Piotr Zielinski

Satrajit Chatterjee, Piotr Zielinski

The generalization mystery in deep learning is the following: Why do over-parameterized neural networks trained with gradient descent (GD) generalize well on real datasets even though they are capable of fitting random datasets of comparable size? Furthermore, from among all solutions that fit the training data, how does GD find one that generalizes well (when such a well-generalizing solution exists)? We argue that the answer to both questions lies in the interaction of the gradients of different examples during training. Intuitively, if the per-example gradients are well-aligned, that is, if they are coherent, then one may expect GD to be (algorithmically) stable, and hence generalize well. We formalize this argument with an easy to compute and interpretable metric for coherence, and show that the metric takes on very different values on real and random datasets for several common vision networks. The theory also explains a number of other phenomena in deep learning, such as why some examples are reliably learned earlier than others, why early stopping works, and why it is possible to learn from noisy labels. Moreover, since the theory provides a causal explanation of how GD finds a well-generalizing solution when one exists, it motivates a class of simple modifications to GD that attenuate memorization and improve generalization. Generalization in deep learning is an extremely broad phenomenon, and therefore, it requires an equally general explanation. We conclude with a survey of alternative lines of attack on this problem, and argue that the proposed approach is the most viable one on this basis.

Sungmin Bae, Piotr Zielinski, Satrajit Chatterjee

Quantizing a Deep Neural Network (DNN) model to be used on a custom accelerator with efficient fixed-point hardware implementations, requires satisfying many stringent hardware-friendly quantization constraints to train the model. We evaluate the two main classes of hardware-friendly quantization methods in the context of weight quantization: the traditional Mean Squared Quantization Error (MSQE)-based methods and the more recent gradient-based methods. We study the two methods on MobileNetV1 and MobileNetV2 using multiple empirical metrics to identify the sources of performance differences between the two classes, namely, sensitivity to outliers and convergence instability of the quantizer scaling factor. Using those insights, we propose various techniques to improve the performance of both quantization methods - they fix the optimization instability issues present in the MSQE-based methods during quantization of MobileNet models and allow us to improve validation performance of the gradient-based methods by 4.0% and 3.3% for MobileNetV1 and MobileNetV2 on ImageNet respectively.

Erwei Wang, James J. Davis, Daniele Moro et al.

The ever-growing computational demands of increasingly complex machine learning models frequently necessitate the use of powerful cloud-based infrastructure for their training. Binary neural networks are known to be promising candidates for on-device inference due to their extreme compute and memory savings over higher-precision alternatives. However, their existing training methods require the concurrent storage of high-precision activations for all layers, generally making learning on memory-constrained devices infeasible. In this article, we demonstrate that the backward propagation operations needed for binary neural network training are strongly robust to quantization, thereby making on-the-edge learning with modern models a practical proposition. We introduce a low-cost binary neural network training strategy exhibiting sizable memory footprint reductions while inducing little to no accuracy loss vs Courbariaux & Bengio's standard approach. These decreases are primarily enabled through the retention of activations exclusively in binary format. Against the latter algorithm, our drop-in replacement sees memory requirement reductions of 3--5$\times$, while reaching similar test accuracy in comparable time, across a range of small-scale models trained to classify popular datasets. We also demonstrate from-scratch ImageNet training of binarized ResNet-18, achieving a 3.78$\times$ memory reduction. Our work is open-source, and includes the Raspberry Pi-targeted prototype we used to verify our modeled memory decreases and capture the associated energy drops. Such savings will allow for unnecessary cloud offloading to be avoided, reducing latency, increasing energy efficiency, and safeguarding end-user privacy.

Wolfgang Lehrach, Daniel Hennes, Miguel Lazaro-Gredilla et al.

