arXiv subjects

Murat A. Erdogdu

Publications and source records attributed to Murat A. Erdogdu.

At least 19 recordsLinked to original sources

arXiv2026

SGD in Multiclass Logistic Regression: Sequential Learning and Scaling Laws

We study the training dynamics of multiclass logistic regression on high-dimensional Gaussian mixture models with a large number of classes and establish precise scaling laws governing the cross-entropy risk under gradient-based optimization. We show that learning proceeds sequentially across classes, from most to least frequent. When the class priors follow a power law distribution, the risk dynamics decompose into three phases: an initial plateau until the first class is learned, a power-law decay regime during which sequential learning occurs, and a final convergence regime. We then analyze how model capacity interacts with optimization under a fixed compute budget. When the effective dimension is restricted via projection onto leading principal components, the risk decomposes into a capacity term (a power law in the retained dimension) and an optimization term (a power law in training time). Optimizing this tradeoff yields a compute-optimal scaling law for logistic regression, with explicit prescriptions for model size and training time as functions of compute. These results extend theoretical scaling laws from linear regression to multiclass classification, while connecting to empirical scaling laws observed in large-scale neural networks.

stat.ML↗

arXiv2026

Post-Training with Policy Gradients: Optimality and the Base Model Barrier

We study post-training linear autoregressive models with outcome and process rewards. Given a context $\boldsymbol{x}$, the model must predict the response $\boldsymbol{y} \in Y^N$, a sequence of length $N$ that satisfies a $γ$ margin condition, an extension of the standard separability to sequences. We prove that on test samples where the base model achieves a non-trivial likelihood $α$, a variant of policy gradient (PG) can achieve likelihood $1 - \varepsilon$ with an essentially minimax optimal number of reward queries $\tilde{O}((α^{-1} + \varepsilon^{-1})/γ^2)$. However, a barrier arises for going beyond the support of the base model. We prove that the overall expected error after post-training with outcome rewards is governed by a property of the base model called the Likelihood Quantile (LQ), and that variants of PG, while minimax optimal, may require a number of reward queries exponential in $N$ to go beyond this support, regardless of the pre-training algorithm. To overcome this barrier, we study post-training with a process reward model, and demonstrate how PG variants in this setting avoid the curse of dimensionality in $N$ via dependence on a token-level LQ. Along the way, we prove that under the margin condition, SGD with adaptive learning rate (LR) achieves a near optimal test error for statistical learning, and PG with adaptive LR achieves a near optimal number of mistakes for online learning while being computationally efficient whenever possible, both of which may be of independent interest.

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arXiv2025

Learning quadratic neural networks in high dimensions: SGD dynamics and scaling laws

We study the optimization and sample complexity of gradient-based training of a two-layer neural network with quadratic activation function in the high-dimensional regime, where the data is generated as $f_*(\boldsymbol{x}) \propto \sum_{j=1}^{r}λ_j σ\left(\langle \boldsymbol{θ_j}, \boldsymbol{x}\rangle\right), \boldsymbol{x} \sim N(0,\boldsymbol{I}_d)$, $σ$ is the 2nd Hermite polynomial, and $\lbrace\boldsymbolθ_j \rbrace_{j=1}^{r} \subset \mathbb{R}^d$ are orthonormal signal directions. We consider the extensive-width regime $r \asymp d^β$ for $β\in [0, 1)$, and assume a power-law decay on the (non-negative) second-layer coefficients $λ_j\asymp j^{-α}$ for $α\geq 0$. We present a sharp analysis of the SGD dynamics in the feature learning regime, for both the population limit and the finite-sample (online) discretization, and derive scaling laws for the prediction risk that highlight the power-law dependencies on the optimization time, sample size, and model width. Our analysis combines a precise characterization of the associated matrix Riccati differential equation with novel matrix monotonicity arguments to establish convergence guarantees for the infinite-dimensional effective dynamics.

