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Tomer Galanti

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17 papers

2 author rows

NeurIPS Conference 2025 Conference Paper

DisCO: Reinforcing Large Reasoning Models with Discriminative Constrained Optimization

Gang Li
Ming Lin
Tomer Galanti
Zhengzhong Tu
Tianbao Yang

The recent success and openness of DeepSeek-R1 have brought widespread attention to Group Relative Policy Optimization (GRPO) as a reinforcement learning method for large reasoning models (LRMs). In this work, we analyze the GRPO objective under a binary reward setting and reveal an inherent limitation of question-level difficulty bias arising from its group relative advantage function. We also identify a connection between GRPO and traditional discriminative methods in supervised learning. Motivated by these insights, we introduce a new Discriminative Constrained Optimization (DisCO) framework for reinforcing LRMs, grounded in the principle of discriminative learning: increasing the scores of positive answers while decreasing those of negative ones. The main differences between DisCO and GRPO and its recent variants are: (1) it replaces the group relative objective with a discriminative objective defined by a scoring function; (2) it abandons clipping-based surrogates in favor of non-clipping RL surrogate objectives used as scoring functions; (3) it employs a simple yet effective constrained optimization approach to enforce the KL divergence constraint. As a result, DisCO offers notable advantages over GRPO and its variants: (i) it completely eliminates difficulty bias by adopting discriminative objectives; (ii) it addresses the entropy instability in GRPO and its variants through the use of non-clipping scoring functions and a constrained optimization approach, yielding long and stable training dynamics; (iii) it allows the incorporation of advanced discriminative learning techniques to address data imbalance, where a significant number of questions have more negative than positive generated answers during training. Our experiments on enhancing the mathematical reasoning capabilities of SFT-finetuned models show that DisCO significantly outperforms GRPO and its improved variants such as DAPO, achieving average gains of 7\% over GRPO and 6\% over DAPO across six benchmark tasks for a 1. 5B model.

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ICLR Conference 2025 Conference Paper

Distributed Speculative Inference (DSI): Speculation Parallelism for Provably Faster Lossless Language Model Inference

Nadav Timor
Jonathan Mamou
Daniel Korat
Moshe Berchansky
Oren Pereg
Moshe Wasserblat
Tomer Galanti
Michal Gordon-Kiwkowitz

This paper introduces *distributed speculative inference (DSI)*, a novel inference algorithm that is provably faster than speculative inference (SI) [leviathan2023, chen2023, miao2024, sun2025, timor2025] and standard autoregressive inference (non-SI). Like other SI algorithms, DSI operates on frozen language models (LMs), requiring no training or architectural modifications, and it preserves the target distribution. Prior studies on SI have demonstrated empirical speedups over non-SI—but rely on sufficiently fast and accurate drafters, which are often unavailable in practice. We identify a gap where SI can be slower than non-SI if drafters are too slow or inaccurate. We close this gap by proving that DSI is faster than both SI and non-SI—given any drafters. DSI is therefore not only faster than SI, but also unlocks the acceleration of LMs for which SI fails. DSI leverages *speculation parallelism (SP)*, a novel type of task parallelism, to orchestrate target and drafter instances that overlap in time, establishing a new foundational tradeoff between computational resources and latency. Our simulations show that DSI is 1.29-1.92x faster than SI in single-node setups for various off-the-shelf LMs and tasks. We open-source all our code.

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ICLR Conference 2025 Conference Paper

Formation of Representations in Neural Networks

Ziyin Liu 0001
Isaac L. Chuang
Tomer Galanti
Tomaso A. Poggio

Understanding neural representations will help open the black box of neural networks and advance our scientific understanding of modern AI systems. However, how complex, structured, and transferable representations emerge in modern neural networks has remained a mystery. Building on previous results, we propose the Canonical Representation Hypothesis (CRH), which posits a set of six alignment relations to universally govern the formation of representations in most hidden layers of a neural network. Under the CRH, the latent representations (R), weights (W), and neuron gradients (G) become mutually aligned during training. This alignment implies that neural networks naturally learn compact representations, where neurons and weights are invariant to task-irrelevant transformations. We then show that the breaking of CRH leads to the emergence of reciprocal power-law relations between R, W, and G, which we refer to as the Polynomial Alignment Hypothesis (PAH). We present a minimal-assumption theory proving that the balance between gradient noise and regularization is crucial for the emergence of the canonical representation. The CRH and PAH lead to an exciting possibility of unifying major key deep learning phenomena, including neural collapse and the neural feature ansatz, in a single framework.

