Maryam Fazel

Samet Oymak, Karthik Mohan, Maryam Fazel et al.

Recovering sparse vectors and low-rank matrices from noisy linear measurements has been the focus of much recent research. Various reconstruction algorithms have been studied, including $\ell_1$ and nuclear norm minimization as well as $\ell_p$ minimization with $p<1$. These algorithms are known to succeed if certain conditions on the measurement map are satisfied. Proofs of robust recovery for matrices have so far been much more involved than in the vector case. In this paper, we show how several robust classes of recovery conditions can be extended from vectors to matrices in a simple and transparent way, leading to the best known restricted isometry and nullspace conditions for matrix recovery. Our results rely on the ability to "vectorize" matrices through the use of a key singular value inequality.

Jingxing Wang, Vasileios Charisopoulos, Maryam Fazel

We study the problem of learning Gaussian mixture models under overparameterization. Prior work has shown that while overparameterization is essential for avoiding spurious local optima and enables global recovery of the ground-truth model using the gradient-EM (expectation-maximization) algorithm, it can dramatically slow down the local rate of convergence. Under certain assumptions on the mixture weights, we show that a standard divergence measure minimized by statistical learning procedures possesses a manifold of slow growth on which the well-known Polyak stepsize reduces the loss geometrically, and design a gradient-based method that converges to minimizers at a locally linear rate. Additionally, we show that our method converges to nearly optimal solutions -- up to a natural misspecification threshold -- for mixtures with arbitrary weights. At a high level, the method alternates between several "short" gradient descent steps that approach the manifold and "long" Polyak steps that contract the distance to minimizers. Our results suggest that slow convergence is not an intrinsic challenge of overparameterization, but can be overcome by exploiting the favorable structure of the loss landscape.

Adhyyan Narang, Omid Sadeghi, Lillian J Ratliff et al. · uw

In the context of online interactive machine learning with combinatorial objectives, we extend purely submodular prior work to more general non-submodular objectives. This includes: (1) those that are additively decomposable into a sum of two terms (a monotone submodular and monotone supermodular term, known as a BP decomposition); and (2) those that are only weakly submodular. In both cases, this allows representing not only competitive (submodular) but also complementary (supermodular) relationships between objects, enhancing this setting to a broader range of applications (e.g., movie recommendations, medical treatments, etc.) where this is beneficial. In the two-term case, moreover, we study not only the more typical monolithic feedback approach but also a novel framework where feedback is available separately for each term. With real-world practicality and scalability in mind, we integrate Nystrom sketching techniques to significantly reduce the computational cost, including for the purely submodular case. In the Gaussian process contextual bandits setting, we show sub-linear theoretical regret bounds in all cases. We also empirically show good applicability to recommendation systems and data subset selection.

Bin Hu, Kaiqing Zhang, Na Li et al.

Gradient-based methods have been widely used for system design and optimization in diverse application domains. Recently, there has been a renewed interest in studying theoretical properties of these methods in the context of control and reinforcement learning. This article surveys some of the recent developments on policy optimization, a gradient-based iterative approach for feedback control synthesis, popularized by successes of reinforcement learning. We take an interdisciplinary perspective in our exposition that connects control theory, reinforcement learning, and large-scale optimization. We review a number of recently-developed theoretical results on the optimization landscape, global convergence, and sample complexity of gradient-based methods for various continuous control problems such as the linear quadratic regulator (LQR), $\mathcal{H}_\infty$ control, risk-sensitive control, linear quadratic Gaussian (LQG) control, and output feedback synthesis. In conjunction with these optimization results, we also discuss how direct policy optimization handles stability and robustness concerns in learning-based control, two main desiderata in control engineering. We conclude the survey by pointing out several challenges and opportunities at the intersection of learning and control.

Vincent Roulet, Siddhartha Srinivasa, Maryam Fazel et al.

Iterative optimization algorithms depend on access to information about the objective function. In a differentiable programming framework, this information, such as gradients, can be automatically derived from the computational graph. We explore how nonlinear control algorithms, often employing linear and/or quadratic approximations, can be effectively cast within this framework. Our approach illuminates shared components and differences between gradient descent, Gauss-Newton, Newton, and differential dynamic programming methods in the context of discrete time nonlinear control. Furthermore, we present line-search strategies and regularized variants of these algorithms, along with a comprehensive analysis of their computational complexities. We study the performance of the aforementioned algorithms on various nonlinear control benchmarks, including autonomous car racing simulations using a simplified car model. All implementations are publicly available in a package coded in a differentiable programming language.

Qiwen Cui, Zhihan Xiong, Maryam Fazel et al.

In this paper, we investigate Nash-regret minimization in congestion games, a class of games with benign theoretical structure and broad real-world applications. We first propose a centralized algorithm based on the optimism in the face of uncertainty principle for congestion games with (semi-)bandit feedback, and obtain finite-sample guarantees. Then we propose a decentralized algorithm via a novel combination of the Frank-Wolfe method and G-optimal design. By exploiting the structure of the congestion game, we show the sample complexity of both algorithms depends only polynomially on the number of players and the number of facilities, but not the size of the action set, which can be exponentially large in terms of the number of facilities. We further define a new problem class, Markov congestion games, which allows us to model the non-stationarity in congestion games. We propose a centralized algorithm for Markov congestion games, whose sample complexity again has only polynomial dependence on all relevant problem parameters, but not the size of the action set.

Mitas Ray, Dmitriy Drusvyatskiy, Maryam Fazel et al.

This paper studies the problem of expected loss minimization given a data distribution that is dependent on the decision-maker's action and evolves dynamically in time according to a geometric decay process. Novel algorithms for both the information setting in which the decision-maker has a first order gradient oracle and the setting in which they have simply a loss function oracle are introduced. The algorithms operate on the same underlying principle: the decision-maker repeatedly deploys a fixed decision over the length of an epoch, thereby allowing the dynamically changing environment to sufficiently mix before updating the decision. The iteration complexity in each of the settings is shown to match existing rates for first and zero order stochastic gradient methods up to logarithmic factors. The algorithms are evaluated on a "semi-synthetic" example using real world data from the SFpark dynamic pricing pilot study; it is shown that the announced prices result in an improvement for the institution's objective (target occupancy), while achieving an overall reduction in parking rates.

Lijun Ding, Dmitriy Drusvyatskiy, Maryam Fazel et al.

