Samet Demir

Samet Demir, Zafer Dogan

Random feature models (RFMs), two-layer networks with a randomly initialized fixed first layer and a trained linear readout, are among the simplest nonlinear predictors. Prior asymptotic analyses in the proportional high-dimensional regime show that, under isotropic data, RFMs reduce to noisy linear models and offer no advantage over classical linear methods such as ridge regression. Yet RFMs frequently outperform linear baselines on structured real data. We show that this tension is explained by a correlation-driven phase transition: under spiked-covariance designs, the interaction between anisotropy and input-label correlation determines whether the RFM behaves as an effectively linear predictor or exhibits genuinely nonlinear gains. Concretely, we establish a universality principle under anisotropy and characterize the RFM generalization error via an equivalent noisy polynomial model. The effective degree of this polynomial, equivalently, which Hermite orders of the activation survive, is governed by the strength of input-label correlation, yielding an explicit boundary in the correlation-spike-magnitude plane. Below the boundary, the RFM collapses to a linear surrogate and can underperform strong linear baselines; above it, higher-order terms persist and the RFM achieves a clear nonlinear advantage. Numerical simulations and real-data experiments corroborate the theory and delineate the transition between these two regimes.

Samet Demir, Zafer Doğan

Random feature model with a nonlinear activation function has been shown to perform asymptotically equivalent to a Gaussian model in terms of training and generalization errors. Analysis of the equivalent model reveals an important yet not fully understood role played by the activation function. To address this issue, we study the "parameters" of the equivalent model to achieve improved generalization performance for a given supervised learning problem. We show that acquired parameters from the Gaussian model enable us to define a set of optimal nonlinearities. We provide two example classes from this set, e.g., second-order polynomial and piecewise linear functions. These functions are optimized to improve generalization performance regardless of the actual form. We experiment with regression and classification problems, including synthetic and real (e.g., CIFAR10) data. Our numerical results validate that the optimized nonlinearities achieve better generalization performance than widely-used nonlinear functions such as ReLU. Furthermore, we illustrate that the proposed nonlinearities also mitigate the so-called double descent phenomenon, which is known as the non-monotonic generalization performance regarding the sample size and the model size.

Samet Demir, Uras Mutlu, Özgur Özdemir

In this work, we tackle the problem of structured text generation, specifically academic paper generation in $\LaTeX{}$, inspired by the surprisingly good results of basic character-level language models. Our motivation is using more recent and advanced methods of language modeling on a more complex dataset of $\LaTeX{}$ source files to generate realistic academic papers. Our first contribution is preparing a dataset with $\LaTeX{}$ source files on recent open-source computer vision papers. Our second contribution is experimenting with recent methods of language modeling and text generation such as Transformer and Transformer-XL to generate consistent $\LaTeX{}$ code. We report cross-entropy and bits-per-character (BPC) results of the trained models, and we also discuss interesting points on some examples of the generated $\LaTeX{}$ code.

Eser Ilke Genc, Samet Demir, Zafer Dogan

Independent Component Analysis (ICA) is a foundational tool for unsupervised representation learning, yet its high-dimensional theory remains largely limited to single-component recovery. We develop an asymptotically exact mean-field theory for multi-component online ICA, capturing the coupling induced by simultaneous learning and orthogonalization. In the high-dimensional limit, the joint empirical distribution of learned estimates and ground-truth components converges to a deterministic process, yielding a closed ODE system for the overlap matrix between learned directions and true components. This characterization reveals a genuinely multi-component, initialization-driven phase structure: a decoupled regime, where estimates align with distinct components and evolve nearly independently, and a competition regime, where overlapping initializations induce orthogonality-driven conflicts, slow reorientation, and delayed convergence. Our steady-state analysis gives explicit learnability boundaries and competition conditions linking step size, data moments, and initialization. These conditions show that larger higher-order moments and competition shrink the stable learning-rate window, increase convergence times, and predict a staircase phenomenon in which the number of recoverable components changes discretely with the learning rate. Experiments on synthetic data and hyperspectral remote sensing data validate the predicted trajectories and phase behavior.

Samet Demir, Zafer Dogan

Many online learning algorithms, including classical online PCA methods, enforce explicit normalization steps that discard the evolving norm of the parameter vector. We show that this norm can in fact encode meaningful information about the underlying statistical structure of the problem, and that exploiting this information leads to improved learning behavior. Motivated by this principle, we introduce Implicitly Normalized Online PCA (INO-PCA), an online PCA algorithm that removes the unit-norm constraint and instead allows the parameter norm to evolve dynamically through a simple regularized update. We prove that in the high-dimensional limit the joint empirical distribution of the estimate and the true component converges to a deterministic measure-valued process governed by a nonlinear PDE. This analysis reveals that the parameter norm obeys a closed-form ODE coupled with the cosine similarity, forming an internal state variable that regulates learning rate, stability, and sensitivity to signal-to-noise ratio (SNR). The resulting dynamics uncover a three-way relationship between the norm, SNR, and optimal step size, and expose a sharp phase transition in steady-state performance. Both theoretically and experimentally, we show that INO-PCA consistently outperforms Oja's algorithm and adapts rapidly in non-stationary environments. Overall, our results demonstrate that relaxing norm constraints can be a principled and effective way to encode and exploit problem-relevant information in online learning algorithms.

