduced structural error to the target backbone structure as measured by TM score and RMSD while using fewer computational resources. Additionally, we demonstrate other advantages of an optimization-based approach to the problem, such as the ability to handle constraints.

q-bio.BM↗

arXiv2023

Local Latent Space Bayesian Optimization over Structured Inputs

Bayesian optimization over the latent spaces of deep autoencoder models (DAEs) has recently emerged as a promising new approach for optimizing challenging black-box functions over structured, discrete, hard-to-enumerate search spaces (e.g., molecules). Here the DAE dramatically simplifies the search space by mapping inputs into a continuous latent space where familiar Bayesian optimization tools can be more readily applied. Despite this simplification, the latent space typically remains high-dimensional. Thus, even with a well-suited latent space, these approaches do not necessarily provide a complete solution, but may rather shift the structured optimization problem to a high-dimensional one. In this paper, we propose LOL-BO, which adapts the notion of trust regions explored in recent work on high-dimensional Bayesian optimization to the structured setting. By reformulating the encoder to function as both an encoder for the DAE globally and as a deep kernel for the surrogate model within a trust region, we better align the notion of local optimization in the latent space with local optimization in the input space. LOL-BO achieves as much as 20 times improvement over state-of-the-art latent space Bayesian optimization methods across six real-world benchmarks, demonstrating that improvement in optimization strategies is as important as developing better DAE models.

cs.LG↗

arXiv2023

Local Bayesian optimization via maximizing probability of descent

Local optimization presents a promising approach to expensive, high-dimensional black-box optimization by sidestepping the need to globally explore the search space. For objective functions whose gradient cannot be evaluated directly, Bayesian optimization offers one solution -- we construct a probabilistic model of the objective, design a policy to learn about the gradient at the current location, and use the resulting information to navigate the objective landscape. Previous work has realized this scheme by minimizing the variance in the estimate of the gradient, then moving in the direction of the expected gradient. In this paper, we re-examine and refine this approach. We demonstrate that, surprisingly, the expected value of the gradient is not always the direction maximizing the probability of descent, and in fact, these directions may be nearly orthogonal. This observation then inspires an elegant optimization scheme seeking to maximize the probability of descent while moving in the direction of most-probable descent. Experiments on both synthetic and real-world objectives show that our method outperforms previous realizations of this optimization scheme and is competitive against other, significantly more complicated baselines.

cs.LG↗

arXiv2022

Markov Chain Score Ascent: A Unifying Framework of Variational Inference with Markovian Gradients

Minimizing the inclusive Kullback-Leibler (KL) divergence with stochastic gradient descent (SGD) is challenging since its gradient is defined as an integral over the posterior. Recently, multiple methods have been proposed to run SGD with biased gradient estimates obtained from a Markov chain. This paper provides the first non-asymptotic convergence analysis of these methods by establishing their mixing rate and gradient variance. To do this, we demonstrate that these methods-which we collectively refer to as Markov chain score ascent (MCSA) methods-can be cast as special cases of the Markov chain gradient descent framework. Furthermore, by leveraging this new understanding, we develop a novel MCSA scheme, parallel MCSA (pMCSA), that achieves a tighter bound on the gradient variance. We demonstrate that this improved theoretical result translates to superior empirical performance.

cs.LG↗

arXiv2022

Extracting or Guessing? Improving Faithfulness of Event Temporal Relation Extraction

In this paper, we seek to improve the faithfulness of TempRel extraction models from two perspectives. The first perspective is to extract genuinely based on contextual description. To achieve this, we propose to conduct counterfactual analysis to attenuate the effects of two significant types of training biases: the event trigger bias and the frequent label bias. We also add tense information into event representations to explicitly place an emphasis on the contextual description. The second perspective is to provide proper uncertainty estimation and abstain from extraction when no relation is described in the text. By parameterization of Dirichlet Prior over the model-predicted categorical distribution, we improve the model estimates of the correctness likelihood and make TempRel predictions more selective. We also employ temperature scaling to recalibrate the model confidence measure after bias mitigation. Through experimental analysis on MATRES, MATRES-DS, and TDDiscourse, we demonstrate that our model extracts TempRel and timelines more faithfully compared to SOTA methods, especially under distribution shifts.

cs.CL↗

arXiv2022

Preconditioning for Scalable Gaussian Process Hyperparameter Optimization

Gaussian process hyperparameter optimization requires linear solves with, and log-determinants of, large kernel matrices. Iterative numerical techniques are becoming popular to scale to larger datasets, relying on the conjugate gradient method (CG) for the linear solves and stochastic trace estimation for the log-determinant. This work introduces new algorithmic and theoretical insights for preconditioning these computations. While preconditioning is well understood in the context of CG, we demonstrate that it can also accelerate convergence and reduce variance of the estimates for the log-determinant and its derivative. We prove general probabilistic error bounds for the preconditioned computation of the log-determinant, log-marginal likelihood and its derivatives. Additionally, we derive specific rates for a range of kernel-preconditioner combinations, showing that up to exponential convergence can be achieved. Our theoretical results enable provably efficient optimization of kernel hyperparameters, which we validate empirically on large-scale benchmark problems. There our approach accelerates training by up to an order of magnitude.

cs.LG↗

arXiv2021

Scaling Gaussian Processes with Derivative Information Using Variational Inference

Gaussian processes with derivative information are useful in many settings where derivative information is available, including numerous Bayesian optimization and regression tasks that arise in the natural sciences. Incorporating derivative observations, however, comes with a dominating $O(N^3D^3)$ computational cost when training on $N$ points in $D$ input dimensions. This is intractable for even moderately sized problems. While recent work has addressed this intractability in the low-$D$ setting, the high-$N$, high-$D$ setting is still unexplored and of great value, particularly as machine learning problems increasingly become high dimensional. In this paper, we introduce methods to achieve fully scalable Gaussian process regression with derivatives using variational inference. Analogous to the use of inducing values to sparsify the labels of a training set, we introduce the concept of inducing directional derivatives to sparsify the partial derivative information of a training set. This enables us to construct a variational posterior that incorporates derivative information but whose size depends neither on the full dataset size $N$ nor the full dimensionality $D$. We demonstrate the full scalability of our approach on a variety of tasks, ranging from a high dimensional stellarator fusion regression task to training graph convolutional neural networks on Pubmed using Bayesian optimization. Surprisingly, we find that our approach can improve regression performance even in settings where only label data is available.

cs.LG↗

arXiv2021

GPyTorch: Blackbox Matrix-Matrix Gaussian Process Inference with GPU Acceleration

Despite advances in scalable models, the inference tools used for Gaussian processes (GPs) have yet to fully capitalize on developments in computing hardware. We present an efficient and general approach to GP inference based on Blackbox Matrix-Matrix multiplication (BBMM). BBMM inference uses a modified batched version of the conjugate gradients algorithm to derive all terms for training and inference in a single call. BBMM reduces the asymptotic complexity of exact GP inference from $O(n^3)$ to $O(n^2)$. Adapting this algorithm to scalable approximations and complex GP models simply requires a routine for efficient matrix-matrix multiplication with the kernel and its derivative. In addition, BBMM uses a specialized preconditioner to substantially speed up convergence. In experiments we show that BBMM effectively uses GPU hardware to dramatically accelerate both exact GP inference and scalable approximations. Additionally, we provide GPyTorch, a software platform for scalable GP inference via BBMM, built on PyTorch.

cs.LG↗