arXiv subjects

Aryan Mokhtari

Publications and source records attributed to Aryan Mokhtari.

At least 55 records · Page 3Linked to original sources

arXiv2023

MAML and ANIL Provably Learn Representations

Recent empirical evidence has driven conventional wisdom to believe that gradient-based meta-learning (GBML) methods perform well at few-shot learning because they learn an expressive data representation that is shared across tasks. However, the mechanics of GBML have remained largely mysterious from a theoretical perspective. In this paper, we prove that two well-known GBML methods, MAML and ANIL, as well as their first-order approximations, are capable of learning common representation among a set of given tasks. Specifically, in the well-known multi-task linear representation learning setting, they are able to recover the ground-truth representation at an exponentially fast rate. Moreover, our analysis illuminates that the driving force causing MAML and ANIL to recover the underlying representation is that they adapt the final layer of their model, which harnesses the underlying task diversity to improve the representation in all directions of interest. To the best of our knowledge, these are the first results to show that MAML and/or ANIL learn expressive representations and to rigorously explain why they do so.

cs.LG↗

arXiv2023

Greedy Pruning with Group Lasso Provably Generalizes for Matrix Sensing

Pruning schemes have been widely used in practice to reduce the complexity of trained models with a massive number of parameters. In fact, several practical studies have shown that if a pruned model is fine-tuned with some gradient-based updates it generalizes well to new samples. Although the above pipeline, which we refer to as pruning + fine-tuning, has been extremely successful in lowering the complexity of trained models, there is very little known about the theory behind this success. In this paper, we address this issue by investigating the pruning + fine-tuning framework on the overparameterized matrix sensing problem with the ground truth $U_\star \in \mathbb{R}^{d \times r}$ and the overparameterized model $U \in \mathbb{R}^{d \times k}$ with $k \gg r$. We study the approximate local minima of the mean square error, augmented with a smooth version of a group Lasso regularizer, $\sum_{i=1}^k \| U e_i \|_2$. In particular, we provably show that pruning all the columns below a certain explicit $\ell_2$-norm threshold results in a solution $U_{\text{prune}}$ which has the minimum number of columns $r$, yet close to the ground truth in training loss. Moreover, in the subsequent fine-tuning phase, gradient descent initialized at $U_{\text{prune}}$ converges at a linear rate to its limit. While our analysis provides insights into the role of regularization in pruning, we also show that running gradient descent in the absence of regularization results in models which {are not suitable for greedy pruning}, i.e., many columns could have their $\ell_2$ norm comparable to that of the maximum. To the best of our knowledge, our results provide the first rigorous insights on why greedy pruning + fine-tuning leads to smaller models which also generalize well.

cs.LG↗

arXiv2023

Accelerated Quasi-Newton Proximal Extragradient: Faster Rate for Smooth Convex Optimization

In this paper, we propose an accelerated quasi-Newton proximal extragradient (A-QPNE) method for solving unconstrained smooth convex optimization problems. With access only to the gradients of the objective, we prove that our method can achieve a convergence rate of ${O}\bigl(\min\{\frac{1}{k^2}, \frac{\sqrt{d\log k}}{k^{2.5}}\}\bigr)$, where $d$ is the problem dimension and $k$ is the number of iterations. In particular, in the regime where $k = {O}(d)$, our method matches the optimal rate of ${O}(\frac{1}{k^2})$ by Nesterov's accelerated gradient (NAG). Moreover, in the the regime where $k = Ω(d \log d)$, it outperforms NAG and converges at a faster rate of ${O}\bigl(\frac{\sqrt{d\log k}}{k^{2.5}}\bigr)$. To the best of our knowledge, this result is the first to demonstrate a provable gain of a quasi-Newton-type method over NAG in the convex setting. To achieve such results, we build our method on a recent variant of the Monteiro-Svaiter acceleration framework and adopt an online learning perspective to update the Hessian approximation matrices, in which we relate the convergence rate of our method to the dynamic regret of a specific online convex optimization problem in the space of matrices.

math.OC↗

arXiv2023

A Conditional Gradient-based Method for Simple Bilevel Optimization with Convex Lower-level Problem

