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Economics Nobelist on causal inference

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Since 2013, Amazon has held an annual internal conference, the Amazon Machine Learning Conference (AMLC), where machine learning practitioners from around the company come together to share their work, teach and learn new techniques, and discuss best practices.

At the third AMLC, in 2015, Guido Imbens, a professor of economics at the Stanford University Graduate School of Business, gave a popular tutorial on causality and machine learning. Nine years and one Nobel Prize for economics later, Imbens — now in his tenth year as an Amazon academic research consultant — was one of the keynote speakers at the 2024 AMLC, held in October.

Guido Imbens, Nobel laureate, professor of economics at the Stanford University Graduate School of Business, and an Amazon academic research consultant for the past 10 years.

In his talk, Imbens discussed causal inference, a mainstay of his research for more than 30 years and the topic that the Nobel committee highlighted in its prize citation. In particular, he considered so-called panel data, in which multiple units — say, products, customers, or geographic regions — and outcomes — say, sales or clicks — are observed at discrete points in time.

Over particular time spans, some units receive a treatment — say, a special product promotion or new environmental regulation — whose effects are reflected in the outcome measurements. Causal inference is the process of determining how much of the change in outcomes over time can be attributed to the treatment. This means adjusting for spurious correlations that result from general trends in the data, which can be inferred from trends among the untreated (control) units.

Imbens began by discussing the value of his work at Amazon. “I started working with people here at Amazon in 2014, and it’s been a real pleasure and a real source of inspiration for my research, interacting with the people here and seeing what kind of problems they’re working on, what kind of questions they have,” he said. “I’ve always found it very useful in my econometric, in my statistics, in my methodological research to talk to people who are using these methods in practice, who are actually working with these things on the ground. So it’s been a real privilege for the last 10 years doing that with the people here at Amazon.”

Panel data

Then, with no further ado, he launched into the substance of his talk. Panel data, he explained, is generally represented by a pair of matrices, whose rows represents units and whose columns represent points in time. In one matrix, the entries represent measurements made on particular units at particular times; the other matrix takes only binary values, which represent whether a given unit was subject to treatment during the corresponding time span.

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Ideally, for a given unit and a given time span, we would run an experiment in which the unit went untreated; then we would back time up and run the experiment again, with the treatment. But of course, time can’t be backed up. So instead, for each treated cell in the matrix, we estimate what the relevant measurement would have been if the treatment hadn’t been applied, and we base that estimate on the outcomes for other units and time periods.

For ease of explanation, Imbens said, he considered the case in which only one unit was treated, for only one time interval: “Once I have methods that work effectively for that case, the particular methods I’m going to suggest extend very naturally to the more-general assignment mechanism,” he said. “This is a very common setup.”

Control estimates

Imbens described five standard methods for estimating what would have been the outcome if a treated unit had been untreated during the same time period. The first method, which is very common in empirical work in economics, is known as known as difference of differences. It involves a regression analysis of all the untreated data up to the treatment period; the regression function can then be used to estimate the outcome for the treated unit if it hadn’t been treated.

The second method is called synthetic control, in which a control version of the treated unit is synthesized as a weighted average of the other control units.

“One of the canonical examples is one where he [Alberto Abadie, an Amazon Scholar, pioneer of synthetic control, and long-time collaborator of Imbens] is interested in estimating the effect of an anti-smoking regulation in California that went into effect in 1989,” Imbens explained. “So he tries to find the convex combination of the other states such that smoking rates for that convex combination match the actual smoking rates in California prior to 1989 — say, 40% Arizona, 30% Utah, 10% Washington and 20% New York. Once he has those weights, he then estimates the counterfactual smoking rate in California.”

A synthetic control estimates a counterfactual control for a treated unit by synthesizing outcomes for untreated units. For instance, smoking rates in California might by synthesized as a convex combination of smoking rates in other states.

The third method, which Imbens and a colleague had proposed in 2016, adds an intercept to the synthetic-control equation; that is, it specifies an output value for the function when all the unit measurements are zero.

The final two methods were variations on difference of differences that added another term to the function to be optimized: a low-rank matrix, which approximates the results of the outcomes matrix at a lower resolution. The first of these variations — the matrix completion method — simply adds the matrix, with a weighting factor, to the standard difference-of-differences function.

