Zhou Lu

Zhou Lu, Wenhan Xia, Sanjeev Arora et al. · princeton

Adaptive gradient methods are the method of choice for optimization in machine learning and used to train the largest deep models. In this paper we study the problem of learning a local preconditioner, that can change as the data is changing along the optimization trajectory. We propose an adaptive gradient method that has provable adaptive regret guarantees vs. the best local preconditioner. To derive this guarantee, we prove a new adaptive regret bound in online learning that improves upon previous adaptive online learning methods. We demonstrate the robustness of our method in automatically choosing the optimal learning rate schedule for popular benchmarking tasks in vision and language domains. Without the need to manually tune a learning rate schedule, our method can, in a single run, achieve comparable and stable task accuracy as a fine-tuned optimizer.

Xinyi Chen, Elad Hazan, Tongyang Li et al. · princeton

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.

Zhou Lu, Nataly Brukhim, Paula Gradu et al. · princeton

In the framework of online convex optimization, most iterative algorithms require the computation of projections onto convex sets, which can be computationally expensive. To tackle this problem HK12 proposed the study of projection-free methods that replace projections with less expensive computations. The most common approach is based on the Frank-Wolfe method, that uses linear optimization computation in lieu of projections. Recent work by GK22 gave sublinear adaptive regret guarantees with projection free algorithms based on the Frank Wolfe approach. In this work we give projection-free algorithms that are based on a different technique, inspired by Mhammedi22, that replaces projections by set-membership computations. We propose a simple lazy gradient-based algorithm with a Minkowski regularization that attains near-optimal adaptive regret bounds. For general convex loss functions we improve previous adaptive regret bounds from $O(T^{3/4})$ to $O(\sqrt{T})$, and further to tight interval dependent bound $\tilde{O}(\sqrt{I})$ where $I$ denotes the interval length. For strongly convex functions we obtain the first poly-logarithmic adaptive regret bounds using a projection-free algorithm.

Udaya Ghai, Zhou Lu, Elad Hazan · princeton

We study an algorithmic equivalence technique between non-convex gradient descent and convex mirror descent. We start by looking at a harder problem of regret minimization in online non-convex optimization. We show that under certain geometric and smoothness conditions, online gradient descent applied to non-convex functions is an approximation of online mirror descent applied to convex functions under reparameterization. In continuous time, the gradient flow with this reparameterization was shown to be exactly equivalent to continuous-time mirror descent by Amid and Warmuth 2020, but theory for the analogous discrete time algorithms is left as an open problem. We prove an $O(T^{\frac{2}{3}})$ regret bound for non-convex online gradient descent in this setting, answering this open problem. Our analysis is based on a new and simple algorithmic equivalence method.

Zhou Lu, Elad Hazan · princeton

In online convex optimization, the player aims to minimize regret, or the difference between her loss and that of the best fixed decision in hindsight over the entire repeated game. Algorithms that minimize (standard) regret may converge to a fixed decision, which is undesirable in changing or dynamic environments. This motivates the stronger metrics of performance, notably adaptive and dynamic regret. Adaptive regret is the maximum regret over any continuous sub-interval in time. Dynamic regret is the difference between the total cost and that of the best sequence of decisions in hindsight. State-of-the-art performance in both adaptive and dynamic regret minimization suffers a computational penalty - typically on the order of a multiplicative factor that grows logarithmically in the number of game iterations. In this paper we show how to reduce this computational penalty to be doubly logarithmic in the number of game iterations, and retain near optimal adaptive and dynamic regret bounds.

Zhou Lu

Human perception of the empirical world involves recognizing the diverse appearances, or 'modalities', of underlying objects. Despite the longstanding consideration of this perspective in philosophy and cognitive science, the study of multimodality remains relatively under-explored within the field of machine learning. Nevertheless, current studies of multimodal machine learning are limited to empirical practices, lacking theoretical foundations beyond heuristic arguments. An intriguing finding from the practice of multimodal learning is that a model trained on multiple modalities can outperform a finely-tuned unimodal model, even on unimodal tasks. This paper provides a theoretical framework that explains this phenomenon, by studying generalization properties of multimodal learning algorithms. We demonstrate that multimodal learning allows for a superior generalization bound compared to unimodal learning, up to a factor of $O(\sqrt{n})$, where $n$ represents the sample size. Such advantage occurs when both connection and heterogeneity exist between the modalities.