Empirical evidence suggests that for a variety of overparameterized nonlinear models, most notably in neural network training, the growth of the loss around a minimizer strongly impacts its performance. Flat minima -- those around which the loss grows slowly -- appear to generalize well. This work takes a step towards understanding this phenomenon by focusing on the simplest class of overparameterized nonlinear models: those arising in low-rank matrix recovery. We analyze overparameterized matrix and bilinear sensing, robust PCA, covariance matrix estimation, and single hidden layer neural networks with quadratic activation functions. In all cases, we show that flat minima, measured by the trace of the Hessian, exactly recover the ground truth under standard statistical assumptions. For matrix completion, we establish weak recovery, although empirical evidence suggests exact recovery holds here as well. We conclude with synthetic experiments that illustrate our findings and discuss the effect of depth on flat solutions.

Haozhe Jiang, Qiwen Cui, Zhihan Xiong et al.

We investigate learning the equilibria in non-stationary multi-agent systems and address the challenges that differentiate multi-agent learning from single-agent learning. Specifically, we focus on games with bandit feedback, where testing an equilibrium can result in substantial regret even when the gap to be tested is small, and the existence of multiple optimal solutions (equilibria) in stationary games poses extra challenges. To overcome these obstacles, we propose a versatile black-box approach applicable to a broad spectrum of problems, such as general-sum games, potential games, and Markov games, when equipped with appropriate learning and testing oracles for stationary environments. Our algorithms can achieve $\widetilde{O}\left(Δ^{1/4}T^{3/4}\right)$ regret when the degree of nonstationarity, as measured by total variation $Δ$, is known, and $\widetilde{O}\left(Δ^{1/5}T^{4/5}\right)$ regret when $Δ$ is unknown, where $T$ is the number of rounds. Meanwhile, our algorithm inherits the favorable dependence on number of agents from the oracles. As a side contribution that may be independent of interest, we show how to test for various types of equilibria by a black-box reduction to single-agent learning, which includes Nash equilibria, correlated equilibria, and coarse correlated equilibria.

Yuzhen Qin, Yingcong Li, Fabio Pasqualetti et al.

The growing interest in complex decision-making and language modeling problems highlights the importance of sample-efficient learning over very long horizons. This work takes a step in this direction by investigating contextual linear bandits where the current reward depends on at most $s$ prior actions and contexts (not necessarily consecutive), up to a time horizon of $h$. In order to avoid polynomial dependence on $h$, we propose new algorithms that leverage sparsity to discover the dependence pattern and arm parameters jointly. We consider both the data-poor ($T<h$) and data-rich ($T\ge h$) regimes, and derive respective regret upper bounds $\tilde O(d\sqrt{sT} +\min\{ q, T\})$ and $\tilde O(\sqrt{sdT})$, with sparsity $s$, feature dimension $d$, total time horizon $T$, and $q$ that is adaptive to the reward dependence pattern. Complementing upper bounds, we also show that learning over a single trajectory brings inherent challenges: While the dependence pattern and arm parameters form a rank-1 matrix, circulant matrices are not isometric over rank-1 manifolds and sample complexity indeed benefits from the sparse reward dependence structure. Our results necessitate a new analysis to address long-range temporal dependencies across data and avoid polynomial dependence on the reward horizon $h$. Specifically, we utilize connections to the restricted isometry property of circulant matrices formed by dependent sub-Gaussian vectors and establish new guarantees that are also of independent interest.

Ruizhe Shi, Minhak Song, Runlong Zhou et al. · tsinghua

We present a fine-grained theoretical analysis of the performance gap between reinforcement learning from human feedback (RLHF) and direct preference optimization (DPO) under a representation gap. Our study decomposes this gap into two sources: an explicit representation gap under exact optimization and an implicit representation gap under finite samples. In the exact optimization setting, we characterize how the relative capacities of the reward and policy model classes influence the final policy qualities. We show that RLHF, DPO, or online DPO can outperform one another depending on type of model mis-specifications. Notably, online DPO can outperform both RLHF and standard DPO when the reward and policy model classes are isomorphic and both mis-specified. In the approximate optimization setting, we provide a concrete construction where the ground-truth reward is implicitly sparse and show that RLHF requires significantly fewer samples than DPO to recover an effective reward model, highlighting a statistical advantage of two-stage learning. Together, these results provide a comprehensive understanding of the performance gap between RLHF and DPO under various settings, and offer practical insights into when each method is preferred.

Sarah Dean, Mihaela Curmei, Lillian J. Ratliff et al.

Numerous online services are data-driven: the behavior of users affects the system's parameters, and the system's parameters affect the users' experience of the service, which in turn affects the way users may interact with the system. For example, people may choose to use a service only for tasks that already works well, or they may choose to switch to a different service. These adaptations influence the ability of a system to learn about a population of users and tasks in order to improve its performance broadly. In this work, we analyze a class of such dynamics -- where users allocate their participation amongst services to reduce the individual risk they experience, and services update their model parameters to reduce the service's risk on their current user population. We refer to these dynamics as \emph{risk-reducing}, which cover a broad class of common model updates including gradient descent and multiplicative weights. For this general class of dynamics, we show that asymptotically stable equilibria are always segmented, with sub-populations allocated to a single learner. Under mild assumptions, the utilitarian social optimum is a stable equilibrium. In contrast to previous work, which shows that repeated risk minimization can result in (Hashimoto et al., 2018; Miller et al., 2021), we find that repeated myopic updates with multiple learners lead to better outcomes. We illustrate the phenomena via a simulated example initialized from real data.

Haozhe Jiang, Qiwen Cui, Zhihan Xiong et al.

This paper investigates when one can efficiently recover an approximate Nash Equilibrium (NE) in offline congestion games. The existing dataset coverage assumption in offline general-sum games inevitably incurs a dependency on the number of actions, which can be exponentially large in congestion games. We consider three different types of feedback with decreasing revealed information. Starting from the facility-level (a.k.a., semi-bandit) feedback, we propose a novel one-unit deviation coverage condition and give a pessimism-type algorithm that can recover an approximate NE. For the agent-level (a.k.a., bandit) feedback setting, interestingly, we show the one-unit deviation coverage condition is not sufficient. On the other hand, we convert the game to multi-agent linear bandits and show that with a generalized data coverage assumption in offline linear bandits, we can efficiently recover the approximate NE. Lastly, we consider a novel type of feedback, the game-level feedback where only the total reward from all agents is revealed. Again, we show the coverage assumption for the agent-level feedback setting is insufficient in the game-level feedback setting, and with a stronger version of the data coverage assumption for linear bandits, we can recover an approximate NE. Together, our results constitute the first study of offline congestion games and imply formal separations between different types of feedback.

Zhihan Xiong, Romain Camilleri, Maryam Fazel et al.

