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Tengyuan LIANG (University of Chicago-Booth School of Business) – "New Thoughts on Adaptivity, Generalization and Interpolation Motivated from Neural Networks"
Time: 2:00 pm – 3:15 pm
Date: 25 th of March 2019
Place: Room 3001.
Tengyuan LIANG (University of Chicago-Booth School of Business) – “New Thoughts on Adaptivity, Generalization and Interpolation Motivated from Neural Networks”
Abstract: Consider the problem: given data pair (x, y) drawn from a population with f_*(x) = E[y|x], specify a neural network and run gradient flow on the weights over time until reaching any stationarity. How does f_t, the function computed by the neural network at time t, relate to f_*, in terms of approximation and representation? What are the provable benefits of the adaptive representation by neural networks compared to the pre-specified fixed basis representation in the classical nonparametric literature? We answer the above questions via a dynamic reproducing kernel Hilbert space (RKHS) approach indexed by the training process of neural networks. We show that when reaching any local stationarity, gradient flow learns an adaptive RKHS representation, and performs the global least squares projection onto the adaptive RKHS, simultaneously. In addition, we prove that as the RKHS is data-adaptive and task-specific, the residual for f˚ lies in a subspace that is smaller than the orthogonal complement of the RKHS, formalizing the representation and approximation benefits of neural networks.
Then we will move to discuss generalization for interpolating methods in RKHS. In the absence of explicit regularization, Kernel “Ridgeless” Regression with nonlinear kernels has the potential to fit the training data perfectly. It has been observed empirically, however, that such interpolated solutions can still generalize well on test data. We isolate a phenomenon of implicit regularization for minimum-norm interpolated solutions which is due to a combination of high dimensionality of the input data, curvature of the kernel function, and favorable geometric properties of the data such as an eigenvalue decay of the empirical covariance and kernel matrices. In addition to deriving a data-dependent upper bound on the out-of-sample error, we present experimental evidence suggesting that the phenomenon occurs in the MNIST dataset.
Cristina BUTUCEA, Alexandre TSYBAKOV, Julie JOSSE, Eric MOULINES, Mathieu ROSENBAUM