Multi-modal contrastive learning as a self-supervised representation learning technique has achieved great success in foundation model training, such as CLIP~\citep{radford2021learning}. In this paper, we study the theoretical properties of the learned representations from multi-modal contrastive learning beyond linear representations and specific data distributions. Our analysis reveals that, enabled by temperature optimization, multi-modal contrastive learning not only maximizes mutual information between modalities but also adapts to intrinsic dimensions of data, which can be much lower than user-specified dimensions for representation vectors. Experiments on both synthetic and real-world datasets demonstrate the ability of contrastive learning to learn low-dimensional and informative representations, bridging theoretical insights and practical performance.

We study nonparametric contextual bandits under batch constraints, where the expected reward for each action is modeled as a smooth function of covariates, and the policy updates are made at the end of each batch of observations. We establish a minimax regret lower bound for this setting and propose a novel batch learning algorithm that achieves the optimal regret (up to logarithmic factors). In essence, our procedure dynamically splits the covariate space into smaller bins, carefully aligning their widths with the batch size. Our theoretical results suggest that for mathematical framework of contextual bandit, a nearly constant number of policy updates can attain optimal regret in the fully online setting.

In this paper, we address the top-$K$ ranking problem with a monotone adversary. We consider the scenario where a comparison graph is randomly generated and the adversary is allowed to add arbitrary edges. The statistician’s goal is then to accurately identify the top-$K$ preferred items based on pairwise comparisons derived from this semi-random comparison graph. The main contribution of this paper is to develop a weighted maximum likelihood estimator (MLE) that achieves near-optimal sample complexity, up to a $\log^2(n)$ factor, where $n$ denotes the number of items under comparison. This is made possible through a combination of analytical and algorithmic innovations. On the analytical front, we provide a refined $\ell_\infty$ error analysis of the weighted MLE that is more explicit and tighter than existing analyses. It relates the $\ell_\infty$ error with the spectral properties of the weighted comparison graph. Motivated by this, our algorithmic innovation involves the development of an SDP-based approach to reweight the semi-random graph and meet specified spectral properties. Additionally, we propose a first-order method based on the Matrix Multiplicative Weight Update (MMWU) framework to solve the resulting SDP efficiently in nearly-linear time in the size of the semi-random comparison graph.

A key challenge of modern machine learning systems is to achieve Out-of-Distribution (OOD) generalization---generalizing to target data whose distribution differs from that of source data. Despite its significant importance, the fundamental question of ``what are the most effective algorithms for OOD generalization'' remains open even under the standard setting of covariate shift. This paper addresses this fundamental question by proving that, surprisingly, classical Maximum Likelihood Estimation (MLE) purely using source data (without any modification) achieves the *minimax* optimality for covariate shift under the *well-specified* setting. That is, *no* algorithm performs better than MLE in this setting (up to a constant factor), justifying MLE is all you need. Our result holds for a very rich class of parametric models, and does not require any boundedness condition on the density ratio. We illustrate the wide applicability of our framework by instantiating it to three concrete examples---linear regression, logistic regression, and phase retrieval. This paper further complement the study by proving that, under the *misspecified setting*, MLE is no longer the optimal choice, whereas Maximum Weighted Likelihood Estimator (MWLE) emerges as minimax optimal in certain scenarios.

We consider estimation of a linear functional of the treatment effect from adaptively collected data. This problem finds a variety of applications including off-policy evaluation in contextual bandits, and estimation of the average treatment effect in causal inference. While a certain class of augmented inverse propensity weighting (AIPW) estimators enjoys desirable asymptotic properties including the semi-parametric efficiency, much less is known about their non-asymptotic theory with adaptively collected data. To fill in the gap, we first present generic upper bounds on the mean-squared error of the class of AIPW estimators that crucially depends on a sequentially weighted error between the treatment effect and its estimates. Motivated by this, we propose a general reduction scheme that allows one to produce a sequence of estimates for the treatment effect via online learning to minimize the sequentially weighted estimation error. To illustrate this, we provide three concrete instantiations in (1) the tabular case; (2) the case of linear function approximation; and (3) the case of general function approximation for the outcome model. We then provide a local minimax lower bound to show the instance-dependent optimality of the AIPW estimator using no-regret online learning algorithms.

