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 $\lbrace \theta_t\rbrace_{t=1}^{T}$ (hence non-stationary), a learner aims to identify the arm with the largest cumulative reward $x_* = \arg\max_{x \in \mathcal{X}} x^\top\sum_{t=1}^T \theta_t$ with high probability. In this setting, it is well-known that i.i.d. sampling arms from the G-optimal design yields a minimax-optimal error probability of $\exp\left(-\Theta\left(T / H_{G}\right)\right)$, 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 \textit{arm-set-dependent} lower bound that, in contrast, holds for any arm set. Motivated by the ideas underlying our lower bound, we propose the \textit{Adjacent-optimal design}, a specialization of the well-known $\mathcal{XY}$-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.

We study the problem of representational 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 in low-rank MDPs, 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.

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 problem and binary classification problems and outline directions for extending this in further work.

We consider gap-dependent regret bounds for episodic MDPs. We show that the Monotonic Value Propagation (MVP) algorithm (Zhang et al. [2024]) achieves a variance-aware gap-dependent regret bound of $$\tilde{O}\left(\left(\sum_{\Delta_h(s,a)>0} \frac{H^2 \log K \land \mathtt{Var}\_{\max}^{\textup{c}}}{\Delta_h(s,a)} +\sum_{\Delta_h(s,a)=0}\frac{ H^2 \land \mathtt{Var}\_{\max}^{\textup{c}}}{\Delta_{\mathrm{min}}} + SAH^4 (S \lor H) \right) \log K\right),$$ where $H$ is the planning horizon, $S$ is the number of states, $A$ is the number of actions, $K$ is the number of episodes, and $\tilde{O}$ hides $\mathsf{poly} \log (S, A, H, 1 / \Delta\_{\mathrm{min}}, 1 / \delta)$ terms. Here, $\Delta_h(s,a) =V_h^* (a) - Q_h^* (s, a)$ represents the suboptimality gap and $\Delta_{\mathrm{min}} := \min_{\Delta_h (s,a) > 0} \Delta_h(s,a)$. The term $\mathtt{Var}\_{\max}^{\textup{c}}$ denotes the maximum conditional total variance, calculated as the maximum over all $(\pi, h, s)$ tuples of the expected total variance under policy $\pi$ conditioned on trajectories visiting state $s$ at step $h$. $\mathtt{Var}\_{\max}^{\textup{c}}$ characterizes the maximum randomness encountered when learning any $(h, s)$ pair. Our result stems from a novel analysis of the weighted sum of the suboptimality gap and can be potentially adapted for other algorithms. To complement the study, we establish a lower bound of $$\Omega \left( \sum_{\Delta_h(s,a)>0} \frac{H^2 \land \mathtt{Var}\_{\max}^{\textup{c}}}{\Delta_h(s,a)}\cdot \log K\right),$$ demonstrating the necessity of dependence on $\mathtt{Var}\_{\max}^{\textup{c}}$ even when the maximum unconditional total variance (without conditioning on $(h, s)$) approaches zero.

A classical approach for solving discrete time nonlinear control on a finite horizon consists in repeatedly minimizing linear quadratic approximations of the original problem around current candidate solutions. While widely popular in many domains, such an approach has mainly been analyzed locally. We provide detailed convergence guarantees to stationary points as well as local linear convergence rates for the Iterative Linear Quadratic Regulator (ILQR) algorithm and its Differential Dynamic Programming (DDP) variant. For problems without costs on control variables, we observe that global convergence to minima can be ensured provided that the linearized discrete time dynamics are surjective, costs on the state variables are gradient dominated. We further detail quadratic local convergence when the costs are self-concordant. We show that surjectivity of the linearized dynamics hold for appropriate discretization schemes given the existence of a feedback linearization scheme. We present complexity bounds of algorithms based on linear quadratic approximations through the lens of generalized Gauss-Newton methods. Our analysis uncovers several convergence phases for regularized generalized Gauss-Newton algorithms.

