We study the Pareto frontier of two archetypal objectives in stochastic bandits, namely, regret minimization (RM) and best arm identification (BAI) with a fixed horizon. It is folklore that the balance between exploitation and exploration is crucial for both RM and BAI, but exploration is more critical in achieving the optimal performance for the latter objective. To make this precise, we first design and analyze the BoBW-lil'UCB$({\gamma})$ algorithm, which achieves order-wise optimal performance for RM or BAI under different values of ${\gamma}$. Complementarily, we show that no algorithm can simultaneously perform optimally for both the RM and BAI objectives. More precisely, we establish non-trivial lower bounds on the regret achievable by any algorithm with a given BAI failure probability. This analysis shows that in some regimes BoBW-lil'UCB$({\gamma})$ achieves Pareto-optimality up to constant or small terms. Numerical experiments further demonstrate that when applied to difficult instances, BoBW-lil'UCB outperforms a close competitor UCB$_{\alpha}$ (Degenne et al., 2019), which is designed for RM and BAI with a fixed confidence.

We propose a nested weighted Tchebycheff Multi-objective Bayesian optimization framework where we build a regression model selection procedure from an ensemble of models, towards better estimation of the uncertain parameters of the weighted-Tchebycheff expensive black-box multi-objective function. In existing work, a weighted Tchebycheff MOBO approach has been demonstrated which attempts to estimate the unknown utopia in formulating acquisition function, through calibration using a priori selected regression model. However, the existing MOBO model lacks flexibility in selecting the appropriate regression models given the guided sampled data and therefore, can under-fit or over-fit as the iterations of the MOBO progress, reducing the overall MOBO performance. As it is too complex to a priori guarantee a best model in general, this motivates us to consider a portfolio of different families of predictive models fitted with current training data, guided by the WTB MOBO; the best model is selected following a user-defined prediction root mean-square-error-based approach. The proposed approach is implemented in optimizing a multi-modal benchmark problem and a thin tube design under constant loading of temperature-pressure, with minimizing the risk of creep-fatigue failure and design cost. Finally, the nested weighted Tchebycheff MOBO model performance is compared with different MOBO frameworks with respect to accuracy in parameter estimation, Pareto-optimal solutions and function evaluation cost. This method is generalized enough to consider different families of predictive models in the portfolio for best model selection, where the overall design architecture allows for solving any high-dimensional (multiple functions) complex black-box problems and can be extended to any other global criterion multi-objective optimization methods where prior knowledge of utopia is required.

Portfolio optimization is a key challenge in finance with the aim of creating portfolios matching the investors' preference. The target distribution approach relying on the Kullback-Leibler [33] or the f-divergence [7] represents one of the most effective forms of achieving this goal. In this paper, we propose to use kernel and optimal transport (KOT) based divergences to tackle the task, which relax the assumptions and the optimization constraints of the previous approaches. In case of the kernel-based maximum mean discrepancy (MMD) we (i) prove the analytic computability of the underlying mean embedding for various target distribution-kernel pairs, (ii) show that such analytic knowledge can lead to faster convergence of MMD estimators, and(iii) extend the results to the unbounded exponential kernel with minimax lower bounds. Numerical experiments demonstrate the improved performance of our KOT estimators both on synthetic and real-world examples.

Gaussian processes (GP) are a widely-adopted tool used to sequentially optimize black-box functions, where evaluations are costly and potentially noisy. Recent works on GP bandits have proposed to move beyond random noise and devise algorithms robust to adversarial attacks. In this paper, we study this problem from the attacker's perspective, proposing various adversarial attack methods with differing assumptions on the attacker's strength and prior information. Our goal is to understand adversarial attacks on GP bandits from both a theoretical and practical perspective. We focus primarily on targeted attacks on the popular GP-UCB algorithm and a related elimination-based algorithm, based on adversarially perturbing the function $f$ to produce another function $\tilde{f}$ whose optima are in some region $\mathcal{R}_{\rm target}$. Based on our theoretical analysis, we devise both white-box attacks (known $f$) and black-box attacks (unknown $f$), with the former including a Subtraction attack and Clipping attack, and the latter including an Aggressive subtraction attack. We demonstrate that adversarial attacks on GP bandits can succeed in forcing the algorithm towards $\mathcal{R}_{\rm target}$ even with a low attack budget, and we compare our attacks' performance and efficiency on several real and synthetic functions.

In this work we introduce a new framework for multi-objective Bayesian optimisation where the multi-objective functions can only be accessed via choice judgements, such as ``I pick options A,B,C among this set of five options A,B,C,D,E''. The fact that the option D is rejected means that there is at least one option among the selected ones A,B,C that I strictly prefer over D (but I do not have to specify which one). We assume that there is a latent vector function f for some dimension $n_e$ which embeds the options into the real vector space of dimension n, so that the choice set can be represented through a Pareto set of non-dominated options. By placing a Gaussian process prior on f and deriving a novel likelihood model for choice data, we propose a Bayesian framework for choice functions learning. We then apply this surrogate model to solve a novel multi-objective Bayesian optimisation from choice data problem.