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arXiv2025

From Information to Generative Exponent: Learning Rate Induces Phase Transitions in SGD

To understand feature learning dynamics in neural networks, recent theoretical works have focused on gradient-based learning of Gaussian single-index models, where the label is a nonlinear function of a latent one-dimensional projection of the input. While the sample complexity of online SGD is determined by the information exponent of the link function, recent works improved this by performing multiple gradient steps on the same sample with different learning rates -- yielding a non-correlational update rule -- and instead are limited by the (potentially much smaller) generative exponent. However, this picture is only valid when these learning rates are sufficiently large. In this paper, we characterize the relationship between learning rate(s) and sample complexity for a broad class of gradient-based algorithms that encapsulates both correlational and non-correlational updates. We demonstrate that, in certain cases, there is a phase transition from an "information exponent regime" with small learning rate to a "generative exponent regime" with large learning rate. Our framework covers prior analyses of one-pass SGD and SGD with batch reuse, while also introducing a new layer-wise training algorithm that leverages a two-timescales approach (via different learning rates for each layer) to go beyond correlational queries without reusing samples or modifying the loss from squared error. Our theoretical study demonstrates that the choice of learning rate is as important as the design of the algorithm in achieving statistical and computational efficiency.

cs.LG↗

arXiv2025

Learning Multi-Index Models with Neural Networks via Mean-Field Langevin Dynamics

We study the problem of learning multi-index models in high-dimensions using a two-layer neural network trained with the mean-field Langevin algorithm. Under mild distributional assumptions on the data, we characterize the effective dimension $d_{\mathrm{eff}}$ that controls both sample and computational complexity by utilizing the adaptivity of neural networks to latent low-dimensional structures. When the data exhibit such a structure, $d_{\mathrm{eff}}$ can be significantly smaller than the ambient dimension. We prove that the sample complexity grows almost linearly with $d_{\mathrm{eff}}$, bypassing the limitations of the information and generative exponents that appeared in recent analyses of gradient-based feature learning. On the other hand, the computational complexity may inevitably grow exponentially with $d_{\mathrm{eff}}$ in the worst-case scenario. Motivated by improving computational complexity, we take the first steps towards polynomial time convergence of the mean-field Langevin algorithm by investigating a setting where the weights are constrained to be on a compact manifold with positive Ricci curvature, such as the hypersphere. There, we study assumptions under which polynomial time convergence is achievable, whereas similar assumptions in the Euclidean setting lead to exponential time complexity.

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arXiv2025

Robust Feature Learning for Multi-Index Models in High Dimensions

Recently, there have been numerous studies on feature learning with neural networks, specifically on learning single- and multi-index models where the target is a function of a low-dimensional projection of the input. Prior works have shown that in high dimensions, the majority of the compute and data resources are spent on recovering the low-dimensional projection; once this subspace is recovered, the remainder of the target can be learned independently of the ambient dimension. However, implications of feature learning in adversarial settings remain unexplored. In this work, we take the first steps towards understanding adversarially robust feature learning with neural networks. Specifically, we prove that the hidden directions of a multi-index model offer a Bayes optimal low-dimensional projection for robustness against $\ell_2$-bounded adversarial perturbations under the squared loss, assuming that the multi-index coordinates are statistically independent from the rest of the coordinates. Therefore, robust learning can be achieved by first performing standard feature learning, then robustly tuning a linear readout layer on top of the standard representations. In particular, we show that adversarially robust learning is just as easy as standard learning. Specifically, the additional number of samples needed to robustly learn multi-index models when compared to standard learning does not depend on dimensionality.

stat.ML↗

arXiv2025

When Do Transformers Outperform Feedforward and Recurrent Networks? A Statistical Perspective

