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Christos Thrampoulidis

Publications and source records attributed to Christos Thrampoulidis.

At least 55 records · Page 3Linked to original sources
arXiv2023

BiSLS/SPS: Auto-tune Step Sizes for Stable Bi-level Optimization

The popularity of bi-level optimization (BO) in deep learning has spurred a growing interest in studying gradient-based BO algorithms. However, existing algorithms involve two coupled learning rates that can be affected by approximation errors when computing hypergradients, making careful fine-tuning necessary to ensure fast convergence. To alleviate this issue, we investigate the use of recently proposed adaptive step-size methods, namely stochastic line search (SLS) and stochastic Polyak step size (SPS), for computing both the upper and lower-level learning rates. First, we revisit the use of SLS and SPS in single-level optimization without the additional interpolation condition that is typically assumed in prior works. For such settings, we investigate new variants of SLS and SPS that improve upon existing suggestions in the literature and are simpler to implement. Importantly, these two variants can be seen as special instances of general family of methods with an envelope-type step-size. This unified envelope strategy allows for the extension of the algorithms and their convergence guarantees to BO settings. Finally, our extensive experiments demonstrate that the new algorithms, which are available in both SGD and Adam versions, can find large learning rates with minimal tuning and converge faster than corresponding vanilla SGD or Adam BO algorithms that require fine-tuning.

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arXiv2023

Symmetric Neural-Collapse Representations with Supervised Contrastive Loss: The Impact of ReLU and Batching

Supervised contrastive loss (SCL) is a competitive and often superior alternative to the cross-entropy loss for classification. While prior studies have demonstrated that both losses yield symmetric training representations under balanced data, this symmetry breaks under class imbalances. This paper presents an intriguing discovery: the introduction of a ReLU activation at the final layer effectively restores the symmetry in SCL-learned representations. We arrive at this finding analytically, by establishing that the global minimizers of an unconstrained features model with SCL loss and entry-wise non-negativity constraints form an orthogonal frame. Extensive experiments conducted across various datasets, architectures, and imbalance scenarios corroborate our finding. Importantly, our experiments reveal that the inclusion of the ReLU activation restores symmetry without compromising test accuracy. This constitutes the first geometry characterization of SCL under imbalances. Additionally, our analysis and experiments underscore the pivotal role of batch selection strategies in representation geometry. By proving necessary and sufficient conditions for mini-batch choices that ensure invariant symmetric representations, we introduce batch-binding as an efficient strategy that guarantees these conditions hold.

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arXiv2023

Engineering the Neural Collapse Geometry of Supervised-Contrastive Loss

Supervised-contrastive loss (SCL) is an alternative to cross-entropy (CE) for classification tasks that makes use of similarities in the embedding space to allow for richer representations. In this work, we propose methods to engineer the geometry of these learnt feature embeddings by modifying the contrastive loss. In pursuit of adjusting the geometry we explore the impact of prototypes, fixed embeddings included during training to alter the final feature geometry. Specifically, through empirical findings, we demonstrate that the inclusion of prototypes in every batch induces the geometry of the learnt embeddings to align with that of the prototypes. We gain further insights by considering a limiting scenario where the number of prototypes far outnumber the original batch size. Through this, we establish a connection to cross-entropy (CE) loss with a fixed classifier and normalized embeddings. We validate our findings by conducting a series of experiments with deep neural networks on benchmark vision datasets.

