Y. Jennifer Sun

Vladimir Feinberg, Xinyi Chen, Y. Jennifer Sun et al. · deepmind, princeton

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.

Elad Hazan, Adam Tauman Kalai, Varun Kanade et al. · princeton

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.

Zhou Lu, Y. Jennifer Sun, Zhiyu Zhang

We study the problem of controlling population dynamics, a class of linear dynamical systems evolving on the probability simplex, from the perspective of online non-stochastic control. While Golowich et.al. 2024 analyzed the fully observable setting, we focus on the more realistic, partially observable case, where only a low-dimensional representation of the state is accessible. In classical non-stochastic control, inputs are set as linear combinations of past disturbances. However, under partial observations, disturbances cannot be directly computed. To address this, Simchowitz et.al. 2020 proposed to construct oblivious signals, which are counterfactual observations with zero control, as a substitute. This raises several challenges in our setting: (1) how to construct oblivious signals under simplex constraints, where zero control is infeasible; (2) how to design a sufficiently expressive convex controller parameterization tailored to these signals; and (3) how to enforce the simplex constraint on control when projections may break the convexity of cost functions. Our main contribution is a new controller that achieves the optimal $\tilde{O}(\sqrt{T})$ regret with respect to a natural class of mixing linear dynamic controllers. To tackle these challenges, we construct signals based on hypothetical observations under a constant control adapted to the simplex domain, and introduce a new controller parameterization that approximates general control policies linear in non-oblivious observations. Furthermore, we employ a novel convex extension surrogate loss, inspired by Lattimore 2024, to bypass the projection-induced convexity issue.

Zhou Lu, Y. Jennifer Sun, Zhiyu Zhang

Focusing on the expert problem in online learning, this paper studies the interpolation of several performance metrics via $φ$-regret minimization, which measures the total loss of an algorithm by its regret with respect to an arbitrary action modification rule $φ$. With $d$ experts and $T\gg d$ rounds in total, we present a single algorithm achieving the instance-adaptive $φ$-regret bound \begin{equation*} \tilde O\left(\min\left\{\sqrt{d-d^{\mathrm{unif}}_φ+1},\sqrt{d-d^{\mathrm{self}}_φ}\right\}\cdot\sqrt{T}\right), \end{equation*} where $d^{\mathrm{unif}}_φ$ is the maximum amount of experts modified identically by $φ$, and $d^{\mathrm{self}}_φ$ is the amount of experts that $φ$ trivially modifies to themselves. By recovering the optimal $O(\sqrt{T\log d})$ external regret bound when $d^{\mathrm{unif}}_φ=d$, the standard $\tilde O(\sqrt{T})$ internal regret bound when $d^{\mathrm{self}}_φ=d-1$ and the optimal $\tilde O(\sqrt{dT})$ swap regret bound in the worst case, we improve upon existing algorithms in the intermediate regimes. In addition, the computational complexity of our algorithm matches that of the standard swap-regret minimization algorithm due to (Blum and Mansour, 2007). Technically, building on the well-known reduction from $φ$-regret minimization to external regret minimization on stochastic matrices, our main idea is to further convert the latter to online linear regression using Haar-wavelet-inspired matrix features. Then, by associating the complexity of each $φ$ instance with its sparsity under the feature representation, we apply techniques from comparator-adaptive online learning to exploit the sparsity in this regression subroutine.

Arun Suggala, Y. Jennifer Sun, Praneeth Netrapalli et al. · princeton

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.

Noah Golowich, Elad Hazan, Zhou Lu et al.

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.

Y. Jennifer Sun, Stephen Newman, Elad Hazan

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.