We propose to view the outcomes of the agents' dynamics as inducing a "meta-game" between the users.

The publicly verifiable record of the randomization is available athttps://www.

Could Ukraine hold a presidential election right now? Will Europe use frozen Russian assets to fund war? How can Ukraine rebuild China ties? 'Ukraine is running out of men, money and time' Five people, including three young children, have been killed in the latest overnight Russian attacks on Ukraine, President Volodymyr Zelenskyy has said, as United States-led efforts to end the nearly war continue to progress at a slow, bogged-down pace. The Ukrainian leader said on Wednesday that a Russian drone had struck a private family home in the town of Bohodukhiv in Ukraine's northeastern Kharkiv region late on Tuesday, killing four and seriously injuring their pregnant mother, the sole survivor.

Figure 1: OurA), theattribX) and labels (Yl) asinput, and attributes (Zn and Za), aswellasYu).

Existing Rademacher complexity bounds for neural networks rely only on norm control of the weight matrices and depend exponentially on depth via a product of the matrix norms. Lower bounds show that this exponential dependence on depth is unavoidable when no additional properties of the training data are considered. We suspect that this conundrum comes from the fact that these bounds depend on the training data only through the margin. In practice, many data-dependent techniques such as Batchnorm improve the generalization performance. For feedforward neural nets as well as RNNs, we obtain tighter Rademacher complexity bounds by considering additional data-dependent properties of the network: the norms of the hidden layers of the network, and the norms of the Jacobians of each layer with respect to all previous layers. Our bounds scale polynomially in depth when these empirical quantities are small, as is usually the case in practice. To obtain these bounds, we develop general tools for augmenting a sequence of functions to make their composition Lipschitz and then covering the augmented functions. Inspired by our theory, we directly regularize the network's Jacobians during training and empirically demonstrate that this improves test performance.