In this paper, we study a class of bilevel optimization problems, also known as simple bilevel optimization, where we minimize a smooth objective function over the optimal solution set of another convex constrained optimization problem. Several iterative methods have been developed for tackling this class of problems. Alas, their convergence guarantees are either asymptotic for the upper-level objective, or the convergence rates are slow and sub-optimal. To address this issue, in this paper, we introduce a novel bilevel optimization method that locally approximates the solution set of the lower-level problem via a cutting plane, and then runs a conditional gradient update to decrease the upper-level objective. When the upper-level objective is convex, we show that our method requires ${\mathcal{O}}(\max\{1/ε_f,1/ε_g\})$ iterations to find a solution that is $ε_f$-optimal for the upper-level objective and $ε_g$-optimal for the lower-level objective. Moreover, when the upper-level objective is non-convex, our method requires ${\mathcal{O}}(\max\{1/ε_f^2,1/(ε_fε_g)\})$ iterations to find an $(ε_f,ε_g)$-optimal solution. We also prove stronger convergence guarantees under the Hölderian error bound assumption on the lower-level problem. To the best of our knowledge, our method achieves the best-known iteration complexity for the considered class of bilevel problems.

math.OC↗

arXiv2023

Exploiting Shared Representations for Personalized Federated Learning

Deep neural networks have shown the ability to extract universal feature representations from data such as images and text that have been useful for a variety of learning tasks. However, the fruits of representation learning have yet to be fully-realized in federated settings. Although data in federated settings is often non-i.i.d. across clients, the success of centralized deep learning suggests that data often shares a global feature representation, while the statistical heterogeneity across clients or tasks is concentrated in the labels. Based on this intuition, we propose a novel federated learning framework and algorithm for learning a shared data representation across clients and unique local heads for each client. Our algorithm harnesses the distributed computational power across clients to perform many local-updates with respect to the low-dimensional local parameters for every update of the representation. We prove that this method obtains linear convergence to the ground-truth representation with near-optimal sample complexity in a linear setting, demonstrating that it can efficiently reduce the problem dimension for each client. This result is of interest beyond federated learning to a broad class of problems in which we aim to learn a shared low-dimensional representation among data distributions, for example in meta-learning and multi-task learning. Further, extensive experimental results show the empirical improvement of our method over alternative personalized federated learning approaches in federated environments with heterogeneous data.

cs.LG↗

arXiv2023

InfoNCE Loss Provably Learns Cluster-Preserving Representations

The goal of contrasting learning is to learn a representation that preserves underlying clusters by keeping samples with similar content, e.g. the ``dogness'' of a dog, close to each other in the space generated by the representation. A common and successful approach for tackling this unsupervised learning problem is minimizing the InfoNCE loss associated with the training samples, where each sample is associated with their augmentations (positive samples such as rotation, crop) and a batch of negative samples (unrelated samples). To the best of our knowledge, it was unanswered if the representation learned by minimizing the InfoNCE loss preserves the underlying data clusters, as it only promotes learning a representation that is faithful to augmentations, i.e., an image and its augmentations have the same representation. Our main result is to show that the representation learned by InfoNCE with a finite number of negative samples is also consistent with respect to clusters in the data, under the condition that the augmentation sets within clusters may be non-overlapping but are close and intertwined, relative to the complexity of the learning function class.

cs.LG↗

arXiv2023

Network Adaptive Federated Learning: Congestion and Lossy Compression

In order to achieve the dual goals of privacy and learning across distributed data, Federated Learning (FL) systems rely on frequent exchanges of large files (model updates) between a set of clients and the server. As such FL systems are exposed to, or indeed the cause of, congestion across a wide set of network resources. Lossy compression can be used to reduce the size of exchanged files and associated delays, at the cost of adding noise to model updates. By judiciously adapting clients' compression to varying network congestion, an FL application can reduce wall clock training time. To that end, we propose a Network Adaptive Compression (NAC-FL) policy, which dynamically varies the client's lossy compression choices to network congestion variations. We prove, under appropriate assumptions, that NAC-FL is asymptotically optimal in terms of directly minimizing the expected wall clock training time. Further, we show via simulation that NAC-FL achieves robust performance improvements with higher gains in settings with positively correlated delays across time.