Zhou Lu

Can a physicist make only a finite number of errors in the eternal quest to uncover the law of nature? This millennium-old philosophical problem, known as inductive inference, lies at the heart of epistemology. Despite its significance to understanding human reasoning, a rigorous justification of inductive inference has remained elusive. At a high level, inductive inference asks whether one can make at most finite errors amidst an infinite sequence of observations, when deducing the correct hypothesis from a given hypothesis class. Historically, the only theoretical guarantee has been that if the hypothesis class is countable, inductive inference is possible, as exemplified by Solomonoff induction for learning Turing machines. In this paper, we provide a tight characterization of inductive inference by establishing a novel link to online learning theory. As our main result, we prove that inductive inference is possible if and only if the hypothesis class is a countable union of online learnable classes, potentially with an uncountable size, no matter the observations are adaptively chosen or iid sampled. Moreover, the same condition is also sufficient and necessary in the agnostic setting, where any hypothesis class meeting this criterion enjoys an $\tilde{O}(\sqrt{T})$ regret bound for any time step $T$, while others require an arbitrarily slow rate of regret. Our main technical tool is a novel non-uniform online learning framework, which may be of independent interest.

Zhou Lu

Human perception inherently operates in a multimodal manner. Similarly, as machines interpret the empirical world, their learning processes ought to be multimodal. The recent, remarkable successes in empirical multimodal learning underscore the significance of understanding this paradigm. Yet, a solid theoretical foundation for multimodal learning has eluded the field for some time. While a recent study by Lu (2023) has shown the superior sample complexity of multimodal learning compared to its unimodal counterpart, another basic question remains: does multimodal learning also offer computational advantages over unimodal learning? This work initiates a study on the computational benefit of multimodal learning. We demonstrate that, under certain conditions, multimodal learning can outpace unimodal learning exponentially in terms of computation. Specifically, we present a learning task that is NP-hard for unimodal learning but is solvable in polynomial time by a multimodal algorithm. Our construction is based on a novel modification to the intersection of two half-spaces problem.

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.

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.

Zhou Lu, Qiuyi Zhang, Xinyi Chen et al. · deepmind, princeton

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.

Daogao Liu, Zhou Lu

Stochastic Gradient Descent (SGD) is among the simplest and most popular methods in optimization. The convergence rate for SGD has been extensively studied and tight analyses have been established for the running average scheme, but the sub-optimality of the final iterate is still not well-understood. shamir2013stochastic gave the best known upper bound for the final iterate of SGD minimizing non-smooth convex functions, which is $O(\log T/\sqrt{T})$ for Lipschitz convex functions and $O(\log T/ T)$ with additional assumption on strongly convexity. The best known lower bounds, however, are worse than the upper bounds by a factor of $\log T$. harvey2019tight gave matching lower bounds but their construction requires dimension $d= T$. It was then asked by koren2020open how to characterize the final-iterate convergence of SGD in the constant dimension setting. In this paper, we answer this question in the more general setting for any $d\leq T$, proving $Ω(\log d/\sqrt{T})$ and $Ω(\log d/T)$ lower bounds for the sub-optimality of the final iterate of SGD in minimizing non-smooth Lipschitz convex and strongly convex functions respectively with standard step size schedules. Our results provide the first general dimension dependent lower bound on the convergence of SGD's final iterate, partially resolving a COLT open question raised by koren2020open. We also present further evidence to show the correct rate in one dimension should be $Θ(1/\sqrt{T})$, such as a proof of a tight $O(1/\sqrt{T})$ upper bound for one-dimensional special cases in settings more general than koren2020open.

Yin Tat Lee, Daogao Liu, Zhou Lu

Differentially private empirical risk minimization (DP-ERM) is a fundamental problem in private optimization. While the theory of DP-ERM is well-studied, as large-scale models become prevalent, traditional DP-ERM methods face new challenges, including (1) the prohibitive dependence on the ambient dimension, (2) the highly non-smooth objective functions, (3) costly first-order gradient oracles. Such challenges demand rethinking existing DP-ERM methodologies. In this work, we show that the regularized exponential mechanism combined with existing samplers can address these challenges altogether: under the standard unconstrained domain and low-rank gradients assumptions, our algorithm can achieve rank-dependent risk bounds for non-smooth convex objectives using only zeroth order oracles, which was not accomplished by prior methods. This highlights the power of sampling in differential privacy. We further construct lower bounds, demonstrating that when gradients are full-rank, there is no separation between the constrained and unconstrained settings. Our lower bound is derived from a general black-box reduction from unconstrained to the constrained domain and an improved lower bound in the constrained setting, which might be of independent interest.

Bohang Zhang, Tianle Cai, Zhou Lu et al.