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The second variation — synthetic difference of differences — weights the distances between the unit-time measurements and the regression curve according to the control units’ similarities to the unit that received the intervention.

“In the context of the smoking example,” Imbens said, “you assign more weight to units that are similar to California, that match California better. So rather than pretending that Delaware or Alaska is very similar to California — other than in their level — you only put weight on states that are very similar to California.”

Drawbacks

Having presented these five methods, Imbens went on to explain what he found wrong with them. The first problem, he said, is that they treat the outcome and treatment matrices as both row (units) and column (points in time) exchangeable. That is, the methods produce the same results whatever the ordering of rows and columns in the matrices.

“The unit exchangeability here seems very reasonable,” Imbens said. “We may have some other covariates, but in principle, there’s nothing that distinguishes these units or suggests treating them in a way that’s different from exchangeable.

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“But for the time dimension, it’s different. You would think that if we’re trying to predict outcomes in 2020, having outcomes measured in 2019 is going to be much more useful than having outcomes measured in 1983. We think that there’s going to be correlation over time that makes predictions based on values from 2019 much more likely to be accurate than predictions based on values from 1983.”

The second problem, Imbens said, is that while the methods work well in the special case he considered, where only a single unit-time pair is treated — and indeed, they work well under any conditions in which the treatment assignments have a clearly discernible structure — they struggle in cases where the treatment assignments are more random. That’s because, with random assignment, units drop in and out of the control group from one time period to the next, making accurate regression analysis difficult.

A new estimator

So Imbens proposed a new estimator, one based on the matrix completion method, but with additional terms that apply two sets of weights to each control unit’s contribution to the regression analysis. The first weight reduces the contribution of a unit measurement according to its distance in time from the measurement of the treated unit — that is, it privileges more recent measurements.

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The second weight reduces the contributions of control unit measurements according to their absolute distance from the measurement of the treated unit. There, the idea is to limit the influence of outliers in sparse datasets — that is, datasets that control units are constantly dropping in and out of.

Imbens then compared the performance of his new estimator to those of the other five, on nine existing datasets that had been chosen to test the accuracy of prior estimators. On eight of the nine datasets, Imbens’s estimator outperformed all five of its predecessors, sometimes by a large margin; on the ninth dataset, it finished a close second to the difference-of-differences approach — which, however, was the last-place finisher on several other datasets.

Root mean squared error of six estimators on nine datasets, normalized to the best-performing dataset. Imbens’s new estimator, the doubly weighted causal panel (DWCP) estimator, outperforms its predecessors, often by a large margin.

“I don’t want to push this as a particular estimator that you should use in all settings,” Imbens explained. “I want to mainly show that even simple changes to existing classes of estimators can actually do substantially better than the previous estimators by incorporating the time dimension in a more uh more satisfactory way.”

For purposes of causal inference, however, the accuracy of an estimator is not the only consideration. The reliability of the estimator — its power, in the statistical sense — also depends on its variance, the degree to which its margin of error deviates from the mean in particular instances. The lower the variance, the more likely the estimator is to provide accurate estimates.

Variance of variance

For the rest of his talk, Imbens discussed methods of estimating the variance of counterfactual estimators. Here things get a little confusing, because the variance estimators themselves display variance. Imbens advocated the use of conditional variance estimators, which hold some variables fixed — in the case of panel data, unit, time, or both — and estimate the variance of the free variables. Counterintuitively, higher-variance variance estimators, Imbens said, offer more power.

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“In general, you should prefer the conditional variance because it adapts more to the particular dataset you’re analyzing,” Imbens explained. “It’s going to give you more power to find the treatment effects. Whereas the marginal variance” — an alternative and widely used method for estimating variance — “has the lowest variance itself, and it’s going to have the lowest power in general for detecting treatment effects.”

Imbens then presented some experimental results using synthetic panel data that indicated that, indeed, in cases where data is heteroskedastic — meaning that the variance of one variable increases with increasing values of the other — variance estimators that themselves use conditional variance have greater statistical power than other estimators.

“There’s clearly more to be done, both in terms of estimation, despite all the work that’s been done in the last couple of years in this area, and in terms of variance estimation,” Imbens concluded. “And where I think the future lies for these models is a combination of the outcome modeling by having something flexible in terms of both factor models as well as weights that ensure that you’re doing the estimation only locally. And we need to do more on variance estimation, keeping in mind both power and validity, with some key role for modeling some of the heteroskedasticity.”