Large Language Models (LLMs) reasoning abilities are increasingly being applied to classical board and card games, but the dominant approach -- involving prompting for direct move generation -- has significant drawbacks. It relies on the model's implicit fragile pattern-matching capabilities, leading to frequent illegal moves and strategically shallow play. Here we introduce an alternative approach: We use the LLM to translate natural language rules and game trajectories into a formal, executable world model represented as Python code. This generated model -- comprising functions for state transition, legal move enumeration, and termination checks -- serves as a verifiable simulation engine for high-performance planning algorithms like Monte Carlo tree search (MCTS). In addition, we prompt the LLM to generate heuristic value functions (to make MCTS more efficient), and inference functions (to estimate hidden states in imperfect information games). Our method offers three distinct advantages compared to directly using the LLM as a policy: (1) Verifiability: The generated CWM serves as a formal specification of the game's rules, allowing planners to algorithmically enumerate valid actions and avoid illegal moves, contingent on the correctness of the synthesized model; (2) Strategic Depth: We combine LLM semantic understanding with the deep search power of classical planners; and (3) Generalization: We direct the LLM to focus on the meta-task of data-to-code translation, enabling it to adapt to new games more easily. We evaluate our agent on 10 different games, of which 4 are novel and created for this paper. 5 of the games are fully observed (perfect information), and 5 are partially observed (imperfect information). We find that our method outperforms or matches Gemini 2.5 Pro in 9 out of the 10 considered games.

Satrajit Chatterjee, Piotr Zielinski

We propose a new metric ($m$-coherence) to experimentally study the alignment of per-example gradients during training. Intuitively, given a sample of size $m$, $m$-coherence is the number of examples in the sample that benefit from a small step along the gradient of any one example on average. We show that compared to other commonly used metrics, $m$-coherence is more interpretable, cheaper to compute ($O(m)$ instead of $O(m^2)$) and mathematically cleaner. (We note that $m$-coherence is closely connected to gradient diversity, a quantity previously used in some theoretical bounds.) Using $m$-coherence, we study the evolution of alignment of per-example gradients in ResNet and Inception models on ImageNet and several variants with label noise, particularly from the perspective of the recently proposed Coherent Gradients (CG) theory that provides a simple, unified explanation for memorization and generalization [Chatterjee, ICLR 20]. Although we have several interesting takeaways, our most surprising result concerns memorization. Naively, one might expect that when training with completely random labels, each example is fitted independently, and so $m$-coherence should be close to 1. However, this is not the case: $m$-coherence reaches much higher values during training (100s), indicating that over-parameterized neural networks find common patterns even in scenarios where generalization is not possible. A detailed analysis of this phenomenon provides both a deeper confirmation of CG, but at the same point puts into sharp relief what is missing from the theory in order to provide a complete explanation of generalization in neural networks.

Piotr Zielinski, Shankar Krishnan, Satrajit Chatterjee

Coherent Gradients (CGH) is a recently proposed hypothesis to explain why over-parameterized neural networks trained with gradient descent generalize well even though they have sufficient capacity to memorize the training set. The key insight of CGH is that, since the overall gradient for a single step of SGD is the sum of the per-example gradients, it is strongest in directions that reduce the loss on multiple examples if such directions exist. In this paper, we validate CGH on ResNet, Inception, and VGG models on ImageNet. Since the techniques presented in the original paper do not scale beyond toy models and datasets, we propose new methods. By posing the problem of suppressing weak gradient directions as a problem of robust mean estimation, we develop a coordinate-based median of means approach. We present two versions of this algorithm, M3, which partitions a mini-batch into 3 groups and computes the median, and a more efficient version RM3, which reuses gradients from previous two time steps to compute the median. Since they suppress weak gradient directions without requiring per-example gradients, they can be used to train models at scale. Experimentally, we find that they indeed greatly reduce overfitting (and memorization) and thus provide the first convincing evidence that CGH holds at scale. We also propose a new test of CGH that does not depend on adding noise to training labels or on suppressing weak gradient directions. Using the intuition behind CGH, we posit that the examples learned early in the training process (i.e., "easy" examples) are precisely those that have more in common with other training examples. Therefore, as per CGH, the easy examples should generalize better amongst themselves than the hard examples amongst themselves. We validate this hypothesis with detailed experiments, and believe that it provides further orthogonal evidence for CGH.