Aryo Lotfi

Emmanuel Abbe, Samy Bengio, Aryo Lotfi et al. · apple-ml

This paper considers the learning of logical (Boolean) functions with a focus on the generalization on the unseen (GOTU) setting, a strong case of out-of-distribution generalization. This is motivated by the fact that the rich combinatorial nature of data in certain reasoning tasks (e.g., arithmetic/logic) makes representative data sampling challenging, and learning successfully under GOTU gives a first vignette of an 'extrapolating' or 'reasoning' learner. We study how different network architectures trained by (S)GD perform under GOTU and provide both theoretical and experimental evidence that for sparse functions and a class of network models including instances of Transformers, random features models, and linear networks, a min-degree-interpolator is learned on the unseen. More specifically, this means an interpolator of the training data that has minimal Fourier mass on the higher degree basis elements. These findings lead to two implications: (1) we provide an explanation to the length generalization problem for Boolean functions (e.g., Anil et al. 2022); (2) we introduce a curriculum learning algorithm called Degree-Curriculum that learns monomials more efficiently by incrementing supports. Finally, we discuss extensions to other models or non-sparse regimes where the min-degree bias may still occur or fade, as well as how it can be potentially corrected when undesirable.

Emmanuel Abbe, Samy Bengio, Elisabetta Cornacchia et al. · apple-ml

This paper considers the Pointer Value Retrieval (PVR) benchmark introduced in [ZRKB21], where a 'reasoning' function acts on a string of digits to produce the label. More generally, the paper considers the learning of logical functions with gradient descent (GD) on neural networks. It is first shown that in order to learn logical functions with gradient descent on symmetric neural networks, the generalization error can be lower-bounded in terms of the noise-stability of the target function, supporting a conjecture made in [ZRKB21]. It is then shown that in the distribution shift setting, when the data withholding corresponds to freezing a single feature (referred to as canonical holdout), the generalization error of gradient descent admits a tight characterization in terms of the Boolean influence for several relevant architectures. This is shown on linear models and supported experimentally on other models such as MLPs and Transformers. In particular, this puts forward the hypothesis that for such architectures and for learning logical functions such as PVR functions, GD tends to have an implicit bias towards low-degree representations, which in turn gives the Boolean influence for the generalization error under quadratic loss.

Emmanuel Abbe, Elisabetta Cornacchia, Aryo Lotfi

Experimental results have shown that curriculum learning, i.e., presenting simpler examples before more complex ones, can improve the efficiency of learning. Some recent theoretical results also showed that changing the sampling distribution can help neural networks learn parities, with formal results only for large learning rates and one-step arguments. Here we show a separation result in the number of training steps with standard (bounded) learning rates on a common sample distribution: if the data distribution is a mixture of sparse and dense inputs, there exists a regime in which a 2-layer ReLU neural network trained by a curriculum noisy-GD (or SGD) algorithm that uses sparse examples first, can learn parities of sufficiently large degree, while any fully connected neural network of possibly larger width or depth trained by noisy-GD on the unordered samples cannot learn without additional steps. We also provide experimental results supporting the qualitative separation beyond the specific regime of the theoretical results.

Ilia Mahrooghi, Aryo Lotfi, Emmanuel Abbe

Reinforcement learning has emerged as a powerful paradigm for unlocking reasoning capabilities in large language models. However, relying on sparse rewards makes this process highly sample-inefficient, as models must navigate vast search spaces with minimal feedback. While classic curriculum learning aims to mitigate this by ordering data based on complexity, the right ordering for a specific model is often unclear. To address this, we propose Goldilocks, a novel teacher-driven data sampling strategy that aims to predict each question's difficulty for the student model. The teacher model selects questions of appropriate difficulty for the student model, i.e., questions that are neither too easy nor too hard (Goldilocks principle), while training the student with GRPO. By leveraging the student's performance on seen samples, the teacher continuously adapts to the student's evolving abilities. On OpenMathReasoning dataset, Goldilocks data sampling improves the performance of models trained with standard GRPO under the same compute budget.

