Barak A. Pearlmutter

Adam D. Cobb, Atılım Güneş Baydin, Barak A. Pearlmutter et al.

This paper introduces a second-order hyperplane search, a novel optimization step that generalizes a second-order line search from a line to a $k$-dimensional hyperplane. This, combined with the forward-mode stochastic gradient method, yields a second-order optimization algorithm that consists of forward passes only, completely avoiding the storage overhead of backpropagation. Unlike recent work that relies on directional derivatives (or Jacobian--Vector Products, JVPs), we use hyper-dual numbers to jointly evaluate both directional derivatives and their second-order quadratic terms. As a result, we introduce forward-mode weight perturbation with Hessian information (FoMoH). We then use FoMoH to develop a novel generalization of line search by extending it to a hyperplane search. We illustrate the utility of this extension and how it might be used to overcome some of the recent challenges of optimizing machine learning models without backpropagation. Our code is open-sourced at https://github.com/SRI-CSL/fomoh.

Alexey Radul, Barak A. Pearlmutter, Jeffrey Mark Siskind

We propose extensions to Fortran which integrate forward and reverse Automatic Differentiation (AD) directly into the programming model. Irrespective of implementation technology, embedding AD constructs directly into the language extends the reach and convenience of AD while allowing abstraction of concepts of interest to scientific-computing practice, such as root finding, optimization, and finding equilibria of continuous games. Multiple different subprograms for these tasks can share common interfaces, regardless of whether and how they use AD internally. A programmer can maximize a function F by calling a library maximizer, XSTAR=ARGMAX(F,X0), which internally constructs derivatives of F by AD, without having to learn how to use any particular AD tool. We illustrate the utility of these extensions by example: programs become much more concise and closer to traditional mathematical notation. A companion paper describes how these extensions can be implemented by a program that generates input to existing Fortran-based AD tools.

Barak A. Pearlmutter, Jeffrey Mark Siskind

We show that, under certain circumstances, it is possible to automatically compute Jacobian-inverse-vector and Jacobian-inverse-transpose-vector products about as efficiently as Jacobian-vector and Jacobian-transpose-vector products. The key insight is to notice that the Jacobian corresponding to the use of one basis function is of a form whose sparsity is invariant to inversion. The main restriction of the method is a constraint on the number of active variables, which suggests a variety of techniques or generalization to allow the constraint to be enforced or relaxed. This technique has the potential to allow the efficient direct calculation of Newton steps as well as other numeric calculations of interest.

Mansura Habiba, Barak A. Pearlmutter, Mehrdad Maleki

Continuous-time series is essential for different modern application areas, e.g. healthcare, automobile, energy, finance, Internet of things (IoT) and other related areas. Different application needs to process as well as analyse a massive amount of data in time series structure in order to determine the data-driven result, for example, financial trend prediction, potential probability of the occurrence of a particular event occurrence identification, patient health record processing and so many more. However, modeling real-time data using a continuous-time series is challenging since the dynamical systems behind the data could be a differential equation. Several research works have tried to solve the challenges of modelling the continuous-time series using different neural network models and approaches for data processing and learning. The existing deep learning models are not free from challenges and limitations due to diversity among different attributes, behaviour, duration of steps, energy, and data sampling rate. This paper has described the general problem domain of time series and reviewed the challenges of modelling the continuous time series. We have presented a comparative analysis of recent developments in deep learning models and their contribution to solving different difficulties of modelling the continuous time series. We have also identified the limitations of the existing neural network model and open issues. The main goal of this review is to understand the recent trend of neural network models used in a different real-world application with continuous-time data.

Thomas Flinkow, Barak A. Pearlmutter, Rosemary Monahan

Extensive research on formal verification of machine learning (ML) systems indicates that learning from data alone often fails to capture underlying background knowledge. A variety of verifiers have been developed to ensure that a machine-learnt model satisfies correctness and safety properties, however, these verifiers typically assume a trained network with fixed weights. ML-enabled autonomous systems are required to not only detect incorrect predictions, but should also possess the ability to self-correct, continuously improving and adapting. A promising approach for creating ML models that inherently satisfy constraints is to encode background knowledge as logical constraints that guide the learning process via so-called differentiable logics. In this research preview, we compare and evaluate various logics from the literature in weakly-supervised contexts, presenting our findings and highlighting open problems for future work. Our experimental results are broadly consistent with results reported previously in literature; however, learning with differentiable logics introduces a new hyperparameter that is difficult to tune and has significant influence on the effectiveness of the logics.