Onat Ure, Samet Demir, Zafer Dogan

We study the effect of high-order statistics of data on the learning dynamics of neural networks (NNs) by using a moment-controllable non-Gaussian data model. Considering the expressivity of two-layer neural networks, we first construct the data model as a generative two-layer NN where the activation function is expanded by using Hermite polynomials. This allows us to achieve interpretable control over high-order cumulants such as skewness and kurtosis through the Hermite coefficients while keeping the data model realistic. Using samples generated from the data model, we perform controlled online learning experiments with a two-layer NN. Our results reveal a moment-wise progression in training: networks first capture low-order statistics such as mean and covariance, and progressively learn high-order cumulants. Finally, we pretrain the generative model on the Fashion-MNIST dataset and leverage the generated samples for further experiments. The results of these additional experiments confirm our conclusions and show the utility of the data model in a real-world scenario. Overall, our proposed approach bridges simplified data assumptions and practical data complexity, which offers a principled framework for investigating distributional effects in machine learning and signal processing.

Samet Demir, Zafer Dogan

Pretrained Transformers excel at in-context learning (ICL), inferring new tasks from only a handful of examples. Yet, their ICL performance can degrade sharply under distribution shift between pretraining and test data, a regime increasingly common in real-world deployments. While recent empirical work hints that adjusting the attention temperature in the softmax can enhance Transformer performance, the attention temperature's role in ICL under distribution shift remains unexplored. This paper provides the first theoretical and empirical study of attention temperature for ICL under distribution shift. Using a simplified but expressive "linearized softmax" framework, we derive closed-form generalization error expressions and prove that shifts in input covariance or label noise substantially impair ICL, but that an optimal attention temperature exists which minimizes this error. We then validate our predictions through extensive simulations on linear regression tasks and large-scale experiments with GPT-2 and LLaMA2-7B on question-answering benchmarks. Our results establish attention temperature as a principled and powerful mechanism for improving the robustness of ICL in pretrained Transformers, advancing theoretical understanding and providing actionable guidance for selecting attention temperature in practice.

Samet Demir, Zafer Dogan

We study the in-context learning (ICL) capabilities of pretrained Transformers in the setting of nonlinear regression. Specifically, we focus on a random Transformer with a nonlinear MLP head where the first layer is randomly initialized and fixed while the second layer is trained. Furthermore, we consider an asymptotic regime where the context length, input dimension, hidden dimension, number of training tasks, and number of training samples jointly grow. In this setting, we show that the random Transformer behaves equivalent to a finite-degree Hermite polynomial model in terms of ICL error. This equivalence is validated through simulations across varying activation functions, context lengths, hidden layer widths (revealing a double-descent phenomenon), and regularization settings. Our results offer theoretical and empirical insights into when and how MLP layers enhance ICL, and how nonlinearity and over-parameterization influence model performance.

Samet Demir, Zafer Dogan

In this work, we study the training and generalization performance of two-layer neural networks (NNs) after one gradient descent step under structured data modeled by Gaussian mixtures. While previous research has extensively analyzed this model under isotropic data assumption, such simplifications overlook the complexities inherent in real-world datasets. Our work addresses this limitation by analyzing two-layer NNs under Gaussian mixture data assumption in the asymptotically proportional limit, where the input dimension, number of hidden neurons, and sample size grow with finite ratios. We characterize the training and generalization errors by leveraging recent advancements in Gaussian universality. Specifically, we prove that a high-order polynomial model performs equivalent to the nonlinear neural networks under certain conditions. The degree of the equivalent model is intricately linked to both the "data spread" and the learning rate employed during one gradient step. Through extensive simulations, we demonstrate the equivalence between the original model and its polynomial counterpart across various regression and classification tasks. Additionally, we explore how different properties of Gaussian mixtures affect learning outcomes. Finally, we illustrate experimental results on Fashion-MNIST classification, indicating that our findings can translate to realistic data.

