arXiv subjects

Tomer Galanti

Publications and source records attributed to Tomer Galanti.

At least 19 recordsLinked to original sources

arXiv2026

Distribution-Aware Programming: Learning Specialized Solvers from Experience

Many optimization problems are solved repeatedly on instances drawn from the same underlying distribution. In this setting, a system can begin with a general-purpose solver and use experience from previous instances to learn a cheaper way to solve future ones. We formalize this as \emph{distribution-aware programming}: samples from an unknown deployment distribution are used to produce executable solver code whose quality and runtime generalize to new instances. A simple analysis shows how sample access interpolates between distribution-oblivious and distribution-informed algorithm design, and when the offline cost of specialization is recovered through lower deployment cost. We instantiate this framework with an LLM agent that proposes structural hypotheses, analyzes training instances, and synthesizes specialized solver code; the LLM is used only before deployment. Across $21$ structured combinatorial-optimization distributions, the synthesized solvers achieve high quality while often replacing generic search or optimization with smaller distribution-specific computations. Against released PACE competition solvers, our method remains competitive while using substantially less runtime: $75$--$125\times$ less on PACE 2025 Dominating Set, $80$--$96\times$ less on Hitting Set, and more than $34{,}000\times$ less on the PACE 2024 OCM exact track. More broadly, the results suggest using AI not only to solve optimization problems, but to learn how recurring distributions should be solved.

cs.AI↗

arXiv2026

Tool Use Reduces Depth-Induced Collapse in OOD Reasoning

Many current paths to more advanced AI depend on the assumption that large language models (LLMs) can generalize learned relationships to solve complex, out-of-distribution (OOD) problems. However, this is not an easy quality to measure. For most real-world problems and benchmarks it suffices to exploit a few memorized subproblems or a small fraction of the available data to produce a correct solution. This is interpolation. Generalization requires the capacity to make use of all available data to solve problems without these shortcuts. To test this quality, we introduce a synthetic Boolean circuit reconstruction benchmark over $GF(2)$. We use an adversarial sampling oracle to block partial-information shortcuts, ensuring that each step requires the integration of all historical context with new evidence. Our evaluations reveal that standalone LLMs fall significantly short of sustained reasoning: as problem depth increases, both small and frontier models exhibit a catastrophic collapse in their ability to accurately predict the next logical step. However, we also demonstrate that tool synthesis provides a remedy to this depth-induced reasoning collapse. When allowed to generate, execute, and iteratively refine code, even small architectures can sustain accurate reasoning over long horizons and rival frontier model performance on this task. These results indicate that synthesizing tools plays a crucial role in generalization, beyond the conventional view of merely augmenting systems with specific capabilities.

cs.AI↗

arXiv2026

LLM Priors for ERM over Programs

We study program-learning methods that are efficient in both samples and computation. Classical learning theory suggests that when the target admits a short program description, for example a short piece of ``Python code'', it can be learned from few examples by ERM over the program class. However, this approach relies on enumerating candidate programs, which is typically exponential in the description length; gradient-based training avoids this explicit search but, for some families of short programs, can require exponentially many samples to succeed. We propose \textsc{LLM-PV}, a propose-and-verify recipe that enables ERM-style selection over a discrete program class without exhaustive enumeration: a pretrained LLM induces a proposal distribution over candidate programs, each proposal is executed and scored on a held-out validation set, and the best program is selected, with no gradient updates or validation feedback used to adapt the sampling distribution. Across algorithmic tasks including parity variants, pattern matching, and primality testing, \textsc{LLM-PV} often recovers the exact underlying rule from a small labeled set and generalizes far beyond the training sequence lengths, while SGD-trained transformers, fine-tuning, in-context learning, and classical ML baselines can fit the training data yet fail to generalize reliably. Together, these results suggest that pretrained LLM priors can serve as effective search biases for ERM, narrowing the gap between statistical and computational efficiency.

cs.LG↗

arXiv2026

Agentic Systems as Boosting Weak Reasoning Models

Can a committee of weak reasoning-model calls reach the performance of much stronger models? We study verifier-backed committee search as inference-time boosting for reasoning language models. The mechanism is not simply that ``more agents help'': samples expose latent correct solutions, while critics and comparators must recover them without access to the hidden verifier. We formalize this view by separating proposal coverage, local identifiability, progress, and diversity. We prove that coverage can be amplified by repeated sampling, but cannot by itself create useful critics or comparators; reliable amplification requires an additional local soundness signal, such as execution, proof checking, type checking, tests, or constraint solving. We give rank-based bounds showing when local selection errors compose into reliable trajectories, and characterize the proposer-side ceiling: oracle best-of-$k$ converges only to the mass of task slices on which the proposal system assigns nonzero useful probability. Empirically, on SWE-bench Verified, a single \texttt{GPT-5.4 nano} proposal solves $67.0\%$ of tasks. Using the same nano model, our critic--comparator orchestration reaches $76.4\%$ with $k=8$ proposals, matching the standalone performance of \texttt{Gemini 3 Pro} and \texttt{Claude Opus 4.5} Thinking and approaching the $79.0\%$ oracle best-of-$8$ upper bound. Thus, many correct patches are already present in weak-model proposal pools; the main challenge is selecting them. The remaining failures are mostly proposal-coverage failures, indicating shared blind spots that stronger selection alone cannot close.