We investigate the fixed-budget best-arm identification (BAI) problem for linear bandits in a potentially non-stationary environment. Given a finite arm set $\mathcal{X}\subset\mathbb{R}^d$, a fixed budget $T$, and an unpredictable sequence of parameters $\left\lbraceθ_t\right\rbrace_{t=1}^{T}$, an algorithm will aim to correctly identify the best arm $x^* := \arg\max_{x\in\mathcal{X}}x^\top\sum_{t=1}^{T}θ_t$ with probability as high as possible. Prior work has addressed the stationary setting where $θ_t = θ_1$ for all $t$ and demonstrated that the error probability decreases as $\exp(-T /ρ^*)$ for a problem-dependent constant $ρ^*$. But in many real-world $A/B/n$ multivariate testing scenarios that motivate our work, the environment is non-stationary and an algorithm expecting a stationary setting can easily fail. For robust identification, it is well-known that if arms are chosen randomly and non-adaptively from a G-optimal design over $\mathcal{X}$ at each time then the error probability decreases as $\exp(-TΔ^2_{(1)}/d)$, where $Δ_{(1)} = \min_{x \neq x^*} (x^* - x)^\top \frac{1}{T}\sum_{t=1}^T θ_t$. As there exist environments where $Δ_{(1)}^2/ d \ll 1/ ρ^*$, we are motivated to propose a novel algorithm $\mathsf{P1}$-$\mathsf{RAGE}$ that aims to obtain the best of both worlds: robustness to non-stationarity and fast rates of identification in benign settings. We characterize the error probability of $\mathsf{P1}$-$\mathsf{RAGE}$ and demonstrate empirically that the algorithm indeed never performs worse than G-optimal design but compares favorably to the best algorithms in the stationary setting.

Shulun Chen, Runlong Zhou, Zihan Zhang et al. · tsinghua

Aligning large language models (LLMs) with human preferences has proven effective for enhancing model capabilities, yet standard preference modeling using the Bradley-Terry model assumes transitivity, overlooking the inherent complexity of human population preferences. Nash learning from human feedback (NLHF) addresses this by framing non-transitive preferences as a two-player zero-sum game, where alignment reduces to finding the Nash equilibrium (NE). However, existing algorithms typically rely on regularization, incurring unavoidable bias when computing the duality gap in the original game. In this work, we provide the first convergence guarantee for Optimistic Multiplicative Weights Update ($\mathtt{OMWU}$) in NLHF, showing that it achieves last-iterate linear convergence after a burn-in phase whenever an NE with full support exists, with an instance-dependent linear convergence rate to the original NE, measured by duality gaps. Compared to prior results in Wei et al. (2020), we do not require the assumption of NE uniqueness. Our analysis identifies a novel marginal convergence behavior, where the probability of rarely played actions grows exponentially from exponentially small values, enabling exponentially better dependence on instance-dependent constants than prior results. Experiments corroborate the theoretical strengths of $\mathtt{OMWU}$ in both tabular and neural policy classes, demonstrating its potential for LLM applications.

Avinandan Bose, Shuyue Stella Li, Faeze Brahman et al.

Cold-start personalization requires inferring user preferences through interaction when no user-specific historical data is available. The core challenge is a routing problem: each task admits dozens of preference dimensions, yet individual users care about only a few, and which ones matter depends on who is asking. With a limited question budget, asking without structure will miss the dimensions that matter. Reinforcement learning is the natural formulation, but in multi-turn settings its terminal reward fails to exploit the factored, per-criterion structure of preference data, and in practice learned policies collapse to static question sequences that ignore user responses. We propose decomposing cold-start elicitation into offline structure learning and online Bayesian inference. Pep (Preference Elicitation with Priors) learns a structured world model of preference correlations offline from complete profiles, then performs training-free Bayesian inference online to select informative questions and predict complete preference profiles, including dimensions never asked about. The framework is modular across downstream solvers and requires only simple belief models. Across medical, mathematical, social, and commonsense reasoning, Pep achieves 80.8% alignment between generated responses and users' stated preferences versus 68.5% for RL, with 3-5x fewer interactions. When two users give different answers to the same question, Pep changes its follow-up 39-62% of the time versus 0-28% for RL. It does so with ~10K parameters versus 8B for RL, showing that the bottleneck in cold-start elicitation is the capability to exploit the factored structure of preference data.

Begoña García Malaxechebarría, Courtney Paquette, Maryam Fazel et al.

Understanding the behavior of stochastic gradient methods is a central problem in modern machine learning. Recent work has highlighted diagonal linear networks as a simplified yet expressive setting for analyzing the optimization and generalization properties of neural models. In this work, we show that in the high-dimensional regime, stochastic gradient descent on diagonal linear networks is well-approximated by continuous dynamics governed by a stochastic differential equation (SDE), which explicitly decouples the drift from the gradient noise. We further derive a deterministic partial differential equation whose solution propagates the relevant state of the iterates and characterizes the time evolution of a broad class of observable statistics, including the risk, curvature, and other metrics for optimality. Finally, we show that, under a suitable parametrization, the stochastic dynamics are globally well posed and converge exponentially fast to zero risk with high probability, yielding a fully explicit non-asymptotic description of their long-time behavior. Numerical simulations corroborate our theoretical findings.

Libin Zhu, Damek Davis, Dmitriy Drusvyatskiy et al.

We study a prototypical situation when a learned predictor can discover useful low-dimensional structure in data, while using fewer samples than are needed for accurate prediction. Specifically, we consider the problem of recovering a multi-index polynomial $f^*(x)=h(Ux)$, with $U\in\mathbb{R}^{r\times d}$ and $r\ll d$, from finitely many data/label pairs. Importantly, the target function depends on input $x$ only through the projection onto an unknown $r$-dimensional central subspace. The algorithm we analyze is appealingly simple: fit kernel ridge regression (KRR) to the data and compute the Average Gradient Outer Product (AGOP) from the fitted predictor. Our main results show that under reasonable assumptions the top $r$-dimensional eigenspace of AGOP provably recovers the central subspace, even in regimes when the prediction error remains large. Specifically, if the target function $f^*$ has degree $p^*$, it is known that $n\asymp d^{p^*}$ samples are necessary for KRR to achieve accurate prediction. In contrast, we show that if a low degree $p$ component of $f^*$ already carries all relevant directions for prediction, subspace recovery occurs in the much lower sample regime $n\asymp d^{p+δ}$ for any $δ\in(0,1)$. Our results thus demonstrate a separation between prediction and representation, and provide an explanation for why iterative kernel methods such as Recursive Feature Machines (RFM) can be sample-efficient in practice.

Minrui Luo, Weihang Xu, Xiang Gao et al.

Gradient descent dynamics on the deep matrix factorization problem is extensively studied as a simplified theoretical model for deep neural networks. Although the convergence theory for two-layer matrix factorization is well-established, no global convergence guarantee for general deep matrix factorization under random initialization has been established to date. To address this gap, we provide a polynomial-time global convergence guarantee for randomly initialized gradient descent on four-layer matrix factorization, given certain conditions on the target matrix and a standard balanced regularization term. Our analysis employs new techniques to show saddle-avoidance properties of gradient decent dynamics, and extends previous theories to characterize the change in eigenvalues of layer weights.