In this paper, we develop a new type of accelerated algorithms to solve some classes of maximally monotone equations as well as monotone inclusions. Instead of using Nesterov's accelerating approach, our methods rely on a so-called Halpern-type fixed-point iteration in [32], and recently exploited by a number of researchers, including [24, 70]. Firstly, we derive a new variant of the anchored extra-gradient scheme in [70] based on Popov's past extra-gradient method to solve a maximally monotone equation $G(x) = 0$. We show that our method achieves the same $\mathcal{O}(1/k)$ convergence rate (up to a constant factor) as in the anchored extra-gradient algorithm on the operator norm $\Vert G(x_k)\Vert$, , but requires only one evaluation of $G$ at each iteration, where $k$ is the iteration counter. Next, we develop two splitting algorithms to approximate a zero point of the sum of two maximally monotone operators. The first algorithm originates from the anchored extra-gradient method combining with a splitting technique, while the second one is its Popov's variant which can reduce the per-iteration complexity. Both algorithms appear to be new and can be viewed as accelerated variants of the Douglas-Rachford (DR) splitting method. They both achieve $\mathcal{O}(1/k)$ rates on the norm $\Vert G_{\gamma}(x_k)\Vert$ of the forward-backward residual operator $G_{\gamma}(\cdot)$ associated with the problem. We also propose a new accelerated Douglas-Rachford splitting scheme for solving this problem which achieves $\mathcal{O}(1/k)$ convergence rate on $\Vert G_{\gamma}(x_k)\Vert$ under only maximally monotone assumptions. Finally, we specify our first algorithm to solve convex-concave minimax problems and apply our accelerated DR scheme to derive a new variant of the alternating direction method of multipliers (ADMM).

Multi-objective optimization (MOO) aims at finding a set of optimal configurations for a given set of objectives. A recent line of work applies MOO methods to the typical Machine Learning (ML) setting, which becomes multi-objective if a model should optimize more than one objective, for instance in fair machine learning. These works also use Multi-Task Learning (MTL) problems to benchmark MOO algorithms treating each task as independent objective. In this work we show that MTL problems do not resemble the characteristics of MOO problems. In particular, MTL losses are not competing in case of a sufficiently expressive single model. As a consequence, a single model can perform just as well as optimizing all objectives with independent models, rendering MOO inapplicable. We provide evidence with extensive experiments on the widely used Multi-Fashion-MNIST datasets. Our results call for new benchmarks to evaluate MOO algorithms for ML. Our code is available at: https://github.com/ruchtem/moo-mtl.

In black-box optimization problems, we aim to maximize an unknown objective function, where the function is only accessible through feedbacks of an evaluation or simulation oracle. In real-life, the feedbacks of such oracles are often noisy and available after some unknown delay that may depend on the computation time of the oracle. Additionally, if the exact evaluations are expensive but coarse approximations are available at a lower cost, the feedbacks can have multi-fidelity. In order to address this problem, we propose a generic extension of hierarchical optimistic tree search (HOO), called ProCrastinated Tree Search (PCTS), that flexibly accommodates a delay and noise-tolerant bandit algorithm. We provide a generic proof technique to quantify regret of PCTS under delayed, noisy, and multi-fidelity feedbacks. Specifically, we derive regret bounds of PCTS enabled with delayed-UCB1 (DUCB1) and delayed-UCB-V (DUCBV) algorithms. Given a horizon $T$, PCTS retains the regret bound of non-delayed HOO for expected delay of $O(\log T)$ and worsens by $O(T^{\frac{1-\alpha}{d+2}})$ for expected delays of $O(T^{1-\alpha})$ for $\alpha \in (0,1]$. We experimentally validate on multiple synthetic functions and hyperparameter tuning problems that PCTS outperforms the state-of-the-art black-box optimization methods for feedbacks with different noise levels, delays, and fidelity.

Autonomous cyber-physical systems (CPS) can improve safety and efficiency for safety-critical applications, but require rigorous testing before deployment. The complexity of these systems often precludes the use of formal verification and real-world testing can be too dangerous during development. Therefore, simulation-based techniques have been developed that treat the system under test as a black box operating in a simulated environment. Safety validation tasks include finding disturbances in the environment that cause the system to fail (falsification), finding the most-likely failure, and estimating the probability that the system fails. Motivated by the prevalence of safety-critical artificial intelligence, this work provides a survey of state-of-the-art safety validation techniques for CPS with a focus on applied algorithms and their modifications for the safety validation problem. We present and discuss algorithms in the domains of optimization, path planning, reinforcement learning, and importance sampling. Problem decomposition techniques are presented to help scale algorithms to large state spaces, which are common for CPS. A brief overview of safety-critical applications is given, including autonomous vehicles and aircraft collision avoidance systems. Finally, we present a survey of existing academic and commercially available safety validation tools.

When learning common skills like driving, beginners usually have domain experts standing by to ensure the safety of the learning process. We formulate such learning scheme under the Expert-in-the-loop Reinforcement Learning where a guardian is introduced to safeguard the exploration of the learning agent. While allowing the sufficient exploration in the uncertain environment, the guardian intervenes under dangerous situations and demonstrates the correct actions to avoid potential accidents. Thus ERL enables both exploration and expert's partial demonstration as two training sources. Following such a setting, we develop a novel Expert Guided Policy Optimization (EGPO) method which integrates the guardian in the loop of reinforcement learning. The guardian is composed of an expert policy to generate demonstration and a switch function to decide when to intervene. Particularly, a constrained optimization technique is used to tackle the trivial solution that the agent deliberately behaves dangerously to deceive the expert into taking over. Offline RL technique is further used to learn from the partial demonstration generated by the expert. Safe driving experiments show that our method achieves superior training and test-time safety, outperforms baselines with a substantial margin in sample efficiency, and preserves the generalizabiliy to unseen environments in test-time. Demo video and source code are available at: https://decisionforce.github.io/EGPO/