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NeurIPS Conference 2025 Conference Paper

Self-Supervised Contrastive Learning is Approximately Supervised Contrastive Learning

Achleshwar Luthra
Tianbao Yang
Tomer Galanti

Despite its empirical success, the theoretical foundations of self-supervised contrastive learning (CL) are not yet fully established. In this work, we address this gap by showing that standard CL objectives implicitly approximate a supervised variant we call the negatives-only supervised contrastive loss (NSCL), which excludes same-class contrasts. We prove that the gap between the CL and NSCL losses vanishes as the number of semantic classes increases, under a bound that is both label-agnostic and architecture-independent. We characterize the geometric structure of the global minimizers of the NSCL loss: the learned representations exhibit augmentation collapse, within-class collapse, and class centers that form a simplex equiangular tight frame. We further introduce a new bound on the few-shot error of linear-probing. This bound depends on two measures of feature variability—within-class dispersion and variation along the line between class centers. We show that directional variation dominates the bound and that the within-class dispersion's effect diminishes as the number of labeled samples increases. These properties enable CL and NSCL-trained representations to support accurate few-shot label recovery using simple linear probes. Finally, we empirically validate our theoretical findings: the gap between CL and NSCL losses decays at a rate of $\mathcal{O}(\frac{1}{\#\text{classes}})$; the two losses are highly correlated; minimizing the CL loss implicitly brings the NSCL loss close to the value achieved by direct minimization; and the proposed few-shot error bound provides a tight estimate of probing performance in practice.

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NeurIPS Conference 2024 Conference Paper

On the Power of Decision Trees in Auto-Regressive Language Modeling

Yulu Gan
Tomer Galanti
Tomaso Poggio
Eran Malach

Originally proposed for handling time series data, Auto-regressive Decision Trees (ARDTs) have not yet been explored for language modeling. This paper delves into both the theoretical and practical applications of ARDTs in this new context. We theoretically demonstrate that ARDTs can compute complex functions, such as simulating automata, Turing machines, and sparse circuits, by leveraging "chain-of-thought" computations. Our analysis provides bounds on the size, depth, and computational efficiency of ARDTs, highlighting their surprising computational power. Empirically, we train ARDTs on simple language generation tasks, showing that they can learn to generate coherent and grammatically correct text on par with a smaller Transformer model. Additionally, we show that ARDTs can be used on top of transformer representations to solve complex reasoning tasks. This research reveals the unique computational abilities of ARDTs, aiming to broaden the architectural diversity in language model development.

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TMLR Journal 2023 Journal Article

Comparative Generalization Bounds for Deep Neural Networks

Tomer Galanti
Liane Galanti
Ido Ben-Shaul

In this work, we investigate the generalization capabilities of deep neural networks. We introduce a novel measure of the effective depth of neural networks, defined as the first layer at which sample embeddings are separable using the nearest-class center classifier. Our empirical results demonstrate that, in standard classification settings, neural networks trained using Stochastic Gradient Descent (SGD) tend to have small effective depths. We also explore the relationship between effective depth, the complexity of the training dataset, and generalization. For instance, we find that the effective depth of a trained neural network increases as the proportion of random labels in the data rises. Finally, we derive a generalization bound by comparing the effective depth of a network with the minimal depth required to fit the same dataset with partially corrupted labels. This bound provides non-vacuous predictions of test performance and is found to be empirically independent of the actual depth of the network.

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TMLR Journal 2023 Journal Article

Exploring the Approximation Capabilities of Multiplicative Neural Networks for Smooth Functions

Ido Ben-Shaul
Tomer Galanti
Shai Dekel

Multiplication layers are a key component in various influential neural network modules, including self-attention and hypernetwork layers. In this paper, we investigate the approximation capabilities of deep neural networks with intermediate neurons connected by simple multiplication operations. We consider two classes of target functions: generalized bandlimited functions, which are frequently used to model real-world signals with finite bandwidth, and Sobolev-Type balls, which are embedded in the Sobolev Space $\mathcal{W}^{r,2}$. Our results demonstrate that multiplicative neural networks can approximate these functions with significantly fewer layers and neurons compared to standard ReLU neural networks, with respect to both input dimension and approximation error. These findings suggest that multiplicative gates can outperform standard feed-forward layers and have potential for improving neural network design.

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ICML Conference 2023 Conference Paper

Feature learning in deep classifiers through Intermediate Neural Collapse

Akshay Rangamani
Marius Lindegaard
Tomer Galanti
Tomaso A. Poggio

In this paper, we conduct an empirical study of the feature learning process in deep classifiers. Recent research has identified a training phenomenon called Neural Collapse (NC), in which the top-layer feature embeddings of samples from the same class tend to concentrate around their means, and the top layer’s weights align with those features. Our study aims to investigate if these properties extend to intermediate layers. We empirically study the evolution of the covariance and mean of representations across different layers and show that as we move deeper into a trained neural network, the within-class covariance decreases relative to the between-class covariance. Additionally, we find that in the top layers, where the between-class covariance is dominant, the subspace spanned by the class means aligns with the subspace spanned by the most significant singular vector components of the weight matrix in the corresponding layer. Finally, we discuss the relationship between NC and Associative Memories (Willshaw et. al. 1969).

