Learning Rate Rewinding (LRR) has been established as a strong variant of Iterative Magnitude Pruning (IMP) to find lottery tickets in deep overparameterized neural networks. While both iterative pruning schemes couple structure and parameter learning, understanding how LRR excels in both aspects can bring us closer to the design of more flexible deep learning algorithms that can optimize diverse sets of sparse architectures. To this end, we conduct experiments that disentangle the effect of mask learning and parameter optimization and how both benefit from overparameterization. The ability of LRR to flip parameter signs early and stay robust to sign perturbations seems to make it not only more effective in mask identification but also in optimizing diverse sets of masks, including random ones. In support of this hypothesis, we prove in a simplified single hidden neuron setting that LRR succeeds in more cases than IMP, as it can escape initially problematic sign configurations.
@inproceedings{gadhikar2024masks,title={Masks, Signs, And Learning Rate Rewinding},author={Gadhikar, Advait Harshal and Burkholz, Rebekka},booktitle={The Twelfth International Conference on Learning Representations},year={2024},url={https://openreview.net/forum?id=qODvxQ8TXW},}
Layer normalization, for which Batch Normalization (BN) is a popular choice, is an integral part of many deep learning architectures and contributes significantly to the learning success. We provide a partial explanation for this phenomenon by proving that training normalization layers alone is already sufficient for universal function approximation if the number of available, potentially random features matches or exceeds the weight parameters of the target networks that can be expressed. Our bound on the number of required features does not only improve on a recent result for fully-connected feed-forward architectures but also applies to CNNs with and without residual connections and almost arbitrary activation functions (which include ReLUs). Our explicit construction of a given target network solves a depth-width trade-off that is driven by architectural constraints and can explain why switching off entire neurons can have representational benefits, as has been observed empirically. To validate our theory, we explicitly match target networks that outperform experimentally obtained networks with trained BN parameters by utilizing a sufficient number of random features.
@inproceedings{burkholz2024batch,title={Batch normalization is sufficient for universal function approximation in {CNN}s},author={Burkholz, Rebekka},booktitle={The Twelfth International Conference on Learning Representations},year={2024},url={https://openreview.net/forum?id=wOSYMHfENq},}
While the expressive power and computational capabilities of graph neural networks (GNNs) have been theoretically studied, their optimization and learning dynamics, in general, remain largely unexplored. Our study undertakes the Graph Attention Network (GAT), a popular GNN architecture in which a node’s neighborhood aggregation is weighted by parameterized attention coefficients. We derive a conservation law of GAT gradient flow dynamics, which explains why a high portion of parameters in GATs with standard initialization struggle to change during training. This effect is amplified in deeper GATs, which perform significantly worse than their shallow counterparts. To alleviate this problem, we devise an initialization scheme that balances the GAT network. Our approach i) allows more effective propagation of gradients and in turn enables trainability of deeper networks, and ii) attains a considerable speedup in training and convergence time in comparison to the standard initialization. Our main theorem serves as a stepping stone to studying the learning dynamics of positive homogeneous models with attention mechanisms.
@inproceedings{mustafa2023are,title={Are {GAT}s Out of Balance?},author={Mustafa, Nimrah and Bojchevski, Aleksandar and Burkholz, Rebekka},booktitle={Thirty-seventh Conference on Neural Information Processing Systems},year={2023},url={https://openreview.net/forum?id=qY7UqLoora},}
Random masks define surprisingly effective sparse neural network models, as has been shown empirically. The resulting sparse networks can often compete with dense architectures and state-of-the-art lottery ticket pruning algorithms, even though they do not rely on computationally expensive prune-train iterations and can be drawn initially without significant computational overhead. We offer a theoretical explanation of how random masks can approximate arbitrary target networks if they are wider by a logarithmic factor in the inverse sparsity 1 / \log(1/\textsparsity). This overparameterization factor is necessary at least for 3-layer random networks, which elucidates the observed degrading performance of random networks at higher sparsity. At moderate to high sparsity levels, however, our results imply that sparser networks are contained within random source networks so that any dense-to-sparse training scheme can be turned into a computationally more efficient sparse-to-sparse one by constraining the search to a fixed random mask. We demonstrate the feasibility of this approach in experiments for different pruning methods and propose particularly effective choices of initial layer-wise sparsity ratios of the random source network. As a special case, we show theoretically and experimentally that random source networks also contain strong lottery tickets.
@inproceedings{pmlr-v202-gadhikar23a,title={Why Random Pruning Is All We Need to Start Sparse},author={Gadhikar, Advait Harshal and Mukherjee, Sohom and Burkholz, Rebekka},booktitle={Proceedings of the 40th International Conference on Machine Learning},pages={10542--10570},year={2023},editor={Krause, Andreas and Brunskill, Emma and Cho, Kyunghyun and Engelhardt, Barbara and Sabato, Sivan and Scarlett, Jonathan},volume={202},series={Proceedings of Machine Learning Research},month={23--29 Jul},publisher={PMLR},url={https://proceedings.mlr.press/v202/gadhikar23a.html},}
The strong lottery ticket hypothesis has highlighted the potential for training deep neural networks by pruning, which has inspired interesting practical and theoretical insights into how neural networks can represent functions. For networks with ReLU activation functions, it has been proven that a target network with depth L can be approximated by the subnetwork of a randomly initialized neural network that has double the target’s depth 2L and is wider by a logarithmic factor. We show that a depth L+1 is sufficient. This result indicates that we can expect to find lottery tickets at realistic, commonly used depths while only requiring logarithmic overparametrization. Our novel construction approach applies to a large class of activation functions and is not limited to ReLUs. Code is available on Github (RelationalML/LT-existence).
@inproceedings{NEURIPS2022_76bf7786,author={Burkholz, Rebekka},booktitle={Advances in Neural Information Processing Systems},editor={Koyejo, S. and Mohamed, S. and Agarwal, A. and Belgrave, D. and Cho, K. and Oh, A.},pages={18707--18720},publisher={Curran Associates, Inc.},title={Most Activation Functions Can Win the Lottery Without Excessive Depth},url={https://papers.nips.cc/paper_files/paper/2022/hash/76bf7786d311217077bc8bb021946cd9-Abstract-Conference.html},volume={35},year={2022}}
The Lottery Ticket Hypothesis continues to have a profound practical impact on the quest for small scale deep neural networks that solve modern deep learning tasks at competitive performance. These lottery tickets are identified by pruning large randomly initialized neural networks with architectures that are as diverse as their applications. Yet, theoretical insights that attest their existence have been mostly focused on deed fully-connected feed forward networks with ReLU activation functions. We prove that also modern architectures consisting of convolutional and residual layers that can be equipped with almost arbitrary activation functions can contain lottery tickets with high probability.
@inproceedings{pmlr-v162-burkholz22a,title={Convolutional and Residual Networks Provably Contain Lottery Tickets},author={Burkholz, Rebekka},booktitle={Proceedings of the 39th International Conference on Machine Learning},pages={2414--2433},year={2022},editor={Chaudhuri, Kamalika and Jegelka, Stefanie and Song, Le and Szepesvari, Csaba and Niu, Gang and Sabato, Sivan},volume={162},series={Proceedings of Machine Learning Research},month={17--23 Jul},publisher={PMLR},url={https://proceedings.mlr.press/v162/burkholz22a.html},}