This paper is concerned with the problem of policy evaluation with linear function approximation in discounted infinite horizon Markov decision processes. We investigate the sample complexities required to guarantee a predefined estimation error of the best linear coefficients for two widely-used policy evaluation algorithms: the temporal difference (TD) learning algorithm and the two-timescale linear TD with gradient correction (TDC) algorithm. In both the on-policy setting, where observations are generated from the target policy, and the off-policy setting, where samples are drawn from a behavior policy potentially different from the target policy, we establish the first sample complexity bound with high-probability convergence guarantee that attains the optimal dependence on the tolerance level. We also exhibit an explicit dependence on problem-related quantities, and show in the on-policy setting that our upper bound matches the minimax lower bound on crucial problem parameters, including the choice of the feature map and the problem dimension.

We propose $\textsf{ScaledGD($\lambda$)}$, a preconditioned gradient descent method to tackle the low-rank matrix sensing problem when the true rank is unknown, and when the matrix is possibly ill-conditioned. Using overparametrized factor representations, $\textsf{ScaledGD($\lambda$)}$ starts from a small random initialization, and proceeds by gradient descent with a specific form of preconditioning with a fixed damping term to combat overparameterization. At the expense of light computational overhead incurred by preconditioners, $\textsf{ScaledGD($\lambda$)}$ is remarkably robust to ill-conditioning compared to vanilla gradient descent ($\mathsf{GD}$). Specifically, we show that, under the Gaussian design, $\textsf{ScaledGD($\lambda$)}$ converges to the true low-rank matrix at a constant linear rate that is independent of the condition number (apart from a short nearly dimension-free burdening period), with near-optimal sample complexity. This significantly improves upon the convergence rate of vanilla $\mathsf{GD}$ which suffers from a polynomial dependency with the condition number. Our work provides evidence on the power of preconditioning in accelerating the convergence without hurting generalization in overparameterized learning.

Reinforcement learning (RL) provides a theoretical framework for continuously improving an agent’s behavior via trial and error. However, efficiently learning policies from scratch can be very difficult, particularly for tasks that present exploration challenges. In such settings, it might be desirable to initialize RL with an existing policy, offline data, or demonstrations. However, naively performing such initialization in RL often works poorly, especially for value-based methods. In this paper, we present a meta algorithm that can use offline data, demonstrations, or a pre-existing policy to initialize an RL policy, and is compatible with any RL approach. In particular, we propose Jump-Start Reinforcement Learning (JSRL), an algorithm that employs two policies to solve tasks: a guide-policy, and an exploration-policy. By using the guide-policy to form a curriculum of starting states for the exploration-policy, we are able to efficiently improve performance on a set of simulated robotic tasks. We show via experiments that it is able to significantly outperform existing imitation and reinforcement learning algorithms, particularly in the small-data regime. In addition, we provide an upper bound on the sample complexity of JSRL and show that with the help of a guide-policy, one can improve the sample complexity for non-optimism exploration methods from exponential in horizon to polynomial.

Matrix completion aims to estimate missing entries in a data matrix, using the assumption of a low-complexity structure (e.g., low-rankness) so that imputation is possible. While many effective estimation algorithms exist in the literature, uncertainty quantification for this problem has proved to be challenging, and existing methods are extremely sensitive to model misspecification. In this work, we propose a distribution-free method for predictive inference in the matrix completion problem. Our method adapts the framework of conformal prediction, which provides prediction intervals with guaranteed distribution-free validity in the setting of regression, to the problem of matrix completion. Our resulting method, conformalized matrix completion (cmc), offers provable predictive coverage regardless of the accuracy of the low-rank model. Empirical results on simulated and real data demonstrate that cmc is robust to model misspecification while matching the performance of existing model-based methods when the model is correct.

This paper is concerned with noisy matrix completion—the problem of recovering a low-rank matrix from partial and noisy entries. Under uniform sampling and incoherence assumptions, we prove that a tuning-free square-root matrix completion estimator (${\mathtt {square{-}root~ MC}}$) achieves optimal statistical performance for solving the noisy matrix completion problem. Similar to the square-root Lasso estimator in high-dimensional linear regression,${\mathtt {square{-}root~ MC}}$does not rely on the knowledge of the size of the noise. While solving${\mathtt {square{-}root~ MC}}$is a convex program, our statistical analysis of${\mathtt {square{-}root~ MC}}$hinges on its intimate connections to a nonconvex rank-constrained estimator.