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arXiv2023

Benign Overfitting in Multiclass Classification: All Roads Lead to Interpolation

The literature on "benign overfitting" in overparameterized models has been mostly restricted to regression or binary classification; however, modern machine learning operates in the multiclass setting. Motivated by this discrepancy, we study benign overfitting in multiclass linear classification. Specifically, we consider the following training algorithms on separable data: (i) empirical risk minimization (ERM) with cross-entropy loss, which converges to the multiclass support vector machine (SVM) solution; (ii) ERM with least-squares loss, which converges to the min-norm interpolating (MNI) solution; and, (iii) the one-vs-all SVM classifier. First, we provide a simple sufficient deterministic condition under which all three algorithms lead to classifiers that interpolate the training data and have equal accuracy. When the data is generated from Gaussian mixtures or a multinomial logistic model, this condition holds under high enough effective overparameterization. We also show that this sufficient condition is satisfied under "neural collapse", a phenomenon that is observed in training deep neural networks. Second, we derive novel bounds on the accuracy of the MNI classifier, thereby showing that all three training algorithms lead to benign overfitting under sufficient overparameterization. Ultimately, our analysis shows that good generalization is possible for SVM solutions beyond the realm in which typical margin-based bounds apply.

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arXiv2023

Fast Convergence in Learning Two-Layer Neural Networks with Separable Data

Normalized gradient descent has shown substantial success in speeding up the convergence of exponentially-tailed loss functions (which includes exponential and logistic losses) on linear classifiers with separable data. In this paper, we go beyond linear models by studying normalized GD on two-layer neural nets. We prove for exponentially-tailed losses that using normalized GD leads to linear rate of convergence of the training loss to the global optimum if the iterates find an interpolating model. This is made possible by showing certain gradient self-boundedness conditions and a log-Lipschitzness property. We also study generalization of normalized GD for convex objectives via an algorithmic-stability analysis. In particular, we show that normalized GD does not overfit during training by establishing finite-time generalization bounds.

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arXiv2023

On the Role of Attention in Prompt-tuning

Prompt-tuning is an emerging strategy to adapt large language models (LLM) to downstream tasks by learning a (soft-)prompt parameter from data. Despite its success in LLMs, there is limited theoretical understanding of the power of prompt-tuning and the role of the attention mechanism in prompting. In this work, we explore prompt-tuning for one-layer attention architectures and study contextual mixture-models where each input token belongs to a context-relevant or -irrelevant set. We isolate the role of prompt-tuning through a self-contained prompt-attention model. Our contributions are as follows: (1) We show that softmax-prompt-attention is provably more expressive than softmax-self-attention and linear-prompt-attention under our contextual data model. (2) We analyze the initial trajectory of gradient descent and show that it learns the prompt and prediction head with near-optimal sample complexity and demonstrate how prompt can provably attend to sparse context-relevant tokens. (3) Assuming a known prompt but an unknown prediction head, we characterize the exact finite sample performance of prompt-attention which reveals the fundamental performance limits and the precise benefit of the context information. We also provide experiments that verify our theoretical insights on real datasets and demonstrate how prompt-tuning enables the model to attend to context-relevant information.

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arXiv2023

Fast Convergence of Random Reshuffling under Over-Parameterization and the Polyak-Łojasiewicz Condition

Modern machine learning models are often over-parameterized and as a result they can interpolate the training data. Under such a scenario, we study the convergence properties of a sampling-without-replacement variant of stochastic gradient descent (SGD) known as random reshuffling (RR). Unlike SGD that samples data with replacement at every iteration, RR chooses a random permutation of data at the beginning of each epoch and each iteration chooses the next sample from the permutation. For under-parameterized models, it has been shown RR can converge faster than SGD under certain assumptions. However, previous works do not show that RR outperforms SGD in over-parameterized settings except in some highly-restrictive scenarios. For the class of Polyak-Łojasiewicz (PL) functions, we show that RR can outperform SGD in over-parameterized settings when either one of the following holds: (i) the number of samples ($n$) is less than the product of the condition number ($κ$) and the parameter ($α$) of a weak growth condition (WGC), or (ii) $n$ is less than the parameter ($ρ$) of a strong growth condition (SGC).

