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Michael W. Mahoney

Publications and source records attributed to Michael W. Mahoney.

At least 127 records · Page 7Linked to original sources
arXiv2023

Multi-scale Local Network Structure Critically Impacts Epidemic Spread and Interventions

Network epidemic simulation holds the promise of enabling fine-grained understanding of epidemic behavior, beyond that which is possible with coarse-grained compartmental models. Key inputs to these epidemic simulations are the networks themselves. However, empirical measurements and samples of realistic interaction networks typically display properties that are challenging to capture with popular synthetic models of networks. Our empirical results show that epidemic spread behavior is very sensitive to a subtle but ubiquitous form of multi-scale local structure that is not present in common baseline models, including (but not limited to) uniform random graph models (Erdos-Renyi), random configuration models (Chung-Lu), etc. Such structure is not necessary to reproduce very simple network statistics, such as degree distributions or triangle closing probabilities. However, we show that this multi-scale local structure impacts, critically, the behavior of more complex network properties, in particular the effect of interventions such as quarantining; and it enables epidemic spread to be halted in realistic interaction networks, even when it cannot be halted in simple synthetic models. Key insights from our analysis include how epidemics on networks with widespread multi-scale local structure are easier to mitigate, as well as characterizing which nodes are ultimately not likely to be infected. We demonstrate that this structure results from more than just local triangle structure in the network, and we illustrate processes based on homophily or social influence and random walks that suggest how this multi-scale local structure arises.

cs.SI↗
arXiv2023

Temperature Balancing, Layer-wise Weight Analysis, and Neural Network Training

Regularization in modern machine learning is crucial, and it can take various forms in algorithmic design: training set, model family, error function, regularization terms, and optimizations. In particular, the learning rate, which can be interpreted as a temperature-like parameter within the statistical mechanics of learning, plays a crucial role in neural network training. Indeed, many widely adopted training strategies basically just define the decay of the learning rate over time. This process can be interpreted as decreasing a temperature, using either a global learning rate (for the entire model) or a learning rate that varies for each parameter. This paper proposes TempBalance, a straightforward yet effective layer-wise learning rate method. TempBalance is based on Heavy-Tailed Self-Regularization (HT-SR) Theory, an approach which characterizes the implicit self-regularization of different layers in trained models. We demonstrate the efficacy of using HT-SR-motivated metrics to guide the scheduling and balancing of temperature across all network layers during model training, resulting in improved performance during testing. We implement TempBalance on CIFAR10, CIFAR100, SVHN, and TinyImageNet datasets using ResNets, VGGs, and WideResNets with various depths and widths. Our results show that TempBalance significantly outperforms ordinary SGD and carefully-tuned spectral norm regularization. We also show that TempBalance outperforms a number of state-of-the-art optimizers and learning rate schedulers.

cs.LG↗
arXiv2023

Learning continuous models for continuous physics

Dynamical systems that evolve continuously over time are ubiquitous throughout science and engineering. Machine learning (ML) provides data-driven approaches to model and predict the dynamics of such systems. A core issue with this approach is that ML models are typically trained on discrete data, using ML methodologies that are not aware of underlying continuity properties. This results in models that often do not capture any underlying continuous dynamics -- either of the system of interest, or indeed of any related system. To address this challenge, we develop a convergence test based on numerical analysis theory. Our test verifies whether a model has learned a function that accurately approximates an underlying continuous dynamics. Models that fail this test fail to capture relevant dynamics, rendering them of limited utility for many scientific prediction tasks; while models that pass this test enable both better interpolation and better extrapolation in multiple ways. Our results illustrate how principled numerical analysis methods can be coupled with existing ML training/testing methodologies to validate models for science and engineering applications.

cs.LG↗
arXiv2023

A PAC-Bayesian Perspective on the Interpolating Information Criterion

Deep learning is renowned for its theory-practice gap, whereby principled theory typically fails to provide much beneficial guidance for implementation in practice. This has been highlighted recently by the benign overfitting phenomenon: when neural networks become sufficiently large to interpolate the dataset perfectly, model performance appears to improve with increasing model size, in apparent contradiction with the well-known bias-variance tradeoff. While such phenomena have proven challenging to theoretically study for general models, the recently proposed Interpolating Information Criterion (IIC) provides a valuable theoretical framework to examine performance for overparameterized models. Using the IIC, a PAC-Bayes bound is obtained for a general class of models, characterizing factors which influence generalization performance in the interpolating regime. From the provided bound, we quantify how the test error for overparameterized models achieving effectively zero training error depends on the quality of the implicit regularization imposed by e.g. the combination of model, optimizer, and parameter-initialization scheme; the spectrum of the empirical neural tangent kernel; curvature of the loss landscape; and noise present in the data.