It is well-known that standard neural networks, even with a high classification accuracy, are vulnerable to small $\ell_\infty$-norm bounded adversarial perturbations. Although many attempts have been made, most previous works either can only provide empirical verification of the defense to a particular attack method, or can only develop a certified guarantee of the model robustness in limited scenarios. In this paper, we seek for a new approach to develop a theoretically principled neural network that inherently resists $\ell_\infty$ perturbations. In particular, we design a novel neuron that uses $\ell_\infty$-distance as its basic operation (which we call $\ell_\infty$-dist neuron), and show that any neural network constructed with $\ell_\infty$-dist neurons (called $\ell_{\infty}$-dist net) is naturally a 1-Lipschitz function with respect to $\ell_\infty$-norm. This directly provides a rigorous guarantee of the certified robustness based on the margin of prediction outputs. We then prove that such networks have enough expressive power to approximate any 1-Lipschitz function with robust generalization guarantee. We further provide a holistic training strategy that can greatly alleviate optimization difficulties. Experimental results show that using $\ell_{\infty}$-dist nets as basic building blocks, we consistently achieve state-of-the-art performance on commonly used datasets: 93.09% certified accuracy on MNIST ($ε=0.3$), 35.42% on CIFAR-10 ($ε=8/255$) and 16.31% on TinyImageNet ($ε=1/255$).

Zhou Lu

In this note we prove a sharp lower bound on the necessary number of nestings of nested absolute-value functions of generalized hinging hyperplanes (GHH) to represent arbitrary CPWL functions. Previous upper bound states that $n+1$ nestings is sufficient for GHH to achieve universal representation power, but the corresponding lower bound was unknown. We prove that $n$ nestings is necessary for universal representation power, which provides an almost tight lower bound. We also show that one-hidden-layer neural networks don't have universal approximation power over the whole domain. The analysis is based on a key lemma showing that any finite sum of periodic functions is either non-integrable or the zero function, which might be of independent interest.

Qinghua Liu, Zhou Lu

Leveraging algorithmic stability to derive sharp generalization bounds is a classic and powerful approach in learning theory. Since Vapnik and Chervonenkis [1974] first formalized the idea for analyzing SVMs, it has been utilized to study many fundamental learning algorithms (e.g., $k$-nearest neighbors [Rogers and Wagner, 1978], stochastic gradient method [Hardt et al., 2016], linear regression [Maurer, 2017], etc). In a recent line of great works by Feldman and Vondrak [2018, 2019] as well as Bousquet et al. [2020b], they prove a high probability generalization upper bound of order $\tilde{\mathcal{O}}(γ+\frac{L}{\sqrt{n}})$ for any uniformly $γ$-stable algorithm and $L$-bounded loss function. Although much progress was achieved in proving generalization upper bounds for stable algorithms, our knowledge of lower bounds is rather limited. In fact, there is no nontrivial lower bound known ever since the study of uniform stability [Bousquet and Elisseeff, 2002], to the best of our knowledge. In this paper we fill the gap by proving a tight generalization lower bound of order $Ω(γ+\frac{L}{\sqrt{n}})$, which matches the best known upper bound up to logarithmic factors

Zhou Lu

We prove a Johns theorem for simplices in $R^d$ with positive dilation factor $d+2$, which improves the previously known $d^2$ upper bound. This bound is tight in view of the $d$ lower bound. Furthermore, we give an example that $d$ isn't the optimal lower bound when $d=2$. Our results answered both questions regarding Johns theorem for simplices with positive dilation raised by \cite{leme2020costly}.

Naman Agarwal, Nataly Brukhim, Elad Hazan et al.

We study the question of how to aggregate controllers for dynamical systems in order to improve their performance. To this end, we propose a framework of boosting for online control. Our main result is an efficient boosting algorithm that combines weak controllers into a provably more accurate one. Empirical evaluation on a host of control settings supports our theoretical findings.

Zhou Lu, Hongming Pu, Feicheng Wang et al.

The expressive power of neural networks is important for understanding deep learning. Most existing works consider this problem from the view of the depth of a network. In this paper, we study how width affects the expressiveness of neural networks. Classical results state that depth-bounded (e.g. depth-$2$) networks with suitable activation functions are universal approximators. We show a universal approximation theorem for width-bounded ReLU networks: width-$(n+4)$ ReLU networks, where $n$ is the input dimension, are universal approximators. Moreover, except for a measure zero set, all functions cannot be approximated by width-$n$ ReLU networks, which exhibits a phase transition. Several recent works demonstrate the benefits of depth by proving the depth-efficiency of neural networks. That is, there are classes of deep networks which cannot be realized by any shallow network whose size is no more than an exponential bound. Here we pose the dual question on the width-efficiency of ReLU networks: Are there wide networks that cannot be realized by narrow networks whose size is not substantially larger? We show that there exist classes of wide networks which cannot be realized by any narrow network whose depth is no more than a polynomial bound. On the other hand, we demonstrate by extensive experiments that narrow networks whose size exceed the polynomial bound by a constant factor can approximate wide and shallow network with high accuracy. Our results provide more comprehensive evidence that depth is more effective than width for the expressiveness of ReLU networks.