arXiv subjects

Elad Hazan

Publications and source records attributed to Elad Hazan.

At least 37 records · Page 2Linked to original sources

arXiv2025

FutureFill: Fast Generation from Convolutional Sequence Models

We address the challenge of efficient auto-regressive generation in sequence prediction models by introducing FutureFill, a general-purpose fast generation method for any sequence prediction algorithm based on convolutional operators. FutureFill reduces generation time from quadratic to quasilinear in the context length. Moreover, when generating from a prompt, it requires a prefill cache whose size grows only with the number of tokens to be generated, often much smaller than the caches required by standard convolutional or attention based models. We validate our theoretical claims with experiments on synthetic tasks and demonstrate substantial efficiency gains when generating from a deep convolutional sequence prediction model.

cs.LG↗

arXiv2025

Efficient Spectral Control of Partially Observed Linear Dynamical Systems

We propose a new method for the problem of controlling linear dynamical systems under partial observation and adversarial disturbances. Our new algorithm, Double Spectral Control (DSC), matches the best known regret guarantees while exponentially improving runtime complexity over previous approaches in its dependence on the system's stability margin. Our key innovation is a two-level spectral approximation strategy, leveraging double convolution with a universal basis of spectral filters, enabling efficient and accurate learning of the best linear dynamical controllers.

cs.LG↗

arXiv2025

A New Approach to Controlling Linear Dynamical Systems

We propose a new method for controlling linear dynamical systems under adversarial disturbances and cost functions. Our algorithm achieves a running time that scales polylogarithmically with the inverse of the stability margin, improving upon prior methods with polynomial dependence maintaining the same regret guarantees. The technique, which may be of independent interest, is based on a novel convex relaxation that approximates linear control policies using spectral filters constructed from the eigenvectors of a specific Hankel matrix.

eess.SY↗

arXiv2024

Provable Length Generalization in Sequence Prediction via Spectral Filtering

We consider the problem of length generalization in sequence prediction. We define a new metric of performance in this setting -- the Asymmetric-Regret -- which measures regret against a benchmark predictor with longer context length than available to the learner. We continue by studying this concept through the lens of the spectral filtering algorithm. We present a gradient-based learning algorithm that provably achieves length generalization for linear dynamical systems. We conclude with proof-of-concept experiments which are consistent with our theory.

cs.LG↗

arXiv2024

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 $κ$-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 $\tildeΩ(T^{2/3})$ regret lower bound, even under the assumption of smooth and quadratic losses.

cs.LG↗

arXiv2024

Adaptive Online Learning of Quantum States

The problem of efficient quantum state learning, also called shadow tomography, aims to comprehend an unknown $d$-dimensional quantum state through POVMs. Yet, these states are rarely static; they evolve due to factors such as measurements, environmental noise, or inherent Hamiltonian state transitions. This paper leverages techniques from adaptive online learning to keep pace with such state changes. The key metrics considered for learning in these mutable environments are enhanced notions of regret, specifically adaptive and dynamic regret. We present adaptive and dynamic regret bounds for online shadow tomography, which are polynomial in the number of qubits and sublinear in the number of measurements. To support our theoretical findings, we include numerical experiments that validate our proposed models.

cs.LG↗

arXiv2024

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 (SSMs) based on learning linear dynamical systems with the spectral filtering algorithm (Hazan et al. (2017)). This gives rise to a novel sequence prediction architecture we call a spectral state space model. Spectral state space models have two primary advantages. First, they have provable robustness properties as their performance depends on neither the spectrum of the underlying dynamics nor the dimensionality of the problem. Second, these models are constructed with fixed convolutional filters that do not require learning while still outperforming SSMs in both theory and practice. The resulting models are evaluated on synthetic dynamical systems and long-range prediction tasks of various modalities. These evaluations support the theoretical benefits of spectral filtering for tasks requiring very long range memory.

cs.LG↗

arXiv2024

Playing Large Games with Oracles and AI Debate

We consider regret minimization in repeated games with a very large number of actions. Such games are inherent in the setting of AI Safety via Debate \cite{irving2018ai}, and more generally games whose actions are language-based. Existing algorithms for online game playing require per-iteration computation polynomial in the number of actions, which can be prohibitive for large games. We thus consider oracle-based algorithms, as oracles naturally model access to AI agents. With oracle access, we characterize when internal and external regret can be minimized efficiently. We give a novel efficient algorithm for simultaneous external and internal regret minimization whose regret depends logarithmically on the number of actions. We conclude with experiments in the setting of AI Safety via Debate that shows the benefit of insights from our algorithmic analysis.

