Randomized classifiers have been shown to provide a promising approach for achieving certified robustness against adversarial attacks in deep learning. However, most existing methods only leverage Gaussian smoothing noise and only work for $\ell_2$ perturbation. We propose a general framework of adversarial certification with non-Gaussian noise and for more general types of attacks, from a unified \functional optimization perspective. Our new framework allows us to identify a key trade-off between accuracy and robustness via designing smoothing distributions, helping to design new families of non-Gaussian smoothing distributions that work more efficiently for different $\ell_p$ settings, including $\ell_1$, $\ell_2$ and $\ell_\infty$ attacks. Our proposed methods achieve better certification results than previous works and provide a new perspective on randomized smoothing certification.

Randomized classifiers have been shown to provide a promising approach for achieving certified robustness against adversarial attacks in deep learning. However, most existing methods only leverage Gaussian smoothing noise and only work for \ell_2 perturbation. We propose a general framework of adversarial certification with non-Gaussian noise and for more general types of attacks, from a unified \functional optimization perspective. Our new framework allows us to identify a key trade-off between accuracy and robustness via designing smoothing distributions, helping to design new families of non-Gaussian smoothing distributions that work more efficiently for different \ell_p settings, including \ell_1, \ell_2 and \ell_\infty attacks. Our proposed methods achieve better certification results than previous works and provide a new perspective on randomized smoothing certification.

Real-time bidding is the new paradigm of programmatic advertising. An advertiser wants to make the intelligent choice of utilizing a \textbf{Demand-Side Platform} to improve the performance of their ad campaigns. Existing approaches are struggling to provide a satisfactory solution for bidding optimization due to stochastic bidding behavior. In this paper, we proposed a multi-agent reinforcement learning architecture for RTB with functional optimization. We designed four agents bidding environment: three Lagrange-multiplier based functional optimization agents and one baseline agent (without any attribute of functional optimization) First, numerous attributes have been assigned to each agent, including biased or unbiased win probability, Lagrange multiplier, and click-through rate. In order to evaluate the proposed RTB strategy's performance, we demonstrate the results on ten sequential simulated auction campaigns. The results show that agents with functional actions and rewards had the most significant average winning rate and winning surplus, given biased and unbiased winning information respectively. The experimental evaluations show that our approach significantly improve the campaign's efficacy and profitability.

Randomized classifiers have been shown to provide a promising approach for achieving certified robustness against adversarial attacks in deep learning. However, most existing methods only leverage Gaussian smoothing noise and only work for $\ell_2$ perturbation. We propose a general framework of adversarial certification with non-Gaussian noise and for more general types of attacks, from a unified functional optimization perspective. Our new framework allows us to identify a key trade-off between accuracy and robustness via designing smoothing distributions, helping to design new families of non-Gaussian smoothing distributions that work more efficiently for different $\ell_p$ settings, including $\ell_1$, $\ell_2$ and $\ell_\infty$ attacks. Our proposed methods achieve better certification results than previous works and provide a new perspective on randomized smoothing certification.

Resource allocation and transceivers in wireless networks are usually designed by solving optimization problems subject to specific constraints, which can be formulated as variable or functional optimization. If the objective and constraint functions of a variable optimization problem can be derived, standard numerical algorithms can be applied for finding the optimal solution, which however incur high computational cost when the dimension of the variable is high. To reduce the on-line computational complexity, learning the optimal solution as a function of the environment's status by deep neural networks (DNNs) is an effective approach. DNNs can be trained under the supervision of optimal solutions, which however, is not applicable to the scenarios without models or for functional optimization where the optimal solutions are hard to obtain. If the objective and constraint functions are unavailable, reinforcement learning can be applied to find the solution of a functional optimization problem, which is however not tailored to optimization problems in wireless networks. In this article, we introduce unsupervised and reinforced-unsupervised learning frameworks for solving both variable and functional optimization problems without the supervision of the optimal solutions. When the mathematical model of the environment is completely known and the distribution of environment's status is known or unknown, we can invoke unsupervised learning algorithm. When the mathematical model of the environment is incomplete, we introduce reinforced-unsupervised learning algorithms that learn the model by interacting with the environment. Our simulation results confirm the applicability of these learning frameworks by taking a user association problem as an example.

Bayesian optimization (BayesOpt) is a derivative-free approach for sequentially optimizing stochastic black-box functions. Standard BayesOpt, which has shown many successes in machine learning applications, assumes a finite dimensional domain which often is a parametric space. The parameter space is defined by the features used in the function approximations which are often selected manually. Therefore, the performance of BayesOpt inevitably depends on the quality of chosen features. This paper proposes a new Bayesian optimization framework that is able to optimize directly on the domain of function spaces. The resulting framework, Bayesian Functional Optimization (BFO), not only extends the application domains of BayesOpt to functional optimization problems but also relaxes the performance dependency on the chosen parameter space. We model the domain of functions as a reproducing kernel Hilbert space (RKHS), and use the notion of Gaussian processes on a real separable Hilbert space. As a result, we are able to define traditional improvement-based (PI and EI) and optimistic acquisition functions (UCB) as functionals. We propose to optimize the acquisition functionals using analytic functional gradients that are also proved to be functions in a RKHS. We evaluate BFO in three typical functional optimization tasks: i) a synthetic functional optimization problem, ii) optimizing activation functions for a multi-layer perceptron neural network, and iii) a reinforcement learning task whose policies are modeled in RKHS.