Tensors, which provide a powerful and flexible model for representing multi-attribute data and multi-way interactions, play an indispensable role in modern data science across various fields in science and engineering. A fundamental task is tensor completion, which aims to faithfully recover the tensor from a small subset of its entries in a statistically and computationally efficient manner. Harnessing the low-rank structure of tensors in the Tucker decomposition, this paper develops a scaled gradient descent (ScaledGD) algorithm to directly recover the tensor factors with tailored spectral initializations, and shows that it provably converges at a linear rate independent of the condition number of the ground truth tensor for tensor completion as soon as the sample size is above the order of $n^{3/2}$ ignoring other parameter dependencies, where $n$ is the dimension of the tensor. To the best of our knowledge, ScaledGD is the first algorithm that achieves near-optimal statistical and computational complexities simultaneously for low-rank tensor completion with the Tucker decomposition. Our algorithm highlights the power of appropriate preconditioning in accelerating nonconvex statistical estimation, where the iteration-varying preconditioners promote desirable invariance properties of the trajectory with respect to the underlying symmetry in low-rank tensor factorization.

An important problem that arises across different applications in signal processing, machine learning, and data science is to reliably estimate a tensor from a small number of measurements that are possibly corrupted. Leveraging the low-rank structure under the Tucker decomposition, we propose a provably efficient algorithm that directly estimates the tensor factors by solving a nonsmooth and nonconvex composite optimization problem that minimizes the least absolute deviation loss. The proposed algorithm—built on subgradient methods—harnesses preconditioners that are designed to be equivariant w.r.t. the low-rank parameterization, and is shown to achieve local linear convergence at a constant rate under the Gaussian design. Numerical experiments are provided to corroborate the superior performance of the proposed algorithm.

We study covariate shift in the context of nonparametric regression. We introduce a new measure of distribution mismatch between the source and target distributions using the integrated ratio of probabilities of balls at a given radius. We use the scaling of this measure with respect to the radius to characterize the minimax rate of estimation over a family of H{ö}lder continuous functions under covariate shift. In comparison to the recently proposed notion of transfer exponent, this measure leads to a sharper rate of convergence and is more fine-grained. We accompany our theory with concrete instances of covariate shift that illustrate this sharp difference.

Tensors, which provide a powerful and flexible model for representing multi-attribute data and multi-way interactions, play an indispensable role in modern data science across various fields in science and engineering. A fundamental task is to faithfully recover the tensor from highly incomplete measurements in a statistically and computationally efficient manner. Harnessing the low-rank structure of tensors in the Tucker decomposition, this paper develops a scaled gradient descent (ScaledGD) algorithm to directly recover the tensor factors with tailored spectral initializations, and shows that it provably converges at a linear rate independent of the condition number of the ground truth tensor for two canonical problems --- tensor completion and tensor regression --- as soon as the sample size is above the order of $n^{3/2}$ ignoring other parameter dependencies, where $n$ is the dimension of the tensor. This leads to an extremely scalable approach to low-rank tensor estimation compared with prior art, which suffers from at least one of the following drawbacks: extreme sensitivity to ill-conditioning, high per-iteration costs in terms of memory and computation, or poor sample complexity guarantees. To the best of our knowledge, ScaledGD is the first algorithm that achieves near-optimal statistical and computational complexities simultaneously for low-rank tensor completion with the Tucker decomposition. Our algorithm highlights the power of appropriate preconditioning in accelerating nonconvex statistical estimation, where the iteration-varying preconditioners promote desirable invariance properties of the trajectory with respect to the underlying symmetry in low-rank tensor factorization.

We study the problem of off-policy evaluation in the multi-armed bandit model with bounded rewards, and develop minimax rate-optimal procedures under three settings. First, when the behavior policy is known, we show that the Switch estimator, a method that alternates between the plug-in and importance sampling estimators, is minimax rate-optimal for all sample sizes. Second, when the behavior policy is unknown, we analyze performance in terms of the competitive ratio, thereby revealing a fundamental gap between the settings of known and unknown behavior policies. When the behavior policy is unknown, any estimator must have mean-squared error larger—relative to the oracle estimator equipped with the knowledge of the behavior policy— by a multiplicative factor proportional to the support size of the target policy. Moreover, we demonstrate that the plug-in approach achieves this worst-case competitive ratio up to a logarithmic factor. Third, we initiate the study of the partial knowledge setting in which it is assumed that the minimum probability taken by the behavior policy is known. We show that the plug-in estimator is optimal for relatively large values of the minimum probability, but is sub-optimal when the minimum probability is low. In order to remedy this gap, we propose a new estimator based on approximation by Chebyshev polynomials that provably achieves the optimal estimation error. Numerical experiments on both simulated and real data corroborate our theoretical findings.

