Anton Schiela

Leonie Kreis, Evelyn Herberg, Frederik Köhne et al.

The training of neural networks requires tedious and often manual tuning of the network architecture. We propose a systematic approach to inserting new layers during the training process. Our method eliminates the need to choose a fixed network size before training, is numerically inexpensive to execute and applicable to various architectures including fully connected feedforward networks, ResNets and CNNs. Our technique borrows ideas from constrained optimization and is based on first-order sensitivity information of the loss function with respect to the virtual parameters that additional layers, if inserted, would offer. In numerical experiments, our proposed sensitivity-based layer insertion technique (SensLI) exhibits improved performance on training loss and test error, compared to training on a fixed architecture, and reduced computational effort in comparison to training the extended architecture from the beginning. Our code is available on https://github.com/mathemml/SensLI.

Roland Herzog, Frederik Köhne, Leonie Kreis et al.

The Frobenius norm is a frequent choice of norm for matrices. In particular, the underlying Frobenius inner product is typically used to evaluate the gradient of an objective with respect to matrix variable, such as those occuring in the training of neural networks. We provide a broader view on the Frobenius norm and inner product for linear maps or matrices, and establish their dependence on inner products in the domain and co-domain spaces. This shows that the classical Frobenius norm is merely one special element of a family of more general Frobenius-type norms. The significant extra freedom furnished by this realization can be used, among other things, to precondition neural network training.

Frederik Köhne, Leonie Kreis, Anton Schiela et al.

This paper proposes a novel approach to adaptive step sizes in stochastic gradient descent (SGD) by utilizing quantities that we have identified as numerically traceable -- the Lipschitz constant for gradients and a concept of the local variance in search directions. Our findings yield a nearly hyperparameter-free algorithm for stochastic optimization, which has provable convergence properties and exhibits truly problem adaptive behavior on classical image classification tasks. Our framework is set in a general Hilbert space and thus enables the potential inclusion of a preconditioner through the choice of the inner product.

Frederik Köhne, Anton Schiela

Averaging, or smoothing, is a fundamental approach to obtain stable, de-noised estimates from noisy observations. In certain scenarios, observations made along trajectories of random dynamical systems are of particular interest. One popular smoothing technique for such a scenario is exponential moving averaging (EMA), which assigns observations a weight that decreases exponentially in their age, thus giving younger observations a larger weight. However, EMA fails to enjoy strong stochastic convergence properties, which stems from the fact that the weight assigned to the youngest observation is constant over time, preventing the noise in the averaged quantity from decreasing to zero. In this work, we consider an adaptation to EMA, which we call $p$-EMA, where the weights assigned to the last observations decrease to zero at a subharmonic rate. We provide stochastic convergence guarantees for this kind of averaging under mild assumptions on the autocorrelations of the underlying random dynamical system. We further discuss the implications of our results for a recently introduced adaptive step size control for Stochastic Gradient Descent (SGD), which uses $p$-EMA for averaging noisy observations.