Theoretical efforts to prove advantages of Transformers in comparison with classical architectures such as feedforward and recurrent neural networks have mostly focused on representational power. In this work, we take an alternative perspective and prove that even with infinite compute, feedforward and recurrent networks may suffer from larger sample complexity compared to Transformers, as the latter can adapt to a form of dynamic sparsity. Specifically, we consider a sequence-to-sequence data generating model on sequences of length $N$, in which the output at each position depends only on $q$ relevant tokens with $q \ll N$, and the positions of these tokens are described in the input prompt. We prove that a single-layer Transformer can learn this model if and only if its number of attention heads is at least $q$, in which case it achieves a sample complexity almost independent of $N$, while recurrent networks require $N^{Ω(1)}$ samples on the same problem. If we simplify this model, recurrent networks may achieve a complexity almost independent of $N$, while feedforward networks still require $N$ samples. Consequently, our proposed sparse retrieval model illustrates a natural hierarchy in sample complexity across these architectures.

stat.ML↗

arXiv2024

On the Efficiency of ERM in Feature Learning

Given a collection of feature maps indexed by a set $\mathcal{T}$, we study the performance of empirical risk minimization (ERM) on regression problems with square loss over the union of the linear classes induced by these feature maps. This setup aims at capturing the simplest instance of feature learning, where the model is expected to jointly learn from the data an appropriate feature map and a linear predictor. We start by studying the asymptotic quantiles of the excess risk of sequences of empirical risk minimizers. Remarkably, we show that when the set $\mathcal{T}$ is not too large and when there is a unique optimal feature map, these quantiles coincide, up to a factor of two, with those of the excess risk of the oracle procedure, which knows a priori this optimal feature map and deterministically outputs an empirical risk minimizer from the associated optimal linear class. We complement this asymptotic result with a non-asymptotic analysis that quantifies the decaying effect of the global complexity of the set $\mathcal{T}$ on the excess risk of ERM, and relates it to the size of the sublevel sets of the suboptimality of the feature maps. As an application of our results, we obtain new guarantees on the performance of the best subset selection procedure in sparse linear regression under general assumptions.

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arXiv2024

Analysis of Langevin Monte Carlo from Poincaré to Log-Sobolev

Classically, the continuous-time Langevin diffusion converges exponentially fast to its stationary distribution $π$ under the sole assumption that $π$ satisfies a Poincaré inequality. Using this fact to provide guarantees for the discrete-time Langevin Monte Carlo (LMC) algorithm, however, is considerably more challenging due to the need for working with chi-squared or Rényi divergences, and prior works have largely focused on strongly log-concave targets. In this work, we provide the first convergence guarantees for LMC assuming that $π$ satisfies either a Latała--Oleszkiewicz or modified log-Sobolev inequality, which interpolates between the Poincaré and log-Sobolev settings. Unlike prior works, our results allow for weak smoothness and do not require convexity or dissipativity conditions.

math.ST↗

arXiv2024

Sampling from the Mean-Field Stationary Distribution

We study the complexity of sampling from the stationary distribution of a mean-field SDE, or equivalently, the complexity of minimizing a functional over the space of probability measures which includes an interaction term. Our main insight is to decouple the two key aspects of this problem: (1) approximation of the mean-field SDE via a finite-particle system, via uniform-in-time propagation of chaos, and (2) sampling from the finite-particle stationary distribution, via standard log-concave samplers. Our approach is conceptually simpler and its flexibility allows for incorporating the state-of-the-art for both algorithms and theory. This leads to improved guarantees in numerous settings, including better guarantees for optimizing certain two-layer neural networks in the mean-field regime. A key technical contribution is to establish a new uniform-in-$N$ log-Sobolev inequality for the stationary distribution of the mean-field Langevin dynamics.

math.ST↗

arXiv2024

Optimal Excess Risk Bounds for Empirical Risk Minimization on $p$-Norm Linear Regression