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arXiv2023

On Generalization of Decentralized Learning with Separable Data

Decentralized learning offers privacy and communication efficiency when data are naturally distributed among agents communicating over an underlying graph. Motivated by overparameterized learning settings, in which models are trained to zero training loss, we study algorithmic and generalization properties of decentralized learning with gradient descent on separable data. Specifically, for decentralized gradient descent (DGD) and a variety of loss functions that asymptote to zero at infinity (including exponential and logistic losses), we derive novel finite-time generalization bounds. This complements a long line of recent work that studies the generalization performance and the implicit bias of gradient descent over separable data, but has thus far been limited to centralized learning scenarios. Notably, our generalization bounds approximately match in order their centralized counterparts. Critical behind this, and of independent interest, is establishing novel bounds on the training loss and the rate-of-consensus of DGD for a class of self-bounded losses. Finally, on the algorithmic front, we design improved gradient-based routines for decentralized learning with separable data and empirically demonstrate orders-of-magnitude of speed-up in terms of both training and generalization performance.

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arXiv2023

Generalization and Stability of Interpolating Neural Networks with Minimal Width

We investigate the generalization and optimization properties of shallow neural-network classifiers trained by gradient descent in the interpolating regime. Specifically, in a realizable scenario where model weights can achieve arbitrarily small training error $ε$ and their distance from initialization is $g(ε)$, we demonstrate that gradient descent with $n$ training data achieves training error $O(g(1/T)^2 /T)$ and generalization error $O(g(1/T)^2 /n)$ at iteration $T$, provided there are at least $m=Ω(g(1/T)^4)$ hidden neurons. We then show that our realizable setting encompasses a special case where data are separable by the model's neural tangent kernel. For this and logistic-loss minimization, we prove the training loss decays at a rate of $\tilde O(1/ T)$ given polylogarithmic number of neurons $m=Ω(\log^4 (T))$. Moreover, with $m=Ω(\log^{4} (n))$ neurons and $T\approx n$ iterations, we bound the test loss by $\tilde{O}(1/n)$. Our results differ from existing generalization outcomes using the algorithmic-stability framework, which necessitate polynomial width and yield suboptimal generalization rates. Central to our analysis is the use of a new self-bounded weak-convexity property, which leads to a generalized local quasi-convexity property for sufficiently parameterized neural-network classifiers. Eventually, despite the objective's non-convexity, this leads to convergence and generalization-gap bounds that resemble those found in the convex setting of linear logistic regression.

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arXiv2023

Towards Better Out-of-Distribution Generalization of Neural Algorithmic Reasoning Tasks

In this paper, we study the OOD generalization of neural algorithmic reasoning tasks, where the goal is to learn an algorithm (e.g., sorting, breadth-first search, and depth-first search) from input-output pairs using deep neural networks. First, we argue that OOD generalization in this setting is significantly different than common OOD settings. For example, some phenomena in OOD generalization of image classifications such as \emph{accuracy on the line} are not observed here, and techniques such as data augmentation methods do not help as assumptions underlying many augmentation techniques are often violated. Second, we analyze the main challenges (e.g., input distribution shift, non-representative data generation, and uninformative validation metrics) of the current leading benchmark, i.e., CLRS \citep{deepmind2021clrs}, which contains 30 algorithmic reasoning tasks. We propose several solutions, including a simple-yet-effective fix to the input distribution shift and improved data generation. Finally, we propose an attention-based 2WL-graph neural network (GNN) processor which complements message-passing GNNs so their combination outperforms the state-of-the-art model by a 3% margin averaged over all algorithms. Our code is available at: \url{https://github.com/smahdavi4/clrs}.

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arXiv2023

On the Implicit Geometry of Cross-Entropy Parameterizations for Label-Imbalanced Data

Various logit-adjusted parameterizations of the cross-entropy (CE) loss have been proposed as alternatives to weighted CE for training large models on label-imbalanced data far beyond the zero train error regime. The driving force behind those designs has been the theory of implicit bias, which for linear(ized) models, explains why they successfully induce bias on the optimization path towards solutions that favor minorities. Aiming to extend this theory to non-linear models, we investigate the implicit geometry of classifiers and embeddings that are learned by different CE parameterizations. Our main result characterizes the global minimizers of a non-convex cost-sensitive SVM classifier for the unconstrained features model, which serves as an abstraction of deep nets. We derive closed-form formulas for the angles and norms of classifiers and embeddings as a function of the number of classes, the imbalance and the minority ratios, and the loss hyperparameters. Using these, we show that logit-adjusted parameterizations can be appropriately tuned to learn symmetric geometries irrespective of the imbalance ratio. We complement our analysis with experiments and an empirical study of convergence accuracy in deep-nets.