stat.ML↗
arXiv2023

Speculative Decoding with Big Little Decoder

The recent emergence of Large Language Models based on the Transformer architecture has enabled dramatic advancements in the field of Natural Language Processing. However, these models have long inference latency, which limits their deployment and makes them prohibitively expensive for various real-time applications. The inference latency is further exacerbated by autoregressive generative tasks, as models need to run iteratively to generate tokens sequentially without leveraging token-level parallelization. To address this, we propose Big Little Decoder (BiLD), a framework that can improve inference efficiency and latency for a wide range of text generation applications. The BiLD framework contains two models with different sizes that collaboratively generate text. The small model runs autoregressively to generate text with a low inference cost, and the large model is only invoked occasionally to refine the small model's inaccurate predictions in a non-autoregressive manner. To coordinate the small and large models, BiLD introduces two simple yet effective policies: (1) the fallback policy that determines when to hand control over to the large model; and (2) the rollback policy that determines when the large model needs to correct the small model's inaccurate predictions. To evaluate our framework across different tasks and models, we apply BiLD to various text generation scenarios encompassing machine translation on IWSLT 2017 De-En and WMT 2014 De-En, and summarization on XSUM and CNN/DailyMail. On an NVIDIA T4 GPU, our framework achieves a speedup of up to 2.12x speedup with minimal generation quality degradation. Furthermore, our framework is fully plug-and-play and can be applied without any modifications in the training process or model architecture. Our code is open-sourced

cs.CL↗
arXiv2023

Learning Physical Models that Can Respect Conservation Laws

Recent work in scientific machine learning (SciML) has focused on incorporating partial differential equation (PDE) information into the learning process. Much of this work has focused on relatively "easy" PDE operators (e.g., elliptic and parabolic), with less emphasis on relatively "hard" PDE operators (e.g., hyperbolic). Within numerical PDEs, the latter problem class requires control of a type of volume element or conservation constraint, which is known to be challenging. Delivering on the promise of SciML requires seamlessly incorporating both types of problems into the learning process. To address this issue, we propose ProbConserv, a framework for incorporating conservation constraints into a generic SciML architecture. To do so, ProbConserv combines the integral form of a conservation law with a Bayesian update. We provide a detailed analysis of ProbConserv on learning with the Generalized Porous Medium Equation (GPME), a widely-applicable parameterized family of PDEs that illustrates the qualitative properties of both easier and harder PDEs. ProbConserv is effective for easy GPME variants, performing well with state-of-the-art competitors; and for harder GPME variants it outperforms other approaches that do not guarantee volume conservation. ProbConserv seamlessly enforces physical conservation constraints, maintains probabilistic uncertainty quantification (UQ), and deals well with shocks and heteroscedasticities. In each case, it achieves superior predictive performance on downstream tasks.

cs.LG↗
arXiv2023

Extensions to the SENSEI In situ Framework for Heterogeneous Architectures

The proliferation of GPUs and accelerators in recent supercomputing systems, so called heterogeneous architectures, has led to increased complexity in execution environments and programming models as well as to deeper memory hierarchies on these systems. In this work, we discuss challenges that arise in in situ code coupling on these heterogeneous architectures. In particular, we present data and execution model extensions to the SENSEI in situ framework that are targeted at the effective use of systems with heterogeneous architectures. We then use these new data and execution model extensions to investigate several in situ placement and execution configurations and to analyze the impact these choices have on overall performance.

cs.DC↗
arXiv2023

Robustifying State-space Models for Long Sequences via Approximate Diagonalization

