Abstract--In this paper, we examine the fundamental performance limitations of online machine learning, by viewing th e online learning problem as a prediction problem with causal side information. T owards this end, we combine the entropic analysis from information theory and the innovations appro ach from prediction theory to derive generic lower bounds on the prediction errors as well as the conditions (in terms of, e.g., d irected information) to achieve the bounds. It is seen in general tha t no specific restrictions have to be imposed on the learning algo rithms or the distributions of the data points for the performance b ounds to be valid. In addition, the cases of supervised learning, s emi-supervised learning, as well as unsupervised learning can a ll be analyzed accordingly. We also investigate the implication s of the results in analyzing the fundamental limits of generalizat ion.

-- In this paper, we study the fundamental performance limitations for generic feedback systems in which both the controller and the plant may be arbitrarily causal while the disturbance can be with any distributions. We also examine the implications of the generic bounds for machine-learning-in-the-loop control; in other words, fundamental limits in general exist to what machine learning elements in feedback loops can achieve. Machine learning techniques are becoming more and more prevalent nowadays in the feedback control of dynamical systems, where system dynamics that are determined by physical laws will play an indispensable role. In this trend, it is becoming more and more critical to be fully aware of the performance limits of the machine learning algorithms that are to be embedded in the feedback loop, especially in scenarios where performance guarantees are required and must be strictly imposed. In conventional performance limitation analysis [1] of feedback systems such as the Bode integral [2], however, specific restrictions on the classes of the controller that can be implemented must be imposed in general. These restrictions would normally render the analysis invalid if machine learning elements such as deep learning or reinforcement learning are to be placed at the position of the controller, as a result of the complexity of the learning algorithms.

AR-NNs are a natural generalization of the classic linear autoregressive AR(p) process (2) See, e.g., Brockwell & Davis (1987) for a comprehensive introduction into AR and ARMA (autoregressive moving average) models.

AR-NNs are a natural generalization of the classic linear autoregressive AR(p) process (2) See, e.g., Brockwell & Davis (1987) for a comprehensive introduction into AR and ARMA (autoregressive moving average) models.

AR-NNs are a natural generalization of the classic linear autoregressive AR(p) process (2) See, e.g., Brockwell & Davis (1987) for a comprehensive introduction into AR and ARMA (autoregressive moving average) models. F. Leisch, A. Trapletti and K. Hornik 268 One of the most central questions in linear time series theory is the stationarity of the model, i.e., whether the probabilistic structure of the series is constant over time or at least asymptotically constant (when not started in equilibrium). Surprisingly, this question has not gained much interest in the NN literature, especially there are-up to our knowledge-no results giving conditions for the stationarity of AR NN models. There are results on the stationarity of Hopfield nets (Wang & Sheng, 1996), but these nets cannot be used to estimate conditional expectations for time series prediction. The rest of this paper is organized as follows: In Section 2 we recall some results from time series analysis and Markov chain theory defining the relationship between a time series and its associated Markov chain. In Section 3 we use these results to establish that standard AR-NN models without shortcut connections are stationary. We also give conditions for AR-NN models with shortcut connections to be stationary. Section 4 examines the NN modeling of an important class of non-stationary to the appendix.time