Stephan Eckstein

Fabian Morelli, Stephan Eckstein

Ensembles of neural networks typically outperform individual networks but incur large computational costs, whereas weight aggregation produces less costly, yet also less accurate, aggregate models. We introduce partial fusion of networks, which interpolates between ensembles and weight aggregation and thus allows for a flexible tradeoff between computational cost and performance. A direct way to achieve this is to extend existing weight aggregation methods based on neuron-level similarity between different networks, where partial fusion then only aggregates weights of neurons which are most similar. We showcase one particular method to jointly identify which neurons are most similar and match them via partial optimal transport. Further, we consider the more general perspective of weight aggregation and partial fusion as generalized pruning of ensemble models, where neurons cannot just be deleted, but also linearly combined. Finally, we show that generalized pruning applied to a single network yields similar benefits as partial fusion by allowing for a tradeoff between isolating, deleting, and linearly combining neurons based on similarity. Our code is available at https://github.com/Fabian-Mor/partial_fusion_nn.

Yibo Yang, Stephan Eckstein, Marcel Nutz et al.

In the theory of lossy compression, the rate-distortion (R-D) function $R(D)$ describes how much a data source can be compressed (in bit-rate) at any given level of fidelity (distortion). Obtaining $R(D)$ for a given data source establishes the fundamental performance limit for all compression algorithms. We propose a new method to estimate $R(D)$ from the perspective of optimal transport. Unlike the classic Blahut--Arimoto algorithm which fixes the support of the reproduction distribution in advance, our Wasserstein gradient descent algorithm learns the support of the optimal reproduction distribution by moving particles. We prove its local convergence and analyze the sample complexity of our R-D estimator based on a connection to entropic optimal transport. Experimentally, we obtain comparable or tighter bounds than state-of-the-art neural network methods on low-rate sources while requiring considerably less tuning and computation effort. We also highlight a connection to maximum-likelihood deconvolution and introduce a new class of sources that can be used as test cases with known solutions to the R-D problem.

Moritz Brösamle, Stephan Eckstein

Existing expressivity results for transformers typically rely on hardmax attention, high precision, and other architectural modifications that disconnect them from the models used in practice. We bridge this gap by analyzing standard transformer decoders with softmax attention and rounding of activations and attention weights, while allowing depth and width to grow logarithmically with the context length. As an intermediate step, we construct hardmax transformers with ternary activations and well-separated attention scores that simulate Turing machines using Chain-of-Thought (CoT). This lets us convert the constructions to equivalent softmax transformers without the unrealistic parameter magnitudes or activation precision that prior approaches would require. Using the same technique, we analyze a recently proposed summarized CoT paradigm and show that it simulates Turing machines more efficiently, with model size scaling logarithmically in a space bound rather than a time bound. We empirically test predictions made by our results on a Sudoku reasoning task and find better alignment with learnability than for prior high-precision results. Our code is available at https://github.com/moritzbroe/transformer-expressivity.

Stephan Eckstein, Armin Iske, Mathias Trabs

In a high-dimensional regression framework, we study consequences of the naive two-step procedure where first the dimension of the input variables is reduced and second, the reduced input variables are used to predict the output variable with kernel regression. In order to analyze the resulting regression errors, a novel stability result for kernel regression with respect to the Wasserstein distance is derived. This allows us to bound errors that occur when perturbed input data is used to fit the regression function. We apply the general stability result to principal component analysis (PCA). Exploiting known estimates from the literature on both principal component analysis and kernel regression, we deduce convergence rates for the two-step procedure. The latter turns out to be particularly useful in a semi-supervised setting.

Beatrice Acciaio, Stephan Eckstein, Songyan Hou

We build a time-causal variational autoencoder (TC-VAE) for robust generation of financial time series data. Our approach imposes a causality constraint on the encoder and decoder networks, ensuring a causal transport from the real market time series to the fake generated time series. Specifically, we prove that the TC-VAE loss provides an upper bound on the causal Wasserstein distance between market distributions and generated distributions. Consequently, the TC-VAE loss controls the discrepancy between optimal values of various dynamic stochastic optimization problems under real and generated distributions. To further enhance the model's ability to approximate the latent representation of the real market distribution, we integrate a RealNVP prior into the TC-VAE framework. Finally, extensive numerical experiments show that TC-VAE achieves promising results on both synthetic and real market data. This is done by comparing real and generated distributions according to various statistical distances, demonstrating the effectiveness of the generated data for downstream financial optimization tasks, as well as showcasing that the generated data reproduces stylized facts of real financial market data.

Luca De Gennaro Aquino, Stephan Eckstein

We study MinMax solution methods for a general class of optimization problems related to (and including) optimal transport. Theoretically, the focus is on fitting a large class of problems into a single MinMax framework and generalizing regularization techniques known from classical optimal transport. We show that regularization techniques justify the utilization of neural networks to solve such problems by proving approximation theorems and illustrating fundamental issues if no regularization is used. We further study the relation to the literature on generative adversarial nets, and analyze which algorithmic techniques used therein are particularly suitable to the class of problems studied in this paper. Several numerical experiments showcase the generality of the setting and highlight which theoretical insights are most beneficial in practice.

Stephan Eckstein

This note shows that, for a fixed Lipschitz constant $L > 0$, one layer neural networks that are $L$-Lipschitz are dense in the set of all $L$-Lipschitz functions with respect to the uniform norm on bounded sets.

Stephan Eckstein, Michael Kupper

This paper presents a widely applicable approach to solving (multi-marginal, martingale) optimal transport and related problems via neural networks. The core idea is to penalize the optimization problem in its dual formulation and reduce it to a finite dimensional one which corresponds to optimizing a neural network with smooth objective function. We present numerical examples from optimal transport, martingale optimal transport, portfolio optimization under uncertainty and generative adversarial networks that showcase the generality and effectiveness of the approach.