Avinandan Bose, Laurent Lessard, Maryam Fazel et al.

The rise of foundation models fine-tuned on human feedback from potentially untrusted users has increased the risk of adversarial data poisoning, necessitating the study of robustness of learning algorithms against such attacks. Existing research on provable certified robustness against data poisoning attacks primarily focuses on certifying robustness for static adversaries who modify a fraction of the dataset used to train the model before the training algorithm is applied. In practice, particularly when learning from human feedback in an online sense, adversaries can observe and react to the learning process and inject poisoned samples that optimize adversarial objectives better than when they are restricted to poisoning a static dataset once, before the learning algorithm is applied. Indeed, it has been shown in prior work that online dynamic adversaries can be significantly more powerful than static ones. We present a novel framework for computing certified bounds on the impact of dynamic poisoning, and use these certificates to design robust learning algorithms. We give an illustration of the framework for the mean estimation and binary classification problems and outline directions for extending this in further work. The code to implement our certificates and replicate our results is available at https://github.com/Avinandan22/Certified-Robustness.

Leo Maynard-Zhang, Zhihan Xiong, Kevin Jamieson et al.

We study the fixed-budget best-arm identification (BAI) problem in non-stationary linear bandits. Concretely, given a fixed time budget $T\in \mathbb{N}$, finite arm set $\mathcal{X} \subset \mathbb{R}^d$, and a potentially adversarial sequence of unknown parameters $\lbrace θ_t\rbrace_{t=1}^{T}$ (hence non-stationary), a learner aims to identify the arm with the largest cumulative reward $x_* = \arg\max_{x \in \mathcal{X}} x^\top\sum_{t=1}^T θ_t$ with high probability. In this setting, it is well-known that uniformly sampling arms from the G-optimal design yields a minimax-optimal error probability of $\exp\left(-Θ\left(T / H_{G}\right)\right)$, where $H_{G}$ scales proportionally with the dimension $d$. However, this notion of complexity is overly pessimistic, as it is derived from a lower bound in which the arm set consists only of the standard basis vectors, thus masking any potential advantages arising from arm sets with richer geometric structure. To address this, we establish an arm-set-dependent lower bound that, in contrast, holds for any arm set. Motivated by the ideas underlying our lower bound, we propose the Adjacent-optimal design, a specialization of the well-known $\mathcal{X}\mathcal{Y}$-optimal design, and develop the $\textsf{Adjacent-BAI}$ algorithm. We prove that the error probability of $\textsf{Adjacent-BAI}$ matches our lower bound up to constants, verifying the tightness of our lower bound, and establishing the arm-set-dependent complexity of this setting.

Avinandan Bose, Simon Shaolei Du, Maryam Fazel

We study the problem of representation transfer in offline Reinforcement Learning (RL), where a learner has access to episodic data from a number of source tasks collected a priori, and aims to learn a shared representation to be used in finding a good policy for a target task. Unlike in online RL where the agent interacts with the environment while learning a policy, in the offline setting there cannot be such interactions in either the source tasks or the target task; thus multi-task offline RL can suffer from incomplete coverage. We propose an algorithm to compute pointwise uncertainty measures for the learnt representation, and establish a data-dependent upper bound for the suboptimality of the learnt policy for the target task. Our algorithm leverages the collective exploration done by source tasks to mitigate poor coverage at some points by a few tasks, thus overcoming the limitation of needing uniformly good coverage for a meaningful transfer by existing offline algorithms. We complement our theoretical results with empirical evaluation on a rich-observation MDP which requires many samples for complete coverage. Our findings illustrate the benefits of penalizing and quantifying the uncertainty in the learnt representation.

Runlong Zhou, Maryam Fazel, Simon S. Du · tsinghua

Reinforcement learning from human feedback (RLHF) has become essential for improving language model capabilities, but traditional approaches rely on the assumption that human preferences follow a transitive Bradley-Terry model. This assumption fails to capture the non-transitive nature of populational human preferences. Nash learning from human feedback (NLHF), targeting non-transitive preferences, is a problem of computing the Nash equilibrium (NE) of the two-player constant-sum game defined by the human preference. We introduce Extragradient preference optimization (EGPO), a novel algorithm for NLHF achieving last-iterate linear convergence to the NE of KL-regularized games and polynomial convergence to the NE of original games, while being robust to noise. Unlike previous approaches that rely on nested optimization, we derive an equivalent implementation using gradients of an online variant of the identity preference optimization (IPO) loss, enabling more faithful implementation for neural networks. Our empirical evaluations demonstrate EGPO's superior performance over baseline methods when training for the same number of epochs, as measured by pairwise win-rates using the ground truth preference. These results validate both the theoretical strengths and practical advantages of EGPO for language model alignment with non-transitive human preferences.

Avinandan Bose, Zhihan Xiong, Yuejie Chi et al.

Personalizing large language models (LLMs) to accommodate diverse user preferences is essential for enhancing alignment and user satisfaction. Traditional reinforcement learning from human feedback (RLHF) approaches often rely on monolithic value representations, limiting their ability to adapt to individual preferences. We introduce a novel framework that leverages low-rank preference modeling to efficiently learn and generalize user-specific reward functions. By representing reward functions in a low-dimensional subspace and modeling individual preferences as weighted combinations of shared basis functions, our approach avoids rigid user categorization while enabling scalability and few-shot adaptation. We validate our method on multiple preference datasets, demonstrating superior generalization to unseen users and improved accuracy in preference prediction tasks.

Yahya Sattar, Yassir Jedra, Maryam Fazel et al.

We consider the problem of learning a realization of a partially observed bilinear dynamical system (BLDS) from noisy input-output data. Given a single trajectory of input-output samples, we provide a finite time analysis for learning the system's Markov-like parameters, from which a balanced realization of the bilinear system can be obtained. Our bilinear system identification algorithm learns the system's Markov-like parameters by regressing the outputs to highly correlated, nonlinear, and heavy-tailed covariates. Moreover, the stability of BLDS depends on the sequence of inputs used to excite the system. These properties, unique to partially observed bilinear dynamical systems, pose significant challenges to the analysis of our algorithm for learning the unknown dynamics. We address these challenges and provide high probability error bounds on our identification algorithm under a uniform stability assumption. Our analysis provides insights into system theoretic quantities that affect learning accuracy and sample complexity. Lastly, we perform numerical experiments with synthetic data to reinforce these insights.

Avinandan Bose, Mihaela Curmei, Daniel L. Jiang et al.

