Navid Azizan

Diffusion and flow-matching have emerged as powerful methodologies for generative modeling, with remarkable success in capturing complex data distributions and enabling flexible guidance at inference time. Many downstream applications, however, demand enforcing hard constraints on generated samples-for example, robot trajectories must avoid obstacles-a requirement that goes beyond simple guidance. Prevailing projection-based approaches constrain the entire sampling path to the constraint manifold, which is overly restrictive and degrades sample quality. In this paper, we introduce a novel framework that reformulates hard-constrained sampling as a trajectory optimization problem. Our key insight is to leverage numerical optimal control to steer the sampling trajectory so that constraints are satisfied precisely at the terminal time. By exploiting the underlying structure of flow-matching models and adopting techniques from model predictive control, we transform this otherwise complex constrained optimization problem into a tractable surrogate that can be solved efficiently and effectively. Furthermore, this trajectory optimization perspective offers significant flexibility beyond mere constraint satisfaction, allowing for the inclusion of integral costs to minimize distribution shift and terminal objectives to further enhance sample quality, all within a unified framework. We provide a control-theoretic analysis of our method, establishing bounds on the approximation error between our tractable surrogate and the ideal formulation. Extensive experiments across diverse domains, including robotics (planning), partial differential equations (boundary control), and vision (text-guided image editing), demonstrate that our algorithm, which we name HardFlow, substantially outperforms existing methods in both constraint satisfaction and sample quality.

DEtection TRansformer (DETR) and its variants have emerged as promising architectures for object detection, offering an end-to-end prediction pipeline. In practice, however, DETRs generate hundreds of predictions that far outnumber the actual objects present in an image. This raises a critical question: which of these predictions could be trusted? This is particularly important for safety-critical applications, such as in autonomous vehicles. Addressing this concern, we provide empirical and theoretical evidence that predictions within the same image play distinct roles, resulting in varying reliability levels. Our analysis reveals that DETRs employ an optimal “specialist strategy”: one prediction per object is trained to be well-calibrated , while the remaining predictions are trained to suppress their foreground confidence to near zero, even when maintaining accurate localization. We show that this strategy emerges as the loss-minimizing solution to the Hungarian matching algorithm, fundamentally shaping DETRs' outputs. While selecting the well-calibrated predictions is ideal, they are unidentifiable at inference time. This means that any post-processing algorithm—used to identify trustworthy predictions—poses a risk of outputting a set of predictions with mixed calibration levels. Therefore, practical deployment necessitates a joint evaluation of both the model's calibration quality and the effectiveness of the post-processing algorithm. However, we demonstrate that existing metrics like average precision and expected calibration error are inadequate for this task. To address this issue, we further introduce Object-level Calibration Error (OCE), which evaluates calibration by aggregating predictions per ground-truth object rather than per prediction. This object-centric design penalizes both retaining suppressed predictions and missed ground truth foreground objects, making OCE suitable for both evaluating models and identifying reliable prediction subsets. Finally, we present a post hoc uncertainty quantification (UQ) framework that predicts per-image model accuracy.

Ensuring the safety of real-world systems is challenging, especially when they rely on learned perception modules to infer the system state from high-dimensional sensor data. These perception modules are vulnerable to epistemic uncertainty, often failing when encountering out-of-distribution (OoD) measurements not seen during training. To address this gap, we introduce ATOM-CBF (Adaptive-To-OoD-Measurement Control Barrier Function), a novel safe control framework that explicitly computes and adapts to the epistemic uncertainty from OoD measurements, without the need for ground-truth labels or information on distribution shifts. Our approach features two key components: (1) an OoD-aware adaptive perception error margin and (2) a safety filter that integrates this adaptive error margin, enabling the filter to adjust its conservatism in real-time. We provide empirical validation in simulations, demonstrating that ATOM-CBF maintains safety for an F1Tenth vehicle with LiDAR scans and a quadruped robot with RGB images.