Bradley T. Baker, Barak A. Pearlmutter, Robyn Miller et al.

Our understanding of learning dynamics of deep neural networks (DNNs) remains incomplete. Recent research has begun to uncover the mathematical principles underlying these networks, including the phenomenon of "Neural Collapse", where linear classifiers within DNNs converge to specific geometrical structures during late-stage training. However, the role of geometric constraints in learning extends beyond this terminal phase. For instance, gradients in fully-connected layers naturally develop a low-rank structure due to the accumulation of rank-one outer products over a training batch. Despite the attention given to methods that exploit this structure for memory saving or regularization, the emergence of low-rank learning as an inherent aspect of certain DNN architectures has been under-explored. In this paper, we conduct a comprehensive study of gradient rank in DNNs, examining how architectural choices and structure of the data effect gradient rank bounds. Our theoretical analysis provides these bounds for training fully-connected, recurrent, and convolutional neural networks. We also demonstrate, both theoretically and empirically, how design choices like activation function linearity, bottleneck layer introduction, convolutional stride, and sequence truncation influence these bounds. Our findings not only contribute to the understanding of learning dynamics in DNNs, but also provide practical guidance for deep learning engineers to make informed design decisions.

Jack A. Kilgallen, Barak A. Pearlmutter, Jeffrey Mark Siskind

In neural-decoding studies, recordings of participants' responses to stimuli are used to train models. In recent years, there has been an explosion of publications detailing applications of innovations from deep-learning research to neural-decoding studies. The data-hungry models used in these experiments have resulted in a demand for increasingly large datasets. Consequently, in some studies, the same stimuli are presented multiple times to each participant to increase the number of trials available for use in model training. However, when a decoding model is trained and subsequently evaluated on responses to the same stimuli, stimulus identity becomes a confounder for accuracy. We term this the repeated-stimulus confound. We identify a susceptible dataset, and 16 publications which report model performance based on evaluation procedures affected by the confound. We conducted experiments using models from the affected studies to investigate the likely extent to which results in the literature have been misreported. Our findings suggest that the decoding accuracies of these models were overestimated by between 4.46-7.42%. Our analysis also indicates that per 1% increase in accuracy under the confound, the magnitude of the overestimation increases by 0.26%. The confound not only results in optimistic estimates of decoding performance, but undermines the validity of several claims made within the affected publications. We conducted further experiments to investigate the implications of the confound in alternative contexts. We found that the same methodology used within the affected studies could also be used to justify an array of pseudoscientific claims, such as the existence of extrasensory perception.

Mrinal Mathur, Mike Doan, Barak Pearlmutter et al.

Transformers evaluated in a single, fixed-depth pass are provably limited in expressive power to the constant-depth circuit class TC0. Running a Transformer autoregressively removes that ceiling -- first in next-token prediction and, more recently, in chain-of-thought reasoning. Both regimes rely on feedback loops that decode internal states into tokens only to re-encode them in subsequent steps. While this "thinking aloud" mirrors human reasoning, biological brains iterate without externalising intermediate states as language. To boost the expressive power of encoder Transformers without resorting to token-level autoregression, we introduce the SELF-Transformer: an encoder layer that iteratively refines its own attention weights to a fixed point. Instead of producing -- in one pass -- the alignment matrix that remixes the input sequence, the SELF-Transformer iteratively updates that matrix internally, scaling test-time computation with input difficulty. This adaptivity yields up to 20\% accuracy gains on encoder-style benchmarks without increasing parameter count, demonstrating that input-adaptive alignment at test time offers substantial benefits for only a modest extra compute budget. Self-Transformers thus recover much of the expressive power of iterative reasoning while preserving the simplicity of pure encoder architectures.

Syed Ali Asadullah Bukhari, Thomas Flinkow, Medet Inkarbekov et al.

The increased reliance of self-driving vehicles on neural networks opens up the challenge of their verification. In this paper we present an experience report, describing a case study which we undertook to explore the design and training of a neural network on a custom dataset for vision-based autonomous navigation. We are particularly interested in the use of machine learning with differentiable logics to obtain networks satisfying basic safety properties by design, guaranteeing the behaviour of the neural network after training. We motivate the choice of a suitable neural network verifier for our purposes and report our observations on the use of neural network verifiers for self-driving systems.

