Deep neural policies have unlocked agile flight for quadcopters, adaptive grasping for manipulators, and reliable navigation for ground robots, yet their millions of weights conflict with the tight memory and real-time constraints of embedded microcontrollers. Second-order pruning methods, such as Optimal Brain Damage (OBD) and its variants, including Optimal Brain Surgeon (OBS) and the recent SparseGPT, compress networks in a single pass by leveraging the local Hessian, achieving far higher sparsity than magnitude thresholding. Despite their success in vision and language, the consequences of such weight removal on closed-loop stability, tracking accuracy, and safety have remained unclear. We present the first mathematically rigorous robustness analysis of second-order pruning in nonlinear discrete-time control. The system evolves under a continuous transition map, while the controller is an $L$-layer multilayer perceptron with ReLU-type activations that are globally 1-Lipschitz. Pruning the weight matrix of layer $k$ replaces $W_k$ with $W_k+δW_k$, producing the perturbed parameter vector $\widehatΘ=Θ+δΘ$ and the pruned policy $π(\cdot;\widehatΘ)$. For every input state $s\in X$ we derive the closed-form inequality $ \|π(s;Θ)-π(s;\widehatΘ)\|_2 \le C_k(s)\,\|δW_k\|_2, $ where the constant $C_k(s)$ depends only on unpruned spectral norms and biases, and can be evaluated in closed form from a single forward pass. The derived bounds specify, prior to field deployment, the maximal admissible pruning magnitude compatible with a prescribed control-error threshold. By linking second-order network compression with closed-loop performance guarantees, our work narrows a crucial gap between modern deep-learning tooling and the robustness demands of safety-critical autonomous systems.

There are many ways to make a deep neural network smaller. In this paper, we focus on three categories of model size reduction: pruning, quantization, and training smaller models from scratch. Quantization means changing model parameters to lower-precision formats, like changing all 32-bit floating point parameters to 16-bit, which results in file size about half as large. Pruning deals with deleting parameters or groups of parameters (like entire neurons) from a trained model to make it smaller (often followed by a fine-tuning round of training, as done in our experiments). Parameter-level pruning (also called unstructured pruning) prunes individual parameters at a time, whereas neuron-level pruning (also called structured pruning) prunes all parameters associated with a given neuron at once. To simplify terminology across multiple methods we use the term'damage' to broadly refer to the undesired impact of removing a node or zeroing a weight on network performance. Different compression methods use different approaches to either estimate damage directly, or rank neurons or weights in order of increasing assumed damage according to some other metric that does not directly evaluate the impact on loss or performance.

We have used information-theoretic ideas to derive a class of prac(cid:173) tical and nearly optimal schemes for adapting the size of a neural network. By removing unimportant weights from a network, sev(cid:173) eral improvements can be expected: better generalization, fewer training examples required, and improved speed of learning and/or classification. The basic idea is to use second-derivative informa(cid:173) tion to make a tradeoff between network complexity and training set error. Experiments confirm the usefulness of the methods on a real-world application.

We extend Optimal Brain Surgeon (OBS) - a second-order method for pruning networks - to allow for general error measures, and explore a reduced computational and storage implementation via a dominant eigenspace decomposition. Simulations on nonlinear, noisy pattern classification problems reveal that OBS does lead to improved generalization, and performs favorably in comparison with Optimal Brain Damage (OBD). We find that the required retraining steps in OBD may lead to inferior generalization, that can be interpreted as due to injecting noise backa result the system. A common technique is to stop training of a largeinto at the minimum validation error. We found that the testnetwork error could be reduced even further by means of OBS (but not OBD) pruning.

Many techniques for model selection in the field of neural networks correspond to well established statistical methods. The method of'stopped training', on the other hand, in which an oversized network is trained until the error on a further validation set of examples deteriorates, then training is stopped, is a true innovation, since model selection doesn't require convergence of the training process. In this paper we show that this performance can be significantly enhanced by extending the'non convergent model selection method' of stopped training to include dynamic topology modifications (dynamic weight pruning) and modified complexity penalty term methods in which the weighting of the penalty term is adjusted during the training process. 1 INTRODUCTION One of the central topics in the field of neural networks is that of model selection. Both the theoretical and practical side of this have been intensively investigated and a vast array of methods have been suggested to perform this task. A widely used class of techniques starts by choosing an'oversized' network architecture then either removing redundant elements based on some measure of saliency (pruning), adding a further term to the cost function penalizing complexity (penalty terms), and finally, observing the error on a further validation set of examples, then stopping training as soon as this performance begins to deteriorate (stopped training).