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An inside look at Meta’s transition from C to Rust on mobile

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Have you ever worked is legacy code? Are you curious what it takes to modernize systems at a massive scale?

Pascal Hartig is joined on the latest Meta Tech Podcast by Elaine and Buping, two software engineers working on a bold project to rewrite the decades-old C code in one of Meta’s core messaging libraries in Rust. It’s an ambitious effort that will transform a central messaging library that is shared across Messenger, Facebook, Instagram, and Meta’s AR/VR platforms.

They discuss taking on a project of this scope – even without a background in Rust, how they’re approaching it, and what it means to optimize for ‘developer happiness.’

Download or listen to the episode below:

You can also find the episode wherever you get your podcasts, including:

The Meta Tech Podcast is a podcast, brought to you by Meta, where we highlight the work Meta’s engineers are doing at every level – from low-level frameworks to end-user features.

Send us feedback on InstagramThreads, or X.

And if you’re interested in learning more about career opportunities at Meta visit the Meta Careers page.





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Amazon Research Awards recipients announced

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Amazon Research Awards (ARA) provides unrestricted funds and AWS Promotional Credits to academic researchers investigating various research topics in multiple disciplines. This cycle, ARA received many excellent research proposals from across the world and today is publicly announcing 73 award recipients who represent 46 universities in 10 countries.

This announcement includes awards funded under five call for proposals during the fall 2024 cycle: AI for Information Security, Automated Reasoning, AWS AI, AWS Cryptography, and Sustainability. Proposals were reviewed for the quality of their scientific content and their potential to impact both the research community and society. Additionally, Amazon encourages the publication of research results, presentations of research at Amazon offices worldwide, and the release of related code under open-source licenses.

Recipients have access to more than 700 Amazon public datasets and can utilize AWS AI/ML services and tools through their AWS Promotional Credits. Recipients also are assigned an Amazon research contact who offers consultation and advice, along with opportunities to participate in Amazon events and training sessions.

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“Automated Reasoning is an important area of research for Amazon, with potential applications across various features and applications to help improve security, reliability, and performance for our customers. Through the ARA program, we collaborate with leading academic researchers to explore challenges in this field,” said Robert Jones, senior principal scientist with the Cloud Automated Reasoning Group. “We were again impressed by the exceptional response to our Automated Reasoning call for proposals this year, receiving numerous high-quality submissions. Congratulations to the recipients! We’re excited to support their work and partner with them as they develop new science and technology in this important area.”

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“At Amazon, we believe that solving the world’s toughest sustainability challenges benefits from both breakthrough scientific research and open and bold collaboration. Through programs like the Amazon Research Awards program, we aim to support academic research that could contribute to our understanding of these complex issues,” said Kommy Weldemariam, Director of Science and Innovation Sustainability. “The selected proposals represent innovative projects that we hope will help advance knowledge in this field, potentially benefiting customers, communities, and the environment.”

ARA funds proposals throughout the year in a variety of research areas. Applicants are encouraged to visit the ARA call for proposals page for more information or send an email to be notified of future open calls.

The tables below list, in alphabetical order by last name, fall 2024 cycle call-for-proposal recipients, sorted by research area.

AI for Information Security

Recipient University Research title
Christopher Amato Northeastern University Multi-Agent Reinforcement Learning Cyber Defense for Securing Cloud Computing Platforms
Bernd Bischl Ludwig Maximilian University of Munich Improving Generative and Foundation Models Reliability via Uncertainty-awareness
Shiqing Ma University Of Massachusetts Amherst LLM and Domain Adaptation for Attack Detection
Alina Oprea Northeastern University Multi-Agent Reinforcement Learning Cyber Defense for Securing Cloud Computing Platforms
Roberto Perdisci University of Georgia ContextADBench: A Comprehensive Benchmark Suite for Contextual Anomaly Detection