State-space models (SSMs) have recently emerged as a framework for learning long-range sequence tasks. An example is the structured state-space sequence (S4) layer, which uses the diagonal-plus-low-rank structure of the HiPPO initialization framework. However, the complicated structure of the S4 layer poses challenges; and, in an effort to address these challenges, models such as S4D and S5 have considered a purely diagonal structure. This choice simplifies the implementation, improves computational efficiency, and allows channel communication. However, diagonalizing the HiPPO framework is itself an ill-posed problem. In this paper, we propose a general solution for this and related ill-posed diagonalization problems in machine learning. We introduce a generic, backward-stable "perturb-then-diagonalize" (PTD) methodology, which is based on the pseudospectral theory of non-normal operators, and which may be interpreted as the approximate diagonalization of the non-normal matrices defining SSMs. Based on this, we introduce the S4-PTD and S5-PTD models. Through theoretical analysis of the transfer functions of different initialization schemes, we demonstrate that the S4-PTD/S5-PTD initialization strongly converges to the HiPPO framework, while the S4D/S5 initialization only achieves weak convergences. As a result, our new models show resilience to Fourier-mode noise-perturbed inputs, a crucial property not achieved by the S4D/S5 models. In addition to improved robustness, our S5-PTD model averages 87.6% accuracy on the Long-Range Arena benchmark, demonstrating that the PTD methodology helps to improve the accuracy of deep learning models.

cs.LG↗
arXiv2023

Monotonicity and Double Descent in Uncertainty Estimation with Gaussian Processes

Despite their importance for assessing reliability of predictions, uncertainty quantification (UQ) measures for machine learning models have only recently begun to be rigorously characterized. One prominent issue is the curse of dimensionality: it is commonly believed that the marginal likelihood should be reminiscent of cross-validation metrics and that both should deteriorate with larger input dimensions. We prove that by tuning hyperparameters to maximize marginal likelihood (the empirical Bayes procedure), the performance, as measured by the marginal likelihood, improves monotonically} with the input dimension. On the other hand, we prove that cross-validation metrics exhibit qualitatively different behavior that is characteristic of double descent. Cold posteriors, which have recently attracted interest due to their improved performance in certain settings, appear to exacerbate these phenomena. We verify empirically that our results hold for real data, beyond our considered assumptions, and we explore consequences involving synthetic covariates.

stat.ML↗
arXiv2023

GEANN: Scalable Graph Augmentations for Multi-Horizon Time Series Forecasting

Encoder-decoder deep neural networks have been increasingly studied for multi-horizon time series forecasting, especially in real-world applications. However, to forecast accurately, these sophisticated models typically rely on a large number of time series examples with substantial history. A rapidly growing topic of interest is forecasting time series which lack sufficient historical data -- often referred to as the ``cold start'' problem. In this paper, we introduce a novel yet simple method to address this problem by leveraging graph neural networks (GNNs) as a data augmentation for enhancing the encoder used by such forecasters. These GNN-based features can capture complex inter-series relationships, and their generation process can be optimized end-to-end with the forecasting task. We show that our architecture can use either data-driven or domain knowledge-defined graphs, scaling to incorporate information from multiple very large graphs with millions of nodes. In our target application of demand forecasting for a large e-commerce retailer, we demonstrate on both a small dataset of 100K products and a large dataset with over 2 million products that our method improves overall performance over competitive baseline models. More importantly, we show that it brings substantially more gains to ``cold start'' products such as those newly launched or recently out-of-stock.

cs.LG↗
arXiv2023

A Heavy-Tailed Algebra for Probabilistic Programming

Despite the successes of probabilistic models based on passing noise through neural networks, recent work has identified that such methods often fail to capture tail behavior accurately, unless the tails of the base distribution are appropriately calibrated. To overcome this deficiency, we propose a systematic approach for analyzing the tails of random variables, and we illustrate how this approach can be used during the static analysis (before drawing samples) pass of a probabilistic programming language compiler. To characterize how the tails change under various operations, we develop an algebra which acts on a three-parameter family of tail asymptotics and which is based on the generalized Gamma distribution. Our algebraic operations are closed under addition and multiplication; they are capable of distinguishing sub-Gaussians with differing scales; and they handle ratios sufficiently well to reproduce the tails of most important statistical distributions directly from their definitions. Our empirical results confirm that inference algorithms that leverage our heavy-tailed algebra attain superior performance across a number of density modeling and variational inference tasks.

stat.ML↗
arXiv2023

Evaluating natural language processing models with generalization metrics that do not need access to any training or testing data