cs.LG↗

arXiv2022

Future Gradient Descent for Adapting the Temporal Shifting Data Distribution in Online Recommendation Systems

One of the key challenges of learning an online recommendation model is the temporal domain shift, which causes the mismatch between the training and testing data distribution and hence domain generalization error. To overcome, we propose to learn a meta future gradient generator that forecasts the gradient information of the future data distribution for training so that the recommendation model can be trained as if we were able to look ahead at the future of its deployment. Compared with Batch Update, a widely used paradigm, our theory suggests that the proposed algorithm achieves smaller temporal domain generalization error measured by a gradient variation term in a local regret. We demonstrate the empirical advantage by comparing with various representative baselines.

cs.LG↗

arXiv2022

How Does the Task Landscape Affect MAML Performance?

Model-Agnostic Meta-Learning (MAML) has become increasingly popular for training models that can quickly adapt to new tasks via one or few stochastic gradient descent steps. However, the MAML objective is significantly more difficult to optimize compared to standard non-adaptive learning (NAL), and little is understood about how much MAML improves over NAL in terms of the fast adaptability of their solutions in various scenarios. We analytically address this issue in a linear regression setting consisting of a mixture of easy and hard tasks, where hardness is related to the rate that gradient descent converges on the task. Specifically, we prove that in order for MAML to achieve substantial gain over NAL, (i) there must be some discrepancy in hardness among the tasks, and (ii) the optimal solutions of the hard tasks must be closely packed with the center far from the center of the easy tasks optimal solutions. We also give numerical and analytical results suggesting that these insights apply to two-layer neural networks. Finally, we provide few-shot image classification experiments that support our insights for when MAML should be used and emphasize the importance of training MAML on hard tasks in practice.

cs.LG↗

arXiv2022

The Power of Adaptivity in SGD: Self-Tuning Step Sizes with Unbounded Gradients and Affine Variance

We study convergence rates of AdaGrad-Norm as an exemplar of adaptive stochastic gradient methods (SGD), where the step sizes change based on observed stochastic gradients, for minimizing non-convex, smooth objectives. Despite their popularity, the analysis of adaptive SGD lags behind that of non adaptive methods in this setting. Specifically, all prior works rely on some subset of the following assumptions: (i) uniformly-bounded gradient norms, (ii) uniformly-bounded stochastic gradient variance (or even noise support), (iii) conditional independence between the step size and stochastic gradient. In this work, we show that AdaGrad-Norm exhibits an order optimal convergence rate of $\mathcal{O}\left(\frac{\mathrm{poly}\log(T)}{\sqrt{T}}\right)$ after $T$ iterations under the same assumptions as optimally-tuned non adaptive SGD (unbounded gradient norms and affine noise variance scaling), and crucially, without needing any tuning parameters. We thus establish that adaptive gradient methods exhibit order-optimal convergence in much broader regimes than previously understood.

stat.ML↗

arXiv2022

Sharpened Quasi-Newton Methods: Faster Superlinear Rate and Larger Local Convergence Neighborhood

Non-asymptotic analysis of quasi-Newton methods have gained traction recently. In particular, several works have established a non-asymptotic superlinear rate of $\mathcal{O}((1/\sqrt{t})^t)$ for the (classic) BFGS method by exploiting the fact that its error of Newton direction approximation approaches zero. Moreover, a greedy variant of BFGS was recently proposed which accelerates its convergence by directly approximating the Hessian, instead of the Newton direction, and achieves a fast local quadratic convergence rate. Alas, the local quadratic convergence of Greedy-BFGS requires way more updates compared to the number of iterations that BFGS requires for a local superlinear rate. This is due to the fact that in Greedy-BFGS the Hessian is directly approximated and the Newton direction approximation may not be as accurate as the one for BFGS. In this paper, we close this gap and present a novel BFGS method that has the best of both worlds in that it leverages the approximation ideas of both BFGS and Greedy-BFGS to properly approximate the Newton direction and the Hessian matrix simultaneously. Our theoretical results show that our method out-performs both BFGS and Greedy-BFGS in terms of convergence rate, while it reaches its quadratic convergence rate with fewer steps compared to Greedy-BFGS. Numerical experiments on various datasets also confirm our theoretical findings.