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arXiv2022

Mirror Descent Maximizes Generalized Margin and Can Be Implemented Efficiently

Driven by the empirical success and wide use of deep neural networks, understanding the generalization performance of overparameterized models has become an increasingly popular question. To this end, there has been substantial effort to characterize the implicit bias of the optimization algorithms used, such as gradient descent (GD), and the structural properties of their preferred solutions. This paper answers an open question in this literature: For the classification setting, what solution does mirror descent (MD) converge to? Specifically, motivated by its efficient implementation, we consider the family of mirror descent algorithms with potential function chosen as the $p$-th power of the $\ell_p$-norm, which is an important generalization of GD. We call this algorithm $p$-$\textsf{GD}$. For this family, we characterize the solutions it obtains and show that it converges in direction to a generalized maximum-margin solution with respect to the $\ell_p$-norm for linearly separable classification. While the MD update rule is in general expensive to compute and perhaps not suitable for deep learning, $p$-$\textsf{GD}$ is fully parallelizable in the same manner as SGD and can be used to train deep neural networks with virtually no additional computational overhead. Using comprehensive experiments with both linear and deep neural network models, we demonstrate that $p$-$\textsf{GD}$ can noticeably affect the structure and the generalization performance of the learned models.

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arXiv2022

FedNest: Federated Bilevel, Minimax, and Compositional Optimization

Standard federated optimization methods successfully apply to stochastic problems with single-level structure. However, many contemporary ML problems -- including adversarial robustness, hyperparameter tuning, and actor-critic -- fall under nested bilevel programming that subsumes minimax and compositional optimization. In this work, we propose \fedblo: A federated alternating stochastic gradient method to address general nested problems. We establish provable convergence rates for \fedblo in the presence of heterogeneous data and introduce variations for bilevel, minimax, and compositional optimization. \fedblo introduces multiple innovations including federated hypergradient computation and variance reduction to address inner-level heterogeneity. We complement our theory with experiments on hyperparameter \& hyper-representation learning and minimax optimization that demonstrate the benefits of our method in practice. Code is available at https://github.com/ucr-optml/FedNest.

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arXiv2022

Imbalance Trouble: Revisiting Neural-Collapse Geometry

Neural Collapse refers to the remarkable structural properties characterizing the geometry of class embeddings and classifier weights, found by deep nets when trained beyond zero training error. However, this characterization only holds for balanced data. Here we thus ask whether it can be made invariant to class imbalances. Towards this end, we adopt the unconstrained-features model (UFM), a recent theoretical model for studying neural collapse, and introduce Simplex-Encoded-Labels Interpolation (SELI) as an invariant characterization of the neural collapse phenomenon. Specifically, we prove for the UFM with cross-entropy loss and vanishing regularization that, irrespective of class imbalances, the embeddings and classifiers always interpolate a simplex-encoded label matrix and that their individual geometries are determined by the SVD factors of this same label matrix. We then present extensive experiments on synthetic and real datasets that confirm convergence to the SELI geometry. However, we caution that convergence worsens with increasing imbalances. We theoretically support this finding by showing that unlike the balanced case, when minorities are present, ridge-regularization plays a critical role in tweaking the geometry. This defines new questions and motivates further investigations into the impact of class imbalances on the rates at which first-order methods converge to their asymptotically preferred solutions.

