Recently, bi-level optimization (BLO) has taken center stage in some very exciting developments in the area of signal processing (SP) and machine learning (ML). Roughly speaking, BLO is a classical optimization problem that involves two levels of hierarchy (i.e., upper and lower levels), wherein obtaining the solution to the upper-level problem requires solving the lower-level one. BLO has become popular largely because it is powerful in modeling problems in SP and ML, among others, that involve optimizing nested objective functions. Prominent applications of BLO range from resource allocation for wireless systems to adversarial machine learning. In this work, we focus on a class of tractable BLO problems that often appear in SP and ML applications. We provide an overview of some basic concepts of this class of BLO problems, such as their optimality conditions, standard algorithms (including their optimization principles and practical implementations), as well as how they can be leveraged to obtain state-of-the-art results for a number of key SP and ML applications. Further, we discuss some recent advances in BLO theory, its implications for applications, and point out some limitations of the state-of-the-art that require significant future research efforts. Overall, we hope that this article can serve to accelerate the adoption of BLO as a generic tool to model, analyze, and innovate on a wide array of emerging SP and ML applications.

In linear distance metric learning, we are given data in one Euclidean metric space and the goal is to find an appropriate linear map to another Euclidean metric space which respects certain distance conditions as much as possible. In this paper, we formalize a simple and elegant method which reduces to a general continuous convex loss optimization problem, and for different noise models we derive the corresponding loss functions. We show that even if the data is noisy, the ground truth linear metric can be learned with any precision provided access to enough samples, and we provide a corresponding sample complexity bound. Moreover, we present an effective way to truncate the learned model to a low-rank model that can provably maintain the accuracy in the loss function and in parameters - the first such results of this type. Several experimental observations on synthetic and real data sets support and inform our theoretical results.

Federated learning (FL) enables distributed clients to collaboratively train a machine learning model without sharing raw data with each other. However, it suffers the leakage of private information from uploading models. In addition, as the model size grows, the training latency increases due to limited transmission bandwidth and the model performance degrades while using differential privacy (DP) protection. In this paper, we propose a gradient sparsification empowered FL framework over wireless channels, in order to improve training efficiency without sacrificing convergence performance. Specifically, we first design a random sparsification algorithm to retain a fraction of the gradient elements in each client's local training, thereby mitigating the performance degradation induced by DP and and reducing the number of transmission parameters over wireless channels. Then, we analyze the convergence bound of the proposed algorithm, by modeling a non-convex FL problem. Next, we formulate a time-sequential stochastic optimization problem for minimizing the developed convergence bound, under the constraints of transmit power, the average transmitting delay, as well as the client's DP requirement. Utilizing the Lyapunov drift-plus-penalty framework, we develop an analytical solution to the optimization problem. Extensive experiments have been implemented on three real life datasets to demonstrate the effectiveness of our proposed algorithm. We show that our proposed algorithms can fully exploit the interworking between communication and computation to outperform the baselines, i.e., random scheduling, round robin and delay-minimization algorithms.

This paper presents an approach, Spectral Dynamics Embedding Control (SDEC), to optimal control for nonlinear stochastic systems. This method leverages an infinite-dimensional feature to linearly represent the state-action value function and exploits finite-dimensional truncation approximation for practical implementation. To characterize the effectiveness of these finite dimensional approximations, we provide an in-depth theoretical analysis to characterize the approximation error induced by the finite-dimension truncation and statistical error induced by finite-sample approximation in both policy evaluation and policy optimization. Our analysis includes two prominent kernel approximation methods: truncations onto random features and Nystrom features. We also empirically test the algorithm and compare the performance with Koopman-based, iLQR, and energy-based methods on a few benchmark problems.

Quantum computing can empower machine learning models by enabling kernel machines to leverage quantum kernels for representing similarity measures between data. Quantum kernels are able to capture relationships in the data that are not efficiently computable on classical devices. However, there is no straightforward method to engineer the optimal quantum kernel for each specific use case. While recent literature has focused on exploiting the potential offered by the presence of symmetries in the data to guide the construction of quantum kernels, we adopt here a different approach, which employs optimization techniques, similar to those used in neural architecture search and AutoML, to automatically find an optimal kernel in a heuristic manner. The algorithm we present constructs a quantum circuit implementing the similarity measure as a combinatorial object, which is evaluated based on a cost function and is then iteratively modified using a meta-heuristic optimization technique. The cost function can encode many criteria ensuring favorable statistical properties of the candidate solution, such as the rank of the Dynamical Lie Algebra. Importantly, our approach is independent of the optimization technique employed. The results obtained by testing our approach on a high-energy physics problem demonstrate that, in the best-case scenario, we can either match or improve testing accuracy with respect to the manual design approach, showing the potential of our technique to deliver superior results with reduced effort.