Atılım Güneş Baydin, Barak A. Pearlmutter, Don Syme et al.

Using backpropagation to compute gradients of objective functions for optimization has remained a mainstay of machine learning. Backpropagation, or reverse-mode differentiation, is a special case within the general family of automatic differentiation algorithms that also includes the forward mode. We present a method to compute gradients based solely on the directional derivative that one can compute exactly and efficiently via the forward mode. We call this formulation the forward gradient, an unbiased estimate of the gradient that can be evaluated in a single forward run of the function, entirely eliminating the need for backpropagation in gradient descent. We demonstrate forward gradient descent in a range of problems, showing substantial savings in computation and enabling training up to twice as fast in some cases.

Mansura Habiba, Barak A. Pearlmutter

Recent work in deep learning focuses on solving physical systems in the Ordinary Differential Equation or Partial Differential Equation. This current work proposed a variant of Convolutional Neural Networks (CNNs) that can learn the hidden dynamics of a physical system using ordinary differential equation (ODEs) systems (ODEs) and Partial Differential Equation systems (PDEs). Instead of considering the physical system such as image, time -series as a system of multiple layers, this new technique can model a system in the form of Differential Equation (DEs). The proposed method has been assessed by solving several steady-state PDEs on irregular domains, including heat equations, Navier-Stokes equations.

Mansura Habiba, Barak A. Pearlmutter

This work proposes a Neural Network model that can control its depth using an iterate-to-fixed-point operator. The architecture starts with a standard layered Network but with added connections from current later to earlier layers, along with a gate to make them inactive under most circumstances. These ``temporal wormhole'' connections create a shortcut that allows the Neural Network to use the information available at deeper layers and re-do earlier computations with modulated inputs. End-to-end training is accomplished by using appropriate calculations for a numeric iterate-to-fixed-point operator. In a typical case, where the ``wormhole'' connections are inactive, this is inexpensive; but when they are active, the network takes a longer time to settle down, and the gradient calculation is also more laborious, with an effect similar to making the network deeper. In contrast to the existing skip-connection concept, this proposed technique enables information to flow up and down in the network. Furthermore, the flow of information follows a fashion that seems analogous to the afferent and efferent flow of information through layers of processing in the brain. We evaluate models that use this novel mechanism on different long-term dependency tasks. The results are competitive with other studies, showing that the proposed model contributes significantly to overcoming traditional deep learning models' vanishing gradient descent problem. At the same time, the training time is significantly reduced, as the ``easy'' input cases are processed more quickly than ``difficult'' ones.

Mansura Habiba, Eoin Brophy, Barak A. Pearlmutter et al.

Continuous medical time series data such as ECG is one of the most complex time series due to its dynamic and high dimensional characteristics. In addition, due to its sensitive nature, privacy concerns and legal restrictions, it is often even complex to use actual data for different medical research. As a result, generating continuous medical time series is a very critical research area. Several research works already showed that the ability of generative adversarial networks (GANs) in the case of continuous medical time series generation is promising. Most medical data generation works, such as ECG synthesis, are mainly driven by the GAN model and its variation. On the other hand, Some recent work on Neural Ordinary Differential Equation (Neural ODE) demonstrates its strength against informative missingness, high dimension as well as dynamic nature of continuous time series. Instead of considering continuous-time series as a discrete-time sequence, Neural ODE can train continuous time series in real-time continuously. In this work, we used Neural ODE based model to generate synthetic sine waves and synthetic ECG. We introduced a new technique to design the generative adversarial network with Neural ODE based Generator and Discriminator. We developed three new models to synthesise continuous medical data. Different evaluation metrics are then used to quantitatively assess the quality of generated synthetic data for real-world applications and data analysis. Another goal of this work is to combine the strength of GAN and Neural ODE to generate synthetic continuous medical time series data such as ECG. We also evaluated both the GAN model and the Neural ODE model to understand the comparative efficiency of models from the GAN and Neural ODE family in medical data synthesis.