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arXiv2022

On how to avoid exacerbating spurious correlations when models are overparameterized

Overparameterized models fail to generalize well in the presence of data imbalance even when combined with traditional techniques for mitigating imbalances. This paper focuses on imbalanced classification datasets, in which a small subset of the population -- a minority -- may contain features that correlate spuriously with the class label. For a parametric family of cross-entropy loss modifications and a representative Gaussian mixture model, we derive non-asymptotic generalization bounds on the worst-group error that shed light on the role of different hyper-parameters. Specifically, we prove that, when appropriately tuned, the recently proposed VS-loss learns a model that is fair towards minorities even when spurious features are strong. On the other hand, alternative heuristics, such as the weighted CE and the LA-loss, can fail dramatically. Compared to previous works, our bounds hold for more general models, they are non-asymptotic, and, they apply even at scenarios of extreme imbalance.

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arXiv2022

Multi-Environment Meta-Learning in Stochastic Linear Bandits

In this work we investigate meta-learning (or learning-to-learn) approaches in multi-task linear stochastic bandit problems that can originate from multiple environments. Inspired by the work of [1] on meta-learning in a sequence of linear bandit problems whose parameters are sampled from a single distribution (i.e., a single environment), here we consider the feasibility of meta-learning when task parameters are drawn from a mixture distribution instead. For this problem, we propose a regularized version of the OFUL algorithm that, when trained on tasks with labeled environments, achieves low regret on a new task without requiring knowledge of the environment from which the new task originates. Specifically, our regret bound for the new algorithm captures the effect of environment misclassification and highlights the benefits over learning each task separately or meta-learning without recognition of the distinct mixture components.

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arXiv2022

AutoBalance: Optimized Loss Functions for Imbalanced Data

Imbalanced datasets are commonplace in modern machine learning problems. The presence of under-represented classes or groups with sensitive attributes results in concerns about generalization and fairness. Such concerns are further exacerbated by the fact that large capacity deep nets can perfectly fit the training data and appear to achieve perfect accuracy and fairness during training, but perform poorly during test. To address these challenges, we propose AutoBalance, a bi-level optimization framework that automatically designs a training loss function to optimize a blend of accuracy and fairness-seeking objectives. Specifically, a lower-level problem trains the model weights, and an upper-level problem tunes the loss function by monitoring and optimizing the desired objective over the validation data. Our loss design enables personalized treatment for classes/groups by employing a parametric cross-entropy loss and individualized data augmentation schemes. We evaluate the benefits and performance of our approach for the application scenarios of imbalanced and group-sensitive classification. Extensive empirical evaluations demonstrate the benefits of AutoBalance over state-of-the-art approaches. Our experimental findings are complemented with theoretical insights on loss function design and the benefits of train-validation split. All code is available open-source.

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arXiv2021

Label-Imbalanced and Group-Sensitive Classification under Overparameterization

The goal in label-imbalanced and group-sensitive classification is to optimize relevant metrics such as balanced error and equal opportunity. Classical methods, such as weighted cross-entropy, fail when training deep nets to the terminal phase of training (TPT), that is training beyond zero training error. This observation has motivated recent flurry of activity in developing heuristic alternatives following the intuitive mechanism of promoting larger margin for minorities. In contrast to previous heuristics, we follow a principled analysis explaining how different loss adjustments affect margins. First, we prove that for all linear classifiers trained in TPT, it is necessary to introduce multiplicative, rather than additive, logit adjustments so that the interclass margins change appropriately. To show this, we discover a connection of the multiplicative CE modification to the cost-sensitive support-vector machines. Perhaps counterintuitively, we also find that, at the start of training, the same multiplicative weights can actually harm the minority classes. Thus, while additive adjustments are ineffective in the TPT, we show that they can speed up convergence by countering the initial negative effect of the multiplicative weights. Motivated by these findings, we formulate the vector-scaling (VS) loss, that captures existing techniques as special cases. Moreover, we introduce a natural extension of the VS-loss to group-sensitive classification, thus treating the two common types of imbalances (label/group) in a unifying way. Importantly, our experiments on state-of-the-art datasets are fully consistent with our theoretical insights and confirm the superior performance of our algorithms. Finally, for imbalanced Gaussian-mixtures data, we perform a generalization analysis, revealing tradeoffs between balanced / standard error and equal opportunity.

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