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We consider networks, trained via stochastic gradient descent to minimize L2 loss, with the training labels perturbed by independent noise at each iteration. We characterize the behavior of the training dynamics near any parameter vector that achieves zero training error, in terms of an implicit regularization term corresponding to the sum over the datapoints, of the squared L2 of the gradient of the model with respect to the parameter vector, evaluated at each data point. This holds for networks of any connectivity, width,depth, and choice of activation function. We interpret this implicit regularization term for three simple settings: matrix sensing, two layer ReLU networks trained on one-dimensional data, and two layer networks with sigmoid activations trained on a single datapoint. For these settings, we show why this new and general implicit regularization effect drives the networks towards "simple" models.
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Model Collapse Demystified: The Case of Regression
The era of proliferation of large language and image generation models begs the question of what happens if models are trained on the synthesized outputs of other models. The phenomenon of "model collapse" refers to the situation whereby as a model is trained recursively on data generated from previous generations of itself over time, its performance degrades until the model eventually becomes completely useless, i.e. the model collapses. In this work, we investigate this phenomenon within the context of high-dimensional regression with Gaussian data, considering both low- and high-dimensional asymptotics. We derive analytical formulas that quantitatively describe this phenomenon in both under-parameterized and over-parameterized regimes. We show how test error increases linearly in the number of model iterations in terms of all problem hyperparameters (covariance spectrum, regularization, label noise level, dataset size) and further isolate how model collapse affects both bias and variance terms in our setup. We show that even in the noise-free case, catastrophic (exponentially fast) model-collapse can happen in the over-parametrized regime. In the special case of polynomial decaying spectral and source conditions, we obtain modified scaling laws which exhibit new crossover phenomena from fast to slow rates. We also propose a simple strategy based on adaptive regularization to mitigate model collapse. Our theoretical results are validated with experiments.
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- Award ID(s):
- 1922658
- PAR ID:
- 10649436
- Publisher / Repository:
- 2024 Conference on Neural Information Processing Systems
- Date Published:
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Accepted Manuscript1.0
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