Eliminating the covariate shift cross domains is one of the common methods to deal with the issue of domain shift in visual unsupervised domain adaptation. However, current alignment methods, especially the prototype based or sample-level based methods neglect the structural properties of the underlying distribution and even break the condition of covariate shift. To relieve the limitations and conflicts, we introduce a novel concept named (virtual) mirror, which represents the equivalent sample in another domain. The equivalent sample pairs, named mirror pairs reflect the natural correspondence of the empirical distributions. Then a mirror loss, which aligns the mirror pairs cross domains, is constructed to enhance the alignment of the domains. The proposed method does not distort the internal structure of the underlying distribution. We also provide theoretical proof that the mirror samples and mirror loss have better asymptotic properties in reducing the domain shift. By applying the virtual mirror and mirror loss to the generic unsupervised domain adaptation model, we achieved consistently superior performance on several mainstream benchmarks.

This work is dedicated to the algorithm design in a competitive framework, with the primary goal of learning a stable equilibrium. We consider the dynamic price competition between two firms operating within an opaque marketplace, where each firm lacks information about its competitor. The demand follows the multinomial logit (MNL) choice model, which depends on the consumers' observed price and their reference price, and consecutive periods in the repeated games are connected by reference price updates. We use the notion of stationary Nash equilibrium (SNE), defined as the fixed point of the equilibrium pricing policy for the single-period game, to simultaneously capture the long-run market equilibrium and stability. We propose the online projected gradient ascent algorithm (OPGA), where the firms adjust prices using the first-order derivatives of their log-revenues that can be obtained from the market feedback mechanism. Despite the absence of typical properties required for the convergence of online games, such as strong monotonicity and variational stability, we demonstrate that under diminishing step-sizes, the price and reference price paths generated by OPGA converge to the unique SNE, thereby achieving the no-regret learning and a stable market. Moreover, with appropriate step-sizes, we prove that this convergence exhibits a rate of O(1/t).

A fundamental question in the theory of reinforcement learning is: suppose the optimal Q-function lies in the linear span of a given ddimensional feature mapping, is sample-efficient reinforcement learning (RL) possible? The recent and remarkable result of Weisz et al. (2020) resolves this question in the negative, providing an exponential (in d) sample size lower bound, which holds even if the agent has access to a generative model of the environment. One may hope that such a lower can be circumvented with an even stronger assumption that there is a constant gap between the optimal Q-value of the best action and that of the second-best action (for all states); indeed, the construction in Weisz et al. (2020) relies on having an exponentially small gap. This work resolves this subsequent question, showing that an exponential sample complexity lower bound still holds even if a constant gap is assumed. Perhaps surprisingly, this result implies an exponential separation between the online RL setting and the generative model setting, where sample-efficient RL is in fact possible in the latter setting with a constant gap. Complementing our negative hardness result, we give two positive results showing that provably sample-efficient RL is possible either under an additional low-variance assumption or under a novel hypercontractivity assumption.

In many machine learning applications on signals and biomedical data, especially electroencephalogram (EEG), one major challenge is the variability of the data across subjects, sessions, and hardware devices. In this work, we propose a new method called Convolutional Monge Mapping Normalization (CMMN), which consists in filtering the signals in order to adapt their power spectrum density (PSD) to a Wasserstein barycenter estimated on training data. CMMN relies on novel closed-form solutions for optimal transport mappings and barycenters and provides individual test time adaptation to new data without needing to retrain a prediction model. Numerical experiments on sleep EEG data show that CMMN leads to significant and consistent performance gains independent from the neural network architecture when adapting between subjects, sessions, and even datasets collected with different hardware. Notably our performance gain is on par with much more numerically intensive Domain Adaptation (DA) methods and can be used in conjunction with those for even better performances.