We investigate the use of information from all second order derivatives of the error function to perfonn network pruning (i.e., removing unimportant weights from a trained network) in order to improve generalization, simplify networks, reduce hardware or storage requirements, increase the speed of further training, and in some cases enable rule extraction. Our method, Optimal Brain Surgeon (OBS), is Significantly better than magnitude-based methods and Optimal Brain Damage [Le Cun, Denker and Sol1a, 1990], which often remove the wrong weights. OBS permits the pruning of more weights than other methods (for the same error on the training set), and thus yields better generalization on test data. Crucial to OBS is a recursion relation for calculating the inverse Hessian matrix HI from training data and structural information of the net. OBS permits a 90%, a 76%, and a 62% reduction in weights over backpropagation with weighL decay on three benchmark MONK's problems [Thrun et aI., 1991]. Of OBS, Optimal Brain Damage, and magnitude-based methods, only OBS deletes the correct weights from a trained XOR network in every case. Finally, whereas Sejnowski and Rosenberg [1987J used 18,000 weights in their NETtalk network, we used OBS to prune a network to just 1560 weights, yielding better generalization.

Many techniques for model selection in the field of neural networks correspond to well established statistical methods. The method of'stopped training', on the other hand, in which an oversized network is trained until the error on a further validation set of examples deteriorates, then training is stopped, is a true innovation, since model selection doesn't require convergence of the training process. In this paper we show that this performance can be significantly enhanced by extending the'non convergent model selection method' of stopped training to include dynamic topology modifications (dynamic weight pruning) and modified complexity penalty term methods in which the weighting of the penalty term is adjusted during the training process. 1 INTRODUCTION One of the central topics in the field of neural networks is that of model selection. Both the theoretical and practical side of this have been intensively investigated and a vast array of methods have been suggested to perform this task. A widely used class of techniques starts by choosing an'oversized' network architecture then either removing redundant elements based on some measure of saliency (pruning), adding a further term to the cost function penalizing complexity (penalty terms), and finally, observing the error on a further validation set of examples, then stopping training as soon as this performance begins to deteriorate (stopped training).

We investigate the use of information from all second order derivatives of the error function to perfonn network pruning (i.e., removing unimportant weights from a trained network) in order to improve generalization, simplify networks, reduce hardware or storage requirements, increase the speed of further training, and in some cases enable rule extraction. Our method, Optimal Brain Surgeon (OBS), is Significantly better than magnitude-based methods and Optimal Brain Damage [Le Cun, Denker and Sol1a, 1990], which often remove the wrong weights. OBS permits the pruning of more weights than other methods (for the same error on the training set), and thus yields better generalization on test data. Crucial to OBS is a recursion relation for calculating the inverse Hessian matrix HI from training data and structural information of the net. OBS permits a 90%, a 76%, and a 62% reduction in weights over backpropagation with weighL decay on three benchmark MONK's problems [Thrun et aI., 1991]. Of OBS, Optimal Brain Damage, and magnitude-based methods, only OBS deletes the correct weights from a trained XOR network in every case. Finally, whereas Sejnowski and Rosenberg [1987J used 18,000 weights in their NETtalk network, we used OBS to prune a network to just 1560 weights, yielding better generalization.

We investigate the use of information from all second order derivatives of the error function to perfonn network pruning (i.e., removing unimportant weights from a trained network) in order to improve generalization, simplify networks, reduce hardware or storage requirements, increase the speed of further training, and in some cases enable rule extraction. Our method, Optimal Brain Surgeon (OBS), is Significantly better than magnitude-based methods and Optimal Brain Damage [Le Cun, Denker and Sol1a, 1990], which often remove the wrong weights. OBS permits the pruning of more weights than other methods (for the same error on the training set), and thus yields better generalization on test data. Crucial to OBS is a recursion relation for calculating the inverse Hessian matrix HI from training data and structural information of the net. OBS permits a 90%, a 76%, and a 62% reduction in weights over backpropagation with weighL decay on three benchmark MONK's problems [Thrun et aI., 1991]. Of OBS, Optimal Brain Damage, and magnitude-based methods, only OBS deletes the correct weights from a trained XOR network in every case. Finally, whereas Sejnowski and Rosenberg [1987J used 18,000 weights in their NETtalk network, we used OBS to prune a network to just 1560 weights, yielding better generalization.

Rep. Germany Abstract Many techniques for model selection in the field of neural networks correspond to well established statistical methods. The method of'stopped training', on the other hand, in which an oversized network is trained until the error on a further validation set of examples deteriorates,then training is stopped, is a true innovation, since model selection doesn't require convergence of the training process. Inthis paper we show that this performance can be significantly enhanced by extending the'nonconvergent model selection method' of stopped training to include dynamic topology modifications (dynamic weight pruning) and modified complexity penalty term methods in which the weighting of the penalty term is adjusted during the training process. 1 INTRODUCTION One of the central topics in the field of neural networks is that of model selection. Both the theoretical and practical side of this have been intensively investigated and a vast array of methods have been suggested to perform this task. A widely used class of techniques starts by choosing an'oversized' network architecture then either removing redundant elements based on some measure of saliency (pruning), adding a further term to the cost function penalizing complexity (penalty terms), and finally, observing the error on a further validation set of examples, then stopping training as soon as this performance begins to deteriorate (stopped training).