Offline reinforcement learning (RL) algorithms seek to learn an optimal policy from a fixed dataset without active data collection. Based on the composition of the offline dataset, two main methods are used: imitation learning which is suitable for expert datasets, and vanilla offline RL which often requires uniform coverage datasets. From a practical standpoint, datasets often deviate from these two extremes and the exact data composition is usually unknown. To bridge this gap, we present a new offline RL framework, called single-policy concentrability, that smoothly interpolates between the two extremes of data composition, hence unifying imitation learning and vanilla offline RL. Under this new framework, we ask: can one develop an algorithm that achieves a minimax optimal rate adaptive to unknown data composition? To address this question, we consider a lower confidence bound (LCB) algorithm developed based on pessimism in the face of uncertainty in offline RL. We study finite-sample properties of LCB as well as information-theoretic limits in multi-armed bandits, contextual bandits, and Markov decision processes (MDPs). Our analysis reveals surprising facts about optimality rates. In particular, in both contextual bandits and RL, LCB achieves a fast convergence rate for nearly-expert datasets, analogous to the one achieved by imitation learning, contrary to the slow rate achieved in offline RL. In contextual bandits, we prove that LCB is adaptively optimal for the entire data composition range, achieving a smooth transition from imitation learning to offline RL. We further show that LCB is almost adaptively optimal in MDPs.

Offline (or batch) reinforcement learning (RL) algorithms seek to learn an optimal policy from a fixed dataset without active data collection. Based on the composition of the offline dataset, two main methods are used: imitation learning which is suitable for expert datasets, and vanilla offline RL which often requires uniform coverage datasets. From a practical standpoint, datasets often deviate from these two extremes and the exact data composition is usually unknown. To bridge this gap, we present a new offline RL framework that smoothly interpolates between the two extremes of data composition, hence unifying imitation learning and vanilla offline RL. The new framework is centered around a weak version of the concentrability coefficient that measures the deviation of the behavior policy from the expert policy alone. Under this new framework, we ask: can one develop an algorithm that achieves a minimax optimal rate adaptive to unknown data composition? To address this question, we consider a lower confidence bound (LCB) algorithm developed based on pessimism in the face of uncertainty in offline RL. We study finite-sample properties of LCB as well as information-theoretic limits in multi-armed bandits, contextual bandits, and Markov decision processes (MDPs). Our analysis reveals surprising facts about optimality rates. In particular, in both contextual bandits and RL, LCB achieves a faster rate of $1/N$ for nearly-expert datasets compared to the usual rate of $1/\sqrt{N}$ in offline RL, where $N$ is the batch dataset sample size. In contextual bandits with at least two contexts, we prove that LCB is adaptively optimal for the entire data composition range, achieving a smooth transition from imitation learning to offline RL. We further show that LCB is almost adaptively optimal in MDPs.

Low-rank matrix estimation is a canonical problem that finds numerous applications in signal processing, machine learning and imaging science. A popular approach in practice is to factorize the matrix into two compact low-rank factors, and then optimize these factors directly via simple iterative methods such as gradient descent and alternating minimization. Despite nonconvexity, recent literatures have shown that these simple heuristics in fact achieve linear convergence when initialized properly for a growing number of problems of interest. However, upon closer examination, existing approaches can still be computationally expensive especially for ill-conditioned matrices: the convergence rate of gradient descent depends linearly on the condition number of the low-rank matrix, while the per-iteration cost of alternating minimization is often prohibitive for large matrices. The goal of this paper is to set forth a competitive algorithmic approach dubbed Scaled Gradient Descent (ScaledGD) which can be viewed as preconditioned or diagonally-scaled gradient descent, where the preconditioners are adaptive and iteration-varying with a minimal computational overhead. With tailored variants for low-rank matrix sensing, robust principal component analysis and matrix completion, we theoretically show that ScaledGD achieves the best of both worlds: it converges linearly at a rate independent of the condition number of the low-rank matrix similar as alternating minimization, while maintaining the low per-iteration cost of gradient descent. Our analysis is also applicable to general loss functions that are restricted strongly convex and smooth over low-rank matrices. To the best of our knowledge, ScaledGD is the first algorithm that provably has such properties over a wide range of low-rank matrix estimation tasks. At the core of our analysis is the introduction of a new distance function that takes account of the preconditioners when measuring the distance between the iterates and the ground truth. Finally, numerical examples are provided to demonstrate the effectiveness of ScaledGD in accelerating the convergence rate of ill-conditioned low-rank matrix estimation in a wide number of applications.

We study the problem of learning mixtures of low-rank models, i.e. reconstructing multiple low-rank matrices from unlabelled linear measurements of each. This problem enriches two widely studied settings - low-rank matrix sensing and mixed linear regression - by bringing latent variables (i.e. unknown labels) and structural priors (i.e. low-rank structures) into consideration. To cope with the non-convexity issues arising from unlabelled heterogeneous data and low-complexity structure, we develop a three-stage meta-algorithm that is guaranteed to recover the unknown matrices with near-optimal sample and computational complexities under Gaussian designs. In addition, the proposed algorithm is provably stable against random noise. We complement the theoretical studies with empirical evidence that confirms the efficacy of our algorithm.