This paper investigates ML systems serving a group of users, with multiple models/services, each aimed at specializing to a sub-group of users. We consider settings where upon deploying a set of services, users choose the one minimizing their personal losses and the learner iteratively learns by interacting with diverse users. Prior research shows that the outcomes of learning dynamics, which comprise both the services' adjustments and users' service selections, hinge significantly on the initialization. However, finding good initializations faces two main challenges: (i) Bandit feedback: Typically, data on user preferences are not available before deploying services and observing user behavior; (ii) Suboptimal local solutions: The total loss landscape (i.e., the sum of loss functions across all users and services) is not convex and gradient-based algorithms can get stuck in poor local minima. We address these challenges with a randomized algorithm to adaptively select a minimal set of users for data collection in order to initialize a set of services. Under mild assumptions on the loss functions, we prove that our initialization leads to a total loss within a factor of the globally optimal total loss with complete user preference data}, and this factor scales logarithmically in the number of services. This result is a generalization of the well-known $k$-means++ guarantee to a broad problem class, which is also of independent interest. The theory is complemented by experiments on real as well as semi-synthetic datasets.

Libin Zhu, Damek Davis, Dmitriy Drusvyatskiy et al.

The impressive practical performance of neural networks is often attributed to their ability to learn low-dimensional data representations and hierarchical structure directly from data. In this work, we argue that these two phenomena are not unique to neural networks, and can be elicited from classical kernel methods. Namely, we show that the derivative of the kernel predictor can detect the influential coordinates with low sample complexity. Moreover, by iteratively using the derivatives to reweight the data and retrain kernel machines, one is able to efficiently learn hierarchical polynomials with finite leap complexity. Numerical experiments illustrate the developed theory.

Yingcong Li, Davoud Ataee Tarzanagh, Ankit Singh Rawat et al.

Linear attention methods offer a compelling alternative to softmax attention due to their efficiency in recurrent decoding. Recent research has focused on enhancing standard linear attention by incorporating gating while retaining its computational benefits. Such Gated Linear Attention (GLA) architectures include competitive models such as Mamba and RWKV. In this work, we investigate the in-context learning capabilities of the GLA model and make the following contributions. We show that a multilayer GLA can implement a general class of Weighted Preconditioned Gradient Descent (WPGD) algorithms with data-dependent weights. These weights are induced by the gating mechanism and the input, enabling the model to control the contribution of individual tokens to prediction. To further understand the mechanics of this weighting, we introduce a novel data model with multitask prompts and characterize the optimization landscape of learning a WPGD algorithm. Under mild conditions, we establish the existence and uniqueness (up to scaling) of a global minimum, corresponding to a unique WPGD solution. Finally, we translate these findings to explore the optimization landscape of GLA and shed light on how gating facilitates context-aware learning and when it is provably better than vanilla linear attention.

Qiwen Cui, Maryam Fazel, Simon S. Du

In multiplayer games, self-interested behavior among the players can harm the social welfare. Tax mechanisms are a common method to alleviate this issue and induce socially optimal behavior. In this work, we take the initial step of learning the optimal tax that can maximize social welfare with limited feedback in congestion games. We propose a new type of feedback named \emph{equilibrium feedback}, where the tax designer can only observe the Nash equilibrium after deploying a tax plan. Existing algorithms are not applicable due to the exponentially large tax function space, nonexistence of the gradient, and nonconvexity of the objective. To tackle these challenges, we design a computationally efficient algorithm that leverages several novel components: (1) a piece-wise linear tax to approximate the optimal tax; (2) extra linear terms to guarantee a strongly convex potential function; (3) an efficient subroutine to find the exploratory tax that can provide critical information about the game. The algorithm can find an $ε$-optimal tax with $O(βF^2/ε)$ sample complexity, where $β$ is the smoothness of the cost function and $F$ is the number of facilities.

Avinandan Bose, Zhihan Xiong, Aadirupa Saha et al.

Reinforcement Learning from Human Feedback (RLHF) is currently the leading approach for aligning large language models with human preferences. Typically, these models rely on extensive offline preference datasets for training. However, offline algorithms impose strict concentrability requirements, which are often difficult to satisfy. On the other hand, while online algorithms can avoid the concentrability issue, pure online exploration could be expensive due to the active preference query cost and real-time implementation overhead. In this paper, we propose a novel approach: Hybrid Preference Optimization (HPO) which combines online exploration with existing offline preferences by relaxing the stringent concentrability conditions for offline exploration, as well as significantly improving the sample efficiency for its online counterpart. We give the first provably optimal theoretical bound for Hybrid RLHF with preference feedback, providing sample complexity bounds for policy optimization with matching lower bounds. Our results yield improved sample efficiency of hybrid RLHF over pure offline and online exploration.

Shuyue Stella Li, Avinandan Bose, Faeze Brahman et al.

Current large language model (LLM) development treats task-solving and preference alignment as separate challenges, optimizing first for objective correctness, then for alignment to aggregated human preferences. This paradigm fails in human-facing applications where solving a problem correctly is insufficient if the response mismatches the user's needs. This challenge intensifies in just-in-time scenarios where no prior user interaction history exists due to cold-start conditions or privacy constraints. LLMs need to identify what they don't know about user preferences, strategically elicit preference values through questioning, then adapt their reasoning processes and responses accordingly -- a complicated chain of cognitive processes which we term personalized reasoning. We introduce PREFDISCO, an evaluation methodology that transforms static benchmarks into interactive personalization tasks using psychologically-grounded personas with sparse preferences. Our framework creates scenarios where identical questions require different reasoning chains depending on user context, as optimal explanation approaches vary by individual expertise and preferences while maintaining factual accuracy. Evaluation of 21 frontier models across 10 tasks reveals 29.0% of naive personalization attempts produce worse preference alignment than generic responses, yet generic responses also fail to serve individual user needs effectively. These findings suggest personalized reasoning requires dedicated development rather than emerging naturally. PREFDISCO establishes personalized reasoning as a measurable research frontier and reveals fundamental limitations in current LLMs' interactive capabilities, providing a foundation for developing systems that can adapt to individual users in education, healthcare, and technical domains where personalization is critical.

Yahya Sattar, Sunmook Choi, Yassir Jedra et al.

We consider the problem of controlling a linear dynamical system from bilinear observations with minimal quadratic cost. Despite the similarity of this problem to standard linear quadratic Gaussian (LQG) control, we show that when the observation model is bilinear, neither does the Separation Principle hold, nor is the optimal controller affine in the estimated state. Moreover, the cost-to-go is non-convex in the control input. Hence, finding an analytical expression for the optimal feedback controller is difficult in general. Under certain settings, we show that the standard LQG controller locally maximizes the cost instead of minimizing it. Furthermore, the optimal controllers (derived analytically) are not unique and are nonlinear in the estimated state. We also introduce a notion of input-dependent observability and derive conditions under which the Kalman filter covariance remains bounded. We illustrate our theoretical results through numerical experiments in multiple synthetic settings.

