Bilevel optimization has become a powerful framework in various machine learning applications including meta-learning, hyperparameter optimization, and network architecture search. There are generally two classes of bilevel optimization formulations for machine learning: 1) problem-based bilevel optimization, whose inner-level problem is formulated as finding a minimizer of a given loss function; and 2) algorithm-based bilevel optimization, whose inner-level solution is an output of a fixed algorithm. For the first class, two popular types of gradient-based algorithms have been proposed for hypergradient estimation via approximate implicit differentiation (AID) and iterative differentiation (ITD). Algorithms for the second class include the popular model-agnostic meta-learning (MAML) and almost no inner loop (ANIL). However, the convergence rate and fundamental limitations of bilevel optimization algorithms have not been well explored. This thesis provides a comprehensive convergence rate analysis for bilevel algorithms in the aforementioned two classes. We further propose principled algorithm designs for bilevel optimization with higher efficiency and scalability. For the problem-based formulation, we provide a convergence rate analysis for AID- and ITD-based bilevel algorithms. We then develop acceleration bilevel algorithms, for which we provide shaper convergence analysis with relaxed assumptions. We also provide the first lower bounds for bilevel optimization, and establish the optimality by providing matching upper bounds under certain conditions. We finally propose new stochastic bilevel optimization algorithms with lower complexity and higher efficiency in practice. For the algorithm-based formulation, we develop a theoretical convergence for general multi-step MAML and ANIL, and characterize the impact of parameter selections and loss geometries on the their complexities.

Discovering the governing laws underpinning physical and chemical phenomena is a key step towards understanding and ultimately controlling systems in science and engineering. We introduce Discovery of Dynamical Systems via Moving Horizon Optimization (DySMHO), a scalable machine learning framework for identifying governing laws in the form of differential equations from large-scale noisy experimental data sets. DySMHO consists of a novel moving horizon dynamic optimization strategy that sequentially learns the underlying governing equations from a large dictionary of basis functions. The sequential nature of DySMHO allows leveraging statistical arguments for eliminating irrelevant basis functions, avoiding overfitting to recover accurate and parsimonious forms of the governing equations. Canonical nonlinear dynamical system examples are used to demonstrate that DySMHO can accurately recover the governing laws, is robust to high levels of measurement noise and that it can handle challenges such as multiple time scale dynamics.

Gradient quantization is an emerging technique in reducing communication costs in distributed learning. Existing gradient quantization algorithms often rely on engineering heuristics or empirical observations, lacking a systematic approach to dynamically quantize gradients. This paper addresses this issue by proposing a novel dynamically quantized SGD (DQ-SGD) framework, enabling us to dynamically adjust the quantization scheme for each gradient descent step by exploring the trade-off between communication cost and convergence error. We derive an upper bound, tight in some cases, of the convergence error for a restricted family of quantization schemes and loss functions. We design our DQ-SGD algorithm via minimizing the communication cost under the convergence error constraints. Finally, through extensive experiments on large-scale natural language processing and computer vision tasks on AG-News, CIFAR-10, and CIFAR-100 datasets, we demonstrate that our quantization scheme achieves better tradeoffs between the communication cost and learning performance than other state-of-the-art gradient quantization methods.

We use Reinforcement Meta Learning to optimize an adaptive guidance system suitable for the approach phase of a gliding hypersonic vehicle. Adaptability is achieved by optimizing over a range of off-nominal flight conditions including perturbation of aerodynamic coefficient parameters, actuator failure scenarios, and sensor noise. The system maps observations directly to commanded bank angle and angle of attack rates. These observations include a velocity field tracking error formulated using parallel navigation, but adapted to work over long trajectories where the Earth's curvature must be taken into account. Minimizing the tracking error keeps the curved space line of sight to the target location aligned with the vehicle's velocity vector. The optimized guidance system will then induce trajectories that bring the vehicle to the target location with a high degree of accuracy at the designated terminal speed, while satisfying heating rate, load, and dynamic pressure constraints. We demonstrate the adaptability of the guidance system by testing over flight conditions that were not experienced during optimization. The guidance system's performance is then compared to that of a linear quadratic regulator tracking an optimal trajectory.