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NeurIPS Conference 2023 Conference Paper

Norm-based Generalization Bounds for Sparse Neural Networks

Tomer Galanti
Mengjia Xu
Liane Galanti
Tomaso Poggio

In this paper, we derive norm-based generalization bounds for sparse ReLU neural networks, including convolutional neural networks. These bounds differ from previous ones because they consider the sparse structure of the neural network architecture and the norms of the convolutional filters, rather than the norms of the (Toeplitz) matrices associated with the convolutional layers. Theoretically, we demonstrate that these bounds are significantly tighter than standard norm-based generalization bounds. Empirically, they offer relatively tight estimations of generalization for various simple classification problems. Collectively, these findings suggest that the sparsity of the underlying target function and the model's architecture plays a crucial role in the success of deep learning.

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NeurIPS Conference 2023 Conference Paper

Reverse Engineering Self-Supervised Learning

Ido Ben-Shaul
Ravid Shwartz-Ziv
Tomer Galanti
Shai Dekel
Yann LeCun

Understanding the learned representation and underlying mechanisms of Self-Supervised Learning (SSL) often poses a challenge. In this paper, we ‘reverse engineer’ SSL, conducting an in-depth empirical analysis of its learned internal representations, encompassing diverse models, architectures, and hyperparameters. Our study reveals an intriguing process within the SSL training: an inherent facilitation of semantic label-based clustering, which is surprisingly driven by the regularization component of the SSL objective. This clustering not only enhances downstream classification, but also compresses the information. We further illustrate that the alignment of the SSL-trained representation is more pronounced with semantic classes rather than random functions. Remarkably, the learned representations align with semantic classes across various hierarchical levels, with this alignment intensifying when going deeper into the network. This ‘reverse engineering’ approach provides valuable insights into the inner mechanism of SSL and their influences on the performance across different class sets.

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ICLR Conference 2022 Conference Paper

On the Role of Neural Collapse in Transfer Learning

Tomer Galanti
András György 0001
Marcus Hutter

We study the ability of foundation models to learn representations for classification that are transferable to new, unseen classes. Recent results in the literature show that representations learned by a single classifier over many classes are competitive on few-shot learning problems with representations learned by special-purpose algorithms designed for such problems. In this paper, we provide an explanation for this behavior based on the recently observed phenomenon that the features learned by overparameterized classification networks show an interesting clustering property, called neural collapse. We demonstrate both theoretically and empirically that neural collapse generalizes to new samples from the training classes, and -- more importantly -- to new classes as well, allowing foundation models to provide feature maps that work well in transfer learning and, specifically, in the few-shot setting.

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CLeaR Conference 2022 Conference Paper

Weakly Supervised Discovery of Semantic Attributes

Ameen Ali Ali
Tomer Galanti
Evgenii Zheltonozhskii
Chaim Baskin
Lior Wolf

We consider the problem of extracting semantic attributes, using only classification labels for supervision. For example, when learning to classify images of birds into species, we would like to observe the emergence of features used by zoologists to classify birds. To tackle this problem, we propose training a neural network with discrete features in the last layer, followed by two heads: a multi-layered perceptron (MLP) and a decision tree. The decision tree utilizes simple binary decision stumps, thus encouraging features to have semantic meaning. We present theoretical analysis, as well as a practical method for learning in the intersection of two hypothesis classes. Compared with various benchmarks, our results show an improved ability to extract a set of features highly correlated with a ground truth set of unseen attributes.

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NeurIPS Conference 2021 Conference Paper

Meta Internal Learning

Raphael Bensadoun
Shir Gur
Tomer Galanti
Lior Wolf

Internal learning for single-image generation is a framework, where a generator is trained to produce novel images based on a single image. Since these models are trained on a single image, they are limited in their scale and application. To overcome these issues, we propose a meta-learning approach that enables training over a collection of images, in order to model the internal statistics of the sample image more effectively. In the presented meta-learning approach, a single-image GAN model is generated given an input image, via a convolutional feedforward hypernetwork $f$. This network is trained over a dataset of images, allowing for feature sharing among different models, and for interpolation in the space of generative models. The generated single-image model contains a hierarchy of multiple generators and discriminators. It is therefore required to train the meta-learner in an adversarial manner, which requires careful design choices that we justify by a theoretical analysis. Our results show that the models obtained are as suitable as single-image GANs for many common image applications, {significantly reduce the training time per image without loss in performance}, and introduce novel capabilities, such as interpolation and feedforward modeling of novel images.