Automated Reasoning

Recipient University Research title
Nada Amin Harvard University LLM-Augmented Semi-Automated Proofs for Interactive Verification
Suguman Bansal Georgia Institute of Technology Certified Inductive Generalization in Reinforcement Learning
Ioana Boureanu University of Surrey Phoebe+: An Automated-Reasoning Tool for Provable Privacy in Cryptographic Systems
Omar Haider Chowdhury Stony Brook University Restricter: An Automatic Tool for Authoring Amazon Cedar Access Control Policies with the Principle of Least Privilege
Stefan Ciobaca Alexandru Ioan Cuza University An Interactive Proof Mode for Dafny
João Ferreira INESC-ID Polyglot Automated Program Repair for Infrastructure as Code
Sicun Gao University Of California, San Diego Monte Carlo Trees with Conflict Models for Proof Search
Mirco Giacobbe University of Birmingham Neural Software Verification
Tobias Grosser University of Cambridge Synthesis-based Symbolic BitVector Simplification for Lean
Ronghui Gu Columbia University Scaling Formal Verification of Security Properties for Unmodified System Software
Alexey Ignatiev Monash University Huub: Next-Gen Lazy Clause Generation
Kenneth McMillan University of Texas At Austin Synthesis of Auxiliary Variables and Invariants for Distributed Protocol Verification
Alexandra Mendes University of Porto Overcoming Barriers to the Adoption of Verification-Aware Languages
Jason Nieh Columbia University Scaling Formal Verification of Security Properties for Unmodified System Software
Rohan Padhye Carnegie Mellon University Automated Synthesis and Evaluation of Property-Based Tests
Nadia Polikarpova University Of California, San Diego Discovering and Proving Critical System Properties with LLMs
Fortunat Rajaona University of Surrey Phoebe+: An Automated-Reasoning Tool for Provable Privacy in Cryptographic Systems
Subhajit Roy Indian Institute of Technology Kanpur Theorem Proving Modulo LLM
Gagandeep Singh University of Illinois At Urbana–Champaign Trustworthy LLM Systems using Formal Contracts
Scott Stoller Stony Brook University Restricter: An Automatic Tool for Authoring Amazon Cedar Access Control Policies with the Principle of Least Privilege
Peter Stuckey Monash University Huub: Next-Gen Lazy Clause Generation
Yulei Sui University of New South Wales Path-Sensitive Typestate Analysis through Sparse Abstract Execution
Nikos Vasilakis Brown University Semantics-Driven Static Analysis for the Unix/Linux Shell
Ping Wang Stevens Institute of Technology Leveraging Large Language Models for Reasoning Augmented Searching on Domain-specific NoSQL Database
John Wawrzynek University of California, Berkeley GPU-Accelerated High-Throughput SAT Sampling

AWS AI

Recipient University Research title
Panagiotis Adamopoulos Emory University Generative AI solutions for The Spillover Effect of Fraudulent Reviews on Product Recommendations
Vikram Adve University of Illinois at Urbana–Champaign Fellini: Differentiable ML Compiler for Full-Graph Optimization for LLM Models
Frances Arnold California Institute of Technology Closed-loop Generative Machine Learning for De Novo Enzyme Discovery and Optimization
Yonatan Bisk Carnegie Mellon University Useful, Safe, and Robust Multiturn Interactions with LLMs
Shiyu Chang University of California, Santa Barbara Cut the Crap: Advancing the Efficient Communication of Multi-Agent Systems via Spatial-Temporal Topology Design and KV Cache Sharing
Yuxin Chen University of Pennsylvania Provable Acceleration of Diffusion Models for Modern Generative AI
Tianlong Chen University of North Carolina at Chapel Hill Cut the Crap: Advancing the Efficient Communication of Multi-Agent Systems via Spatial-Temporal Topology Design and KV Cache Sharing
Mingyu Ding University of North Carolina at Chapel Hill Aligning Long Videos and Language as Long-Horizon World Models
Nikhil Garg Cornell University Market Design for Responsible Multi-agent LLMs
Jessica Hullman Northwestern University Human-Aligned Uncertainty Quantification in High Dimensions
Christopher Jermaine Rice University Fast, Trusted AI Using the EINSUMMABLE Compiler
Yunzhu Li Columbia University Physics-Informed Foundation Models Through Embodied Interactions
Pattie Maes Massachusetts Institute of Technology Understanding How LLM Agents Deviate from Human Choices
Sasa Misailovic University of Illinois at Urbana–Champaign Fellini: Differentiable ML Compiler for Full-Graph Optimization for LLM Models
Kristina Monakhova Cornell University Trustworthy extreme imaging for science using interpretable uncertainty quantification
Todd Mowry Carnegie Mellon University Efficient LLM Serving on Trainium via Kernel Generation
Min-hwan Oh Seoul National University Mutually Beneficial Interplay Between Selection Fairness and Context Diversity in Contextual Bandits
Patrick Rebeschini University of Oxford Optimal Regularization for LLM Alignment
Jose Renau University of California, Santa Cruz Verification Constrained Hardware Optimization using Intelligent Design Agentic Programming
Vilma Todri Emory University Generative AI solutions for The Spillover Effect of Fraudulent Reviews on Product Recommendations
Aravindan Vijayaraghavan Northwestern University Human-Aligned Uncertainty Quantification in High Dimensions
Wei Yang University of Texas at Dallas Optimizing RISC-V Compilers with RISC-LLM and Syntax Parsing
Huaxiu Yao University of North Carolina at Chapel Hill Aligning Long Videos and Language as Long-Horizon World Models
Amy Zhang University of Washington Tools for Governing AI Agent Autonomy
Ruqi Zhang Purdue University Efficient Test-time Alignment for Large Language Models and Large Multimodal Models
Zheng Zhang Rutgers University-New Brunswick AlphaQC: An AI-powered Quantum Circuit Optimizer and Denoiser

