Sharper Guarantees for Learning Neural Network Classifiers with Gradient Methods

In this paper, we study the data-dependent convergence and generalization behavior of gradient methods for neural networks with smooth activation. Our first result is a novel bound on the excess risk of deep networks trained by the logistic loss, via an alogirthmic stability analysis. Compared to previous works, our results improve upon the shortcomings of the well-established Rademacher complexity-based bounds. Importantly, the bounds we derive in this paper are tighter, hold even for neural networks of small width, do not scale unfavorably with width, are algorithm-dependent, and consequently capture the role of initialization on the sample complexity of gradient descent for deep nets. Specialized to noiseless data separable with margin $\gamma$ by neural tangent kernel (NTK) features of a network of width $\Omega(\poly(\log(n)))$, we show the test-error rate to be $e^{O(L)}/{\gamma^2 n}$, where $n$ is the training set size and $L$ denotes the number of hidden layers. This is an improvement in the test loss bound compared to previous works while maintaining the poly-logarithmic width conditions. We further investigate excess risk bounds for deep nets trained with noisy data, establishing that under a polynomial condition on the network width, gradient descent can achieve the optimal excess risk. Finally, we show that a large step-size significantly improves upon the NTK regime's results in classifying the XOR distribution. In particular, we show for a one-hidden-layer neural network of constant width $m$ with quadratic activation and standard Gaussian initialization that mini-batch SGD with linear sample complexity and with a large step-size $\eta=m$ reaches the perfect test accuracy after only $\ceil{\log(d)}$ iterations, where $d$ is the data dimension.

DARE the Extreme: Revisiting Delta-Parameter Pruning For Fine-Tuned Models

Storing open-source fine-tuned models separately introduces redundancy and increases response times in applications utilizing multiple models. Delta-parameter pruning (DPP), particularly the random drop and rescale (DARE) method proposed by Yu et al., addresses this by pruning the majority of delta parameters--the differences between fine-tuned and pre-trained model weights--while typically maintaining minimal performance loss. However, DARE fails when either the pruning rate or the magnitude of the delta parameters is large. We highlight two key reasons for this failure: (1) an excessively large rescaling factor as pruning rates increase, and (2) high mean and variance in the delta parameters. To push DARE's limits, we introduce DAREx (DARE the eXtreme), which features two algorithmic improvements: (1) DAREx-q, a rescaling factor modification that significantly boosts performance at high pruning rates (e.g., >30 % on COLA and SST2 for encoder models, with even greater gains in decoder models), and (2) DAREx-L2, which combines DARE with AdamR, an in-training method that applies appropriate delta regularization before DPP. We also demonstrate that DAREx-q can be seamlessly combined with vanilla parameter-efficient fine-tuning techniques like LoRA and can facilitate structural DPP. Additionally, we revisit the application of importance-based pruning techniques within DPP, demonstrating that they outperform random-based methods when delta parameters are large. Through this comprehensive study, we develop a pipeline for selecting the most appropriate DPP method under various practical scenarios.

Implicit Geometry of Next-token Prediction: From Language Sparsity Patterns to Model Representations

Next-token prediction (NTP) over large text corpora has become the go-to paradigm to train large language models. Yet, it remains unclear how NTP influences the mapping of linguistic patterns to geometric properties of the resulting model representations. We frame training of large language models as soft-label classification over sparse probabilistic label vectors, coupled with an analytical approximation that allows unrestricted generation of context embeddings. This approach links NTP training to rank-constrained, nuclear-norm regularized optimization in the logit domain, offering a framework for analyzing the geometry of word and context embeddings. In large embedding spaces, we find that NTP implicitly favors learning logits with a sparse plus low-rank structure. While the sparse component captures the co-occurrence frequency of context-word pairs, the orthogonal low-rank component, which becomes dominant as training progresses, depends solely on the sparsity pattern of the co-occurrence matrix. Consequently, when projected onto an appropriate subspace, representations of contexts that are followed by the same set of next-tokens collapse, a phenomenon we term subspace-collapse. We validate our findings on synthetic and small-scale real language datasets. Finally, we outline potential research directions aimed at deepening the understanding of NTP's influence on the learning of linguistic patterns and regularities.

Upper and lower memory capacity bounds of transformers for next-token prediction

Given a sequence of tokens, such as words, the task of next-token prediction is to predict the next-token conditional probability distribution. Decoder-only transformers have become effective models for this task, but their properties are still not fully understood. In particular, the largest number of distinct context sequences that a decoder-only transformer can interpolate next-token distributions for has not been established. To fill this gap, we prove upper and lower bounds on this number, which are equal up to a multiplicative constant. We prove these bounds in the general setting where next-token distributions can be arbitrary as well as the empirical setting where they are calculated from a finite number of document sequences. Our lower bounds are for one-layer transformers and our proofs highlight an important injectivity property satisfied by self-attention. Furthermore, we provide numerical evidence that the minimal number of parameters for memorization is sufficient for being able to train the model to the entropy lower bound.

Supervised Contrastive Representation Learning: Landscape Analysis with Unconstrained Features

Recent findings reveal that over-parameterized deep neural networks, trained beyond zero training-error, exhibit a distinctive structural pattern at the final layer, termed as Neural-collapse (NC). These results indicate that the final hidden-layer outputs in such networks display minimal within-class variations over the training set. While existing research extensively investigates this phenomenon under cross-entropy loss, there are fewer studies focusing on its contrastive counterpart, supervised contrastive (SC) loss. Through the lens of NC, this paper employs an analytical approach to study the solutions derived from optimizing the SC loss. We adopt the unconstrained features model (UFM) as a representative proxy for unveiling NC-related phenomena in sufficiently over-parameterized deep networks. We show that, despite the non-convexity of SC loss minimization, all local minima are global minima. Furthermore, the minimizer is unique (up to a rotation). We prove our results by formalizing a tight convex relaxation of the UFM. Finally, through this convex formulation, we delve deeper into characterizing the properties of global solutions under label-imbalanced training data.

