Chaining Meets Chain Rule: Multilevel Entropic Regularization and Training of Neural Nets

We derive generalization and excess risk bounds for neural nets using a family of complexity measures based on a multilevel relative entropy. The bounds are obtained by introducing the notion of generated hierarchical coverings of neural nets and by using the technique of chaining mutual information introduced in Asadi et al. NeurIPS'18. The resulting bounds are algorithm-dependent and exploit the multilevel structure of neural nets. This, in turn, leads to an empirical risk minimization problem with a multilevel entropic regularization. The minimization problem is resolved by introducing a multi-scale generalization of the celebrated Gibbs posterior distribution, proving that the derived distribution achieves the unique minimum. This leads to a new training procedure for neural nets with performance guarantees, which exploits the chain rule of relative entropy rather than the chain rule of derivatives (as in backpropagation). To obtain an efficient implementation of the latter, we further develop a multilevel Metropolis algorithm simulating the multi-scale Gibbs distribution, with an experiment for a two-layer neural net on the MNIST data set.Comment: 30 pages, 3 figure

Optimal No-regret Learning in Repeated First-price Auctions

We study online learning in repeated first-price auctions with censored feedback, where a bidder, only observing the winning bid at the end of each auction, learns to adaptively bid in order to maximize her cumulative payoff. To achieve this goal, the bidder faces a challenging dilemma: if she wins the bid--the only way to achieve positive payoffs--then she is not able to observe the highest bid of the other bidders, which we assume is iid drawn from an unknown distribution. This dilemma, despite being reminiscent of the exploration-exploitation trade-off in contextual bandits, cannot directly be addressed by the existing UCB or Thompson sampling algorithms in that literature, mainly because contrary to the standard bandits setting, when a positive reward is obtained here, nothing about the environment can be learned. In this paper, by exploiting the structural properties of first-price auctions, we develop the first learning algorithm that achieves $O(\sqrt{T}\log^2 T)$ regret bound when the bidder's private values are stochastically generated. We do so by providing an algorithm on a general class of problems, which we call monotone group contextual bandits, where the same regret bound is established under stochastically generated contexts. Further, by a novel lower bound argument, we characterize an $\Omega(T^{2/3})$ lower bound for the case where the contexts are adversarially generated, thus highlighting the impact of the contexts generation mechanism on the fundamental learning limit. Despite this, we further exploit the structure of first-price auctions and develop a learning algorithm that operates sample-efficiently (and computationally efficiently) in the presence of adversarially generated private values. We establish an $O(\sqrt{T}\log^3 T)$ regret bound for this algorithm, hence providing a complete characterization of optimal learning guarantees for this problem

Online learning with kernel losses

We present a generalization of the adversarial linear bandits framework, where the underlying losses are kernel functions (with an associated reproducing kernel Hilbert space) rather than linear functions. We study a version of the exponential weights algorithm and bound its regret in this setting. Under conditions on the eigendecay of the kernel we provide a sharp characterization of the regret for this algorithm. When we have polynomial eigendecay $\mu_j \le \mathcal{O}(j^{-\beta})$, we find that the regret is bounded by $\mathcal{R}_n \le \mathcal{O}(n^{\beta/(2(\beta-1))})$; while under the assumption of exponential eigendecay $\mu_j \le \mathcal{O}(e^{-\beta j })$, we get an even tighter bound on the regret $\mathcal{R}_n \le \mathcal{O}(n^{1/2}\log(n)^{1/2})$. We also study the full information setting when the underlying losses are kernel functions and present an adapted exponential weights algorithm and a conditional gradient descent algorithm.Comment: 40 pages, 4 figure