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Online Forecast Combination for Dependent Heterogeneous Data
This paper studies a procedure to combine individual forecasts that achieve theoretical optimal performance. The results apply to a wide variety of loss functions and no conditions are imposed on the data sequences and the individual forecasts apart from a tail condition. The theoretical results show that the bounds are also valid in the case of time varying combination weights, under specific conditions on the nature of time variation. Some experimental evidence to confirm the results is provided
Learning Linear Dynamical Systems via Spectral Filtering
We present an efficient and practical algorithm for the online prediction of
discrete-time linear dynamical systems with a symmetric transition matrix. We
circumvent the non-convex optimization problem using improper learning:
carefully overparameterize the class of LDSs by a polylogarithmic factor, in
exchange for convexity of the loss functions. From this arises a
polynomial-time algorithm with a near-optimal regret guarantee, with an
analogous sample complexity bound for agnostic learning. Our algorithm is based
on a novel filtering technique, which may be of independent interest: we
convolve the time series with the eigenvectors of a certain Hankel matrix.Comment: Published as a conference paper at NIPS 201
Generalization Error Bounds of Gradient Descent for Learning Over-parameterized Deep ReLU Networks
Empirical studies show that gradient-based methods can learn deep neural
networks (DNNs) with very good generalization performance in the
over-parameterization regime, where DNNs can easily fit a random labeling of
the training data. Very recently, a line of work explains in theory that with
over-parameterization and proper random initialization, gradient-based methods
can find the global minima of the training loss for DNNs. However, existing
generalization error bounds are unable to explain the good generalization
performance of over-parameterized DNNs. The major limitation of most existing
generalization bounds is that they are based on uniform convergence and are
independent of the training algorithm. In this work, we derive an
algorithm-dependent generalization error bound for deep ReLU networks, and show
that under certain assumptions on the data distribution, gradient descent (GD)
with proper random initialization is able to train a sufficiently
over-parameterized DNN to achieve arbitrarily small generalization error. Our
work sheds light on explaining the good generalization performance of
over-parameterized deep neural networks.Comment: 27 pages. This version simplifies the proof and improves the
presentation in Version 3. In AAAI 202
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