Online Control in Population Dynamics

The study of population dynamics originated with early sociological works (Malthus, 1872) but has since extended into many fields, including biology, epidemiology, evolutionary game theory, and economics. Most studies on population dynamics focus on the problem of prediction rather than control. Existing mathematical models for population control are often restricted to specific, noise-free dynamics, while real-world population changes can be complex and adversarial. To address this gap, we propose a new framework based on the paradigm of online control. We first characterize a set of linear dynamical systems that can naturally model evolving populations. We then give an efficient gradient-based controller for these systems, with near-optimal regret bounds with respect to a broad class of linear policies. Our empirical evaluations demonstrate the effectiveness of the proposed algorithm for population control even in non-linear models such as SIR and replicator dynamics.

Second Order Methods for Bandit Optimization and Control

Bandit convex optimization (BCO) is a general framework for online decision making under uncertainty. While tight regret bounds for general convex losses have been established, existing algorithms achieving these bounds have prohibitive computational costs for high dimensional data. In this paper, we propose a simple and practical BCO algorithm inspired by the online Newton step algorithm. We show that our algorithm achieves optimal (in terms of horizon) regret bounds for a large class of convex functions that we call $\kappa$-convex. This class contains a wide range of practically relevant loss functions including linear, quadratic, and generalized linear models. In addition to optimal regret, this method is the most efficient known algorithm for several well-studied applications including bandit logistic regression. Furthermore, we investigate the adaptation of our second-order bandit algorithm to online convex optimization with memory. We show that for loss functions with a certain affine structure, the extended algorithm attains optimal regret. This leads to an algorithm with optimal regret for bandit LQR/LQG problems under a fully adversarial noise model, thereby resolving an open question posed in \citep{gradu2020non} and \citep{sun2023optimal}. Finally, we show that the more general problem of BCO with (non-affine) memory is harder. We derive a $\tilde{\Omega}(T^{2/3})$ regret lower bound, even under the assumption of smooth and quadratic losses.

Adaptive Regret for Bandits Made Possible: Two Queries Suffice

Fast changing states or volatile environments pose a significant challenge to online optimization, which needs to perform rapid adaptation under limited observation. In this paper, we give query and regret optimal bandit algorithms under the strict notion of strongly adaptive regret, which measures the maximum regret over any contiguous interval $I$. Due to its worst-case nature, there is an almost-linear $\Omega(|I|^{1-\epsilon})$ regret lower bound, when only one query per round is allowed [Daniely el al, ICML 2015]. Surprisingly, with just two queries per round, we give Strongly Adaptive Bandit Learner (StABL) that achieves $\tilde{O}(\sqrt{n|I|})$ adaptive regret for multi-armed bandits with $n$ arms. The bound is tight and cannot be improved in general. Our algorithm leverages a multiplicative update scheme of varying stepsizes and a carefully chosen observation distribution to control the variance. Furthermore, we extend our results and provide optimal algorithms in the bandit convex optimization setting. Finally, we empirically demonstrate the superior performance of our algorithms under volatile environments and for downstream tasks, such as algorithm selection for hyperparameter optimization.

Chain of LoRA: Efficient Fine-tuning of Language Models via Residual Learning

Fine-tuning is the primary methodology for tailoring pre-trained large language models to specific tasks. As the model's scale and the diversity of tasks expand, parameter-efficient fine-tuning methods are of paramount importance. One of the most widely used family of methods is low-rank adaptation (LoRA) and its variants. LoRA encodes weight update as the product of two low-rank matrices. Despite its advantages, LoRA falls short of full-parameter fine-tuning in terms of generalization error for certain tasks. We introduce Chain of LoRA (COLA), an iterative optimization framework inspired by the Frank-Wolfe algorithm, to bridge the gap between LoRA and full parameter fine-tuning, without incurring additional computational costs or memory overheads. COLA employs a residual learning procedure where it merges learned LoRA modules into the pre-trained language model parameters and re-initilize optimization for new born LoRA modules. We provide theoretical convergence guarantees as well as empirical results to validate the effectiveness of our algorithm. Across various models (OPT and llama-2) and seven benchmarking tasks, we demonstrate that COLA can consistently outperform LoRA without additional computational or memory costs.