Multi-agent learning faces a fundamental tension: leveraging distributed collaboration without sacrificing the personalization needed for diverse agents. This tension intensifies when aiming for full personalization while adapting to unknown heterogeneity levels -- gaining collaborative speedup when agents are similar, without performance degradation when they are different. Embracing the challenge, we propose personalized collaborative learning (PCL), a novel framework for heterogeneous agents to collaboratively learn personalized solutions with seamless adaptivity. Through carefully designed bias correction and importance correction mechanisms, our method AffPCL robustly handles both environment and objective heterogeneity. We prove that AffPCL reduces sample complexity over independent learning by a factor of $\max\{n^{-1}, δ\}$, where $n$ is the number of agents and $δ\in[0,1]$ measures their heterogeneity. This affinity-based acceleration automatically interpolates between the linear speedup of federated learning in homogeneous settings and the baseline of independent learning, without requiring prior knowledge of the system. Our analysis further reveals that an agent may obtain linear speedup even by collaborating with arbitrarily dissimilar agents, unveiling new insights into personalization and collaboration in the high heterogeneity regime.

Decision-making in building energy systems critically depends on the predictive accuracy of relevant time-series models. In scenarios lacking extensive data from a target building, foundation models (FMs) represent a promising technology that can leverage prior knowledge from vast and diverse pre-training datasets to construct accurate probabilistic predictors for use in decision-making tools. This paper investigates the applicability and fine-tuning strategies of time-series foundation models (TSFMs) in building energy forecasting. We analyze both full fine-tuning and parameter-efficient fine-tuning approaches, particularly low-rank adaptation (LoRA), by using real-world data from a commercial net-zero energy building to capture signals such as room occupancy, carbon emissions, plug loads, and HVAC energy consumption. Our analysis reveals that the zero-shot predictive performance of TSFMs is generally suboptimal. To address this shortcoming, we demonstrate that employing either full fine-tuning or parameter-efficient fine-tuning significantly enhances forecasting accuracy, even with limited historical data. Notably, fine-tuning with low-rank adaptation (LoRA) substantially reduces computational costs without sacrificing accuracy. Furthermore, fine-tuned TSFMs consistently outperform state-of-the-art deep forecasting models (e.g., temporal fusion transformers) in accuracy, robustness, and generalization across varying building zones and seasonal conditions. These results underline the efficacy of TSFMs for practical, data-constrained building energy management systems, enabling improved decision-making in pursuit of energy efficiency and sustainability.

Model merging, a method that combines the parameters and embeddings of multiple fine-tuned large language models (LLMs), offers a promising approach to enhance model performance across various tasks while maintaining computational efficiency. This paper introduces Activation-Informed Merging (AIM), a technique that integrates the information from the activation space of LLMs into the merging process to improve performance and robustness. AIM is designed as a flexible, complementary solution that is applicable to any existing merging method. It aims to preserve critical weights from the base model, drawing on principles from continual learning (CL) and model compression. Utilizing a task-agnostic calibration set, AIM selectively prioritizes essential weights during merging. We empirically demonstrate that AIM significantly enhances the performance of merged models across multiple benchmarks. Our findings suggest that considering the activation-space information can provide substantial advancements in the model merging strategies for LLMs, with up to a 40% increase in benchmark performance.

Transformer-based models demonstrate a remarkable ability for in-context learning (ICL), where they can adapt to unseen tasks from a few prompt examples without parameter updates. Recent research has illuminated how Transformers perform ICL, showing that the optimal linear self-attention (LSA) mechanism can implement one step of gradient descent for linear least-squares objectives when trained on random linear regression tasks. Building on this, we investigate ICL for nonlinear function classes. We first prove that LSA is inherently incapable of outperforming linear predictors on nonlinear tasks, underscoring why prior solutions cannot readily extend to these problems. To overcome this limitation, we analyze a Transformer block consisting of LSA and feed-forward layers inspired by the gated linear units (GLU), which is a standard component of modern Transformers. We show that this block achieves nonlinear ICL by implementing one step of gradient descent on a polynomial kernel regression loss. Furthermore, our analysis reveals that the expressivity of a single block is inherently limited by its dimensions. We then show that a deep Transformer can overcome this bottleneck by distributing the computation of richer kernel functions across multiple blocks, performing block-coordinate descent in a high-dimensional feature space that a single block cannot represent. Our findings highlight that the feed-forward layers provide a crucial and scalable mechanism by which Transformers can express nonlinear representations for ICL.