Implicit Bias of Next-Token Prediction

Next-token prediction (NTP), the go-to training paradigm in training large language models, involves predicting the next token in a sequence. Departing from traditional one-hot classification, in NTP, multiple tokens with varying frequencies follow each given context. This work frames NTP training as cross-entropy minimization over distinct contexts, each associated with a sparse empirical probability vector across a finite vocabulary. It then addresses the following question: do gradient-based optimizers exhibit a bias towards solutions with specific structure as the NTP training loss reaches its lower bound (entropy)? Specifically, for linear NTP models trained using gradient descent (GD), we make the following contributions: Firstly, we determine NTP-separability conditions on the data, under which GD can attain its lower bound. We also demonstrate that these conditions hold under overparameterization. Secondly, we establish that the parameters of GD projected onto an appropriate data subspace converge to the unique solution of a system of linear equations, which requires the logits' difference of in-support tokens to be equal to the log-ratio of their respective probabilities. Meanwhile, on the orthogonal subspace, the parameters diverge and converge in the direction of the solution of a max-margin quadratic program, minimizing the Euclidean norm of parameters satisfying the \NTP-separability conditions. Akin to prior research on implicit bias of one-hot classification, our work opens exciting avenues for future research that can lead to better understanding optimization, generalization and robustness principles of models trained with NTP.

Content Conditional Debiasing for Fair Text Embedding

Mitigating biases in machine learning models has gained increasing attention in Natural Language Processing (NLP). Yet, only a few studies focus on fair text embeddings, which are crucial yet challenging for real-world applications. In this paper, we propose a novel method for learning fair text embeddings. We achieve fairness while maintaining utility trade-off by ensuring conditional independence between sensitive attributes and text embeddings conditioned on the content. Specifically, we enforce that embeddings of texts with different sensitive attributes but identical content maintain the same distance toward the embedding of their corresponding neutral text. Furthermore, we address the issue of lacking proper training data by using Large Language Models (LLMs) to augment texts into different sensitive groups. Our extensive evaluations demonstrate that our approach effectively improves fairness while preserving the utility of embeddings, representing a pioneering effort in achieving conditional independence for fair text embeddings.

Implicit Bias and Fast Convergence Rates for Self-attention

Self-attention, the core mechanism of transformers, distinguishes them from traditional neural networks and drives their outstanding performance. Towards developing the fundamental optimization principles of self-attention, we investigate the implicit bias of gradient descent (GD) in training a self-attention layer with fixed linear decoder in binary classification. Drawing inspiration from the study of GD in linear logistic regression over separable data, recent work demonstrates that as the number of iterations $t$ approaches infinity, the key-query matrix $W_t$ converges locally (with respect to the initialization direction) to a hard-margin SVM solution $W_{mm}$. Our work enhances this result in four aspects. Firstly, we identify non-trivial data settings for which convergence is provably global, thus shedding light on the optimization landscape. Secondly, we provide the first finite-time convergence rate for $W_t$ to $W_{mm}$, along with quantifying the rate of sparsification in the attention map. Thirdly, through an analysis of normalized GD and Polyak step-size, we demonstrate analytically that adaptive step-size rules can accelerate the convergence of self-attention. Additionally, we remove the restriction of prior work on a fixed linear decoder. Our results reinforce the implicit-bias perspective of self-attention and strengthen its connections to implicit-bias in linear logistic regression, despite the intricate non-convex nature of the former.

Class-attribute Priors: Adapting Optimization to Heterogeneity and Fairness Objective

Modern classification problems exhibit heterogeneities across individual classes: Each class may have unique attributes, such as sample size, label quality, or predictability (easy vs difficult), and variable importance at test-time. Without care, these heterogeneities impede the learning process, most notably, when optimizing fairness objectives. Confirming this, under a gaussian mixture setting, we show that the optimal SVM classifier for balanced accuracy needs to be adaptive to the class attributes. This motivates us to propose CAP: An effective and general method that generates a class-specific learning strategy (e.g. hyperparameter) based on the attributes of that class. This way, optimization process better adapts to heterogeneities. CAP leads to substantial improvements over the naive approach of assigning separate hyperparameters to each class. We instantiate CAP for loss function design and post-hoc logit adjustment, with emphasis on label-imbalanced problems. We show that CAP is competitive with prior art and its flexibility unlocks clear benefits for fairness objectives beyond balanced accuracy. Finally, we evaluate CAP on problems with label noise as well as weighted test objectives to showcase how CAP can jointly adapt to different heterogeneities.

Heterogeneous Federated Learning with Group-Aware Prompt Tuning

Transformers have achieved remarkable success in various machine-learning tasks, prompting their widespread adoption. In this paper, we explore their application in the context of federated learning (FL), with a particular focus on heterogeneous scenarios where individual clients possess diverse local datasets. To meet the computational and communication demands of FL, we leverage pre-trained Transformers and use an efficient prompt-tuning strategy. Our strategy introduces the concept of learning both shared and group prompts, enabling the acquisition of universal knowledge and group-specific knowledge simultaneously. Additionally, a prompt selection module assigns personalized group prompts to each input, aligning the global model with the data distribution of each client. This approach allows us to train a single global model that can automatically adapt to various local client data distributions without requiring local fine-tuning. In this way, our proposed method effectively bridges the gap between global and personalized local models in Federated Learning and surpasses alternative approaches that lack the capability to adapt to previously unseen clients. The effectiveness of our approach is rigorously validated through extensive experimentation and ablation studies.