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UAI Conference 2021 Conference Paper

On random kernels of residual architectures

Etai Littwin
Tomer Galanti
Lior Wolf

We analyze the finite corrections to the neural tangent kernel (NTK) of residual and densely connected networks, as a function of both depth and width. Surprisingly, our analysis reveals that given a fixed depth, residual networks provide the best tradeoff between the parameter complexity and the coefficient of variation (normalized variance), followed by densely connected networks and vanilla MLPs. While in networks that do not use skip connections, convergence to the NTK requires one to fix the depth, while increasing the layers’ width. Our findings show that in ResNets, convergence to the NTK may occur when depth and width simultaneously tend to infinity, provided with a proper initialization. In DenseNets, however, the convergence of the NTK to its limit as the width tends to infinity is guaranteed, at a rate that is independent of both the depth and scale of the weights. Our experiments validate the theoretical results and demonstrate the advantage of deep ResNets and DenseNets for kernel regression with random gradient features.

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JMLR Journal 2021 Journal Article

Risk Bounds for Unsupervised Cross-Domain Mapping with IPMs

Tomer Galanti
Sagie Benaim
Lior Wolf

The recent empirical success of unsupervised cross-domain mapping algorithms, in mapping between two domains that share common characteristics, is not well-supported by theoretical justifications. This lacuna is especially troubling, given the clear ambiguity in such mappings. We work with adversarial training methods based on integral probability metrics (IPMs) and derive a novel risk bound, which upper bounds the risk between the learned mapping $h$ and the target mapping $y$, by a sum of three terms: (i) the risk between $h$ and the most distant alternative mapping that was learned by the same cross-domain mapping algorithm, (ii) the minimal discrepancy between the target domain and the domain obtained by applying a hypothesis $h^*$ on the samples of the source domain, where $h^*$ is a hypothesis selectable by the same algorithm, and (iii) an approximation error term that decreases as the capacity of the class of discriminators increases and is empirically shown to be small. The bound is directly related to Occam's razor and encourages the selection of the minimal architecture that supports a small mapping discrepancy. The bound leads to multiple algorithmic consequences, including a method for hyperparameter selection and early stopping in cross-domain mapping. [abs] [ pdf ][ bib ] [ code ] &copy JMLR 2021. ( edit, beta )

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NeurIPS Conference 2020 Conference Paper

On Infinite-Width Hypernetworks

Etai Littwin
Tomer Galanti
Lior Wolf
Greg Yang

{\em Hypernetworks} are architectures that produce the weights of a task-specific {\em primary network}. A notable application of hypernetworks in the recent literature involves learning to output functional representations. In these scenarios, the hypernetwork learns a representation corresponding to the weights of a shallow MLP, which typically encodes shape or image information. While such representations have seen considerable success in practice, they remain lacking in the theoretical guarantees in the wide regime of the standard architectures. In this work, we study wide over-parameterized hypernetworks. We show that unlike typical architectures, infinitely wide hypernetworks do not guarantee convergence to a global minima under gradient descent. We further show that convexity can be achieved by increasing the dimensionality of the hypernetwork's output, to represent wide MLPs. In the dually infinite-width regime, we identify the functional priors of these architectures by deriving their corresponding GP and NTK kernels, the latter of which we refer to as the {\em hyperkernel}. As part of this study, we make a mathematical contribution by deriving tight bounds on high order Taylor expansion terms of standard fully connected ReLU networks.

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NeurIPS Conference 2020 Conference Paper

On the Modularity of Hypernetworks

Tomer Galanti
Lior Wolf

In the context of learning to map an input $I$ to a function $h_I: \mathcal{X}\to \mathbb{R}$, two alternative methods are compared: (i) an embedding-based method, which learns a fixed function in which $I$ is encoded as a conditioning signal $e(I)$ and the learned function takes the form $h_I(x) = q(x, e(I))$, and (ii) hypernetworks, in which the weights $\theta_I$ of the function $h_I(x) = g(x; \theta_I)$ are given by a hypernetwork $f$ as $\theta_I=f(I)$. In this paper, we define the property of modularity as the ability to effectively learn a different function for each input instance $I$. For this purpose, we adopt an expressivity perspective of this property and extend the theory of~\cite{devore} and provide a lower bound on the complexity (number of trainable parameters) of neural networks as function approximators, by eliminating the requirements for the approximation method to be robust. Our results are then used to compare the complexities of $q$ and $g$, showing that under certain conditions and when letting the functions $e$ and $f$ be as large as we wish, $g$ can be smaller than $q$ by orders of magnitude. This sheds light on the modularity of hypernetworks in comparison with the embedding-based method. Besides, we show that for a structured target function, the overall number of trainable parameters in a hypernetwork is smaller by orders of magnitude than the number of trainable parameters of a standard neural network and an embedding method.

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