Selecting suitable architecture parameters and training hyperparameters is essential for enhancing machine learning (ML) model performance. Several recent empirical studies conduct large-scale correlational analysis on neural networks (NNs) to search for effective \emph{generalization metrics} that can guide this type of model selection. Effective metrics are typically expected to correlate strongly with test performance. In this paper, we expand on prior analyses by examining generalization-metric-based model selection with the following objectives: (i) focusing on natural language processing (NLP) tasks, as prior work primarily concentrates on computer vision (CV) tasks; (ii) considering metrics that directly predict \emph{test error} instead of the \emph{generalization gap}; (iii) exploring metrics that do not need access to data to compute. From these objectives, we are able to provide the first model selection results on large pretrained Transformers from Huggingface using generalization metrics. Our analyses consider (I) hundreds of Transformers trained in different settings, in which we systematically vary the amount of data, the model size and the optimization hyperparameters, (II) a total of 51 pretrained Transformers from eight families of Huggingface NLP models, including GPT2, BERT, etc., and (III) a total of 28 existing and novel generalization metrics. Despite their niche status, we find that metrics derived from the heavy-tail (HT) perspective are particularly useful in NLP tasks, exhibiting stronger correlations than other, more popular metrics. To further examine these metrics, we extend prior formulations relying on power law (PL) spectral distributions to exponential (EXP) and exponentially-truncated power law (E-TPL) families.

cs.CL↗
arXiv2023

Constrained Optimization via Exact Augmented Lagrangian and Randomized Iterative Sketching

We consider solving equality-constrained nonlinear, nonconvex optimization problems. This class of problems appears widely in a variety of applications in machine learning and engineering, ranging from constrained deep neural networks, to optimal control, to PDE-constrained optimization. We develop an adaptive inexact Newton method for this problem class. In each iteration, we solve the Lagrangian Newton system inexactly via a randomized iterative sketching solver, and select a suitable stepsize by performing line search on an exact augmented Lagrangian merit function. The randomized solvers have advantages over deterministic linear system solvers by significantly reducing per-iteration flops complexity and storage cost, when equipped with suitable sketching matrices. Our method adaptively controls the accuracy of the randomized solver and the penalty parameters of the exact augmented Lagrangian, to ensure that the inexact Newton direction is a descent direction of the exact augmented Lagrangian. This allows us to establish a global almost sure convergence. We also show that a unit stepsize is admissible locally, so that our method exhibits a local linear convergence. Furthermore, we prove that the linear convergence can be strengthened to superlinear convergence if we gradually sharpen the adaptive accuracy condition on the randomized solver. We demonstrate the superior performance of our method on benchmark nonlinear problems in CUTEst test set, constrained logistic regression with data from LIBSVM, and a PDE-constrained problem.

math.OC↗
arXiv2023

A Three-regime Model of Network Pruning

Recent work has highlighted the complex influence training hyperparameters, e.g., the number of training epochs, can have on the prunability of machine learning models. Perhaps surprisingly, a systematic approach to predict precisely how adjusting a specific hyperparameter will affect prunability remains elusive. To address this gap, we introduce a phenomenological model grounded in the statistical mechanics of learning. Our approach uses temperature-like and load-like parameters to model the impact of neural network (NN) training hyperparameters on pruning performance. A key empirical result we identify is a sharp transition phenomenon: depending on the value of a load-like parameter in the pruned model, increasing the value of a temperature-like parameter in the pre-pruned model may either enhance or impair subsequent pruning performance. Based on this transition, we build a three-regime model by taxonomizing the global structure of the pruned NN loss landscape. Our model reveals that the dichotomous effect of high temperature is associated with transitions between distinct types of global structures in the post-pruned model. Based on our results, we present three case-studies: 1) determining whether to increase or decrease a hyperparameter for improved pruning; 2) selecting the best model to prune from a family of models; and 3) tuning the hyperparameter of the Sharpness Aware Minimization method for better pruning performance.

stat.ML↗
arXiv2023

When are ensembles really effective?