math.OC↗

arXiv2022

Straggler-Resilient Personalized Federated Learning

Federated Learning is an emerging learning paradigm that allows training models from samples distributed across a large network of clients while respecting privacy and communication restrictions. Despite its success, federated learning faces several challenges related to its decentralized nature. In this work, we develop a novel algorithmic procedure with theoretical speedup guarantees that simultaneously handles two of these hurdles, namely (i) data heterogeneity, i.e., data distributions can vary substantially across clients, and (ii) system heterogeneity, i.e., the computational power of the clients could differ significantly. Our method relies on ideas from representation learning theory to find a global common representation using all clients' data and learn a user-specific set of parameters leading to a personalized solution for each client. Furthermore, our method mitigates the effects of stragglers by adaptively selecting clients based on their computational characteristics and statistical significance, thus achieving, for the first time, near optimal sample complexity and provable logarithmic speedup. Experimental results support our theoretical findings showing the superiority of our method over alternative personalized federated schemes in system and data heterogeneous environments.

cs.LG↗

arXiv2022

FedAvg with Fine Tuning: Local Updates Lead to Representation Learning

The Federated Averaging (FedAvg) algorithm, which consists of alternating between a few local stochastic gradient updates at client nodes, followed by a model averaging update at the server, is perhaps the most commonly used method in Federated Learning. Notwithstanding its simplicity, several empirical studies have illustrated that the output model of FedAvg, after a few fine-tuning steps, leads to a model that generalizes well to new unseen tasks. This surprising performance of such a simple method, however, is not fully understood from a theoretical point of view. In this paper, we formally investigate this phenomenon in the multi-task linear representation setting. We show that the reason behind generalizability of the FedAvg's output is its power in learning the common data representation among the clients' tasks, by leveraging the diversity among client data distributions via local updates. We formally establish the iteration complexity required by the clients for proving such result in the setting where the underlying shared representation is a linear map. To the best of our knowledge, this is the first such result for any setting. We also provide empirical evidence demonstrating FedAvg's representation learning ability in federated image classification with heterogeneous data.

cs.LG↗

arXiv2022

Provably Private Distributed Averaging Consensus: An Information-Theoretic Approach

In this work, we focus on solving a decentralized consensus problem in a private manner. Specifically, we consider a setting in which a group of nodes, connected through a network, aim at computing the mean of their local values without revealing those values to each other. The distributed consensus problem is a classic problem that has been extensively studied and its convergence characteristics are well-known. Alas, state-of-the-art consensus methods build on the idea of exchanging local information with neighboring nodes which leaks information about the users' local values. We propose an algorithmic framework that is capable of achieving the convergence limit and rate of classic consensus algorithms while keeping the users' local values private. The key idea of our proposed method is to carefully design noisy messages that are passed from each node to its neighbors such that the consensus algorithm still converges precisely to the average of local values, while a minimum amount of information about local values is leaked. We formalize this by precisely characterizing the mutual information between the private message of a node and all the messages that another adversary collects over time. We prove that our method is capable of preserving users' privacy for any network without a so-called "generalized leaf", and formalize the trade-off between privacy and convergence time. Unlike many private algorithms, any desired accuracy is achievable by our method, and the required level of privacy only affects the convergence time.