Yiran Zhang, Weihang Xu, Mo Zhou et al.

Score matching has become a central training objective in modern generative modeling, particularly in diffusion models, where it is used to learn high-dimensional data distributions through the estimation of score functions. Despite its empirical success, the theoretical understanding of the optimization behavior of score matching, particularly in over-parameterized regimes, remains limited. In this work, we study gradient descent for training over-parameterized models to learn a single Gaussian distribution. Specifically, we use a student model with $n$ learnable parameters and train it on data generated from a single ground-truth Gaussian using the population score matching objective. We analyze the optimization dynamics under multiple regimes. When the noise scale is sufficiently large, we prove a global convergence result for gradient descent. In the low-noise regime, we identify the existence of a stationary point, highlighting the difficulty of proving global convergence in this case. Nevertheless, we show convergence under certain initialization conditions: when the parameters are initialized to be exponentially small, gradient descent ensures convergence of all parameters to the ground truth. We further prove that without the exponentially small initialization, the parameters may not converge to the ground truth. Finally, we consider the case where parameters are randomly initialized from a Gaussian distribution far from the ground truth. We prove that, with high probability, only one parameter converges while the others diverge, yet the loss still converges to zero with a $1/τ$ rate, where $τ$ is the number of iterations. We also establish a nearly matching lower bound on the convergence rate in this regime. This is the first work to establish global convergence guarantees for Gaussian mixtures with at least three components under the score matching framework.

Sunmook Choi, Yahya Sattar, Yassir Jedra et al.

We study a nonstationary bandit problem where rewards depend on both actions and latent states, the latter governed by unknown linear dynamics. Crucially, the state dynamics also depend on the actions, resulting in tension between short-term and long-term rewards. We propose an explore-then-commit algorithm for a finite horizon $T$. During the exploration phase, random Rademacher actions enable estimation of the Markov parameters of the linear dynamics, which characterize the action-reward relationship. In the commit phase, the algorithm uses the estimated parameters to design an optimized action sequence for long-term reward. Our proposed algorithm achieves $\tilde{\mathcal{O}}(T^{2/3})$ regret. Our analysis handles two key challenges: learning from temporally correlated rewards, and designing action sequences with optimal long-term reward. We address the first challenge by providing near-optimal sample complexity and error bounds for system identification using bilinear rewards. We address the second challenge by proving an equivalence with indefinite quadratic optimization over a hypercube, a known NP-hard problem. We provide a sub-optimality guarantee for this problem, enabling our regret upper bound. Lastly, we propose a semidefinite relaxation with Goemans-Williamson rounding as a practical approach.

Weihang Xu, Maryam Fazel, Simon S. Du

We study the gradient Expectation-Maximization (EM) algorithm for Gaussian Mixture Models (GMM) in the over-parameterized setting, where a general GMM with $n>1$ components learns from data that are generated by a single ground truth Gaussian distribution. While results for the special case of 2-Gaussian mixtures are well-known, a general global convergence analysis for arbitrary $n$ remains unresolved and faces several new technical barriers since the convergence becomes sub-linear and non-monotonic. To address these challenges, we construct a novel likelihood-based convergence analysis framework and rigorously prove that gradient EM converges globally with a sublinear rate $O(1/\sqrt{t})$. This is the first global convergence result for Gaussian mixtures with more than $2$ components. The sublinear convergence rate is due to the algorithmic nature of learning over-parameterized GMM with gradient EM. We also identify a new emerging technical challenge for learning general over-parameterized GMM: the existence of bad local regions that can trap gradient EM for an exponential number of steps.

Omid Sadeghi, Maryam Fazel

We study a generalization of the online binary prediction with expert advice framework where at each round, the learner is allowed to pick $m\geq 1$ experts from a pool of $K$ experts and the overall utility is a modular or submodular function of the chosen experts. We focus on the setting in which experts act strategically and aim to maximize their influence on the algorithm's predictions by potentially misreporting their beliefs about the events. Among others, this setting finds applications in forecasting competitions where the learner seeks not only to make predictions by aggregating different forecasters but also to rank them according to their relative performance. Our goal is to design algorithms that satisfy the following two requirements: 1) $\textit{Incentive-compatible}$: Incentivize the experts to report their beliefs truthfully, and 2) $\textit{No-regret}$: Achieve sublinear regret with respect to the true beliefs of the best fixed set of $m$ experts in hindsight. Prior works have studied this framework when $m=1$ and provided incentive-compatible no-regret algorithms for the problem. We first show that a simple reduction of our problem to the $m=1$ setting is neither efficient nor effective. Then, we provide algorithms that utilize the specific structure of the utility functions to achieve the two desired goals.

Yue Sun, Samet Oymak, Maryam Fazel

This paper studies the problem of identifying low-order linear systems via Hankel nuclear norm regularization. Hankel regularization encourages the low-rankness of the Hankel matrix, which maps to the low-orderness of the system. We provide novel statistical analysis for this regularization and carefully contrast it with the unregularized ordinary least-squares (OLS) estimator. Our analysis leads to new bounds on estimating the impulse response and the Hankel matrix associated with the linear system. We first design an input excitation and show that Hankel regularization enables one to recover the system using optimal number of observations in the true system order and achieve strong statistical estimation rates. Surprisingly, we demonstrate that the input design indeed matters, by showing that intuitive choices such as i.i.d. Gaussian input leads to provably sub-optimal sample complexity. To better understand the benefits of regularization, we also revisit the OLS estimator. Besides refining existing bounds, we experimentally identify when regularized approach improves over OLS: (1) For low-order systems with slow impulse-response decay, OLS method performs poorly in terms of sample complexity, (2) Hankel matrix returned by regularization has a more clear singular value gap that ease identification of the system order, (3) Hankel regularization is less sensitive to hyperparameter choice. Finally, we establish model selection guarantees through a joint train-validation procedure where we tune the regularization parameter for near-optimal estimation.

Yue Sun, Adhyyan Narang, Halil Ibrahim Gulluk et al.