Ensembling has a long history in statistical data analysis, with many impactful applications. However, in many modern machine learning settings, the benefits of ensembling are less ubiquitous and less obvious. We study, both theoretically and empirically, the fundamental question of when ensembling yields significant performance improvements in classification tasks. Theoretically, we prove new results relating the \emph{ensemble improvement rate} (a measure of how much ensembling decreases the error rate versus a single model, on a relative scale) to the \emph{disagreement-error ratio}. We show that ensembling improves performance significantly whenever the disagreement rate is large relative to the average error rate; and that, conversely, one classifier is often enough whenever the disagreement rate is low relative to the average error rate. On the way to proving these results, we derive, under a mild condition called \emph{competence}, improved upper and lower bounds on the average test error rate of the majority vote classifier. To complement this theory, we study ensembling empirically in a variety of settings, verifying the predictions made by our theory, and identifying practical scenarios where ensembling does and does not result in large performance improvements. Perhaps most notably, we demonstrate a distinct difference in behavior between interpolating models (popular in current practice) and non-interpolating models (such as tree-based methods, where ensembling is popular), demonstrating that ensembling helps considerably more in the latter case than in the former.

stat.ML↗
arXiv2023

Learning differentiable solvers for systems with hard constraints

We introduce a practical method to enforce partial differential equation (PDE) constraints for functions defined by neural networks (NNs), with a high degree of accuracy and up to a desired tolerance. We develop a differentiable PDE-constrained layer that can be incorporated into any NN architecture. Our method leverages differentiable optimization and the implicit function theorem to effectively enforce physical constraints. Inspired by dictionary learning, our model learns a family of functions, each of which defines a mapping from PDE parameters to PDE solutions. At inference time, the model finds an optimal linear combination of the functions in the learned family by solving a PDE-constrained optimization problem. Our method provides continuous solutions over the domain of interest that accurately satisfy desired physical constraints. Our results show that incorporating hard constraints directly into the NN architecture achieves much lower test error when compared to training on an unconstrained objective.

cs.LG↗
arXiv2023

End-to-end codesign of Hessian-aware quantized neural networks for FPGAs and ASICs

We develop an end-to-end workflow for the training and implementation of co-designed neural networks (NNs) for efficient field-programmable gate array (FPGA) and application-specific integrated circuit (ASIC) hardware. Our approach leverages Hessian-aware quantization (HAWQ) of NNs, the Quantized Open Neural Network Exchange (QONNX) intermediate representation, and the hls4ml tool flow for transpiling NNs into FPGA and ASIC firmware. This makes efficient NN implementations in hardware accessible to nonexperts, in a single open-sourced workflow that can be deployed for real-time machine learning applications in a wide range of scientific and industrial settings. We demonstrate the workflow in a particle physics application involving trigger decisions that must operate at the 40 MHz collision rate of the CERN Large Hadron Collider (LHC). Given the high collision rate, all data processing must be implemented on custom ASIC and FPGA hardware within a strict area and latency. Based on these constraints, we implement an optimized mixed-precision NN classifier for high-momentum particle jets in simulated LHC proton-proton collisions.

cs.LG↗
arXiv2023

Randomized Numerical Linear Algebra : A Perspective on the Field With an Eye to Software

Randomized numerical linear algebra - RandNLA, for short - concerns the use of randomization as a resource to develop improved algorithms for large-scale linear algebra computations. The origins of contemporary RandNLA lay in theoretical computer science, where it blossomed from a simple idea: randomization provides an avenue for computing approximate solutions to linear algebra problems more efficiently than deterministic algorithms. This idea proved fruitful in the development of scalable algorithms for machine learning and statistical data analysis applications. However, RandNLA's true potential only came into focus upon integration with the fields of numerical analysis and "classical" numerical linear algebra. Through the efforts of many individuals, randomized algorithms have been developed that provide full control over the accuracy of their solutions and that can be every bit as reliable as algorithms that might be found in libraries such as LAPACK. Recent years have even seen the incorporation of certain RandNLA methods into MATLAB, the NAG Library, NVIDIA's cuSOLVER, and SciKit-Learn. For all its success, we believe that RandNLA has yet to realize its full potential. In particular, we believe the scientific community stands to benefit significantly from suitably defined "RandBLAS" and "RandLAPACK" libraries, to serve as standards conceptually analogous to BLAS and LAPACK. This 200-page monograph represents a step toward defining such standards. In it, we cover topics spanning basic sketching, least squares and optimization, low-rank approximation, full matrix decompositions, leverage score sampling, and sketching data with tensor product structures (among others). Much of the provided pseudo-code has been tested via publicly available MATLAB and Python implementations.

math.NA↗