cs.MA↗

arXiv2021

Non-asymptotic Superlinear Convergence of Standard Quasi-Newton Methods

In this paper, we study and prove the non-asymptotic superlinear convergence rate of the Broyden class of quasi-Newton algorithms which includes the Davidon--Fletcher--Powell (DFP) method and the Broyden--Fletcher--Goldfarb--Shanno (BFGS) method. The asymptotic superlinear convergence rate of these quasi-Newton methods has been extensively studied in the literature, but their explicit finite-time local convergence rate is not fully investigated. In this paper, we provide a finite-time (non-asymptotic) convergence analysis for Broyden quasi-Newton algorithms under the assumptions that the objective function is strongly convex, its gradient is Lipschitz continuous, and its Hessian is Lipschitz continuous at the optimal solution. We show that in a local neighborhood of the optimal solution, the iterates generated by both DFP and BFGS converge to the optimal solution at a superlinear rate of $(1/k)^{k/2}$, where $k$ is the number of iterations. We also prove a similar local superlinear convergence result holds for the case that the objective function is self-concordant. Numerical experiments on several datasets confirm our explicit convergence rate bounds. Our theoretical guarantee is one of the first results that provide a non-asymptotic superlinear convergence rate for quasi-Newton methods.

math.OC↗

arXiv2021

On the Convergence Theory of Debiased Model-Agnostic Meta-Reinforcement Learning

We consider Model-Agnostic Meta-Learning (MAML) methods for Reinforcement Learning (RL) problems, where the goal is to find a policy using data from several tasks represented by Markov Decision Processes (MDPs) that can be updated by one step of stochastic policy gradient for the realized MDP. In particular, using stochastic gradients in MAML update steps is crucial for RL problems since computation of exact gradients requires access to a large number of possible trajectories. For this formulation, we propose a variant of the MAML method, named Stochastic Gradient Meta-Reinforcement Learning (SG-MRL), and study its convergence properties. We derive the iteration and sample complexity of SG-MRL to find an $ε$-first-order stationary point, which, to the best of our knowledge, provides the first convergence guarantee for model-agnostic meta-reinforcement learning algorithms. We further show how our results extend to the case where more than one step of stochastic policy gradient method is used at test time. Finally, we empirically compare SG-MRL and MAML in several deep RL environments.

cs.LG↗

arXiv2021

Generalization of Model-Agnostic Meta-Learning Algorithms: Recurring and Unseen Tasks

In this paper, we study the generalization properties of Model-Agnostic Meta-Learning (MAML) algorithms for supervised learning problems. We focus on the setting in which we train the MAML model over $m$ tasks, each with $n$ data points, and characterize its generalization error from two points of view: First, we assume the new task at test time is one of the training tasks, and we show that, for strongly convex objective functions, the expected excess population loss is bounded by ${\mathcal{O}}(1/mn)$. Second, we consider the MAML algorithm's generalization to an unseen task and show that the resulting generalization error depends on the total variation distance between the underlying distributions of the new task and the tasks observed during the training process. Our proof techniques rely on the connections between algorithmic stability and generalization bounds of algorithms. In particular, we propose a new definition of stability for meta-learning algorithms, which allows us to capture the role of both the number of tasks $m$ and number of samples per task $n$ on the generalization error of MAML.

cs.LG↗

arXiv2021

Minimax Optimization: The Case of Convex-Submodular

Minimax optimization has been central in addressing various applications in machine learning, game theory, and control theory. Prior literature has thus far mainly focused on studying such problems in the continuous domain, e.g., convex-concave minimax optimization is now understood to a significant extent. Nevertheless, minimax problems extend far beyond the continuous domain to mixed continuous-discrete domains or even fully discrete domains. In this paper, we study mixed continuous-discrete minimax problems where the minimization is over a continuous variable belonging to Euclidean space and the maximization is over subsets of a given ground set. We introduce the class of convex-submodular minimax problems, where the objective is convex with respect to the continuous variable and submodular with respect to the discrete variable. Even though such problems appear frequently in machine learning applications, little is known about how to address them from algorithmic and theoretical perspectives. For such problems, we first show that obtaining saddle points are hard up to any approximation, and thus introduce new notions of (near-) optimality. We then provide several algorithmic procedures for solving convex and monotone-submodular minimax problems and characterize their convergence rates, computational complexity, and quality of the final solution according to our notions of optimally. Our proposed algorithms are iterative and combine tools from both discrete and continuous optimization. Finally, we provide numerical experiments to showcase the effectiveness of our purposed methods.

math.OC↗