Leading approaches to algorithmic fairness and policy-induced distribution shift are often misaligned with long-term objectives in sequential settings. We aim to correct these shortcomings by ensuring that both the objective and fairness constraints account for policy-induced distribution shift. First, we motivate this problem using an example in which individuals subject to algorithmic predictions modulate their willingness to participate with the policy maker. Fairness in this example is measured by the variance of group participation rates. Next, we develop a method for solving the resulting constrained, non-linear optimization problem and prove that this method converges to a fair, locally optimal policy given first-order information. Finally, we experimentally validate our claims in a semi-synthetic setting.

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 representation disparity and high overall loss with a single learner (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.

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 $\left\lbrace\theta_t\right\rbrace_{t=1}^{T}$, an algorithm will aim to correctly identify the best arm $x^* := \arg\max_{x\in\mathcal{X}}x^\top\sum_{t=1}^{T}\theta_t$ with probability as high as possible. Prior work has addressed the stationary setting where $\theta_t = \theta_1$ for all $t$ and demonstrated that the error probability decreases as $\exp(-T /\rho^*)$ for a problem-dependent constant $\rho^*$. 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 $\exp(-T\Delta^2_{(1)}/d)$, where $\Delta_{(1)} = \min_{x \neq x^*} (x^* - x)^\top \frac{1}{T}\sum_{t=1}^T \theta_t$. As there exist environments where $\Delta_{(1)}^2/ d \ll 1/ \rho^*$, we are motivated to propose a novel algorithm P1-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 P1-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.

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 $\widetilde{O}\left(\Delta^{1/4}T^{3/4}\right)$ regret when the degree of nonstationarity, as measured by total variation $\Delta$, is known, and $\widetilde{O}\left(\Delta^{1/5}T^{4/5}\right)$ regret when $\Delta$ 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.

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 initial conditions. However, finding good initial conditions faces two main challenges: (i) \emph{Bandit feedback:} Typically, data on user preferences are not available before deploying services and observing user behavior; (ii) \emph{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 \textit{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.

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 $\epsilon$-optimal tax with $O(\beta F^2/\epsilon)$ sample complexity, where $\beta$ is the smoothness of the cost function and $F$ is the number of facilities.

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.

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 improve the computational complexity, 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.

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) regimes and derive respective regret upper bounds O(d square-root(sT) +min(q, T) and O( square-root(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.

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) \emph{Incentive-compatible}: Incentivize the experts to report their beliefs truthfully, and 2) \emph{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.

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 are generally computationally difficult to find since they are solutions of non-monotone games. 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.

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 deploys a fixed decision repeatedly 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.

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 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.

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.

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 $\Omega\left(H\sqrt{SAT}\right)$ 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.

We present an algorithm for computing approximate ℓp Lewis weights to high precision. Given a full-rank A ∊ ℝm × n with m ≥ n and a scalar p > 2, our algorithm computes ∊-approximate ℓp Lewis weights of A in Õp(log(1/∊)) iterations; the cost of each iteration is linear in the input size plus the cost of computing the leverage scores of DA for diagonal D ∊ ℝm × m. Prior to our work, such a computational complexity was known only for p ∊ (0,4) [CP15], and combined with this result, our work yields the first polylogarithmic-depth polynomial-work algorithm for the problem of computing ℓp Lewis weights to high precision for all constant p > 0. An important consequence of this result is also the first polylogarithmic-depth polynomial-work algorithm for computing a nearly optimal self-concordant barrier for a polytope.

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. In particular, at each time step a DR-submodular utility function and a constraint vector, i.i.d. generated from an unknown distribution, are revealed after committing to an action and we aim to maximize the overall utility while the expected cumulative resource consumption is below a fixed budget. 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.

Numerous tasks in machine learning and artificial intelligence have been modeled as submodular maximization problems. These problems usually involve sensitive data about individuals, and in addition to maximizing the utility, privacy concerns should be considered. In this paper, we study the general framework of non-negative monotone submodular maximization subject to matroid or knapsack constraints in both offline and online settings. For the offline setting, we propose a differentially private $(1-\frac{\kappa}{e})$-approximation algorithm, where $\kappa\in[0,1]$ is the total curvature of the submodular set function, which improves upon prior works in terms of approximation guarantee and query complexity under the same privacy budget. In the online setting, we propose the first differentially private algorithm, and we specify the conditions under which the regret bound scales as $Ø(\sqrt{T})$, i.e., privacy could be ensured while maintaining the same regret bound as the optimal regret guarantee in the non-private setting.