Multi-agent reinforcement learning (MARL) has achieved notable success in cooperative tasks, demonstrating impressive performance and scalability. However, deploying MARL agents in real-world applications presents critical safety challenges. Current safe MARL algorithms are largely based on the constrained Markov decision process (CMDP) framework, which enforces constraints only on discounted cumulative costs and lacks an all-time safety assurance. Moreover, these methods often overlook the feasibility issue (the system will inevitably violate state constraints within certain regions of the constraint set), resulting in either suboptimal performance or increased constraint violations. To address these challenges, we propose a novel theoretical framework for safe MARL with $\textit{state-wise}$ constraints, where safety requirements are enforced at every state the agents visit. To resolve the feasibility issue, we leverage a control-theoretic notion of the feasible region, the controlled invariant set (CIS), characterized by the safety value function. We develop a multi-agent method for identifying CISs, ensuring convergence to a Nash equilibrium on the safety value function. By incorporating CIS identification into the learning process, we introduce a multi-agent dual policy iteration algorithm that guarantees convergence to a generalized Nash equilibrium in state-wise constrained cooperative Markov games, achieving an optimal balance between feasibility and performance. Furthermore, for practical deployment in complex high-dimensional systems, we propose $\textit{Multi-Agent Dual Actor-Critic}$ (MADAC), a safe MARL algorithm that approximates the proposed iteration scheme within the deep RL paradigm. Empirical evaluations on safe MARL benchmarks demonstrate that MADAC consistently outperforms existing methods, delivering much higher rewards while reducing constraint violations.

Attention mechanisms have revolutionized several domains of artificial intelligence, such as natural language processing and computer vision, by enabling models to selectively focus on relevant parts of the input data. While recent work has characterized the optimization dynamics of gradient descent (GD) in attention-based models and the structural properties of its preferred solutions, less is known about more general optimization algorithms such as mirror descent (MD). In this paper, we investigate the convergence properties and implicit biases of a family of MD algorithms tailored for softmax attention mechanisms, with the potential function chosen as the $p$-th power of the $\ell_p$-norm. Specifically, we show that these algorithms converge in direction to a generalized hard-margin SVM with an $\ell_p$-norm objective when applied to a classification problem using a softmax attention model. Notably, our theoretical results reveal that the convergence rate is comparable to that of traditional GD in simpler models, despite the highly nonlinear and nonconvex nature of the present problem. Additionally, we delve into the joint optimization dynamics of the key-query matrix and the decoder, establishing conditions under which this complex joint optimization converges to their respective hard-margin SVM solutions. Lastly, our numerical experiments on real data demonstrate that MD algorithms improve generalization over standard GD and excel in optimal token selection.

Incorporating prior knowledge or specifications of input-output relationships into machine learning models has attracted significant attention, as it enhances generalization from limited data and yields conforming outputs. However, most existing approaches use soft constraints by penalizing violations through regularization, which offers no guarantee of constraint satisfaction, especially on inputs far from the training distribution--an essential requirement in safety-critical applications. On the other hand, imposing hard constraints on neural networks may hinder their representational power, adversely affecting performance. To address this, we propose HardNet, a practical framework for constructing neural networks that inherently satisfy hard constraints without sacrificing model capacity. Unlike approaches that modify outputs only at inference time, HardNet enables end-to-end training with hard constraint guarantees, leading to improved performance. To the best of our knowledge, HardNet is the first method that enables efficient and differentiable enforcement of more than one input-dependent inequality constraint. It allows unconstrained optimization of the network parameters using standard algorithms by appending a differentiable closed-form enforcement layer to the network's output. Furthermore, we show that HardNet retains neural networks' universal approximation capabilities. We demonstrate its versatility and effectiveness across various applications: learning with piecewise constraints, learning optimization solvers with guaranteed feasibility, and optimizing control policies in safety-critical systems.