cs.AI↗

arXiv2026

Directional Neural Collapse Explains Few-Shot Transfer in Self-Supervised Learning

Frozen self-supervised representations often transfer well with only a few labels across many semantic tasks. We argue that a single geometric quantity, \emph{directional} CDNV (decision-axis variance), sits at the core of two favorable behaviors: strong few-shot transfer within a task, and low interference across many tasks. We show that both emerge when variability \emph{along} class-separating directions is small. First, we prove sharp non-asymptotic multiclass generalization bounds for downstream classification whose leading term is the directional CDNV. The bounds include finite-shot corrections that cleanly separate intrinsic decision-axis variability from centroid-estimation error. Second, we link decision-axis collapse to multitask geometry: for independent balanced labelings, small directional CDNV across tasks forces the corresponding decision axes to be nearly orthogonal, helping a single representation support many tasks with minimal interference. Empirically, across SSL objectives, directional CDNV collapses during pretraining even when classical CDNV remains large, and our bounds closely track few-shot error at practical shot sizes. Additionally, on synthetic multitask data, we verify that SSL learns representations whose induced decision axes are nearly orthogonal. The code and project page of the paper are available at [\href{https://dlfundamentals.github.io/directional-neural-collapse/}{project page}].

cs.LG↗

arXiv2026

DisCO: Reinforcing Large Reasoning Models with Discriminative Constrained Optimization

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. 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. 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.

cs.LG↗

arXiv2025

Self-Supervised Contrastive Learning is Approximately Supervised Contrastive Learning

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. The code and project page of the paper are available at [\href{https://github.com/DLFundamentals/understanding-ssl}{code}, \href{https://dlfundamentals.github.io/ssl-is-approximately-sl/}{project page}].

cs.LG↗

arXiv2025

Scalable Principal-Agent Contract Design via Gradient-Based Optimization

We study a bilevel \emph{max-max} optimization framework for principal-agent contract design, in which a principal chooses incentives to maximize utility while anticipating the agent's best response. This problem, central to moral hazard and contract theory, underlies applications ranging from market design to delegated portfolio management, hedge fund fee structures, and executive compensation. While linear-quadratic models such as Holmstr"om-Milgrom admit closed-form solutions, realistic environments with nonlinear utilities, stochastic dynamics, or high-dimensional actions generally do not. We introduce a generic algorithmic framework that removes this reliance on closed forms. Our method adapts modern machine learning techniques for bilevel optimization -- using implicit differentiation with conjugate gradients (CG) -- to compute hypergradients efficiently through Hessian-vector products, without ever forming or inverting Hessians. In benchmark CARA-Normal (Constant Absolute Risk Aversion with Gaussian distribution of uncertainty) environments, the approach recovers known analytical optima and converges reliably from random initialization. More broadly, because it is matrix-free, variance-reduced, and problem-agnostic, the framework extends naturally to complex nonlinear contracts where closed-form solutions are unavailable, such as sigmoidal wage schedules (logistic pay), relative-performance/tournament compensation with common shocks, multi-task contracts with vector actions and heterogeneous noise, and CARA-Poisson count models with $\mathbb{E}[X\mid a]=e^{a}$. This provides a new computational tool for contract design, enabling systematic study of models that have remained analytically intractable.

cs.LG↗

arXiv2025

On the Alignment Between Supervised and Self-Supervised Contrastive Learning

Self-supervised contrastive learning (CL) has achieved remarkable empirical success, often producing representations that rival supervised pre-training on downstream tasks. Recent theory explains this by showing that the CL loss closely approximates a supervised surrogate, Negatives-Only Supervised Contrastive Learning (NSCL) loss, as the number of classes grows. Yet this loss-level similarity leaves an open question: {\em Do CL and NSCL also remain aligned at the representation level throughout training, not just in their objectives?} We address this by analyzing the representation alignment of CL and NSCL models trained under shared randomness (same initialization, batches, and augmentations). First, we show that their induced representations remain similar: specifically, we prove that the similarity matrices of CL and NSCL stay close under realistic conditions. Our bounds provide high-probability guarantees on alignment metrics such as centered kernel alignment (CKA) and representational similarity analysis (RSA), and they clarify how alignment improves with more classes, higher temperatures, and its dependence on batch size. In contrast, we demonstrate that parameter-space coupling is inherently unstable: divergence between CL and NSCL weights can grow exponentially with training time. Finally, we validate these predictions empirically, showing that CL-NSCL alignment strengthens with scale and temperature, and that NSCL tracks CL more closely than other supervised objectives. This positions NSCL as a principled bridge between self-supervised and supervised learning. Our code and project page are available at [\href{https://github.com/DLFundamentals/understanding_ssl_v2}{code}, \href{https://dlfundamentals.github.io/cl-nscl-representation-alignment/}{project page}].

