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We study the smoothness of the solution of the directed chain stochastic differential equations, where each process is affected by its neighborhood process in an infinite directed chain graph, introduced by Detering et al. (2020). Because of the auxiliary process in the chain-like structure, classic methods of Malliavin derivatives are not directly applicable. Namely, we cannot make a connection between the Malliavin derivative and the first order derivative of the state process. It turns out that the partial Malliavin derivatives can be used here to fix this problem. 

Real-world data can be multimodal distributed, e.g., data describing the opinion divergence in a community, the interspike interval distribution of neurons, and the oscillators' natural frequencies. Generating multimodal distributed real-world data has become a challenge to existing generative adversarial networks (GANs). For example, it is often observed that Neural SDEs have only demonstrated successful performance mainly in generating unimodal time series datasets. In this paper, we propose a novel time series generator, named directed chain GANs (DC-GANs), which inserts a time series dataset (called a neighborhood process of the directed chain or input) into the drift and diffusion coefficients of the directed chain SDEs with distributional constraints. DC-GANs can generate new time series of the same distribution as the neighborhood process, and the neighborhood process will provide the key step in learning and generating multimodal distributed time series. The proposed DC-GANs are examined on four datasets, including two stochastic models from social sciences and computational neuroscience, and two real-world datasets on stock prices and energy consumption. To our best knowledge, DC-GANs are the first work that can generate multimodal time series data and consistently outperforms state-of-the-art benchmarks with respect to measures of distribution, data similarity, and predictive ability. 

Abstract Signature is an infinite graded sequence of statistics known to characterize geometric rough paths. While the use of the signature in machine learning is successful in low-dimensional cases, it suffers from the curse of dimensionality in high-dimensional cases, as the number of features in the truncated signature transform grows exponentially fast. With the idea of Convolutional Neural Network, we propose a novel neural network to address this problem. Our model reduces the number of features efficiently in a data-dependent way. Some empirical experiments including high-dimensional financial time series classification and natural language processing are provided to support our convolutional signature model. 

We study the problem of reinforcement learning (RL) with low (policy) switching cost {—} a problem well-motivated by real-life RL applications in which deployments of new policies are costly and the number of policy updates must be low. In this paper, we propose a new algorithm based on stage-wise exploration and adaptive policy elimination that achieves a regret of $$\widetilde{O}(\sqrt{H^4S^2AT})$$ while requiring a switching cost of $$O(HSA \log\log T)$$. This is an exponential improvement over the best-known switching cost $$O(H^2SA\log T)$$ among existing methods with $$\widetilde{O}(\mathrm{poly}(H,S,A)\sqrt{T})$$ regret. In the above, $S,A$ denotes the number of states and actions in an $$H$$-horizon episodic Markov Decision Process model with unknown transitions, and $$T$$ is the number of steps. As a byproduct of our new techniques, we also derive a reward-free exploration algorithm with a switching cost of $O(HSA)$. Furthermore, we prove a pair of information-theoretical lower bounds which say that (1) Any no-regret algorithm must have a switching cost of $$\Omega(HSA)$$; (2) Any $$\widetilde{O}(\sqrt{T})$$ regret algorithm must incur a switching cost of $$\Omega(HSA\log\log T)$$. Both our algorithms are thus optimal in their switching costs.