Mehrdad Maleki, Mansura Habiba, Barak A. Pearlmutter

There is an analogy between the ResNet (Residual Network) architecture for deep neural networks and an Euler solver for an ODE. The transformation performed by each layer resembles an Euler step in solving an ODE. We consider the Heun Method, which involves a single predictor-corrector cycle, and complete the analogy, building a predictor-corrector variant of ResNet, which we call a HeunNet. Just as Heun's method is more accurate than Euler's, experiments show that HeunNet achieves high accuracy with low computational (both training and test) time compared to both vanilla recurrent neural networks and other ResNet variants.

Bradley T. Baker, Aashis Khanal, Vince D. Calhoun et al.

Although distributed machine learning has opened up many new and exciting research frontiers, fragmentation of models and data across different machines, nodes, and sites still results in considerable communication overhead, impeding reliable training in real-world contexts. The focus on gradients as the primary shared statistic during training has spawned a number of intuitive algorithms for distributed deep learning; however, gradient-centric training of large deep neural networks (DNNs) tends to be communication-heavy, often requiring additional adaptations such as sparsity constraints, compression, quantization, and more, to curtail bandwidth. We introduce an innovative, communication-friendly approach for training distributed DNNs, which capitalizes on the outer-product structure of the gradient as revealed by the mechanics of auto-differentiation. The exposed structure of the gradient evokes a new class of distributed learning algorithm, which is naturally more communication-efficient than full gradient sharing. Our approach, called distributed auto-differentiation (dAD), builds off a marriage of rank-based compression and the innate structure of the gradient as an outer-product. We demonstrate that dAD trains more efficiently than other state of the art distributed methods on modern architectures, such as transformers, when applied to large-scale text and imaging datasets. The future of distributed learning, we determine, need not be dominated by gradient-centric algorithms.

Mansura Habiba, Barak A. Pearlmutter

Neural differential equations are a promising new member in the neural network family. They show the potential of differential equations for time series data analysis. In this paper, the strength of the ordinary differential equation (ODE) is explored with a new extension. The main goal of this work is to answer the following questions: (i)~can ODE be used to redefine the existing neural network model? (ii)~can Neural ODEs solve the irregular sampling rate challenge of existing neural network models for a continuous time series, i.e., length and dynamic nature, (iii)~how to reduce the training and evaluation time of existing Neural ODE systems? This work leverages the mathematical foundation of ODEs to redesign traditional RNNs such as Long Short-Term Memory (LSTM) and Gated Recurrent Unit (GRU). The main contribution of this paper is to illustrate the design of two new ODE-based RNN models (GRU-ODE model and LSTM-ODE) which can compute the hidden state and cell state at any point of time using an ODE solver. These models reduce the computation overhead of hidden state and cell state by a vast amount. The performance evaluation of these two new models for learning continuous time series with irregular sampling rate is then demonstrated. Experiments show that these new ODE based RNN models require less training time than Latent ODEs and conventional Neural ODEs. They can achieve higher accuracy quickly, and the design of the neural network is simpler than, previous neural ODE systems.

Mansura Habiba, Barak A. Pearlmutter

Informative missingness is unavoidable in the digital processing of continuous time series, where the value for one or more observations at different time points are missing. Such missing observations are one of the major limitations of time series processing using deep learning. Practical applications, e.g., sensor data, healthcare, weather, generates data that is in truth continuous in time, and informative missingness is a common phenomenon in these datasets. These datasets often consist of multiple variables, and often there are missing values for one or many of these variables. This characteristic makes time series prediction more challenging, and the impact of missing input observations on the accuracy of the final output can be significant. A recent novel deep learning model called GRU-D is one early attempt to address informative missingness in time series data. On the other hand, a new family of neural networks called Neural ODEs (Ordinary Differential Equations) are natural and efficient for processing time series data which is continuous in time. In this paper, a deep learning model is proposed that leverages the effective imputation of GRU-D, and the temporal continuity of Neural ODEs. A time series classification task performed on the PhysioNet dataset demonstrates the performance of this architecture.

Sébastien Geeraert, Charles-Albert Lehalle, Barak Pearlmutter et al.