Mohammad Hossein Amani, Aryo Lotfi, Nicolas Mario Baldwin et al. · apple-ml

We propose that reinforcement learning (RL) from partial expert demonstrations is not merely a training heuristic, but a promising framework for solving complex sequence generation tasks. Supervised fine-tuning (SFT) relies on dense ground-truth labels, which become increasingly costly as sequence length grows. RL, on the other hand, struggles with sparse rewards and a combinatorially large output space. We address this by introducing adaptive backtracking (AdaBack), a per-sample curriculum learning algorithm that reveals only a partial prefix of the target output during training. The supervision length is adjusted dynamically for each sample based on the model's past reward signal, allowing it to incrementally learn to complete reasoning chains by conditioning on correct partial solutions. We investigate this intermediate regime between SFT and RL and argue that per-sample curriculum learning is more than a trade-off between efficiency and generality, it can succeed in tasks with long sequences of latent dependencies where SFT and RL both fail to generalize. Using a synthetic task with latent parity constraints, we show that our adaptive curriculum over partial answers reliably solves problems that are otherwise intractable. On mathematical reasoning benchmarks (MATH, GSM8k), we find that curriculum learning enables models to solve problems that RL alone cannot, acquiring new reasoning capabilities through incremental exposure to partial solutions.

Eran Malach, Omid Saremi, Sinead Williamson et al.

State Space Models (SSMs) have become the leading alternative to Transformers for sequence modeling. Their primary advantage is efficiency in long-context and long-form generation, enabled by fixed-size memory and linear scaling of computational complexity. We begin this work by showing a simple theoretical result stating that SSMs cannot accurately solve any ``truly long-form'' generation problem (in a sense we formally define), undermining their main competitive advantage. However, we show that this limitation can be mitigated by allowing SSMs interactive access to external tools. In fact, we show that given the right choice of tool access and problem-dependent training data, SSMs can learn to solve any tractable problem and generalize to arbitrary problem length/complexity (i.e., achieve length generalization). Following our theoretical finding, we demonstrate that tool-augmented SSMs achieve remarkable length generalization on a variety of arithmetic, reasoning, and coding tasks. These findings highlight SSMs as a potential efficient alternative to Transformers in interactive tool-based and agentic settings.

Emmanuel Abbe, Samy Bengio, Aryo Lotfi et al.

Can Transformers predict new syllogisms by composing established ones? More generally, what type of targets can be learned by such models from scratch? Recent works show that Transformers can be Turing-complete in terms of expressivity, but this does not address the learnability objective. This paper puts forward the notion of 'globality degree' of a target distribution to capture when weak learning is efficiently achievable by regular Transformers. This measure shows a contrast with the expressivity results of Transformers captured by $TC^0/TC^1$ classes (further studied here), since the globality relates to correlations with the more limited $NC^0$ class. We show here experimentally and theoretically under additional assumptions that distributions with high globality cannot be learned efficiently. In particular, syllogisms cannot be composed on long chains. Further, we develop scratchpad techniques and show that: (i) agnostic scratchpads cannot break the globality barrier, (ii) educated scratchpads can break the globality with intermediate steps, although not all such scratchpads can generalize out-of-distribution (OOD), (iii) a notion of 'inductive scratchpad', that composes the prior information more efficiently, can both break the globality barrier and improve the OOD generalization. In particular, some of our inductive scratchpads can achieve length generalizations of up to $6\times$ for some arithmetic tasks depending on the input formatting.

Sina Hajimiri, Aryo Lotfi, Mahdieh Soleymani Baghshah

In recent years, extending variational autoencoder's framework to learn disentangled representations has received much attention. We address this problem by proposing a framework capable of disentangling class-related and class-independent factors of variation in data. Our framework employs an attention mechanism in its latent space in order to improve the process of extracting class-related factors from data. We also deal with the multimodality of data distribution by utilizing mixture models as learnable prior distributions, as well as incorporating the Bhattacharyya coefficient in the objective function to prevent highly overlapping mixtures. Our model's encoder is further trained in a semi-supervised manner, with a small fraction of labeled data, to improve representations' interpretability. Experiments show that our framework disentangles class-related and class-independent factors of variation and learns interpretable features. Moreover, we demonstrate our model's performance with quantitative and qualitative results on various datasets.