An overarching goal in machine learning is to build a generalizable model with few samples. To this end, overparameterization has been the subject of immense interest to explain the generalization ability of deep nets even when the size of the dataset is smaller than that of the model. While the prior literature focuses on the classical supervised setting, this paper aims to demystify overparameterization for meta-learning. Here we have a sequence of linear-regression tasks and we ask: (1) Given earlier tasks, what is the optimal linear representation of features for a new downstream task? and (2) How many samples do we need to build this representation? This work shows that surprisingly, overparameterization arises as a natural answer to these fundamental meta-learning questions. Specifically, for (1), we first show that learning the optimal representation coincides with the problem of designing a task-aware regularization to promote inductive bias. We leverage this inductive bias to explain how the downstream task actually benefits from overparameterization, in contrast to prior works on few-shot learning. For (2), we develop a theory to explain how feature covariance can implicitly help reduce the sample complexity well below the degrees of freedom and lead to small estimation error. We then integrate these findings to obtain an overall performance guarantee for our meta-learning algorithm. Numerical experiments on real and synthetic data verify our insights on overparameterized meta-learning.

Adhyyan Narang, Evan Faulkner, Dmitriy Drusvyatskiy et al.

Learning problems commonly exhibit an interesting feedback mechanism wherein the population data reacts to competing decision makers' actions. This paper formulates a new game theoretic framework for this phenomenon, called "multi-player performative prediction". We focus on two distinct solution concepts, namely (i) performatively stable equilibria and (ii) Nash equilibria of the game. The latter equilibria are arguably more informative, but can be found efficiently only when the game is monotone. We show that under mild assumptions, the performatively stable equilibria can be found efficiently by a variety of algorithms, including repeated retraining and the repeated (stochastic) gradient method. We then establish transparent sufficient conditions for strong monotonicity of the game and use them to develop algorithms for finding Nash equilibria. We investigate derivative free methods and adaptive gradient algorithms wherein each player alternates between learning a parametric description of their distribution and gradient steps on the empirical risk. Synthetic and semi-synthetic numerical experiments illustrate the results.

Omid Sadeghi, Maryam Fazel

Continuous DR-submodular functions are a class of functions that satisfy the Diminishing Returns (DR) property, which implies that they are concave along non-negative directions. Existing works have studied monotone continuous DR-submodular maximization subject to a convex constraint and have proposed efficient algorithms with approximation guarantees. However, in many applications, e.g., computing the stability number of a graph and mean-field inference for probabilistic log-submodular models, the DR-submodular function has the additional property of being \emph{strongly} concave along non-negative directions that could be utilized for obtaining faster convergence rates. In this paper, we first introduce and characterize the class of \emph{strongly DR-submodular} functions and show how such a property implies strong concavity along non-negative directions. Then, we study $L$-smooth monotone strongly DR-submodular functions that have bounded curvature, and we show how to exploit such additional structure to obtain algorithms with improved approximation guarantees and faster convergence rates for the maximization problem. In particular, we propose the SDRFW algorithm that matches the provably optimal $1-\frac{c}{e}$ approximation ratio after only $\lceil\frac{L}μ\rceil$ iterations, where $c\in[0,1]$ and $μ\geq 0$ are the curvature and the strong DR-submodularity parameter. Furthermore, we study the Projected Gradient Ascent (PGA) method for this problem and provide a refined analysis of the algorithm with an improved $\frac{1}{1+c}$ approximation ratio and a linear convergence rate. Given that both algorithms require knowledge of the smoothness parameter $L$, we provide a \emph{novel} characterization of $L$ for DR-submodular functions showing that in many cases, computing $L$ could be formulated as a convex problem, i.e., a geometric program, that could be solved efficiently.

Romain Camilleri, Zhihan Xiong, Maryam Fazel et al.

This work considers the problem of selective-sampling for best-arm identification. Given a set of potential options $\mathcal{Z}\subset\mathbb{R}^d$, a learner aims to compute with probability greater than $1-δ$, $\arg\max_{z\in \mathcal{Z}} z^{\top}θ_{\ast}$ where $θ_{\ast}$ is unknown. At each time step, a potential measurement $x_t\in \mathcal{X}\subset\mathbb{R}^d$ is drawn IID and the learner can either choose to take the measurement, in which case they observe a noisy measurement of $x^{\top}θ_{\ast}$, or to abstain from taking the measurement and wait for a potentially more informative point to arrive in the stream. Hence the learner faces a fundamental trade-off between the number of labeled samples they take and when they have collected enough evidence to declare the best arm and stop sampling. The main results of this work precisely characterize this trade-off between labeled samples and stopping time and provide an algorithm that nearly-optimally achieves the minimal label complexity given a desired stopping time. In addition, we show that the optimal decision rule has a simple geometric form based on deciding whether a point is in an ellipse or not. Finally, our framework is general enough to capture binary classification improving upon previous works.

Omid Sadeghi, Prasanna Raut, Maryam Fazel

In this paper, we consider an online optimization problem over $T$ rounds where at each step $t\in[T]$, the algorithm chooses an action $x_t$ from the fixed convex and compact domain set $\mathcal{K}$. A utility function $f_t(\cdot)$ is then revealed and the algorithm receives the payoff $f_t(x_t)$. This problem has been previously studied under the assumption that the utilities are adversarially chosen monotone DR-submodular functions and $\mathcal{O}(\sqrt{T})$ regret bounds have been derived. We first characterize the class of strongly DR-submodular functions and then, we derive regret bounds for the following new online settings: $(1)$ $\{f_t\}_{t=1}^T$ are monotone strongly DR-submodular and chosen adversarially, $(2)$ $\{f_t\}_{t=1}^T$ are monotone submodular (while the average $\frac{1}{T}\sum_{t=1}^T f_t$ is strongly DR-submodular) and chosen by an adversary but they arrive in a uniformly random order, $(3)$ $\{f_t\}_{t=1}^T$ are drawn i.i.d. from some unknown distribution $f_t\sim \mathcal{D}$ where the expected function $f(\cdot)=\mathbb{E}_{f_t\sim\mathcal{D}}[f_t(\cdot)]$ is monotone DR-submodular. For $(1)$, we obtain the first logarithmic regret bounds. In terms of the second framework, we show that it is possible to obtain similar logarithmic bounds with high probability. Finally, for the i.i.d. model, we provide algorithms with $\tilde{\mathcal{O}}(\sqrt{T})$ stochastic regret bound, both in expectation and with high probability. Experimental results demonstrate that our algorithms outperform the previous techniques in the aforementioned three settings.

Zhihan Xiong, Ruoqi Shen, Qiwen Cui et al.

We study algorithms using randomized value functions for exploration in reinforcement learning. This type of algorithms enjoys appealing empirical performance. We show that when we use 1) a single random seed in each episode, and 2) a Bernstein-type magnitude of noise, we obtain a worst-case $\widetilde{O}\left(H\sqrt{SAT}\right)$ regret bound for episodic time-inhomogeneous Markov Decision Process where $S$ is the size of state space, $A$ is the size of action space, $H$ is the planning horizon and $T$ is the number of interactions. This bound polynomially improves all existing bounds for algorithms based on randomized value functions, and for the first time, matches the $Ω\left(H\sqrt{SAT}\right)$ lower bound up to logarithmic factors. Our result highlights that randomized exploration can be near-optimal, which was previously achieved only by optimistic algorithms. To achieve the desired result, we develop 1) a new clipping operation to ensure both the probability of being optimistic and the probability of being pessimistic are lower bounded by a constant, and 2) a new recursive formula for the absolute value of estimation errors to analyze the regret.