Automatic differentiation is involved for long in applied mathematics as an alternative to finite difference to improve the accuracy of numerical computation of derivatives. Each time a numerical minimization is involved, automatic differentiation can be used. In between formal derivation and standard numerical schemes, this approach is based on software solutions applying mechanically the chain rule to obtain an exact value for the desired derivative. It has a cost in memory and cpu consumption. For participants of financial markets (banks, insurances, financial intermediaries, etc), computing derivatives is needed to obtain the sensitivity of its exposure to well-defined potential market moves. It is a way to understand variations of their balance sheets in specific cases. Since the 2008 crisis, regulation demand to compute this kind of exposure to many different case, to be sure market participants are aware and ready to face a wide spectrum of configurations. This paper shows how automatic differentiation provides a partial answer to this recent explosion of computation to perform. One part of the answer is a straightforward application of Adjoint Algorithmic Differentiation (AAD), but it is not enough. Since financial sensitivities involves specific functions and mix differentiation with Monte-Carlo simulations, dedicated tools and associated theoretical results are needed. We give here short introductions to typical cases arising when one use AAD on financial markets.

Atılım Güneş Baydin, Barak A. Pearlmutter, Jeffrey Mark Siskind

The deep learning community has devised a diverse set of methods to make gradient optimization, using large datasets, of large and highly complex models with deeply cascaded nonlinearities, practical. Taken as a whole, these methods constitute a breakthrough, allowing computational structures which are quite wide, very deep, and with an enormous number and variety of free parameters to be effectively optimized. The result now dominates much of practical machine learning, with applications in machine translation, computer vision, and speech recognition. Many of these methods, viewed through the lens of algorithmic differentiation (AD), can be seen as either addressing issues with the gradient itself, or finding ways of achieving increased efficiency using tricks that are AD-related, but not provided by current AD systems. The goal of this paper is to explain not just those methods of most relevance to AD, but also the technical constraints and mindset which led to their discovery. After explaining this context, we present a "laundry list" of methods developed by the deep learning community. Two of these are discussed in further mathematical detail: a way to dramatically reduce the size of the tape when performing reverse-mode AD on a (theoretically) time-reversible process like an ODE integrator; and a new mathematical insight that allows for the implementation of a stochastic Newton's method.

Jeffrey Mark Siskind, Barak A. Pearlmutter

Heretofore, automatic checkpointing at procedure-call boundaries, to reduce the space complexity of reverse mode, has been provided by systems like Tapenade. However, binomial checkpointing, or treeverse, has only been provided in Automatic Differentiation (AD) systems in special cases, e.g., through user-provided pragmas on DO loops in Tapenade, or as the nested taping mechanism in adol-c for time integration processes, which requires that user code be refactored. We present a framework for applying binomial checkpointing to arbitrary code with no special annotation or refactoring required. This is accomplished by applying binomial checkpointing directly to a program trace. This trace is produced by a general-purpose checkpointing mechanism that is orthogonal to AD.

Atilim Gunes Baydin, Barak A. Pearlmutter

Automatic differentiation---the mechanical transformation of numeric computer programs to calculate derivatives efficiently and accurately---dates to the origin of the computer age. Reverse mode automatic differentiation both antedates and generalizes the method of backwards propagation of errors used in machine learning. Despite this, practitioners in a variety of fields, including machine learning, have been little influenced by automatic differentiation, and make scant use of available tools. Here we review the technique of automatic differentiation, describe its two main modes, and explain how it can benefit machine learning practitioners. To reach the widest possible audience our treatment assumes only elementary differential calculus, and does not assume any knowledge of linear algebra.

Vamsi K. Potluru, Sergey M. Plis, Jonathan Le Roux et al.

Nonnegative matrix factorization (NMF) has become a ubiquitous tool for data analysis. An important variant is the sparse NMF problem which arises when we explicitly require the learnt features to be sparse. A natural measure of sparsity is the L$_0$ norm, however its optimization is NP-hard. Mixed norms, such as L$_1$/L$_2$ measure, have been shown to model sparsity robustly, based on intuitive attributes that such measures need to satisfy. This is in contrast to computationally cheaper alternatives such as the plain L$_1$ norm. However, present algorithms designed for optimizing the mixed norm L$_1$/L$_2$ are slow and other formulations for sparse NMF have been proposed such as those based on L$_1$ and L$_0$ norms. Our proposed algorithm allows us to solve the mixed norm sparsity constraints while not sacrificing computation time. We present experimental evidence on real-world datasets that shows our new algorithm performs an order of magnitude faster compared to the current state-of-the-art solvers optimizing the mixed norm and is suitable for large-scale datasets.