Adaptive control achieves concurrent parameter learning and stable control under uncertainties that are linearly parameterized with known nonlinear features. Nonetheless, it is often difficult to obtain such nonlinear features. To address this difficulty, recent progress has been made in integrating meta-learning with adaptive control to learn such nonlinear features from data. However, these meta-learning-based control methods rely on classical adaptation laws using gradient descent, which is confined to the Euclidean geometry. In this paper, we propose a novel method that combines meta-learning and adaptation laws based on mirror descent, a popular generalization of gradient descent, which takes advantage of the potentially non-Euclidean geometry of the parameter space. In our approach, meta-learning not only learns the nonlinear features but also searches for a suitable mirror-descent potential function that optimizes control performance. Through numerical simulations, we demonstrate the effectiveness of the proposed method in learning efficient representations and real-time tracking control performance under uncertain dynamics.

This paper delves into the problem of one-pass learning, where the objective is to train a model on each datapoint in a stream while maintaining performance on past data without retraining on them. An existing approach to this problem in the context of overparameterized (underdetermined) models is Orthogonal Recursive Fitting (ORFit), which fits every new data point while maintaining predictions on previous datapoints by ensuring that parameter updates are orthogonal to the directions that are critical for past data (i.e., the direction of gradient of the model output with respect to the parameters, for those data). For overparameterized linear models, when initialized at zero, ORFit obtains the parameter vector that perfectly fits the data and has the minimum <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\ell^{2}$</tex> -norm, among the infinitely many perfectly fitting parameter vectors. To generalize this and gain control over the selection of desired parameters, in this paper, we introduce Projected Orthogonal Recursive Fitting (Π-ORFit). We begin by characterizing all parameters that can precisely fit data in general vector-output linear models, employing a formalism based on nullspace projector matrices. This framework yields an alternative derivation of ORFit. Building on this, we further extend ORFit to learn a desired parameter by incorporating a Bregman projection into the update rule. Importantly, we show that the resulting parameter minimizes the potential function that defines the Bregman projection at each update step, enabling the selection of a desired parameter among the (infinitely many) candidates consistent with the data. We provide numerical experiments that validate our analytical findings and underscore the practical significance of this generalized approach.

Existing differentially private (DP) synthetic data generation mechanisms typically assume a single-source table. In practice, data is often distributed across multiple tables with relationships across tables. In this paper, we introduce the first-of-its-kind algorithm that can be combined with any existing DP mechanisms to generate synthetic relational databases. Our algorithm iteratively refines the relationship between individual synthetic tables to minimize their approximation errors in terms of low-order marginal distributions while maintaining referential integrity. This algorithm eliminates the need to flatten a relational database into a master table (saving space), operates efficiently (saving time), and scales effectively to high-dimensional data. We provide both DP and theoretical utility guarantees for our algorithm. Through numerical experiments on real-world datasets, we demonstrate the effectiveness of our method in preserving fidelity to the original data.

Recent advances in learning-based control leverage deep function approximators, such as neural networks, to model the evolution of controlled dynamical systems over time. However, the problem of learning a dynamics model and a stabilizing controller persists, since the synthesis of a stabilizing feedback law for known nonlinear systems is a difficult task, let alone for complex parametric representations that must be fit to data. To this end, we propose Control with Inherent Lyapunov Stability (CoILS), a method for jointly learning parametric representations of a nonlinear dynamics model and a stabilizing controller from data. To do this, our approach simultaneously learns a parametric Lyapunov function which intrinsically constrains the dynamics model to be stabilizable by the learned controller. In addition to the stabilizability of the learned dynamics guaranteed by our novel construction, we show that the learned controller stabilizes the true dynamics under certain assumptions on the fidelity of the learned dynamics. Finally, we demonstrate the efficacy of CoILs on a variety of simulated nonlinear dynamical systems.