We study the performance of empirical risk minimization on the $p$-norm linear regression problem for $p \in (1, \infty)$. We show that, in the realizable case, under no moment assumptions, and up to a distribution-dependent constant, $O(d)$ samples are enough to exactly recover the target. Otherwise, for $p \in [2, \infty)$, and under weak moment assumptions on the target and the covariates, we prove a high probability excess risk bound on the empirical risk minimizer whose leading term matches, up to a constant that depends only on $p$, the asymptotically exact rate. We extend this result to the case $p \in (1, 2)$ under mild assumptions that guarantee the existence of the Hessian of the risk at its minimizer.

math.ST↗

arXiv2024

Minimax Linear Regression under the Quantile Risk

We study the problem of designing minimax procedures in linear regression under the quantile risk. We start by considering the realizable setting with independent Gaussian noise, where for any given noise level and distribution of inputs, we obtain the exact minimax quantile risk for a rich family of error functions and establish the minimaxity of OLS. This improves on the known lower bounds for the special case of square error, and provides us with a lower bound on the minimax quantile risk over larger sets of distributions. Under the square error and a fourth moment assumption on the distribution of inputs, we show that this lower bound is tight over a larger class of problems. Specifically, we prove a matching upper bound on the worst-case quantile risk of a variant of the recently proposed min-max regression procedure, thereby establishing its minimaxity, up to absolute constants. We illustrate the usefulness of our approach by extending this result to all $p$-th power error functions for $p \in (2, \infty)$. Along the way, we develop a generic analogue to the classical Bayesian method for lower bounding the minimax risk when working with the quantile risk, as well as a tight characterization of the quantiles of the smallest eigenvalue of the sample covariance matrix.

math.ST↗

arXiv2024

Beyond Labeling Oracles: What does it mean to steal ML models?

Model extraction attacks are designed to steal trained models with only query access, as is often provided through APIs that ML-as-a-Service providers offer. Machine Learning (ML) models are expensive to train, in part because data is hard to obtain, and a primary incentive for model extraction is to acquire a model while incurring less cost than training from scratch. Literature on model extraction commonly claims or presumes that the attacker is able to save on both data acquisition and labeling costs. We thoroughly evaluate this assumption and find that the attacker often does not. This is because current attacks implicitly rely on the adversary being able to sample from the victim model's data distribution. We thoroughly research factors influencing the success of model extraction. We discover that prior knowledge of the attacker, i.e., access to in-distribution data, dominates other factors like the attack policy the adversary follows to choose which queries to make to the victim model API. Our findings urge the community to redefine the adversarial goals of ME attacks as current evaluation methods misinterpret the ME performance.

cs.LG↗

arXiv2024

Pruning is Optimal for Learning Sparse Features in High-Dimensions

While it is commonly observed in practice that pruning networks to a certain level of sparsity can improve the quality of the features, a theoretical explanation of this phenomenon remains elusive. In this work, we investigate this by demonstrating that a broad class of statistical models can be optimally learned using pruned neural networks trained with gradient descent, in high-dimensions. We consider learning both single-index and multi-index models of the form $y = σ^*(\boldsymbol{V}^{\top} \boldsymbol{x}) + ε$, where $σ^*$ is a degree-$p$ polynomial, and $\boldsymbol{V} \in \mathbbm{R}^{d \times r}$ with $r \ll d$, is the matrix containing relevant model directions. We assume that $\boldsymbol{V}$ satisfies a certain $\ell_q$-sparsity condition for matrices and show that pruning neural networks proportional to the sparsity level of $\boldsymbol{V}$ improves their sample complexity compared to unpruned networks. Furthermore, we establish Correlational Statistical Query (CSQ) lower bounds in this setting, which take the sparsity level of $\boldsymbol{V}$ into account. We show that if the sparsity level of $\boldsymbol{V}$ exceeds a certain threshold, training pruned networks with a gradient descent algorithm achieves the sample complexity suggested by the CSQ lower bound. In the same scenario, however, our results imply that basis-independent methods such as models trained via standard gradient descent initialized with rotationally invariant random weights can provably achieve only suboptimal sample complexity.