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.

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 $1-\delta$, $\arg\max_{z\in \mathcal{Z}} z^{\top}\theta_{\ast}$ where $\theta_{\ast}$ 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 $x^{\top}\theta_{\ast}$, 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.

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.

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. In this online setting, a sequence of monotone DR-submodular objective functions and linear budget functions arrive over time and assuming a limited total budget, the goal is to take actions at each time, before observing the utility and budget function arriving at that round, to achieve sub-linear regret bound while the total budget violation is sub-linear as well. Prior work has shown that achieving sub-linear regret and total budget violation simultaneously is impossible if the utility and budget functions are chosen adversarially. Therefore, we modify the notion of regret by comparing the agent against 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, if $W$ is sub-linear in $T$, we show that it is possible to obtain sub-linear bounds for both the regret and the total budget violation.

In this paper, we consider an online optimization problem in which the reward functions are DR-submodular, and in addition to maximizing the total reward, the sequence of decisions must satisfy some convex constraints on average. Specifically, at each round $t\in\{1,\dots,T\}$, upon committing to an action $x_t$, a DR-submodular utility function $f_t(\cdot)$ and a convex constraint function $g_t(\cdot)$ are revealed, and the goal is to maximize the overall utility while ensuring the average of the constraint functions $\frac{1}{T}\sum_{t=1}^T g_t(x_t)$ is non-positive. Such cumulative constraints arise naturally in applications where the average resource consumption is required to remain below a prespecified threshold. We study this problem under an adversarial model and a stochastic model for the convex constraints, where the functions $g_t$ can vary arbitrarily or according to an i.i.d. process over time slots $t\in\{1,\dots,T\}$, respectively. We propose a single algorithm which achieves sub-linear (with respect to $T$) regret as well as sub-linear constraint violation bounds in both settings, without prior knowledge of the regime. Prior works have studied this problem in the special case of linear constraint functions. Our results not only improve upon the existing bounds under linear cumulative constraints, but also give the first sub-linear bounds for general convex long-term constraints.

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

Direct policy gradient methods for reinforcement learning and continuous control problems are a popular approach for a variety of reasons: 1) they are easy to implement without explicit knowledge of the underlying model, 2) they are an “end-to-end” approach, directly optimizing the performance metric of interest, 3) they inherently allow for richly parameterized policies. A notable drawback is that even in the most basic continuous control problem (that of linear quadratic regulators), these methods must solve a non-convex optimization problem, where little is understood about their efficiency from both computational and statistical perspectives. In contrast, system identification and model based planning in optimal control theory have a much more solid theoretical footing, where much is known with regards to their computational and statistical properties. This work bridges this gap showing that (model free) policy gradient methods globally converge to the optimal solution and are efficient (polynomially so in relevant problem dependent quantities) with regards to their sample and computational complexities.

We analyze an estimator based on the Bregman divergence for recovery of structured models from additive noise. The estimator can be seen as a regularized maximum likelihood estimator for an exponential family where the natural parameter is assumed to be structured. For all such Bregman denoising estimators, we provide an error bound for a natural associated error measure. Our error bound makes it possible to analyze a wide range of estimators, such as those in proximal denoising and inverse covariance matrix estimation, in a unified manner. In the case of proximal denoising, we exactly recover the existing tight normalized mean squared error bounds. In sparse precision matrix estimation, our bounds provide optimal scaling with interpretable constants in terms of the associated error measure.

Phaseless super-resolution is the problem of reconstructing a signal from its low-frequency Fourier magnitude measurements. It is the combination of two classic signal processing problems: phase retrieval and super-resolution. Due to the absence of phase and high-frequency measurements, additional information is required in order to be able to uniquely reconstruct the signal of interest. In this work, we use masks to introduce redundancy in the phaseless measurements. We develop an analysis framework for this setup, and use it to show that any super-resolution algorithm can be seamlessly extended to solve phaseless superresolution (up to a global phase), when measurements are obtained using a certain set of masks. In particular, we focus our attention on a robust semidefinite relaxation-based algorithm, and provide reconstruction guarantees. Numerical simulations complement our theoretical analysis.