AWS Cryptography

Recipient University Research title
Alexandra Boldyreva Georgia Institute of Technology Quantifying Information Leakage in Searchable Encryption Protocols
Maria Eichlseder Graz University of Technology, Austria SALAD – Systematic Analysis of Lightweight Ascon-based Designs
Venkatesan Guruswami University of California, Berkeley Obfuscation, Proof Systems, and Secure Computation: A Research Program on Cryptography at the Simons Institute for the Theory of Computing
Joseph Jaeger Georgia Institute of Technology Analyzing Chat Encryption for Group Messaging
Aayush Jain Carnegie Mellon Large Scale Multiparty Silent Preprocessing for MPC from LPN
Huijia Lin University of Washington Large Scale Multiparty Silent Preprocessing for MPC from LPN
Hamed Nemati KTH Royal Institute of Technology Trustworthy Automatic Verification of Side-Channel Countermeasures for Binary Cryptographic Programs using the HoIBA libary
Karl Palmskog KTH Royal Institute of Technology Trustworthy Automatic Verification of Side-Channel Countermeasures for Binary Cryptographic Programs using the HoIBA libary
Chris Peikert University of Michigan, Ann Arbor Practical Third-Generation FHE and Bootstrapping
Dimitrios Skarlatos Carnegie Mellon University Scale-Out FHE LLMs on GPUs
Vinod Vaikuntanathan Massachusetts Institute of Technology Can Quantum Computers (Really) Factor?
Daniel Wichs Northeastern University Obfuscation, Proof Systems, and Secure Computation: A Research Program on Cryptography at the Simons Institute for the Theory of Computing
David Wu University Of Texas At Austin Fast Private Information Retrieval and More using Homomorphic Encryption

Sustainability

Recipient University Research title
Meeyoung Cha Max Planck Institute Forest-Blossom (Flossom): A New Framework for Sustaining Forest Biodiversity Through Outcome-Driven Remote Sensing Monitoring
Jingrui He University of Illinois at Urbana–Champaign Foundation Model Enabled Earth’s Ecosystem Monitoring
Pedro Lopes University of Chicago AI-powered Tools that Enable Engineers to Make & Re-make Sustainable Hardware
Cheng Yaw Low Max Planck Institute Forest-Blossom (Flossom): A New Framework for Sustaining Forest Biodiversity Through Outcome-Driven Remote Sensing Monitoring





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Independent evaluations demonstrate Nova Premier’s safety

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AI safety is a priority at Amazon. Our investment in safe, transparent, and responsible AI (RAI) includes collaboration with the global community and policymakers. We are members of and collaborate with organizations such as the Frontier Model Forum, the Partnership on AI, and other forums organized by government agencies such as the National Institute of Standards and Technology (NIST). Consistent with Amazon’s endorsement of the Korea Frontier AI Safety Commitments, we published our Frontier Model Safety Framework earlier this year.

Amazon Nova Premier’s guardrails help prevent generation of unsafe content.