The study of market equilibria is central to economic theory, particularly in efficiently allocating scarce resources. However, the computation of equilibrium prices at which the supply of goods matches their demand typically relies on having access to complete information on private attributes of agents, e.g., suppliers' cost functions, which are often unavailable in practice. Motivated by this practical consideration, we consider the problem of setting equilibrium prices in the incomplete information setting wherein a market operator seeks to satisfy the customer demand for a commodity by purchasing the required amount from competing suppliers with privately known cost functions unknown to the market operator. In this incomplete information setting, we consider the online learning problem of learning equilibrium prices over time while jointly optimizing three performance metrics-unmet demand, cost regret, and payment regret-pertinent in the context of equilibrium pricing over a horizon of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$T$</tex> periods. In the general setting when suppliers' cost functions are time-varying, we show that no online algorithm can achieve sublinear regret on all three metrics. Thus, we consider the setting when suppliers' cost functions are fixed and develop algorithms that achieve a regret of (i) O(log log T) when the customer demand is constant over time and (ii) <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$O$</tex>(log log T) when the demand is variable over time.

We consider the fundamental problem of solving a large-scale system of linear equations. In particular, we consider the setting where a taskmaster intends to solve the system in a distributed/federated fashion with the help of a set of machines, who each have a subset of the equations. Although there exist several approaches for solving this problem, missing is a rigorous comparison between the convergence rates of the projection-based methods and those of the optimization- based ones. In this paper, we analyze and compare these two classes of algorithms with a particular focus on the most efficient method from each class, namely, the recently proposed Accelerated Projection-Based Consensus (APC) [1] and the Distributed Heavy-Ball Method (D-HBM). To this end, we first propose a geometric notion of data heterogeneity called angular heterogeneity and discuss its generality. Using this notion, we bound and compare the convergence rates of the studied algorithms and capture the effects of both cross-machine and local data heterogeneity on these quantities. Our analysis results in a number of novel insights besides showing that APC is the most efficient method in realistic scenarios where there is a large data heterogeneity. Our numerical analyses validate our theoretical results.

When machine learning models are trained on synthetic data and then deployed on real data, there is often a performance drop due to the distribution shift between synthetic and real data. In this paper, we introduce a new ensemble strategy for training downstream models, with the goal of enhancing their performance when used on real data. We generate multiple synthetic datasets by applying a differential privacy (DP) mechanism several times in parallel and then ensemble the downstream models trained on these datasets. While each synthetic dataset might deviate more from the real data distribution, they collectively increase sample diversity. This may enhance the robustness of downstream models against distribution shifts. Our extensive experiments reveal that while ensembling does not enhance downstream performance (compared with training a single model) for models trained on synthetic data generated by marginal-based or workload-based DP mechanisms, our proposed ensemble strategy does improve the performance for models trained using GAN-based DP mechanisms in terms of both accuracy and calibration of downstream models.

Inspired by the remarkable success of large neural networks, there has been significant interest in understanding the generalization performance of over-parameterized models. Substantial efforts have been invested in characterizing how optimization algorithms impact generalization through their "preferred" solutions, a phenomenon commonly referred to as implicit regularization. In particular, it has been argued that gradient descent (GD) induces an implicit $\ell_2$-norm regularization in regression and classification problems. However, the implicit regularization of different algorithms are confined to either a specific geometry or a particular class of learning problems, indicating a gap in a general approach for controlling the implicit regularization. To address this, we present a unified approach using mirror descent (MD), a notable generalization of GD, to control implicit regularization in both regression and classification settings. More specifically, we show that MD with the general class of homogeneous potential functions converges in direction to a generalized maximum-margin solution for linear classification problems, thereby answering a long-standing question in the classification setting. Further, we show that MD can be implemented efficiently and enjoys fast convergence under suitable conditions. Through comprehensive experiments, we demonstrate that MD is a versatile method to produce learned models with different regularizers, which in turn have different generalization performances.