Let X* be a n1× n2matrix with entries in F2and rank r1, n2) (often r ≪ min(n1, n2)). We consider the problem of reconstructing X* given only a subset of its entries. This problem has recently found numerous applications, most notably in network and index coding, where finding optimal linear codes (over some field Fq) can be reduced to finding the minimum rank completion of a matrix with a subset of revealed entries. The problem of matrix completion over reals also has many applications and in recent years several polynomial-time algorithms with provable recovery guarantees have been developed. However, to date, such algorithms do not exist in the finite-field case. We propose a linear algebraic algorithm, based on inferring low-weight relations among the rows and columns of X*, to attempt to complete X* given a random subset of its entries. We establish conditions on the row and column spaces of X* under which the algorithm runs in polynomial time (in the size of X*) and can successfully complete X* with high probability from a vanishing fraction of its entries. We then propose a linear programming-based extension of our basic algorithm, and evaluate it empirically.

The Stochastic Block Model (SBM) is a widely used random graph model for networks with communities. Despite the recent burst of interest in community detection under the SBM from statistical and computational points of view, there are still gaps in understanding the fundamental limits of recovery. In this paper, we consider the SBM in its full generality, where there is no restriction on the number and sizes of communities or how they grow with the number of nodes, as well as on the connectivity probabilities inside or across communities. For such stochastic block models, we provide guarantees for exact recovery via a semidefinite program as well as upper and lower bounds on SBM parameters for exact recoverability. Our results exploit the tradeoffs among the various parameters of heterogenous SBM and provide recovery guarantees for many new interesting SBM configurations.

Online optimization covers problems such as online resource allocation, online bipartite matching, adwords (a central problem in e-commerce and advertising), and adwords with separable concave returns. We analyze the worst case competitive ratio of two primal-dual algorithms for a class of online convex (conic) optimization problems that contains the previous examples as special cases defined on the positive orthant. We derive a sufficient condition on the objective function that guarantees a constant worst case competitive ratio (greater than or equal to $\frac{1}{2}$) for monotone objective functions. We provide new examples of online problems on the positive orthant % and the positive semidefinite cone that satisfy the sufficient condition. We show how smoothing can improve the competitive ratio of these algorithms, and in particular for separable functions, we show that the optimal smoothing can be derived by solving a convex optimization problem. This result allows us to directly optimize the competitive ratio bound over a class of smoothing functions, and hence design effective smoothing customized for a given cost function.

Graphical models provide a rich framework for summarizing the dependencies among variables. The graphical lasso approach attempts to learn the structure of a Gaussian graphical model (GGM) by maximizing the log likelihood of the data, subject to an l1 penalty on the elements of the inverse covariance matrix. Most algorithms for solving the graphical lasso problem do not scale to a very large number of variables. Furthermore, the learned network structure is hard to interpret. To overcome these challenges, we propose a novel GGM structure learning method that exploits the fact that for many real-world problems we have prior knowledge that certain edges are unlikely to be present. For example, in gene regulatory networks, a pair of genes that does not participate together in any of the cellular processes, typically referred to as pathways, is less likely to be connected. In computer vision applications in which each variable corresponds to a pixel, each variable is likely to be connected to the nearby variables. In this paper, we propose the pathway graphical lasso, which learns the structure of a GGM subject to pathway-based constraints. In order to solve this problem, we decompose the network into smaller parts, and use a message-passing algorithm in order to communicate among the subnetworks. Our algorithm has orders of magnitude improvement in run time compared to the state-of-the-art optimization methods for the graphical lasso problem that were modified to handle pathway-based constraints.