Halil Ibrahim Gulluk, Yue Sun, Samet Oymak et al.

Constructing good representations is critical for learning complex tasks in a sample efficient manner. In the context of meta-learning, representations can be constructed from common patterns of previously seen tasks so that a future task can be learned quickly. While recent works show the benefit of subspace-based representations, such results are limited to linear-regression tasks. This work explores a more general class of nonlinear tasks with applications ranging from binary classification, generalized linear models and neural nets. We prove that subspace-based representations can be learned in a sample-efficient manner and provably benefit future tasks in terms of sample complexity. Numerical results verify the theoretical predictions in classification and neural-network regression tasks.

Mitas Ray, Omid Sadeghi, Lillian J. Ratliff et al.

We study the problem of online resource allocation, where multiple customers arrive sequentially and the seller must irrevocably allocate resources to each incoming customer while also facing a procurement cost for the total allocation. Assuming resource procurement follows an a priori known marginally increasing cost function, the objective is to maximize the reward obtained from fulfilling the customers' requests sans the cumulative procurement cost. We analyze the competitive ratio of a primal-dual algorithm in this setting, and develop an optimization framework for synthesizing a surrogate function for the procurement cost function to be used by the algorithm, in order to improve the competitive ratio of the primal-dual algorithm. Our first design method focuses on polynomial procurement cost functions and uses the optimal surrogate function to provide a more refined bound than the state of the art. Our second design method uses quasiconvex optimization to find optimal design parameters for a general class of procurement cost functions. Numerical examples are used to illustrate the design techniques. We conclude by extending the analysis to devise a posted pricing mechanism in which the algorithm does not require the customers' preferences to be revealed.

Prasanna Sanjay Raut, Omid Sadeghi, Maryam Fazel

In this paper, we consider online continuous DR-submodular maximization with linear stochastic long-term constraints. Compared to the prior work on online submodular maximization, our setting introduces the extra complication of stochastic linear constraint functions that are i.i.d. generated at each round. To be precise, at step $t\in\{1,\dots,T\}$, a DR-submodular utility function $f_t(\cdot)$ and a constraint vector $p_t$, i.i.d. generated from an unknown distribution with mean $p$, are revealed after committing to an action $x_t$ and we aim to maximize the overall utility while the expected cumulative resource consumption $\sum_{t=1}^T \langle p,x_t\rangle$ is below a fixed budget $B_T$. Stochastic long-term constraints arise naturally in applications where there is a limited budget or resource available and resource consumption at each step is governed by stochastically time-varying environments. We propose the Online Lagrangian Frank-Wolfe (OLFW) algorithm to solve this class of online problems. We analyze the performance of the OLFW algorithm and we obtain sub-linear regret bounds as well as sub-linear cumulative constraint violation bounds, both in expectation and with high probability.

Omid Sadeghi, Maryam Fazel

In this paper, we study a class of online optimization problems with long-term budget constraints where the objective functions are not necessarily concave (nor convex) but they instead satisfy the Diminishing Returns (DR) property. Specifically, a sequence of monotone DR-submodular objective functions $\{f_t(x)\}_{t=1}^T$ and monotone linear budget functions $\{\langle p_t,x \rangle \}_{t=1}^T$ arrive over time and assuming a total targeted budget $B_T$, the goal is to choose points $x_t$ at each time $t\in\{1,\dots,T\}$, without knowing $f_t$ and $p_t$ on that step, to achieve sub-linear regret bound while the total budget violation $\sum_{t=1}^T \langle p_t,x_t \rangle -B_T$ is sub-linear as well. Prior work has shown that achieving sub-linear regret is impossible if the budget functions are chosen adversarially. Therefore, we modify the notion of regret by comparing the agent against a $(1-\frac{1}{e})$-approximation to the best fixed decision in hindsight which satisfies the budget constraint proportionally over any window of length $W$. We propose the Online Saddle Point Hybrid Gradient (OSPHG) algorithm to solve this class of online problems. For $W=T$, we recover the aforementioned impossibility result. However, when $W=o(T)$, we show that it is possible to obtain sub-linear bounds for both the $(1-\frac{1}{e})$-regret and the total budget violation.

Omid Sadeghi, Reza Eghbali, Maryam Fazel

In this paper, we study a certain class of online optimization problems, where the goal is to maximize a function that is not necessarily concave and satisfies the Diminishing Returns (DR) property under budget constraints. We analyze a primal-dual algorithm, called the Generalized Sequential algorithm, and we obtain the first bound on the competitive ratio of online monotone DR-submodular function maximization subject to linear packing constraints which matches the known tight bound in the special case of linear objective function.

Yue Sun, Nicolas Flammarion, Maryam Fazel

We consider minimizing a nonconvex, smooth function $f$ on a Riemannian manifold $\mathcal{M}$. We show that a perturbed version of Riemannian gradient descent algorithm converges to a second-order stationary point (and hence is able to escape saddle points on the manifold). The rate of convergence depends as $1/ε^2$ on the accuracy $ε$, which matches a rate known only for unconstrained smooth minimization. The convergence rate depends polylogarithmically on the manifold dimension $d$, hence is almost dimension-free. The rate also has a polynomial dependence on the parameters describing the curvature of the manifold and the smoothness of the function. While the unconstrained problem (Euclidean setting) is well-studied, our result is the first to prove such a rate for nonconvex, manifold-constrained problems.

Amin Jalali, Adel Javanmard, Maryam Fazel

Prior knowledge on properties of a target model often come as discrete or combinatorial descriptions. This work provides a unified computational framework for defining norms that promote such structures. More specifically, we develop associated tools for optimization involving such norms given only the orthogonal projection oracle onto the non-convex set of desired models. As an example, we study a norm, which we term the doubly-sparse norm, for promoting vectors with few nonzero entries taking only a few distinct values. We further discuss how the K-means algorithm can serve as the underlying projection oracle in this case and how it can be efficiently represented as a quadratically constrained quadratic program. Our motivation for the study of this norm is regularized regression in the presence of rare features which poses a challenge to various methods within high-dimensional statistics, and in machine learning in general. The proposed estimation procedure is designed to perform automatic feature selection and aggregation for which we develop statistical bounds. The bounds are general and offer a statistical framework for norm-based regularization. The bounds rely on novel geometric quantities on which we attempt to elaborate as well.