Felix Petersen

Nina Shvetsova, Felix Petersen, Anna Kukleva et al. · ibm-research, mit

Contrastive learning has become an important tool in learning representations from unlabeled data mainly relying on the idea of minimizing distance between positive data pairs, e.g., views from the same images, and maximizing distance between negative data pairs, e.g., views from different images. This paper proposes a new variation of the contrastive learning objective, Group Ordering Constraints (GroCo), that leverages the idea of sorting the distances of positive and negative pairs and computing the respective loss based on how many positive pairs have a larger distance than the negative pairs, and thus are not ordered correctly. To this end, the GroCo loss is based on differentiable sorting networks, which enable training with sorting supervision by matching a differentiable permutation matrix, which is produced by sorting a given set of scores, to a respective ground truth permutation matrix. Applying this idea to groupwise pre-ordered inputs of multiple positive and negative pairs allows introducing the GroCo loss with implicit emphasis on strong positives and negatives, leading to better optimization of the local neighborhood. We evaluate the proposed formulation on various self-supervised learning benchmarks and show that it not only leads to improved results compared to vanilla contrastive learning but also shows competitive performance to comparable methods in linear probing and outperforms current methods in k-NN performance.

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.

Debarghya Mukherjee, Felix Petersen, Mikhail Yurochkin et al.

Many instances of algorithmic bias are caused by distributional shifts. For example, machine learning (ML) models often perform worse on demographic groups that are underrepresented in the training data. In this paper, we leverage this connection between algorithmic fairness and distribution shifts to show that algorithmic fairness interventions can help ML models overcome distribution shifts, and that domain adaptation methods (for overcoming distribution shifts) can mitigate algorithmic biases. In particular, we show that (i) enforcing suitable notions of individual fairness (IF) can improve the out-of-distribution accuracy of ML models under the covariate shift assumption and that (ii) it is possible to adapt representation alignment methods for domain adaptation to enforce individual fairness. The former is unexpected because IF interventions were not developed with distribution shifts in mind. The latter is also unexpected because representation alignment is not a common approach in the individual fairness literature.

Felix Petersen

Classic algorithms and machine learning systems like neural networks are both abundant in everyday life. While classic computer science algorithms are suitable for precise execution of exactly defined tasks such as finding the shortest path in a large graph, neural networks allow learning from data to predict the most likely answer in more complex tasks such as image classification, which cannot be reduced to an exact algorithm. To get the best of both worlds, this thesis explores combining both concepts leading to more robust, better performing, more interpretable, more computationally efficient, and more data efficient architectures. The thesis formalizes the idea of algorithmic supervision, which allows a neural network to learn from or in conjunction with an algorithm. When integrating an algorithm into a neural architecture, it is important that the algorithm is differentiable such that the architecture can be trained end-to-end and gradients can be propagated back through the algorithm in a meaningful way. To make algorithms differentiable, this thesis proposes a general method for continuously relaxing algorithms by perturbing variables and approximating the expectation value in closed form, i.e., without sampling. In addition, this thesis proposes differentiable algorithms, such as differentiable sorting networks, differentiable renderers, and differentiable logic gate networks. Finally, this thesis presents alternative training strategies for learning with algorithms.

Jiaming Liu, Felix Petersen, Yunhe Gao et al.

Adversarial diffusion and diffusion-inversion methods have advanced unpaired image-to-image translation, but each faces key limitations. Adversarial approaches require target-domain adversarial loss during training, which can limit generalization to unseen data, while diffusion-inversion methods often produce low-fidelity translations due to imperfect inversion into noise-latent representations. In this work, we propose the Self-Supervised Semantic Bridge (SSB), a versatile framework that integrates external semantic priors into diffusion bridge models to enable spatially faithful translation without cross-domain supervision. Our key idea is to leverage self-supervised visual encoders to learn representations that are invariant to appearance changes but capture geometric structure, forming a shared latent space that conditions the diffusion bridges. Extensive experiments show that SSB outperforms strong prior methods for challenging medical image synthesis in both in-domain and out-of-domain settings, and extends easily to high-quality text-guided editing.

Joshua Kazdan, Hao Sun, Jiaqi Han et al.

Diffusion models have a tendency to exactly replicate their training data, especially when trained on small datasets. Most prior work has sought to mitigate this problem by imposing differential privacy constraints or masking parts of the training data, resulting in a notable substantial decrease in image quality. We present CPSample, a method that modifies the sampling process to prevent training data replication while preserving image quality. CPSample utilizes a classifier that is trained to overfit on random binary labels attached to the training data. CPSample then uses classifier guidance to steer the generation process away from the set of points that can be classified with high certainty, a set that includes the training data. CPSample achieves FID scores of 4.97 and 2.97 on CIFAR-10 and CelebA-64, respectively, without producing exact replicates of the training data. Unlike prior methods intended to guard the training images, CPSample only requires training a classifier rather than retraining a diffusion model, which is computationally cheaper. Moreover, our technique provides diffusion models with greater robustness against membership inference attacks, wherein an adversary attempts to discern which images were in the model's training dataset. We show that CPSample behaves like a built-in rejection sampler, and we demonstrate its capabilities to prevent mode collapse in Stable Diffusion.