Real-time adaptation is imperative to the control of robots operating in complex, dynamic environments. Adaptive control laws can endow even nonlinear systems with good trajectory tracking performance, provided that any uncertain dynamics terms are linearly parameterizable with known nonlinear features. However, it is often difficult to specify such features a priori, such as for aerodynamic disturbances on rotorcraft or interaction forces between a manipulator arm and various objects. In this paper, we turn to data-driven modeling with neural networks to learn, offline from past data, an adaptive controller with an internal parametric model of these nonlinear features. Our key insight is that we can better prepare the controller for deployment with control-oriented meta-learning of features in closed-loop simulation, rather than regression-oriented meta-learning of features to fit input-output data. Specifically, we meta-learn the adaptive controller with closed-loop tracking simulation as the base-learner and the average tracking error as the meta-objective. With both fully actuated and underactuated nonlinear planar rotorcraft subject to wind, we demonstrate that our adaptive controller outperforms other controllers trained with regression-oriented meta-learning when deployed in closed-loop for trajectory tracking control.

Self-supervised learning models extract general-purpose representations from data. Quantifying the reliability of these representations is crucial, as many downstream models rely on them as input for their own tasks. To this end, we introduce a formal definition of representation reliability: the representation for a given test point is considered to be reliable if the downstream models built on top of that representation can consistently generate accurate predictions for that test point. However, accessing downstream data to quantify the representation reliability is often infeasible or restricted due to privacy concerns. We propose an ensemble-based method for estimating the representation reliability without knowing the downstream tasks a priori. Our method is based on the concept of neighborhood consistency across distinct pre-trained representation spaces. The key insight is to find shared neighboring points as anchors to align these representation spaces before comparing them. We demonstrate through comprehensive numerical experiments that our method effectively captures the representation reliability with a high degree of correlation, achieving robust and favorable performance compared with baseline methods.

When machine learning models are trained continually on a sequence of tasks, they are often liable to forget what they learned on previous tasks--a phenomenon known as catastrophic forgetting. Proposed solutions to catastrophic forgetting tend to involve storing information about past tasks, meaning that memory usage is a chief consideration in determining their practicality. This paper develops a memory-efficient solution to catastrophic forgetting using the idea of matrix sketching, in the context of a simple continual learning algorithm known as orthogonal gradient descent (OGD). OGD finds weight updates that aim to preserve performance on prior datapoints, using gradients of the model on those datapoints. However, since the memory cost of storing prior model gradients grows with the runtime of the algorithm, OGD is ill-suited to continual learning over long time horizons. To address this problem, we propose SketchOGD. SketchOGD employs an online sketching algorithm to compress model gradients as they are encountered into a matrix of a fixed, user-determined size. In contrast to existing memory-efficient variants of OGD, SketchOGD runs online without the need for advance knowledge of the total number of tasks, is simple to implement, and is more amenable to analysis. We provide theoretical guarantees on the approximation error of the relevant sketches under a novel metric suited to the downstream task of OGD. Experimentally, we find that SketchOGD tends to outperform current state-of-the-art variants of OGD given a fixed memory budget.

We consider the problem of solving a large-scale system of linear equations in a distributed or federated manner by a taskmaster and a set of machines, each possessing a subset of the equations. We provide a comprehensive comparison of two well-known classes of algorithms used to solve this problem: projection-based methods and optimization-based methods. First, we introduce a novel geometric notion of data heterogeneity called angular heterogeneity and discuss its generality. Using this notion, we characterize the optimal convergence rates of the most prominent algorithms from each class, capturing the effects of the number of machines, the number of equations, and that of both cross-machine and local data heterogeneity on these rates. Our analysis establishes the superiority of Accelerated Projected Consensus in realistic scenarios with significant data heterogeneity and offers several insights into how angular heterogeneity affects the efficiency of the methods studied. Additionally, we develop distributed algorithms for the efficient computation of the proposed angular heterogeneity metrics. Our extensive numerical analyses validate and complement our theoretical results.

