arXiv subjects

Christos Thrampoulidis

Publications and source records attributed to Christos Thrampoulidis.

At least 37 records · Page 2Linked to original sources

arXiv2025

Leveraging Online Olympiad-Level Math Problems for LLMs Training and Contamination-Resistant Evaluation

Advances in Large Language Models (LLMs) have sparked interest in their ability to solve Olympiad-level math problems. However, the training and evaluation of these models are constrained by the limited size and quality of available datasets, as creating large-scale data for such advanced problems requires extensive effort from human experts. In addition, current benchmarks are prone to contamination, leading to unreliable evaluations. In this paper, we present an automated pipeline that leverages the rich resources of the Art of Problem Solving (AoPS) forum, which predominantly features Olympiad-level problems and community-driven solutions. Using open-source LLMs, we develop a method to extract question-answer pairs from the forum, resulting in AoPS-Instruct, a dataset of more than 600,000 high-quality QA pairs. Our experiments demonstrate that fine-tuning LLMs on AoPS-Instruct improves their reasoning abilities across various benchmarks. Moreover, we build an automatic pipeline that introduces LiveAoPSBench, an evolving evaluation set with timestamps, derived from the latest forum data, providing a contamination-resistant benchmark for assessing LLM performance. Notably, we observe a significant decline in LLM performance over time, suggesting their success on older examples may stem from pre-training exposure rather than true reasoning ability. Our work presents a scalable approach to creating and maintaining large-scale, high-quality datasets for advanced math reasoning, offering valuable insights into the capabilities and limitations of LLMs in this domain. Our benchmark and code is available at https://github.com/DSL-Lab/aops

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arXiv2025

Next-token prediction capacity: general upper bounds and a lower bound for transformers

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 multi-head decoder-only 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.

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arXiv2025

In-Context Occam's Razor: How Transformers Prefer Simpler Hypotheses on the Fly

In-context learning (ICL) enables transformers to adapt to new tasks through contextual examples without parameter updates. While existing research has typically studied ICL in fixed-complexity environments, practical language models encounter tasks spanning diverse complexity levels. This paper investigates how transformers navigate hierarchical task structures where higher-complexity categories can perfectly represent any pattern generated by simpler ones. We design well-controlled testbeds based on Markov chains and linear regression that reveal transformers not only identify the appropriate complexity level for each task but also accurately infer the corresponding parameters--even when the in-context examples are compatible with multiple complexity hypotheses. Notably, when presented with data generated by simpler processes, transformers consistently favor the least complex sufficient explanation. We theoretically explain this behavior through a Bayesian framework, demonstrating that transformers effectively implement an in-context Bayesian Occam's razor by balancing model fit against complexity penalties. We further ablate on the roles of model size, training mixture distribution, inference context length, and architecture. Finally, we validate this Occam's razor-like inductive bias on a pretrained GPT-4 model with Boolean-function tasks as case study, suggesting it may be inherent to transformers trained on diverse task distributions.

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arXiv2025

Thumb on the Scale: Optimal Loss Weighting in Last Layer Retraining

While machine learning models become more capable in discriminative tasks at scale, their ability to overcome biases introduced by training data has come under increasing scrutiny. Previous results suggest that there are two extremes of parameterization with very different behaviors: the population (underparameterized) setting where loss weighting is optimal and the separable overparameterized setting where loss weighting is ineffective at ensuring equal performance across classes. This work explores the regime of last layer retraining (LLR) in which the unseen limited (retraining) data is frequently inseparable and the model proportionately sized, falling between the two aforementioned extremes. We show, in theory and practice, that loss weighting is still effective in this regime, but that these weights \emph{must} take into account the relative overparameterization of the model.

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arXiv2025

On the Effect of Negative Gradient in Group Relative Deep Reinforcement Optimization

Reinforcement learning (RL) has become popular in enhancing the reasoning capabilities of large language models (LLMs), with Group Relative Policy Optimization (GRPO) emerging as a widely used algorithm in recent systems. Despite GRPO's widespread adoption, we identify a previously unrecognized phenomenon we term Lazy Likelihood Displacement (LLD), wherein the likelihood of correct responses marginally increases or even decreases during training. This behavior mirrors a recently discovered misalignment issue in Direct Preference Optimization (DPO), attributed to the influence of negative gradients. We provide a theoretical analysis of GRPO's learning dynamic, identifying the source of LLD as the naive penalization of all tokens in incorrect responses with the same strength. To address this, we develop a method called NTHR, which downweights penalties on tokens contributing to the LLD. Unlike prior DPO-based approaches, NTHR takes advantage of GRPO's group-based structure, using correct responses as anchors to identify influential tokens. Experiments on math reasoning benchmarks demonstrate that NTHR effectively mitigates LLD, yielding consistent performance gains across models ranging from 0.5B to 3B parameters.