Spectral State Space Models

This paper studies sequence modeling for prediction tasks with long range dependencies. We propose a new formulation for state space models based on learning linear dynamical systems with the spectral filtering algorithm [HSZ17]. This gives rise to a novel sequence prediction architecture we call spectral state space models. The resulting models are evaluated on synthetic dynamical systems. These evaluations support the theoretical benefits of spectral filtering for tasks requiring very long range memory.

AI safety by debate via regret minimization

We consider the setting of AI safety by debate as a repeated game. We consider the question of efficient regret minimization in this setting, when the players are either AIs or humans, equipped with access to computationally superior AIs. In such a setting, we characterize when internal and external regret can be minimized efficiently. We conclude with conditions in which a sequence of strategies converges to a correlated equilibrium.

An Efficient Interior-Point Method for Online Convex Optimization

A new algorithm for regret minimization in online convex optimization is described. The regret of the algorithm after $T$ time periods is $O(\sqrt{T \log T})$ - which is the minimum possible up to a logarithmic term. In addition, the new algorithm is adaptive, in the sense that the regret bounds hold not only for the time periods $1,\ldots,T$ but also for every sub-interval $s,s+1,\ldots,t$. The running time of the algorithm matches that of newly introduced interior point algorithms for regret minimization: in $n$-dimensional space, during each iteration the new algorithm essentially solves a system of linear equations of order $n$, rather than solving some constrained convex optimization problem in $n$ dimensions and possibly many constraints.

Online Learning for Obstacle Avoidance

We approach the fundamental problem of obstacle avoidance for robotic systems via the lens of online learning. In contrast to prior work that either assumes worst-case realizations of uncertainty in the environment or a stationary stochastic model of uncertainty, we propose a method that is efficient to implement and provably grants instance-optimality with respect to perturbations of trajectories generated from an open-loop planner (in the sense of minimizing worst-case regret). The resulting policy adapts online to realizations of uncertainty and provably compares well with the best obstacle avoidance policy in hindsight from a rich class of policies. The method is validated in simulation on a dynamical system environment and compared to baseline open-loop planning and robust Hamilton- Jacobi reachability techniques. Further, it is implemented on a hardware example where a quadruped robot traverses a dense obstacle field and encounters input disturbances due to time delays, model uncertainty, and dynamics nonlinearities.

Online Nonstochastic Model-Free Reinforcement Learning

In this work, we explore robust model-free reinforcement learning algorithms for environments that may be dynamic or even adversarial. Conventional state-based policies fail to accommodate the challenge imposed by the presence of unmodeled disturbances in such settings. Additionally, optimizing linear state-based policies pose obstacle for efficient optimization, leading to nonconvex objectives even in benign environments like linear dynamical systems. Drawing inspiration from recent advancements in model-based control, we introduce a novel class of policies centered on disturbance signals. We define several categories of these signals, referred to as pseudo-disturbances, and corresponding policy classes based on them. We provide efficient and practical algorithms for optimizing these policies. Next, we examine the task of online adaptation of reinforcement learning agents to adversarial disturbances. Our methods can be integrated with any black-box model-free approach, resulting in provable regret guarantees if the underlying dynamics is linear. We evaluate our method over different standard RL benchmarks and demonstrate improved robustness.

Optimal Rates for Bandit Nonstochastic Control

Linear Quadratic Regulator (LQR) and Linear Quadratic Gaussian (LQG) control are foundational and extensively researched problems in optimal control. We investigate LQR and LQG problems with semi-adversarial perturbations and time-varying adversarial bandit loss functions. The best-known sublinear regret algorithm of~\cite{gradu2020non} has a $T^{\frac{3}{4}}$ time horizon dependence, and its authors posed an open question about whether a tight rate of $\sqrt{T}$ could be achieved. We answer in the affirmative, giving an algorithm for bandit LQR and LQG which attains optimal regret (up to logarithmic factors) for both known and unknown systems. A central component of our method is a new scheme for bandit convex optimization with memory, which is of independent interest.