The architecture of a deep neural network is defined explicitly in terms of the number of layers, the width of each layer and the general network topology. Existing optimisation frameworks neglect this information in favour of implicit architectural information (e.g. second-order methods) or architecture-agnostic distance functions (e.g. mirror descent). Meanwhile, the most popular optimiser in practice, Adam, is based on heuristics. This paper builds a new framework for deriving optimisation algorithms that explicitly leverage neural architecture. The theory extends mirror descent to non-convex composite objective functions: the idea is to transform a Bregman divergence to account for the non-linear structure of neural architecture. Working through the details for deep fully-connected networks yields automatic gradient descent: a first-order optimiser without any hyperparameters. Automatic gradient descent trains both fully-connected and convolutional networks out-of-the-box and at ImageNet scale. A PyTorch implementation is available at https://github.com/jxbz/agd and also in Appendix B. Overall, the paper supplies a rigorous theoretical foundation for a next-generation of architecture-dependent optimisers that work automatically and without hyperparameters.

The computation of equilibrium prices at which the supply of goods matches their demand typically relies on complete information on agents' private attributes, e.g., suppliers' cost functions, which are often unavailable in practice. Motivated by this practical consideration, we consider the problem of learning equilibrium prices over a horizon of $T$ periods in the incomplete information setting wherein a market operator seeks to satisfy the customer demand for a commodity by purchasing it from competing suppliers with cost functions unknown to the operator. We first consider the setting when suppliers' cost functions are fixed and develop algorithms that, on three pertinent regret metrics, simultaneously achieve a regret of $O(1)$ when the customer demand is constant over time, and $O(\sqrt{T})$ when the demand varies over time. In the setting when the suppliers' cost functions vary over time, we demonstrate that, in general, no online algorithm can achieve sublinear regret on all three metrics. Thus, we consider an augmented setting wherein the operator has access to hints/contexts that reflect the variation in the cost functions and propose an algorithm with sublinear regret in this augmented setting. Finally, we present numerical experiments that validate our results and discuss various model extensions.

Recent advances in learning-based control leverage deep function approximators, such as neural networks, to model the evolution of controlled dynamical systems over time. However, the problem of learning a dynamics model and a stabilizing controller persists, since the synthesis of a stabilizing feedback law for known nonlinear systems is a difficult task, let alone for complex parametric representations that must be fit to data. To this end, we propose Control with Inherent Lyapunov Stability (CoILS), a method for jointly learning parametric representations of a nonlinear dynamics model and a stabilizing controller from data. To do this, our approach simultaneously learns a parametric Lyapunov function which intrinsically constrains the dynamics model to be stabilizable by the learned controller. In addition to the stabilizability of the learned dynamics guaranteed by our novel construction, we show that the learned controller stabilizes the true dynamics under certain assumptions on the fidelity of the learned dynamics. Finally, we demonstrate the efficacy of CoILS on a variety of simulated nonlinear dynamical systems.

Even for known nonlinear dynamical systems, feedback controller synthesis is a difficult problem that often requires leveraging the particular structure of the dynamics to induce a stable closed-loop system. For general nonlinear models, including those fit to data, there may not be enough known structure to reliably synthesize a stabilizing feedback controller. In this paper, we discuss a state-dependent nonlinear tracking controller formulation based on a state-dependent Riccati equation for general nonlinear control-affine systems. This formulation depends on a nonlinear factorization of the system of vector fields defining the control-affine dynamics, which always exists under mild smoothness assumptions. We propose a method for learning this factorization from a finite set of data. On a variety of simulated nonlinear dynamical systems, we empirically demonstrate the efficacy of learned versions of this controller in stable trajectory tracking. Alongside our learning method, we evaluate recent ideas in jointly learning a controller and stabilizability certificate for known dynamical systems; we show experimentally that such methods can be frail in comparison.