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arXiv2025

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.

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arXiv2025

Implicit Bias and Fast Convergence Rates for Self-attention

We study the fundamental optimization principles of self-attention, the defining mechanism of transformers, by analyzing the implicit bias of gradient-based optimizers in training a self-attention layer with a linear decoder in binary classification. Building on prior studies in linear logistic regression, recent findings demonstrate that the key-query matrix $W_t$ from gradient-descent (GD) converges in direction towards $W_{mm}$, which maximizes the margin between optimal and non-optimal tokens across sequences. However, this convergence is local, dependent on initial conditions, only holds asymptotically as the number of iterations increases, and leaves questions about the potential benefits of adaptive step-size rules unaddressed. To bridge this gap, we first establish scenarios for which convergence is provably \emph{global}. We then analyze two adaptive step-size strategies: normalized GD and Polyak step-size, demonstrating \emph{finite-time} convergence rates for $W_t$ to $W_{mm}$, and quantifying the sparsification rate of the attention map. These findings not only show that these strategies can accelerate parameter convergence over standard GD in a non-convex setting but also deepen the understanding of the implicit bias in self-attention, linking it more closely to the phenomena observed in linear logistic regression despite its intricate non-convex nature.

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arXiv2025

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.

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arXiv2024

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 $γ$ by neural tangent kernel (NTK) features of a network of width $Ω(\text{poly}(\log(n)))$, we show the test-error rate to be $e^{O(L)}/{γ^2 n}$, 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 $η=m$ reaches the perfect test accuracy after only $\ceil{\log(d)}$ iterations, where $d$ is the data dimension.

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arXiv2024

Implicit Optimization Bias of Next-Token Prediction in Linear Models

We initiate an investigation into the optimization properties of next-token prediction (NTP), the dominant training paradigm for modern language models. Specifically, we study the structural properties of the solutions selected by gradient-based optimizers among the many possible minimizers of the NTP objective. By framing NTP as cross-entropy minimization across distinct contexts, each tied with a sparse conditional probability distribution across a finite vocabulary of tokens, we introduce "NTP-separability conditions" that enable reaching the data-entropy lower bound. With this setup, and focusing on linear models with fixed context embeddings, we characterize the optimization bias of gradient descent (GD): Within the data subspace defined by the sparsity patterns of distinct contexts, GD selects parameters that equate the logits' differences of in-support tokens to their log-odds. In the orthogonal subspace, the GD parameters diverge in norm and select the direction that maximizes a margin specific to NTP. These findings extend previous research on implicit bias in one-hot classification to the NTP setting, highlighting key differences and prompting further research into the optimization and generalization properties of NTP, irrespective of the specific architecture used to generate the context embeddings.

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arXiv2024

On the Optimization and Generalization of Multi-head Attention

The training and generalization dynamics of the Transformer's core mechanism, namely the Attention mechanism, remain under-explored. Besides, existing analyses primarily focus on single-head attention. Inspired by the demonstrated benefits of overparameterization when training fully-connected networks, we investigate the potential optimization and generalization advantages of using multiple attention heads. Towards this goal, we derive convergence and generalization guarantees for gradient-descent training of a single-layer multi-head self-attention model, under a suitable realizability condition on the data. We then establish primitive conditions on the initialization that ensure realizability holds. Finally, we demonstrate that these conditions are satisfied for a simple tokenized-mixture model. We expect the analysis can be extended to various data-model and architecture variations.

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arXiv2024

LLM-Assisted Content Conditional Debiasing for Fair Text Embedding

Mitigating biases in machine learning models has become an increasing concern in Natural Language Processing (NLP), particularly in developing fair text embeddings, which are crucial yet challenging for real-world applications like search engines. In response, this paper proposes a novel method for learning fair text embeddings. First, we define a novel content-conditional equal distance (CCED) fairness for text embeddings, ensuring content-conditional independence between sensitive attributes and text embeddings. Building on CCED, we introduce a content-conditional debiasing (CCD) loss to ensure that embeddings of texts with different sensitive attributes but identical content maintain the same distance from the embedding of their corresponding neutral text. Additionally, we tackle the issue of insufficient training data by using Large Language Models (LLMs) with instructions to fairly augment texts into different sensitive groups. Our extensive evaluations show that our approach effectively enhances fairness while maintaining the utility of embeddings. Furthermore, our augmented dataset, combined with the CCED metric, serves as an new benchmark for evaluating fairness.