Information-based Bayesian optimization (BO) algorithms have achieved state-of-the-art performance in optimizing a black-box objective function. However, they usually require several approximations or simplifying assumptions (without clearly understanding their effects on the BO performance) and/or their generalization to batch BO is computationally unwieldy, especially with an increasing batch size. To alleviate these issues, this paper presents a novel trusted-maximizers entropy search (TES) acquisition function: It measures how much an input query contributes to the information gain on the maximizer over a finite set of trusted maximizers, i.e., inputs optimizing functions that are sampled from the Gaussian process posterior belief of the objective function. Evaluating TES requires either only a stochastic approximation with sampling or a deterministic approximation with expectation propagation, both of which are investigated and empirically evaluated using synthetic benchmark objective functions and real-world optimization problems, e.g., hyperparameter tuning of a convolutional neural network and synthesizing 'physically realizable' faces to fool a black-box face recognition system. Though TES can naturally be generalized to a batch variant with either approximation, the latter is amenable to be scaled to a much larger batch size in our experiments.

The derivative defines the rate at which one variable changes with respect to another. It is an important concept that comes in extremely useful in many applications: in everyday life, the derivative can tell you at which speed you are driving, or help you predict fluctuations on the stock market; in machine learning, derivatives are important for function optimization. This tutorial will explore different applications of derivatives, starting with the more familiar ones before moving to machine learning. We will be taking a closer look at what the derivatives tell us about the different functions we are studying. In this tutorial, you will discover different applications of derivatives.

A solution that is only reliable under favourable conditions is hardly a safe solution. Min Max Optimization is an approach that returns optima that are robust against worst case conditions. We propose algorithms that perform Min Max Optimization in a setting where the function that should be optimized is not known a priori and hence has to be learned by experiments. Therefore we extend the Bayesian Optimization setting, which is tailored to maximization problems, to Min Max Optimization problems. While related work extends the two acquisition functions Expected Improvement and Gaussian Process Upper Confidence Bound; we extend the two acquisition functions Entropy Search and Knowledge Gradient. These acquisition functions are able to gain knowledge about the optimum instead of just looking for points that are supposed to be optimal. In our evaluation we show that these acquisition functions allow for better solutions - converging faster to the optimum than the benchmark settings.

Swarm optimization algorithms are widely used for feature selection before data mining and machine learning applications. The metaheuristic nature-inspired feature selection approaches are used for single-objective optimization tasks, though the major problem is their frequent premature convergence, leading to weak contribution to data mining. In this paper, we propose a novel feature selection algorithm named Reinforced Swarm Optimization (RSO) leveraging some of the existing problems in feature selection. This algorithm embeds the widely used Bee Swarm Optimization (BSO) algorithm along with Reinforcement Learning (RL) to maximize the reward of a superior search agent and punish the inferior ones. This hybrid optimization algorithm is more adaptive and robust with a good balance between exploitation and exploration of the search space. The proposed method is evaluated on 25 widely known UCI datasets containing a perfect blend of balanced and imbalanced data. The obtained results are compared with several other popular and recent feature selection algorithms with similar classifier configurations. The experimental outcome shows that our proposed model outperforms BSO in 22 out of 25 instances (88%). Moreover, experimental results also show that RSO performs the best among all the methods compared in this paper in 19 out of 25 cases (76%), establishing the superiority of our proposed method.

Recommender systems (RSs) employ user-item feedback, e.g., ratings, to match customers to personalized lists of products. Approaches to top-k recommendation mainly rely on Learning-To-Rank algorithms and, among them, the most widely adopted is Bayesian Personalized Ranking (BPR), which bases on a pair-wise optimization approach. Recently, BPR has been found vulnerable against adversarial perturbations of its model parameters. Adversarial Personalized Ranking (APR) mitigates this issue by robustifying BPR via an adversarial training procedure. The empirical improvements of APR's accuracy performance on BPR have led to its wide use in several recommender models. However, a key overlooked aspect has been the beyond-accuracy performance of APR, i.e., novelty, coverage, and amplification of popularity bias, considering that recent results suggest that BPR, the building block of APR, is sensitive to the intensification of biases and reduction of recommendation novelty. In this work, we model the learning characteristics of the BPR and APR optimization frameworks to give mathematical evidence that, when the feedback data have a tailed distribution, APR amplifies the popularity bias more than BPR due to an unbalanced number of received positive updates from short-head items. Using matrix factorization (MF), we empirically validate the theoretical results by performing preliminary experiments on two public datasets to compare BPR-MF and APR-MF performance on accuracy and beyond-accuracy metrics. The experimental results consistently show the degradation of novelty and coverage measures and a worrying amplification of bias.

I'm very glad to have opportunity to teach you one of the most popular and powerful optimization algorithms in this course. If you search FireFly optimization algorithm in google scholar, it could be seen that there are many vast range of papers has been published by implementing this optimization algorithm in different fields of science. In this course, after presenting the mathematical concept of each part of the considered optimization algorithm, I write its code immediately in matlab. All of the written codes are available, however, I strongly suggest to write the codes with me. Notice that, if you don't have matlab or you know another programming language, don't worry at all.