We consider the problem of recovering low-rank matrices from random rank-one measurements, which spans numerous applications including covariance sketching, phase retrieval, quantum state tomography, and learning shallow polynomial neural networks, among others. Our approach is to directly estimate the low-rank factor by minimizing a nonconvex least-squares loss function via vanilla gradient descent, following a tailored spectral initialization. When the true rank is bounded by a constant, this algorithm is guaranteed to converge to the ground truth (up to global ambiguity) with near-optimal sample complexity and computational complexity. To the best of our knowledge, this is the first guarantee that achieves near-optimality in both metrics. In particular, the key enabler of near-optimal computational guarantees is an implicit regularization phenomenon: without explicit regularization, both spectral initialization and the gradient descent iterates automatically stay within a region incoherent with the measurement vectors. This feature allows one to employ much more aggressive step sizes compared with the ones suggested in prior literature, without the need of sample splitting.

We consider the problem of recovering low-rank matrices from random rank-one measurements, which spans numerous applications including phase retrieval, quantum state tomography, and learning shallow neural networks with quadratic activations, among others. Our approach is to directly estimate the low-rank factor by minimizing a nonconvex least-squares loss function via vanilla gradient descent, following a tailored spectral initialization. When the true rank is small, this algorithm is guaranteed to converge to the ground truth (up to global ambiguity) with near-optimal sample and computational complexities with respect to the problem size. To the best of our knowledge, this is the first theoretical guarantee that achieves near optimality in both metrics. In particular, the key enabler of near-optimal computational guarantees is an implicit regularization phenomenon: without explicit regularization, both spectral initialization and the gradient descent iterates automatically stay within a region incoherent with the measurement vectors. This feature allows one to employ much more aggressive step sizes compared with the ones suggested in prior literature, without the need of sample splitting.

Recent years have seen a flurry of activities in designing provably efficient nonconvex optimization procedures for solving statistical estimation problems. For various problems like phase retrieval or low-rank matrix completion, state-of-the-art nonconvex procedures require proper regularization (e.g. trimming, regularized cost, projection) in order to guarantee fast convergence. When it comes to vanilla procedures such as gradient descent, however, prior theory either recommends highly conservative learning rates to avoid overshooting, or completely lacks performance guarantees. This paper uncovers a striking phenomenon in several nonconvex problems: even in the absence of explicit regularization, gradient descent follows a trajectory staying within a basin that enjoys nice geometry, consisting of points incoherent with the sampling mechanism. This “implicit regularization” feature allows gradient descent to proceed in a far more aggressive fashion without overshooting, which in turn results in substantial computational savings. Focusing on two statistical estimation problems, i.e. solving random quadratic systems of equations and low-rank matrix completion, we establish that gradient descent achieves near-optimal statistical and computational guarantees without explicit regularization. As a byproduct, for noisy matrix completion, we demonstrate that gradient descent enables optimal control of both entrywise and spectral-norm errors.

Similarity search is a fundamental problem in network analysis and can be applied in many applications, such as collaborator recommendation in coauthor networks, friend recommendation in social networks, and relation prediction in medical information networks. In this article, we propose a sampling-based method using random paths to estimate the similarities based on both common neighbors and structural contexts efficiently in very large homogeneous or heterogeneous information networks. We give a theoretical guarantee that the sampling size depends on the error-bound ε, the confidence level (1-δ), and the path length T of each random walk. We perform an extensive empirical study on a Tencent microblogging network of 1,000,000,000 edges. We show that our algorithm can return top- k similar vertices for any vertex in a network 300× faster than the state-of-the-art methods. We develop a prototype system of recommending similar authors to demonstrate the effectiveness of our method.

Estimating similarity between vertices is a fundamental issue in network analysis across various domains, such as social networks and biological networks. Methods based on common neighbors and structural contexts have received much attention. However, both categories of methods are difficult to scale up to handle large networks (with billions of nodes). In this paper, we propose a sampling method that provably and accurately estimates the similarity between vertices. The algorithm is based on a novel idea of random path. Specifically, given a network, we perform R random walks, each starting from a randomly picked vertex and walking T steps. Theoretically, the algorithm guarantees that the sampling size R = O(2ε-2 log2 T) depends on the error-bound ε, the confidence level (1 -- δ), and the path length T of each random walk.