Christian Borgelt

Felix Petersen, Christian Borgelt, Hilde Kuehne et al. · ibm-research, mit

Recently, research has increasingly focused on developing efficient neural network architectures. In this work, we explore logic gate networks for machine learning tasks by learning combinations of logic gates. These networks comprise logic gates such as "AND" and "XOR", which allow for very fast execution. The difficulty in learning logic gate networks is that they are conventionally non-differentiable and therefore do not allow training with gradient descent. Thus, to allow for effective training, we propose differentiable logic gate networks, an architecture that combines real-valued logics and a continuously parameterized relaxation of the network. The resulting discretized logic gate networks achieve fast inference speeds, e.g., beyond a million images of MNIST per second on a single CPU core.

Felix Petersen, Hilde Kuehne, Christian Borgelt et al. · ibm-research, mit

The top-k classification accuracy is one of the core metrics in machine learning. Here, k is conventionally a positive integer, such as 1 or 5, leading to top-1 or top-5 training objectives. In this work, we relax this assumption and optimize the model for multiple k simultaneously instead of using a single k. Leveraging recent advances in differentiable sorting and ranking, we propose a differentiable top-k cross-entropy classification loss. This allows training the network while not only considering the top-1 prediction, but also, e.g., the top-2 and top-5 predictions. We evaluate the proposed loss function for fine-tuning on state-of-the-art architectures, as well as for training from scratch. We find that relaxing k does not only produce better top-5 accuracies, but also leads to top-1 accuracy improvements. When fine-tuning publicly available ImageNet models, we achieve a new state-of-the-art for these models.

Felix Petersen, Christian Borgelt, Hilde Kuehne et al. · ibm-research, mit

Differentiable sorting algorithms allow training with sorting and ranking supervision, where only the ordering or ranking of samples is known. Various methods have been proposed to address this challenge, ranging from optimal transport-based differentiable Sinkhorn sorting algorithms to making classic sorting networks differentiable. One problem of current differentiable sorting methods is that they are non-monotonic. To address this issue, we propose a novel relaxation of conditional swap operations that guarantees monotonicity in differentiable sorting networks. We introduce a family of sigmoid functions and prove that they produce differentiable sorting networks that are monotonic. Monotonicity ensures that the gradients always have the correct sign, which is an advantage in gradient-based optimization. We demonstrate that monotonic differentiable sorting networks improve upon previous differentiable sorting methods.

Felix Petersen, Bastian Goldluecke, Christian Borgelt et al.

In this work, we present and study a generalized family of differentiable renderers. We discuss from scratch which components are necessary for differentiable rendering and formalize the requirements for each component. We instantiate our general differentiable renderer, which generalizes existing differentiable renderers like SoftRas and DIB-R, with an array of different smoothing distributions to cover a large spectrum of reasonable settings. We evaluate an array of differentiable renderer instantiations on the popular ShapeNet 3D reconstruction benchmark and analyze the implications of our results. Surprisingly, the simple uniform distribution yields the best overall results when averaged over 13 classes; in general, however, the optimal choice of distribution heavily depends on the task.

Felix Petersen, Hilde Kuehne, Christian Borgelt et al.

With the increasing inference cost of machine learning models, there is a growing interest in models with fast and efficient inference. Recently, an approach for learning logic gate networks directly via a differentiable relaxation was proposed. Logic gate networks are faster than conventional neural network approaches because their inference only requires logic gate operators such as NAND, OR, and XOR, which are the underlying building blocks of current hardware and can be efficiently executed. We build on this idea, extending it by deep logic gate tree convolutions, logical OR pooling, and residual initializations. This allows scaling logic gate networks up by over one order of magnitude and utilizing the paradigm of convolution. On CIFAR-10, we achieve an accuracy of 86.29% using only 61 million logic gates, which improves over the SOTA while being 29x smaller.

Felix Petersen, Aashwin Mishra, Hilde Kuehne et al.

