Behrad Moniri

Behrad Moniri, Donghwan Lee, Hamed Hassani et al.

Feature learning is thought to be one of the fundamental reasons for the success of deep neural networks. It is rigorously known that in two-layer fully-connected neural networks under certain conditions, one step of gradient descent on the first layer can lead to feature learning; characterized by the appearance of a separated rank-one component -- spike -- in the spectrum of the feature matrix. However, with a constant gradient descent step size, this spike only carries information from the linear component of the target function and therefore learning non-linear components is impossible. We show that with a learning rate that grows with the sample size, such training in fact introduces multiple rank-one components, each corresponding to a specific polynomial feature. We further prove that the limiting large-dimensional and large sample training and test errors of the updated neural networks are fully characterized by these spikes. By precisely analyzing the improvement in the training and test errors, we demonstrate that these non-linear features can enhance learning.

Donghwan Lee, Behrad Moniri, Xinmeng Huang et al.

Evaluating the performance of machine learning models under distribution shift is challenging, especially when we only have unlabeled data from the shifted (target) domain, along with labeled data from the original (source) domain. Recent work suggests that the notion of disagreement, the degree to which two models trained with different randomness differ on the same input, is a key to tackle this problem. Experimentally, disagreement and prediction error have been shown to be strongly connected, which has been used to estimate model performance. Experiments have led to the discovery of the disagreement-on-the-line phenomenon, whereby the classification error under the target domain is often a linear function of the classification error under the source domain; and whenever this property holds, disagreement under the source and target domain follow the same linear relation. In this work, we develop a theoretical foundation for analyzing disagreement in high-dimensional random features regression; and study under what conditions the disagreement-on-the-line phenomenon occurs in our setting. Experiments on CIFAR-10-C, Tiny ImageNet-C, and Camelyon17 are consistent with our theory and support the universality of the theoretical findings.

Behrad Moniri, Hamed Hassani

We study feature learning in two-layer neural networks within the linear-width regime, where the number of hidden neurons, sample size, and input dimension scale proportionally. While recent work has analyzed feature learning via a single step of gradient descent, such updates are fundamentally limited: they are approximately rank-one, capturing only a single direction, and require the target function to have an information exponent of one. In this paper, we go beyond one-step updates to provide a full characterization of the features learned during the second step of gradient descent with step-sizes $η_1 \asymp N^{α_1}$ and $η_2 \asymp N^{α_2}$ for $α_1, α_2 \in [0,0.5)$. We derive a sharp spectral characterization of the updated weights, demonstrating they behave as a spiked random matrix with multiple outliers, each corresponding to a learned direction. We show that the number of these outliers is determined by the scaling parameters $α_1$ and $α_2$ through $\lfloor \frac{α_2}{1/2 - α_1} \rfloor$. Furthermore, by analyzing the alignment between these learned directions and the target function, we identify a qualitative gap between training with independent versus reused batches. While independent batches restrict learning to directions with an information exponent of one, batch reuse enables the second update to capture directions even when the information exponent exceeds one, under the condition that $α_1, α_2$ are chosen properly. This confirms that the benefits of batch reuse, previously observed in finite-width regimes, persist in the high-dimensional linear-width limit. By characterizing these early-phase spectral transitions, our work establishes a tractable mathematical framework for studying optimization and feature learning phenomenology in modern overparameterized networks.

Thomas T. Zhang, Behrad Moniri, Ansh Nagwekar et al.

Layer-wise preconditioning methods are a family of memory-efficient optimization algorithms that introduce preconditioners per axis of each layer's weight tensors. These methods have seen a recent resurgence, demonstrating impressive performance relative to entry-wise ("diagonal") preconditioning methods such as Adam(W) on a wide range of neural network optimization tasks. Complementary to their practical performance, we demonstrate that layer-wise preconditioning methods are provably necessary from a statistical perspective. To showcase this, we consider two prototypical models, linear representation learning and single-index learning, which are widely used to study how typical algorithms efficiently learn useful features to enable generalization. In these problems, we show SGD is a suboptimal feature learner when extending beyond ideal isotropic inputs $\mathbf{x} \sim \mathsf{N}(\mathbf{0}, \mathbf{I})$ and well-conditioned settings typically assumed in prior work. We demonstrate theoretically and numerically that this suboptimality is fundamental, and that layer-wise preconditioning emerges naturally as the solution. We further show that standard tools like Adam preconditioning and batch-norm only mildly mitigate these issues, supporting the unique benefits of layer-wise preconditioning.