cs.GT↗

arXiv2024

Online Control in Population Dynamics

The study of population dynamics originated with early sociological works 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 control in population dynamics 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 control in population dynamics even for non-linear models such as SIR and replicator dynamics.

cs.LG↗

arXiv2024

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 $Ω(|I|^{1-ε})$ 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.

cs.LG↗

arXiv2024

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.

cs.LG↗

arXiv2023

Partial Matrix Completion

The matrix completion problem aims to reconstruct a low-rank matrix based on a revealed set of possibly noisy entries. Prior works consider completing the entire matrix with generalization error guarantees. However, the completion accuracy can be drastically different over different entries. This work establishes a new framework of partial matrix completion, where the goal is to identify a large subset of the entries that can be completed with high confidence. We propose an efficient algorithm with the following provable guarantees. Given access to samples from an unknown and arbitrary distribution, it guarantees: (a) high accuracy over completed entries, and (b) high coverage of the underlying distribution. We also consider an online learning variant of this problem, where we propose a low-regret algorithm based on iterative gradient updates. Preliminary empirical evaluations are included.

cs.LG↗

arXiv2023

A Nonstochastic Control Approach to Optimization

Selecting the best hyperparameters for a particular optimization instance, such as the learning rate and momentum, is an important but nonconvex problem. As a result, iterative optimization methods such as hypergradient descent lack global optimality guarantees in general. We propose an online nonstochastic control methodology for mathematical optimization. First, we formalize the setting of meta-optimization, an online learning formulation of learning the best optimization algorithm from a class of methods. The meta-optimization problem over gradient-based methods can be framed as a feedback control problem over the choice of hyperparameters, including the learning rate, momentum, and the preconditioner. Although the original optimal control problem is nonconvex, we show how recent methods from online nonstochastic control using convex relaxations can be used to overcome the challenge of nonconvexity, and obtain regret guarantees against the best offline solution. This guarantees that in meta-optimization, given a sequence of optimization problems, we can learn a method that attains convergence comparable to that of the best optimization method in hindsight from a class of methods.

cs.LG↗

arXiv2023

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.

cs.RO↗

arXiv2023

Online Nonstochastic Model-Free Reinforcement Learning

We investigate robust model-free reinforcement learning algorithms designed for environments that may be dynamic or even adversarial. Traditional state-based policies often struggle to accommodate the challenges imposed by the presence of unmodeled disturbances in such settings. Moreover, optimizing linear state-based policies pose an 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, which we term pseudo-disturbances, and develop 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 in the face of adversarial disturbances. Our methods seamlessly integrate with any black-box model-free approach, yielding provable regret guarantees when dealing with linear dynamics. These regret guarantees unconditionally improve the best-known results for bandit linear control in having no dependence on the state-space dimension. We evaluate our method over various standard RL benchmarks and demonstrate improved robustness.

cs.LG↗

arXiv2023

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.

cs.LG↗

arXiv2023

Sketchy: Memory-efficient Adaptive Regularization with Frequent Directions

Adaptive regularization methods that exploit more than the diagonal entries exhibit state of the art performance for many tasks, but can be prohibitive in terms of memory and running time. We find the spectra of the Kronecker-factored gradient covariance matrix in deep learning (DL) training tasks are concentrated on a small leading eigenspace that changes throughout training, motivating a low-rank sketching approach. We describe a generic method for reducing memory and compute requirements of maintaining a matrix preconditioner using the Frequent Directions (FD) sketch. While previous approaches have explored applying FD for second-order optimization, we present a novel analysis which allows efficient interpolation between resource requirements and the degradation in regret guarantees with rank $k$: in the online convex optimization (OCO) setting over dimension $d$, we match full-matrix $d^2$ memory regret using only $dk$ memory up to additive error in the bottom $d-k$ eigenvalues of the gradient covariance. Further, we show extensions of our work to Shampoo, resulting in a method competitive in quality with Shampoo and Adam, yet requiring only sub-linear memory for tracking second moments.

stat.ML↗

arXiv2023

Introduction to Online Convex Optimization

This manuscript portrays optimization as a process. In many practical applications the environment is so complex that it is infeasible to lay out a comprehensive theoretical model and use classical algorithmic theory and mathematical optimization. It is necessary as well as beneficial to take a robust approach, by applying an optimization method that learns as one goes along, learning from experience as more aspects of the problem are observed. This view of optimization as a process has become prominent in varied fields and has led to some spectacular success in modeling and systems that are now part of our daily lives.

cs.LG↗