This paper describes a new exponential language model that decomposes the model parameters into one or more low-rank matrices that learn regularities in the training data and one or more sparse matrices that learn exceptions (e.g., keywords). The low-rank matrices induce continuous-space representations of words and histories. The sparse matrices learn multi-word lexical items and topic/domain idiosyncrasies. This model generalizes the standard ℓ1-regularized exponential language model, and has an efficient accelerated first-order training algorithm. Language modeling experiments show that the approach is useful in scenarios with limited training data, including low resource languages and domain adaptation.

Recovering structured models (e.g., sparse or group-sparse vectors, low-rank matrices) given a few linear observations have been well-studied recently. In various applications in signal processing and machine learning, the model of interest is structured in several ways, for example, a matrix that is simultaneously sparse and low rank. Often norms that promote the individual structures are known, and allow for recovery using an order-wise optimal number of measurements (e.g., 11 norm for sparsity, nuclear norm for matrix rank). Hence, it is reasonable to minimize a combination of such norms. We show that, surprisingly, using multiobjective optimization with these norms can do no better, orderwise, than exploiting only one of the structures, thus revealing a fundamental limitation in sample complexity. This result suggests that to fully exploit the multiple structures, we need an entirely new convex relaxation. Further, specializing our results to the case of sparse and low-rank matrices, we show that a nonconvex formulation recovers the model from very few measurements (on the order of the degrees of freedom), whereas the convex problem combining the 11 and nuclear norms requires many more measurements, illustrating a gap between the performance of the convex and nonconvex recovery problems. Our framework applies to arbitrary structure-inducing norms as well as to a wide range of measurement ensembles. This allows us to give sample complexity bounds for problems such as sparse phase retrieval and low-rank tensor completion.

Word usage is influenced by diverse factors, including topic, genre and various speaker/author characteristics. To characterize these aspects of language, we introduce the “Multi-Factor Sparse Plus Low Rank” exponential language model, which allows supervised joint training of arbitrary overlapping factor-specific model components. This flexible architecture has the advantage of being highly interpretable. The elements of sparse parameter matrices can be viewed as factor-dependent corrections (e.g. topic- or speaker-dependent phenomena). In topic modeling experiments on conversational telephone speech, we obtain modest perplexity reductions over an n-gram baseline and demonstrate topic-dependent keyword extraction that leads to a 13% (absolute) improvement in precision over TFIDF. We also show how keywords can be jointly learned for speakers, roles and topics in a study of Supreme Court oral arguments.

Random Access Compressed Sensing (RACS) is an efficient method for data gathering from a network of distributed sensors with limited resources. RACS relies on integrating random sensing with the communication architecture, and achieves overall efficiency in terms of the energy per bit of information successfully delivered. To address realistic deployment conditions, we consider data gathering over a fading and noisy communication channel. We provide a framework for system design under various fading conditions, and quantify the bandwidth and energy requirements of RACS in fading. We show that for most practical values of the signal to noise ratio, energy utilization is higher in a fading channel than it is in a non-fading channel, while the minimum required bandwidth is lower. Finally, we demonstrate the savings in the overall energy and the bandwidth requirements of RACS compared to a conventional TDMA scheme. We show that considerable gains in energy -on the order of 10 dB- are achievable, as well as a reduction in the required bandwidth, e.g., 2.5-fold decrease in the bandwidth for a network of 4000 nodes.

This work introduces a new maximum entropy language model that decomposes the model parameters into a low rank component that learns regularities in the training data and a sparse component that learns exceptions (e.g. multiword expressions). The low rank component corresponds to a continuous-space language model. This model generalizes the standard ℓ1regularized maximum entropy model, and has an efficient accelerated first-order training algorithm. In conversational speech language modeling experiments, we see perplexity reductions

In the human body tracking based on video sequences, the pose estimation of the upper/lower limbs is the most challenging task since the limbs possess most variations of motions and are easily occluded. In this work, we present a sophisticated scheme to track the human limbs. First, the tracking is formulated as a constrained optimization problem with multiple kernels. The color features of the upper/lower limbs are used as the control variables in the objective function. Moreover, the inequality constraints are imposed to control the angle between the arm/forearm or upper/lower legs during tracking. Finally, the gradient projection algorithm is adopted to solve the optimization problem with inequality constraints. The proposed scheme is implemented and experimented on HumanEva dataset and self-recorded video sequences including tracking of arm/forearm and upper/lower legs.

