Itai Shapira

Nikhil Vyas, Depen Morwani, Rosie Zhao et al.

There is growing evidence of the effectiveness of Shampoo, a higher-order preconditioning method, over Adam in deep learning optimization tasks. However, Shampoo's drawbacks include additional hyperparameters and computational overhead when compared to Adam, which only updates running averages of first- and second-moment quantities. This work establishes a formal connection between Shampoo (implemented with the 1/2 power) and Adafactor -- a memory-efficient approximation of Adam -- showing that Shampoo is equivalent to running Adafactor in the eigenbasis of Shampoo's preconditioner. This insight leads to the design of a simpler and computationally efficient algorithm: $\textbf{S}$hampo$\textbf{O}$ with $\textbf{A}$dam in the $\textbf{P}$reconditioner's eigenbasis (SOAP). With regards to improving Shampoo's computational efficiency, the most straightforward approach would be to simply compute Shampoo's eigendecomposition less frequently. Unfortunately, as our empirical results show, this leads to performance degradation that worsens with this frequency. SOAP mitigates this degradation by continually updating the running average of the second moment, just as Adam does, but in the current (slowly changing) coordinate basis. Furthermore, since SOAP is equivalent to running Adam in a rotated space, it introduces only one additional hyperparameter (the preconditioning frequency) compared to Adam. We empirically evaluate SOAP on language model pre-training with 360m and 660m sized models. In the large batch regime, SOAP reduces the number of iterations by over 40% and wall clock time by over 35% compared to AdamW, with approximately 20% improvements in both metrics compared to Shampoo. An implementation of SOAP is available at https://github.com/nikhilvyas/SOAP.

Hadi Khalaf, Serena L. Wang, Daniel Halpern et al. · harvard

The standard way to evaluate language models on subjective tasks is through pairwise comparisons: an annotator chooses the "better" of two responses to a prompt. Leaderboards aggregate these comparisons into a single Bradley-Terry (BT) ranking, forcing heterogeneous preferences into a total order and violating basic social-choice desiderata. In contrast, social choice theory provides an alternative approach called maximal lotteries, which aggregates pairwise preferences without imposing any assumptions on their structure. However, we show that maximal lotteries are highly sensitive to preference heterogeneity and can favor models that severely underperform on specific tasks or user subpopulations. We introduce robust lotteries that optimize worst-case performance under plausible shifts in the preference data. On large-scale preference datasets, robust lotteries provide more reliable win rate guarantees across the annotator distribution and recover a stable set of top-performing models. By moving from rankings to pluralistic sets of winners, robust lotteries offer a principled step toward an ecosystem of complementary AI systems that serve the full spectrum of human preferences.

Itai Shapira · harvard

This study explores the number of neurons required for a Rectified Linear Unit (ReLU) neural network to approximate multivariate monomials. We establish an exponential lower bound on the complexity of any shallow network approximating the product function over a general compact domain. We also demonstrate this lower bound doesn't apply to normalized Lipschitz monomials over the unit cube. These findings suggest that shallow ReLU networks experience the curse of dimensionality when expressing functions with a Lipschitz parameter scaling with the dimension of the input, and that the expressive power of neural networks is more dependent on their depth rather than overall complexity.

Luise Ge, Daniel Halpern, Evi Micha et al. · harvard

In the context of reinforcement learning from human feedback (RLHF), the reward function is generally derived from maximum likelihood estimation of a random utility model based on pairwise comparisons made by humans. The problem of learning a reward function is one of preference aggregation that, we argue, largely falls within the scope of social choice theory. From this perspective, we can evaluate different aggregation methods via established axioms, examining whether these methods meet or fail well-known standards. We demonstrate that both the Bradley-Terry-Luce Model and its broad generalizations fail to meet basic axioms. In response, we develop novel rules for learning reward functions with strong axiomatic guarantees. A key innovation from the standpoint of social choice is that our problem has a linear structure, which greatly restricts the space of feasible rules and leads to a new paradigm that we call linear social choice.