During the development of the Nova Premier model, we conducted a comprehensive evaluation to assess its performance and safety. This included testing on both internal and public benchmarks and internal/automated and third-party red-teaming exercises. Once the final model was ready, we prioritized obtaining unbiased, third-party evaluations of the model’s robustness against RAI controls. In this post, we outline the key findings from these evaluations, demonstrating the strength of our testing approach and Amazon Premier’s standing as a safe model. Specifically, we cover our evaluations with two third-party evaluators: PRISM AI and ActiveFence.

Evaluation of Nova Premier against PRISM AI

PRISM Eval’s Behavior Elicitation Tool (BET) dynamically and systematically stress-tests AI models’ safety guardrails. The methodology focuses on measuring how many adversarial attempts (steps) it takes to get a model to generate harmful content across several key risk dimensions. The central metric is “steps to elicit” — the number of increasingly sophisticated prompting attempts required before a model generates an inappropriate response. A higher number of steps indicates stronger safety measures, as the model is more resistant to manipulation. The PRISM risk dimensions (inspired by the MLCommons AI Safety Benchmarks) include CBRNE weapons, violent crimes, non-violent crimes, defamation, and hate, amongst several others.

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Using the BET Eval tool and its V1.0 metric, which is tailored toward non-reasoning models, we compared the recently released Nova models (Pro and Premier) to the latest models in the same class: Claude (3.5 v2 and 3.7 non-reasoning) and Llama4 Maverick, all available through Amazon Bedrock. PRISM BET conducts black-box evaluations (where model developers don’t have access to the test prompts) of models integrated with their API. The evaluation conducted with BET Eval MAX, PRISM’s most comprehensive/aggressive testing suite, revealed significant variations in safety against malicious instructions. Nova models demonstrated superior overall safety performance, with an average of 43 steps for Premier and 52 steps for Pro, compared to 37.7 for Claude 3.5 v2 and fewer than 12 steps for other models in the comparison set (namely, 9.9 for Claude3.7, 11.5 for Claude 3.7 thinking, and 6.5 for Maverick). This higher step count suggests that on average, Nova’s safety guardrails are more sophisticated and harder to circumvent through adversarial prompting. The figure below presents the number of steps per harm category evaluated through BET Eval MAX.

Results of tests using PRISM’s BET Eval MAX testing suite.

The PRISM evaluation provides valuable insights into the relative safety of different Amazon Bedrock models. Nova’s strong performance, particularly in hate speech and defamation resistance, represents meaningful progress in AI safety. However, the results also highlight the ongoing challenge of building truly robust safety measures into AI systems. As the field continues to evolve, frameworks like BET will play an increasingly important role in benchmarking and improving AI safety. As a part of this collaboration Nicolas Miailhe, CEO of PRISM Eval, said, “It’s incredibly rewarding for us to see Nova outperforming strong baselines using the BET Eval MAX; our aim is to build a long-term partnership toward safer-by-design models and to make BET available to various model providers.” Organizations deploying AI systems should carefully consider these safety metrics when selecting models for their applications.

Manual red teaming with ActiveFence

The AI safety & security company ActiveFence benchmarked Nova Premier on Bedrock on prompts distributed across Amazon’s eight core RAI categories. ActiveFence also evaluated Claude 3.7 (non-reasoning mode) and GPT 4.1 API on the same set. The flag rate on Nova Premier was lower than that on the other two models, indicating that Nova Premier is the safest of the three.

Model 3P Flag Rate [↓ is better]
Nova Premier 12.0%
Sonnet 3.7 (non-reasoning) 20.6%
GPT4.1 API 22.4%

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“Our role is to think like an adversary but act in service of safety,” said Guy Paltieli from ActiveFence. “By conducting a blind stress test of Nova Premier under realistic threat scenarios, we helped evaluate its security posture in support of Amazon’s broader responsible-AI goals, ensuring the model could be deployed with greater confidence.”

These evaluations conducted with PRISM and ActiveFence give us confidence in the strength of our guardrails and our ability to protect our customers’ safety when they use our models. While these evaluations demonstrate strong safety performance, we recognize that AI safety is an ongoing challenge requiring continuous improvement. These assessments represent a point-in-time snapshot, and we remain committed to regular testing and enhancement of our safety measures. No AI system can guarantee perfect safety in all scenarios, which is why we maintain monitoring and response systems after deployment.

Acknowledgments: Vincent Ponzo, Elyssa Vincent





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