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.

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.

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, Moritz Schubotz, Andre Greiner-Petter et al.

We tackle the problem of neural machine translation of mathematical formulae between ambiguous presentation languages and unambiguous content languages. Compared to neural machine translation on natural language, mathematical formulae have a much smaller vocabulary and much longer sequences of symbols, while their translation requires extreme precision to satisfy mathematical information needs. In this work, we perform the tasks of translating from LaTeX to Mathematica as well as from LaTeX to semantic LaTeX. While recurrent, recursive, and transformer networks struggle with preserving all contained information, we find that convolutional sequence-to-sequence networks achieve 95.1% and 90.7% exact matches, respectively.

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, Debarghya Mukherjee, Yuekai Sun et al.

Post-processing in algorithmic fairness is a versatile approach for correcting bias in ML systems that are already used in production. The main appeal of post-processing is that it avoids expensive retraining. In this work, we propose general post-processing algorithms for individual fairness (IF). We consider a setting where the learner only has access to the predictions of the original model and a similarity graph between individuals, guiding the desired fairness constraints. We cast the IF post-processing problem as a graph smoothing problem corresponding to graph Laplacian regularization that preserves the desired "treat similar individuals similarly" interpretation. Our theoretical results demonstrate the connection of the new objective function to a local relaxation of the original individual fairness. Empirically, our post-processing algorithms correct individual biases in large-scale NLP models such as BERT, while preserving accuracy.

Felix Petersen, Bastian Goldluecke, Oliver Deussen et al.

Reconstructing the 3D geometry of an object from an image is a major challenge in computer vision. Recently introduced differentiable renderers can be leveraged to learn the 3D geometry of objects from 2D images, but those approaches require additional supervision to enable the renderer to produce an output that can be compared to the input image. This can be scene information or constraints such as object silhouettes, uniform backgrounds, material, texture, and lighting. In this paper, we propose an approach that enables a differentiable rendering-based learning of 3D objects from images with backgrounds without the need for silhouette supervision. Instead of trying to render an image close to the input, we propose an adversarial style-transfer and domain adaptation pipeline that allows to translate the input image domain to the rendered image domain. This allows us to directly compare between a translated image and the differentiable rendering of a 3D object reconstruction in order to train the 3D object reconstruction network. We show that the approach learns 3D geometry from images with backgrounds and provides a better performance than constrained methods for single-view 3D object reconstruction on this task.

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.

Felix Petersen, Amit H. Bermano, Oliver Deussen et al.

The long-coveted task of reconstructing 3D geometry from images is still a standing problem. In this paper, we build on the power of neural networks and introduce Pix2Vex, a network trained to convert camera-captured images into 3D geometry. We present a novel differentiable renderer ($DR$) as a forward validation means during training. Our key insight is that $DR$s produce images of a particular appearance, different from typical input images. Hence, we propose adding an image-to-image translation component, converting between these rendering styles. This translation closes the training loop, while allowing to use minimal supervision only, without needing any 3D model as ground truth. Unlike state-of-the-art methods, our $DR$ is $C^\infty$ smooth and thus does not display any discontinuities at occlusions or dis-occlusions. Through our novel training scheme, our network can train on different types of images, where previous work can typically only train on images of a similar appearance to those rendered by a $DR$.

Felix Petersen, Moritz Schubotz, Bela Gipp

While it has become common to perform automated translations on natural language, performing translations between different representations of mathematical formulae has thus far not been possible. We implemented the first translator for mathematical formulae based on recursive neural networks. We chose recursive neural networks because mathematical formulae inherently include a structural encoding. In our implementation, we developed new techniques and topologies for recursive tree-to-tree neural networks based on multi-variate multi-valued Long Short-Term Memory cells. We propose a novel approach for mini-batch training that utilizes clustering and tree traversal. We evaluate our translator and analyze the behavior of our proposed topologies and techniques based on a translation from generic LaTeX to the semantic LaTeX notation. We use the semantic LaTeX notation from the Digital Library for Mathematical Formulae and the Digital Repository for Mathematical Formulae at the National Institute for Standards and Technology. We find that a simple heuristics-based clustering algorithm outperforms the conventional clustering algorithms on the task of clustering binary trees of mathematical formulae with respect to their topology. Furthermore, we find a mask for the loss function, which can prevent the neural network from finding a local minimum of the loss function. Given our preliminary results, a complete translation from formula to formula is not yet possible. However, we achieved a prediction accuracy of 47.05% for predicting symbols at the correct position and an accuracy of 92.3% when ignoring the predicted position. Concluding, our work advances the field of recursive neural networks by improving the training speed and quality of training. In the future, we will work towards a complete translation allowing a machine-interpretation of LaTeX formulae.