We present a novel approach to non-convex optimization with certificates, which handles smooth functions on the hypercube or on the torus. Unlike traditional methods that rely on algebraic properties, our algorithm exploits the regularity of the target function intrinsic in the decay of its Fourier spectrum. By defining a tractable family of models, we allow at the same time to obtain precise certificates and to leverage the advanced and powerful computational techniques developed to optimize neural networks. In this way the scalability of our approach is naturally enhanced by parallel computing with GPUs. Our approach, when applied to the case of polynomials of moderate dimensions but with thousands of coefficients, outperforms the state-of-the-art optimization methods with certificates, as the ones based on Lasserre's hierarchy, addressing problems intractable for the competitors.

Canonical correlation analysis (CCA) is a classic statistical method for discovering latent co-variation that underpins two or more observed random vectors. Several extensions and variations of CCA have been proposed that have strengthened our capabilities in terms of revealing common random factors from multiview datasets. In this work, we first revisit the most recent deterministic extensions of deep CCA and highlight the strengths and limitations of these state-of-the-art methods. Some methods allow trivial solutions, while others can miss weak common factors. Others overload the problem by also seeking to reveal what is not common among the views -- i.e., the private components that are needed to fully reconstruct each view. The latter tends to overload the problem and its computational and sample complexities. Aiming to improve upon these limitations, we design a novel and efficient formulation that alleviates some of the current restrictions. The main idea is to model the private components as conditionally independent given the common ones, which enables the proposed compact formulation. In addition, we also provide a sufficient condition for identifying the common random factors. Judicious experiments with synthetic and real datasets showcase the validity of our claims and the effectiveness of the proposed approach.

In this paper, we revisit the bilevel optimization problem, in which the upper-level objective function is generally nonconvex and the lower-level objective function is strongly convex. Although this type of problem has been studied extensively, it still remains an open question how to achieve an ${O}(\epsilon^{-1.5})$ sample complexity in Hessian/Jacobian-free stochastic bilevel optimization without any second-order derivative computation. To fill this gap, we propose a novel Hessian/Jacobian-free bilevel optimizer named FdeHBO, which features a simple fully single-loop structure, a projection-aided finite-difference Hessian/Jacobian-vector approximation, and momentum-based updates. Theoretically, we show that FdeHBO requires ${O}(\epsilon^{-1.5})$ iterations (each using ${O}(1)$ samples and only first-order gradient information) to find an $\epsilon$-accurate stationary point. As far as we know, this is the first Hessian/Jacobian-free method with an ${O}(\epsilon^{-1.5})$ sample complexity for nonconvex-strongly-convex stochastic bilevel optimization.

Focusing on stochastic programming (SP) with covariate information, this paper proposes an empirical risk minimization (ERM) method embedded within a nonconvex piecewise affine decision rule (PADR), which aims to learn the direct mapping from features to optimal decisions. We establish the nonasymptotic consistency result of our PADR-based ERM model for unconstrained problems and asymptotic consistency result for constrained ones. To solve the nonconvex and nondifferentiable ERM problem, we develop an enhanced stochastic majorization-minimization algorithm and establish the asymptotic convergence to (composite strong) directional stationarity along with complexity analysis. We show that the proposed PADR-based ERM method applies to a broad class of nonconvex SP problems with theoretical consistency guarantees and computational tractability. Our numerical study demonstrates the superior performance of PADR-based ERM methods compared to state-of-the-art approaches under various settings, with significantly lower costs, less computation time, and robustness to feature dimensions and nonlinearity of the underlying dependency.

As we approach the operation of magnetic confinement burning-plasma experiments [3, 4] and as reactor design studies are becoming widespread in the fusion energy community [5-8], the need for reliable, fast and accurate physics models of the confined plasma becomes essential to realize economically-attractive fusion energy. Particularly, accurate physics models for the transport of energy, particles and momentum in the confined plasma are exceptionally important. Fusion power production and reactor efficiency strongly depend on the core pressure gradients that are attained within the operational space of the fusion device. These gradients in kinetic profiles are determined by a balance of the energy, particle and torque input and turbulent and collisional transport processes. The nonlinear gyrokinetic framework for turbulent transport [9, 10] is the gold standard for a rigorous description of micro-turbulence in the plasma core.