Random Access Compressed Sensing (RACS) is an efficient method for data telemetry from a network of distributed sensors deployed in a challenging field environment with limited resources. RACS relies on integrating sensing with the communication architecture, in order to achieve overall efficiency in terms of the energy per bit of information successfully delivered to the fusion center. Targeting realistic deployment conditions, we consider data gathering over a Ricean fading channel and in the presence of communication noise. We provide a framework for system design and study the energy and bandwidth requirements of the network. We then show that compared to a conventional TDMA network with ARQ, RACS achieves significant energy and bandwidth savings. For example, for a network of 4000 nodes, we observe 10 dB gain in the energy as well as a 2.5-fold reduction in the required bandwidth.

We consider estimation of multiple high-dimensional Gaussian graphical models corresponding to a single set of nodes under several distinct conditions. We assume that most aspects of the networks are shared, but that there are some structured differences between them. Specifically, the network differences are generated from node perturbations: a few nodes are perturbed across networks, and most or all edges stemming from such nodes differ between networks. This corresponds to a simple model for the mechanism underlying many cancers, in which the gene regulatory network is disrupted due to the aberrant activity of a few specific genes. We propose to solve this problem using the structured joint graphical lasso, a convex optimization problem that is based upon the use of a novel symmetric overlap norm penalty, which we solve using an alternating directions method of multipliers algorithm. Our proposal is illustrated on synthetic data and on an application to brain cancer gene expression data.

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 ℓ1and nuclear norm minimization as well as ℓpminimization with p <; 1. These algorithms are known to succeed if certain conditions on the measurement map are satisfied. Proofs for the recovery of matrices have so far been much more involved than in the vector case. In this paper, we show how several 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.

Inspired by the theory of compressed sensing and employing random channel access, we propose a distributed energy-efficient sensor network scheme denoted by Random Access Compressed Sensing (RACS). The proposed scheme is suitable for long-term deployment of large underwater networks, in which saving energy and bandwidth is of crucial importance. During each frame, a randomly chosen subset of nodes participate in the sensing process, then share the channel using random access. Due to the nature of random access, packets may collide at the fusion center. To account for the packet loss that occurs due to collisions, the network design employs the concept of sufficient sensing probability. With this probability, sufficiently many data packets - as required for field reconstruction based on compressed sensing - are to be received. The RACS scheme prolongs network life-time while employing a simple and distributed scheme which eliminates the need for scheduling.

Several language model smoothing techniques are available that are effective for a variety of tasks; however, training with small data sets is still difficult. This letter introduces the low rank language model, which uses a low rank tensor representation of joint probability distributions for parameter-tying and optimizes likelihood under a rank constraint. It obtains lower perplexity than standard smoothing techniques when the training set is small and also leads to perplexity reduction when used in domain adaptation via interpolation with a general, out-of-domain model.

The problem of minimizing the rank of a matrix subject to linear equality constraints arises in applications in machine learning, dimensionality reduction, and control theory, and is known to be NP-hard. A popular heuristic minimizes the nuclear norm (sum of the singular values) of the matrix instead of the rank, and was recently shown to give an exact solution in several scenarios. In this paper, we present a new analysis for this heuristic based on a property of the nullspace of the operator defining the constraints, called the spherical section property. We give conditions for the exact recovery of all matrices up to a certain rank, and show that these conditions hold with high probability for operators generated from random Gaussian ensembles. Our analysis provides simpler proofs than existing isometry-based methods, as well as robust recovery results when the matrix is not exactly low-rank.

The problem of recovering a low-rank matrix consistent with noisy linear measurements is a fundamental problem with applications in machine learning, statistics, and control. Reweighted trace minimization, which extends and improves upon the popular nuclear norm heuristic, has been used as an iterative heuristic for this problem. In this paper, we present theoretical guarantees for the reweighted trace heuristic. We quantify its improvement over nuclear norm minimization by proving tighter bounds on the recovery error for low-rank matrices with noisy measurements. Our analysis is based on the Restricted Isometry Property (RIP) and extends some recent results from Compressed Sensing. As a second contribution, we improve the existing RIP recovery results for the nuclear norm heuristic, and show that recovery happens under a weaker assumption on the RIP constants.