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arXiv2024

Memory capacity of two layer neural networks with smooth activations

Determining the memory capacity of two layer neural networks with $m$ hidden neurons and input dimension $d$ (i.e., $md+2m$ total trainable parameters), which refers to the largest size of general data the network can memorize, is a fundamental machine learning question. For activations that are real analytic at a point and, if restricting to a polynomial there, have sufficiently high degree, we establish a lower bound of $\lfloor md/2\rfloor$ and optimality up to a factor of approximately $2$. All practical activations, such as sigmoids, Heaviside, and the rectified linear unit (ReLU), are real analytic at a point. Furthermore, the degree condition is mild, requiring, for example, that $\binom{k+d-1}{d-1}\ge n$ if the activation is $x^k$. Analogous prior results were limited to Heaviside and ReLU activations -- our result covers almost everything else. In order to analyze general activations, we derive the precise generic rank of the network's Jacobian, which can be written in terms of Hadamard powers and the Khatri-Rao product. Our analysis extends classical linear algebraic facts about the rank of Hadamard powers. Overall, our approach differs from prior works on memory capacity and holds promise for extending to deeper models and other architectures.

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arXiv2024

Memorization Capacity of Multi-Head Attention in Transformers

Transformers have become the go-to architecture for language and vision tasks, yet their theoretical properties, especially memorization capacity, remain elusive. This paper investigates the memorization abilities of multi-head attention mechanisms, examining how many example sequences they can memorize, as a function of the number of heads and sequence length. Motivated by experimental findings on vision transformers, we introduce novel assumptions about the linear independence of input data, distinct from the commonly used general-position assumption. Under these assumptions, we demonstrate that an attention layer with $H$ heads, dimension $d$, and context size $n < d$, featuring $Θ(Hd^2)$ parameters, can memorize $Ω(Hn)$ examples. Our analysis sheds light on how different attention heads handle various example sequences, aided by the softmax operator's saturation property. We validate our findings through experiments on synthetic data.

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arXiv2024

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.

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arXiv2024

Unlocking the Potential of Prompt-Tuning in Bridging Generalized and Personalized Federated Learning

Vision Transformers (ViT) and Visual Prompt Tuning (VPT) achieve state-of-the-art performance with improved efficiency in various computer vision tasks. This suggests a promising paradigm shift of adapting pre-trained ViT models to Federated Learning (FL) settings. However, the challenge of data heterogeneity among FL clients presents a significant hurdle in effectively deploying ViT models. Existing Generalized FL (GFL) and Personalized FL (PFL) methods have limitations in balancing performance across both global and local data distributions. In this paper, we present a novel algorithm, SGPT, that integrates GFL and PFL approaches by employing a unique combination of both shared and group-specific prompts. This design enables SGPT to capture both common and group-specific features. A key feature of SGPT is its prompt selection module, which facilitates the training of a single global model capable of automatically adapting to diverse local client data distributions without the need for local fine-tuning. To effectively train the prompts, we utilize block coordinate descent (BCD), learning from common feature information (shared prompts), and then more specialized knowledge (group prompts) iteratively. Theoretically, we justify that learning the proposed prompts can reduce the gap between global and local performance. Empirically, we conduct experiments on both label and feature heterogeneity settings in comparison with state-of-the-art baselines, along with extensive ablation studies, to substantiate the superior performance of SGPT.

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arXiv2024

Transformers as Support Vector Machines

Since its inception in "Attention Is All You Need", transformer architecture has led to revolutionary advancements in NLP. The attention layer within the transformer admits a sequence of input tokens $X$ and makes them interact through pairwise similarities computed as softmax$(XQK^\top X^\top)$, where $(K,Q)$ are the trainable key-query parameters. In this work, we establish a formal equivalence between the optimization geometry of self-attention and a hard-margin SVM problem that separates optimal input tokens from non-optimal tokens using linear constraints on the outer-products of token pairs. This formalism allows us to characterize the implicit bias of 1-layer transformers optimized with gradient descent: (1) Optimizing the attention layer with vanishing regularization, parameterized by $(K,Q)$, converges in direction to an SVM solution minimizing the nuclear norm of the combined parameter $W=KQ^\top$. Instead, directly parameterizing by $W$ minimizes a Frobenius norm objective. We characterize this convergence, highlighting that it can occur toward locally-optimal directions rather than global ones. (2) Complementing this, we prove the local/global directional convergence of gradient descent under suitable geometric conditions. Importantly, we show that over-parameterization catalyzes global convergence by ensuring the feasibility of the SVM problem and by guaranteeing a benign optimization landscape devoid of stationary points. (3) While our theory applies primarily to linear prediction heads, we propose a more general SVM equivalence that predicts the implicit bias with nonlinear heads. Our findings are applicable to arbitrary datasets and their validity is verified via experiments. We also introduce several open problems and research directions. We believe these findings inspire the interpretation of transformers as a hierarchy of SVMs that separates and selects optimal tokens.

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arXiv2024

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.

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