Itai Shapira, Gerdus Benade, Ariel D. Procaccia

Large language models often exhibit increased sycophantic behavior after preference-based post-training, showing a stronger tendency to affirm a user's stated or implied belief even when this conflicts with factual accuracy or sound judgment. We present a formal analysis of how alignment from human feedback can increase this failure mode by identifying an explicit amplification mechanism that causally links optimization against a learned reward to bias in the human preference data used for alignment. We show that the direction of behavioral drift is determined by a covariance under the base policy between endorsing the belief signal in the prompt and the learned reward, and that the first-order effect reduces to a simple mean-gap condition. We then analyze reward learning from pairwise comparisons under random utility models like Bradley-Terry and characterize when bias in human annotators' preferences induces this reward gap. Next, we propose a training-time intervention designed to neutralize the amplification mechanism itself. Among all post-trained policies that prevent sycophantic behavior from increasing, we characterize the unique policy closest in KL divergence to the unconstrained post-trained policy, and derive the corresponding minimal reward correction as a closed-form agreement penalty. Computational experiments find that reward gaps are common and cause behavioral drift in all the configurations considered.

Daniel Halpern, Evi Micha, Ariel D. Procaccia et al. · harvard

Current alignment pipelines presume a single, universal notion of desirable behavior. However, human preferences often diverge across users, contexts, and cultures. As a result, disagreement collapses into the majority signal and minority perspectives are discounted. To address this, we propose reflecting diverse human preferences through a distribution over multiple reward functions, each inducing a distinct aligned policy. The distribution is learned directly from pairwise preference without annotator identifiers or predefined groups. Instead, annotator disagreements are treated as informative soft labels. Our central criterion is pairwise calibration: for every pair of candidate responses, the proportion of reward functions preferring one response matches the fraction of annotators with that preference. We prove that even a small outlier-free ensemble can accurately represent diverse preference distributions. Empirically, we introduce and validate a practical training heuristic to learn such ensembles, and demonstrate its effectiveness through improved calibration, implying a more faithful representation of pluralistic values.

Depen Morwani, Itai Shapira, Nikhil Vyas et al.

Shampoo, a second-order optimization algorithm which uses a Kronecker product preconditioner, has recently garnered increasing attention from the machine learning community. The preconditioner used by Shampoo can be viewed either as an approximation of the Gauss--Newton component of the Hessian or the covariance matrix of the gradients maintained by Adagrad. We provide an explicit and novel connection between the $\textit{optimal}$ Kronecker product approximation of these matrices and the approximation made by Shampoo. Our connection highlights a subtle but common misconception about Shampoo's approximation. In particular, the $\textit{square}$ of the approximation used by the Shampoo optimizer is equivalent to a single step of the power iteration algorithm for computing the aforementioned optimal Kronecker product approximation. Across a variety of datasets and architectures we empirically demonstrate that this is close to the optimal Kronecker product approximation. Additionally, for the Hessian approximation viewpoint, we empirically study the impact of various practical tricks to make Shampoo more computationally efficient (such as using the batch gradient and the empirical Fisher) on the quality of Hessian approximation.

Sara Fish, Paul Gölz, David C. Parkes et al.

The mathematical study of voting, social choice theory, has traditionally only been applicable to choices among a few predetermined alternatives, but not to open-ended decisions such as collectively selecting a textual statement. We introduce generative social choice, a design methodology for open-ended democratic processes that combines the rigor of social choice theory with the capability of large language models to generate text and extrapolate preferences. Our framework divides the design of AI-augmented democratic processes into two components: first, proving that the process satisfies representation guarantees when given access to oracle queries; second, empirically validating that these queries can be approximately implemented using a large language model. We apply this framework to the problem of summarizing free-form opinions into a proportionally representative slate of opinion statements; specifically, we develop a democratic process with representation guarantees and use this process to portray the opinions of participants in a survey about abortion policy. In a trial with 100 representative US residents, we find that 84 out of 100 participants feel "excellently" or "exceptionally" represented by the slate of five statements we extracted.