We propose a new approach for propagating stable probability distributions through neural networks. Our method is based on local linearization, which we show to be an optimal approximation in terms of total variation distance for the ReLU non-linearity. This allows propagating Gaussian and Cauchy input uncertainties through neural networks to quantify their output uncertainties. To demonstrate the utility of propagating distributions, we apply the proposed method to predicting calibrated confidence intervals and selective prediction on out-of-distribution data. The results demonstrate a broad applicability of propagating distributions and show the advantages of our method over other approaches such as moment matching.

Christina Halmich, Lucas Höschler, Christoph Schranz et al.

The integration of machine learning and deep learning has transformed data analytics in biomechanics, enabled by extensive wearable sensor data. However, the field faces challenges such as limited large-scale datasets and high data acquisition costs, which hinder the development of robust algorithms. Data augmentation techniques show promise in addressing these issues, but their application to biomechanical time-series data requires comprehensive evaluation. This scoping review investigates data augmentation methods for time-series data in the biomechanics domain. It analyzes current approaches for augmenting and generating time-series datasets, evaluates their effectiveness, and offers recommendations for applying these techniques in biomechanics. Four databases, PubMed, IEEE Xplore, Scopus, and Web of Science, were searched for studies published between 2013 and 2024. Following PRISMA-ScR guidelines, a two-stage screening identified 21 relevant publications. Results show that there is no universally preferred method for augmenting biomechanical time-series data; instead, methods vary based on study objectives. A major issue identified is the absence of soft tissue artifacts in synthetic data, leading to discrepancies referred to as the synthetic gap. Moreover, many studies lack proper evaluation of augmentation methods, making it difficult to assess their effects on model performance and data quality. This review highlights the critical role of data augmentation in addressing limited dataset availability and improving model generalization in biomechanics. Tailoring augmentation strategies to the characteristics of biomechanical data is essential for advancing predictive modeling. A better understanding of how different augmentation methods impact data quality and downstream tasks will be key to developing more effective and realistic techniques.

Felix Petersen, Christian Borgelt, Tobias Sutter et al.

When training neural networks with custom objectives, such as ranking losses and shortest-path losses, a common problem is that they are, per se, non-differentiable. A popular approach is to continuously relax the objectives to provide gradients, enabling learning. However, such differentiable relaxations are often non-convex and can exhibit vanishing and exploding gradients, making them (already in isolation) hard to optimize. Here, the loss function poses the bottleneck when training a deep neural network. We present Newton Losses, a method for improving the performance of existing hard to optimize losses by exploiting their second-order information via their empirical Fisher and Hessian matrices. Instead of training the neural network with second-order techniques, we only utilize the loss function's second-order information to replace it by a Newton Loss, while training the network with gradient descent. This makes our method computationally efficient. We apply Newton Losses to eight differentiable algorithms for sorting and shortest-paths, achieving significant improvements for less-optimized differentiable algorithms, and consistent improvements, even for well-optimized differentiable algorithms.

Raoul H. Kutil, Georg Zimmermann, Barbara Strasser-Kirchweger et al.

Mental health disorders, particularly cognitive disorders defined by deficits in cognitive abilities, are described in detail in the DSM-5, which includes definitions and examples of signs and symptoms. A simplified, machine-actionable representation was developed to assess the similarity and separability of these disorders, but it is not suited for the most complex cases. Generating or applying a full binary matrix for similarity calculations is infeasible due to the vast number of symptom combinations. This research develops an alternative representation that links the narrative form of the DSM-5 with the binary matrix representation and enables automated generation of valid symptom combinations. Using a strict pre-defined format of lists, sets, and numbers with slight variations, complex diagnostic pathways involving numerous symptom combinations can be represented. This format, called the symptom profile generator (or simply generator), provides a readable, adaptable, and comprehensive alternative to a binary matrix while enabling easy generation of symptom combinations (profiles). Cognitive disorders, which typically involve multiple diagnostic criteria with several symptoms, can thus be expressed as lists of generators. Representing several psychotic disorders in generator form and generating all symptom combinations showed that matrix representations of complex disorders become too large to manage. The MPCS (maximum pairwise cosine similarity) algorithm cannot handle matrices of this size, prompting the development of a profile reduction method using targeted generator manipulation to find specific MPCS values between disorders. The generators allow easier creation of binary representations for large matrices and make it possible to calculate specific MPCS cases between complex disorders through conditional generators.