Samet Demir, Zafer Dogan

Pretrained Transformers demonstrate remarkable in-context learning (ICL) capabilities, enabling them to adapt to new tasks from demonstrations without parameter updates. However, theoretical studies often rely on simplified architectures (e.g., omitting MLPs), data models (e.g., linear regression with isotropic inputs), and single-source training, limiting their relevance to realistic settings. In this work, we study ICL in pretrained Transformers with nonlinear MLP heads on nonlinear tasks drawn from multiple data sources with heterogeneous input, task, and noise distributions. We analyze a model where the MLP comprises two layers, with the first layer trained via a single gradient step and the second layer fully optimized. Under high-dimensional asymptotics, we prove that such models are equivalent in ICL error to structured polynomial predictors, leveraging results from the theory of Gaussian universality and orthogonal polynomials. This equivalence reveals that nonlinear MLPs meaningfully enhance ICL performance, particularly on nonlinear tasks, compared to linear baselines. It also enables a precise analysis of data mixing effects: we identify key properties of high-quality data sources (low noise, structured covariances) and show that feature learning emerges only when the task covariance exhibits sufficient structure. These results are validated empirically across various activation functions, model sizes, and data distributions. Finally, we experiment with a real-world scenario involving multilingual sentiment analysis where each language is treated as a different source. Our experimental results for this case exemplify how our findings extend to real-world cases. Overall, our work advances the theoretical foundations of ICL in Transformers and provides actionable insight into the role of architecture and data in ICL.

Yigit E. Yildirim, Samet Demir, Zafer Dogan

Empirical Risk Minimization (ERM) is a foundational framework for supervised learning but primarily optimizes average-case performance, often neglecting fairness and robustness considerations. Tilted Empirical Risk Minimization (TERM) extends ERM by introducing an exponential tilt hyperparameter $t$ to balance average-case accuracy with worst-case fairness and robustness. However, in online or streaming settings where data arrive one sample at a time, the classical TERM objective degenerates to standard ERM, losing tilt sensitivity. We address this limitation by proposing an online TERM formulation that removes the logarithm from the classical objective, preserving tilt effects without additional computational or memory overhead. This formulation enables a continuous trade-off controlled by $t$, smoothly interpolating between ERM ($t \to 0$), fairness emphasis ($t > 0$), and robustness to outliers ($t < 0$). We empirically validate online TERM on two representative streaming tasks: robust linear regression with adversarial outliers and minority-class detection in binary classification. Our results demonstrate that negative tilting effectively suppresses outlier influence, while positive tilting improves recall with minimal impact on precision, all at per-sample computational cost equivalent to ERM. Online TERM thus recovers the full robustness-fairness spectrum of classical TERM in an efficient single-sample learning regime.

M. Oguzhan Gultekin, Samet Demir, Zafer Dogan

We investigate the impact of high-order moments on the learning dynamics of an online Independent Component Analysis (ICA) algorithm under a high-dimensional data model composed of a weighted sum of two non-Gaussian random variables. This model allows precise control of the input moment structure via a weighting parameter. Building on an existing ordinary differential equation (ODE)-based analysis in the high-dimensional limit, we demonstrate that as the high-order moments increase, the algorithm exhibits slower convergence and demands both a lower learning rate and greater initial alignment to achieve informative solutions. Our findings highlight the algorithm's sensitivity to the statistical structure of the input data, particularly its moment characteristics. Furthermore, the ODE framework reveals a critical learning rate threshold necessary for learning when moments approach their maximum. These insights motivate future directions in moment-aware initialization and adaptive learning rate strategies to counteract the degradation in learning speed caused by high non-Gaussianity, thereby enhancing the robustness and efficiency of ICA in complex, high-dimensional settings.

Samet Demir, Hasan Ferit Eniser, Alper Sen

Testing Deep Neural Network (DNN) models has become more important than ever with the increasing usage of DNN models in safety-critical domains such as autonomous cars. The traditional approach of testing DNNs is to create a test set, which is a random subset of the dataset about the problem of interest. This kind of approach is not enough for testing most of the real-world scenarios since these traditional test sets do not include corner cases, while a corner case input is generally considered to introduce erroneous behaviors. Recent works on adversarial input generation, data augmentation, and coverage-guided fuzzing (CGF) have provided new ways to extend traditional test sets. Among those, CGF aims to produce new test inputs by fuzzing existing ones to achieve high coverage on a test adequacy criterion (i.e. coverage criterion). Given that the subject test adequacy criterion is a well-established one, CGF can potentially find error inducing inputs for different underlying reasons. In this paper, we propose a novel CGF solution for structural testing of DNNs. The proposed fuzzer employs Monte Carlo Tree Search to drive the coverage-guided search in the pursuit of achieving high coverage. Our evaluation shows that the inputs generated by our method result in higher coverage than the inputs produced by the previously introduced coverage-guided fuzzing techniques.