Behrad Moniri, Hamed Hassani

In this paper we study the asymptotics of linear regression in settings with non-Gaussian covariates where the covariates exhibit a linear dependency structure, departing from the standard assumption of independence. We model the covariates using stochastic processes with spatio-temporal covariance and analyze the performance of ridge regression in the high-dimensional proportional regime, where the number of samples and feature dimensions grow proportionally. A Gaussian universality theorem is proven, demonstrating that the asymptotics are invariant under replacing the non-Gaussian covariates with Gaussian vectors preserving mean and covariance, for which tools from random matrix theory can be used to derive precise characterizations of the estimation error. The estimation error is characterized by a fixed-point equation involving the spectral properties of the spatio-temporal covariance matrices, enabling efficient computation. We then study optimal regularization, overparameterization, and the double descent phenomenon in the context of dependent data. Simulations validate our theoretical predictions, shedding light on how dependencies influence estimation error and the choice of regularization parameters.

Behrad Moniri, Hamed Hassani

Weak-to-strong generalization, where a student model trained on imperfect labels generated by a weaker teacher nonetheless surpasses that teacher, has been widely observed but the mechanisms that enable it have remained poorly understood. In this paper, through a theoretical analysis of simple models, we uncover three core mechanisms that can drive this phenomenon. First, by analyzing ridge regression, we study the interplay between the teacher and student regularization and prove that a student can compensate for a teacher's under-regularization and achieve lower test error. We also analyze the role of the parameterization regime of the models. Second, by analyzing weighted ridge regression, we show that a student model with a regularization structure more aligned to the target, can outperform its teacher. Third, in a nonlinear multi-index setting, we demonstrate that a student can learn easy, task-specific features from the teacher while leveraging its own broader pre-training to learn hard-to-learn features that the teacher cannot capture.

Behrad Moniri, Hamed Hassani, Edgar Dobriban

Large Language Models (LLMs) are rapidly evolving and impacting various fields, necessitating the development of effective methods to evaluate and compare their performance. Most current approaches for performance evaluation are either based on fixed, domain-specific questions that lack the flexibility required in many real-world applications, or rely on human input, making them unscalable. To address these issues, we propose an automated benchmarking framework based on debates between LLMs, judged by another LLM. This method assesses not only domain knowledge, but also skills such as argumentative reasoning and inconsistency recognition. We evaluate the performance of various state-of-the-art LLMs using the debate framework and achieve rankings that align closely with popular rankings based on human input, eliminating the need for costly human crowdsourcing.

Hassan Hafez-Kolahi, Behrad Moniri, Shohreh Kasaei

Two main concepts studied in machine learning theory are generalization gap (difference between train and test error) and excess risk (difference between test error and the minimum possible error). While information-theoretic tools have been used extensively to study the generalization gap of learning algorithms, the information-theoretic nature of excess risk has not yet been fully investigated. In this paper, some steps are taken toward this goal. We consider the frequentist problem of minimax excess risk as a zero-sum game between the algorithm designer and the world. Then, we argue that it is desirable to modify this game in a way that the order of play can be swapped. We then prove that, under some regularity conditions, if the world and designer can play randomly the duality gap is zero and the order of play can be changed. In this case, a Bayesian problem surfaces in the dual representation. This makes it possible to utilize recent information-theoretic results on minimum excess risk in Bayesian learning to provide bounds on the minimax excess risk. We demonstrate the applicability of the results by providing information theoretic insight on two important classes of problems: classification when the hypothesis space has finite VC-dimension, and regularized least squares.

Hassan Hafez-Kolahi, Behrad Moniri, Shohreh Kasaei et al.

In parametric Bayesian learning, a prior is assumed on the parameter $W$ which determines the distribution of samples. In this setting, Minimum Excess Risk (MER) is defined as the difference between the minimum expected loss achievable when learning from data and the minimum expected loss that could be achieved if $W$ was observed. In this paper, we build upon and extend the recent results of (Xu & Raginsky, 2020) to analyze the MER in Bayesian learning and derive information-theoretic bounds on it. We formulate the problem as a (constrained) rate-distortion optimization and show how the solution can be bounded above and below by two other rate-distortion functions that are easier to study. The lower bound represents the minimum possible excess risk achievable by any process using $R$ bits of information from the parameter $W$. For the upper bound, the optimization is further constrained to use $R$ bits from the training set, a setting which relates MER to information-theoretic bounds on the generalization gap in frequentist learning. We derive information-theoretic bounds on the difference between these upper and lower bounds and show that they can provide order-wise tight rates for MER under certain conditions. This analysis gives more insight into the information-theoretic nature of Bayesian learning as well as providing novel bounds.