Felix Petersen, Christian Borgelt, Stefano Ermon

We consider the training of the first layer of vision models and notice the clear relationship between pixel values and gradient update magnitudes: the gradients arriving at the weights of a first layer are by definition directly proportional to (normalized) input pixel values. Thus, an image with low contrast has a smaller impact on learning than an image with higher contrast, and a very bright or very dark image has a stronger impact on the weights than an image with moderate brightness. In this work, we propose performing gradient descent on the embeddings produced by the first layer of the model. However, switching to discrete inputs with an embedding layer is not a reasonable option for vision models. Thus, we propose the conceptual procedure of (i) a gradient descent step on first layer activations to construct an activation proposal, and (ii) finding the optimal weights of the first layer, i.e., those weights which minimize the squared distance to the activation proposal. We provide a closed form solution of the procedure and adjust it for robust stochastic training while computing everything efficiently. Empirically, we find that TrAct (Training Activations) speeds up training by factors between 1.25x and 4x while requiring only a small computational overhead. We demonstrate the utility of TrAct with different optimizers for a range of different vision models including convolutional and transformer architectures.

Felix Petersen, Tobias Sutter, Christian Borgelt et al.

We present ISAAC (Input-baSed ApproximAte Curvature), a novel method that conditions the gradient using selected second-order information and has an asymptotically vanishing computational overhead, assuming a batch size smaller than the number of neurons. We show that it is possible to compute a good conditioner based on only the input to a respective layer without a substantial computational overhead. The proposed method allows effective training even in small-batch stochastic regimes, which makes it competitive to first-order as well as second-order methods.

Felix Petersen, Christian Borgelt, Hilde Kuehne et al.

The integration of algorithmic components into neural architectures has gained increased attention recently, as it allows training neural networks with new forms of supervision such as ordering constraints or silhouettes instead of using ground truth labels. Many approaches in the field focus on the continuous relaxation of a specific task and show promising results in this context. But the focus on single tasks also limits the applicability of the proposed concepts to a narrow range of applications. In this work, we build on those ideas to propose an approach that allows to integrate algorithms into end-to-end trainable neural network architectures based on a general approximation of discrete conditions. To this end, we relax these conditions in control structures such as conditional statements, loops, and indexing, so that resulting algorithms are smoothly differentiable. To obtain meaningful gradients, each relevant variable is perturbed via logistic distributions and the expectation value under this perturbation is approximated. We evaluate the proposed continuous relaxation model on four challenging tasks and show that it can keep up with relaxations specifically designed for each individual task.

Felix Petersen, Christian Borgelt, Hilde Kuehne et al.

Sorting and ranking supervision is a method for training neural networks end-to-end based on ordering constraints. That is, the ground truth order of sets of samples is known, while their absolute values remain unsupervised. For that, we propose differentiable sorting networks by relaxing their pairwise conditional swap operations. To address the problems of vanishing gradients and extensive blurring that arise with larger numbers of layers, we propose mapping activations to regions with moderate gradients. We consider odd-even as well as bitonic sorting networks, which outperform existing relaxations of the sorting operation. We show that bitonic sorting networks can achieve stable training on large input sets of up to 1024 elements.

Felix Petersen, Christian Borgelt, Oliver Deussen

Artificial neural networks revolutionized many areas of computer science in recent years since they provide solutions to a number of previously unsolved problems. On the other hand, for many problems, classic algorithms exist, which typically exceed the accuracy and stability of neural networks. To combine these two concepts, we present a new kind of neural networks$-$algorithmic neural networks (AlgoNets). These networks integrate smooth versions of classic algorithms into the topology of neural networks. A forward AlgoNet includes algorithmic layers into existing architectures while a backward AlgoNet can solve inverse problems without or with only weak supervision. In addition, we present the $\texttt{algonet}$ package, a PyTorch based library that includes, inter alia, a smoothly evaluated programming language, a smooth 3D mesh renderer, and smooth sorting algorithms.