cs.LG↗

arXiv2025

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

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.

cs.DC↗

arXiv2025

Formation of Representations in Neural Networks

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.

cs.LG↗

arXiv2024

SGD and Weight Decay Secretly Minimize the Rank of Your Neural Network

We investigate the inherent bias of Stochastic Gradient Descent (SGD) toward learning low-rank weight matrices during the training of deep neural networks. Our results demonstrate that training with mini-batch SGD and weight decay induces a bias toward rank minimization in the weight matrices. Specifically, we show both theoretically and empirically that this bias becomes more pronounced with smaller batch sizes, higher learning rates, or stronger weight decay. Additionally, we predict and empirically confirm that weight decay is essential for this bias to occur. Unlike previous literature, our analysis does not rely on assumptions about the data, convergence, or optimality of the weight matrices, making it applicable to a wide range of neural network architectures of any width or depth. Finally, we empirically explore the connection between this bias and generalization, finding that it has a marginal effect on the test performance.

cs.LG↗

arXiv2024

The Fair Language Model Paradox

Large Language Models (LLMs) are widely deployed in real-world applications, yet little is known about their training dynamics at the token level. Evaluation typically relies on aggregated training loss, measured at the batch level, which overlooks subtle per-token biases arising from (i) varying token-level dynamics and (ii) structural biases introduced by hyperparameters. While weight decay is commonly used to stabilize training, we reveal that it silently introduces performance biases detectable only at the token level. In fact, we empirically show across different dataset sizes, model architectures and sizes ranging from 270M to 3B parameters that as weight decay increases, low-frequency tokens are disproportionately depreciated. This is particularly concerning, as these neglected low-frequency tokens represent the vast majority of the token distribution in most languages, calling for novel regularization techniques that ensure fairness across all available tokens.

cs.CL↗

arXiv2024

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

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.

cs.CL↗

arXiv2024

Type-II Saddles and Probabilistic Stability of Stochastic Gradient Descent

Characterizing and understanding the dynamics of stochastic gradient descent (SGD) around saddle points remains an open problem. We first show that saddle points in neural networks can be divided into two types, among which the Type-II saddles are especially difficult to escape from because the gradient noise vanishes at the saddle. The dynamics of SGD around these saddles are thus to leading order described by a random matrix product process, and it is thus natural to study the dynamics of SGD around these saddles using the notion of probabilistic stability and the related Lyapunov exponent. Theoretically, we link the study of SGD dynamics to well-known concepts in ergodic theory, which we leverage to show that saddle points can be either attractive or repulsive for SGD, and its dynamics can be classified into four different phases, depending on the signal-to-noise ratio in the gradient close to the saddle.

cs.LG↗

arXiv2023

Generalization Bounds for Few-Shot Transfer Learning with Pretrained Classifiers

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. We offer a theoretical explanation for this behavior based on the recently discovered phenomenon of class-feature-variability collapse, that is, that during the training of deep classification networks the feature embeddings of samples belonging to the same class tend to concentrate around their class means. More specifically, we show that the few-shot error of the learned feature map on new classes (defined as the classification error of the nearest class-center classifier using centers learned from a small number of random samples from each new class) is small in case of class-feature-variability collapse, under the assumption that the classes are selected independently from a fixed distribution. This suggests that foundation models can provide feature maps that are transferable to new downstream tasks, even with very few samples; to our knowledge, this is the first performance bound for transfer-learning that is non-vacuous in the few-shot setting.

cs.LG↗

arXiv2023

Centered Self-Attention Layers

The self-attention mechanism in transformers and the message-passing mechanism in graph neural networks are repeatedly applied within deep learning architectures. We show that this application inevitably leads to oversmoothing, i.e., to similar representations at the deeper layers for different tokens in transformers and different nodes in graph neural networks. Based on our analysis, we present a correction term to the aggregating operator of these mechanisms. Empirically, this simple term eliminates much of the oversmoothing problem in visual transformers, obtaining performance in weakly supervised segmentation that surpasses elaborate baseline methods that introduce multiple auxiliary networks and training phrases. In graph neural networks, the correction term enables the training of very deep architectures more effectively than many recent solutions to the same problem.

cs.LG↗

arXiv2023

Reverse Engineering Self-Supervised Learning

Self-supervised learning (SSL) is a powerful tool in machine learning, but understanding the learned representations and their underlying mechanisms remains a challenge. This paper presents an in-depth empirical analysis of SSL-trained representations, encompassing diverse models, architectures, and hyperparameters. Our study reveals an intriguing aspect of the SSL training process: it inherently facilitates the clustering of samples with respect to semantic labels, which is surprisingly driven by the SSL objective's regularization term. This clustering process not only enhances downstream classification but also compresses the data information. Furthermore, we establish that SSL-trained representations align more closely with semantic classes rather than random classes. Remarkably, we show that learned representations align with semantic classes across various hierarchical levels, and this alignment increases during training and when moving deeper into the network. Our findings provide valuable insights into SSL's representation learning mechanisms and their impact on performance across different sets of classes.

cs.LG↗