Building an audio-visual data corpus is one significant step in audio-visual research. One of the most challenging tasks in computer science is computer-aided speech therapy and language learning. Developing computer applications for training and rehabilitation of the handicapped and helping the hearing and speaking-impaired by facial speech synthesis are among the most helpful, state-of-the-art roles of computer technology in today's human-machine interacting systems. To date, there have been no audio-visual corpora in Persian language, in that it makes it difficult or even impossible for researchers to carry out studies in the area. This paper gives an indication of the collected Persian audio-visual data corpus. AVA is a comprehensive, systematic collection of both continuous speech and isolated spoken utterances in Persian language. The goal of this project is to facilitate audio-visual research in the language through this data corpus which is available upon request.

Power control mitigates interference and maintains required QoS levels in cellular wireless networks. An important class of distributed power control (DPC) was proposed by Foschini and Miljanic in 1993, with many variants developed since. Almost all related work focuses on the equilibrium and asymptotic convergence properties. However, for many applications transient behavior is more important. If a link's SIR drops below a critical threshold for too long, the connections over this link will be dropped, rendering the entire concept of equilibrium resource allocation meaningless. This paper proposes a systematic approach to the analysis of transient properties of DPC algorithms, in particular Foschini-Miljanic, based on tools from control theory. Analytically, we present a sufficient condition to ensure that after links reach their minimum SIR levels, their SIR requirements can be guaranteed for future time steps. Computationally, we pose this problem as verifying the invariance of certain regions in the SIR space, which for the basic DPC algorithm can be cast as a Linear Program (LP). Furthermore, using insights gained from the analysis, we propose a preliminary design framework for new iterative power control schemes.

Network Utility Maximization (NUM) has significantly extended the classical network flow problem and provided an emerging framework to design resource allocation algorithms such as TCP congestion control and to understand layering as optimization decomposition. We present a summary of very recent results in the theory and applications of NUM. We show new distributed algorithms that converge to the globally optimal rate allocation for NUM problems with nonconcave utility functions representing inelastic flows, with coupled utility functions representing interference effects or hybrid social-selfish utilities, and with rate-reliability tradeoff through adaptive channel coding in the physical layer. We conclude by discussing how do different decompositions of a generalized NUM problem correspond to different layering architectures.

In this letter, we propose a method to reduce the peak-to-mean-envelope-power ratio (PMEPR) of multicarrier signals by modifying the constellation. For M-ary phase-shift keying constellations, we minimize the maximum of the multicarrier signal over the sign and amplitude of each subcarrier. In order to find an efficient solution to the aforementioned nonconvex optimization problem, we present a suboptimal solution by first optimizing over the signs, and then optimizing over the amplitudes given the signs. We prove that the minimization of the maximum of a continuous multicarrier signal over the amplitude of each subcarrier can be written as a convex optimization problem with linear matrix inequality constraints. We also generalize the idea to other constellations such as 16-quadrature amplitude modulation. Simulation results show that by an average power increase of 0.21 dB, and not sending information over the sign of each subcarrier, PMEPR can be decreased by 5.1 dB for a system with 128 subcarriers.

In this paper, we propose a method to reduce the peak to mean envelope power ratio (PMEPR) of multicarrier signals by modifying the constellation. For MPSK constellations, we minimize the maximum of the multicarrier signal over the sign and amplitude of each subcarrier. In order to find an efficient solution to the aforementioned non-convex optimization problem, we present a suboptimal solution by first optimizing over the signs using the result of M. Sharif et al. (2003), and then optimizing over the amplitudes given the signs. We prove that the minimization of the maximum of a multicarrier signal over the amplitude of each subcarrier can be written as a convex optimization problem with linear matrix inequality constraints. We also generalize the idea to other constellations such as 16QAM. Simulation results show that by an average power increase of 0.21 db and not sending information over the sign of each subcarrier, PMEPR can be decreased by 5.1 db for a system with 128 subcarriers.