stat.ML↗

arXiv2024

A Separation in Heavy-Tailed Sampling: Gaussian vs. Stable Oracles for Proximal Samplers

We study the complexity of heavy-tailed sampling and present a separation result in terms of obtaining high-accuracy versus low-accuracy guarantees i.e., samplers that require only $O(\log(1/\varepsilon))$ versus $Ω(\text{poly}(1/\varepsilon))$ iterations to output a sample which is $\varepsilon$-close to the target in $χ^2$-divergence. Our results are presented for proximal samplers that are based on Gaussian versus stable oracles. We show that proximal samplers based on the Gaussian oracle have a fundamental barrier in that they necessarily achieve only low-accuracy guarantees when sampling from a class of heavy-tailed targets. In contrast, proximal samplers based on the stable oracle exhibit high-accuracy guarantees, thereby overcoming the aforementioned limitation. We also prove lower bounds for samplers under the stable oracle and show that our upper bounds cannot be fundamentally improved.

math.ST↗

arXiv2023

Distributional Model Equivalence for Risk-Sensitive Reinforcement Learning

We consider the problem of learning models for risk-sensitive reinforcement learning. We theoretically demonstrate that proper value equivalence, a method of learning models which can be used to plan optimally in the risk-neutral setting, is not sufficient to plan optimally in the risk-sensitive setting. We leverage distributional reinforcement learning to introduce two new notions of model equivalence, one which is general and can be used to plan for any risk measure, but is intractable; and a practical variation which allows one to choose which risk measures they may plan optimally for. We demonstrate how our framework can be used to augment any model-free risk-sensitive algorithm, and provide both tabular and large-scale experiments to demonstrate its ability.

cs.LG↗

arXiv2023

Gradient-Based Feature Learning under Structured Data

Recent works have demonstrated that the sample complexity of gradient-based learning of single index models, i.e. functions that depend on a 1-dimensional projection of the input data, is governed by their information exponent. However, these results are only concerned with isotropic data, while in practice the input often contains additional structure which can implicitly guide the algorithm. In this work, we investigate the effect of a spiked covariance structure and reveal several interesting phenomena. First, we show that in the anisotropic setting, the commonly used spherical gradient dynamics may fail to recover the true direction, even when the spike is perfectly aligned with the target direction. Next, we show that appropriate weight normalization that is reminiscent of batch normalization can alleviate this issue. Further, by exploiting the alignment between the (spiked) input covariance and the target, we obtain improved sample complexity compared to the isotropic case. In particular, under the spiked model with a suitably large spike, the sample complexity of gradient-based training can be made independent of the information exponent while also outperforming lower bounds for rotationally invariant kernel methods.

stat.ML↗

arXiv2023

Towards a Complete Analysis of Langevin Monte Carlo: Beyond Poincaré Inequality

Langevin diffusions are rapidly convergent under appropriate functional inequality assumptions. Hence, it is natural to expect that with additional smoothness conditions to handle the discretization errors, their discretizations like the Langevin Monte Carlo (LMC) converge in a similar fashion. This research program was initiated by Vempala and Wibisono (2019), who established results under log-Sobolev inequalities. Chewi et al. (2022) extended the results to handle the case of Poincaré inequalities. In this paper, we go beyond Poincaré inequalities, and push this research program to its limit. We do so by establishing upper and lower bounds for Langevin diffusions and LMC under weak Poincaré inequalities that are satisfied by a large class of densities including polynomially-decaying heavy-tailed densities (i.e., Cauchy-type). Our results explicitly quantify the effect of the initializer on the performance of the LMC algorithm. In particular, we show that as the tail goes from sub-Gaussian, to sub-exponential, and finally to Cauchy-like, the dependency on the initial error goes from being logarithmic, to polynomial, and then finally to being exponential. This three-step phase transition is in particular unavoidable as demonstrated by our lower bounds, clearly defining the boundaries of LMC.

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