Math Speaks All Languages: Enhancing LLM Problem-Solving Across Multilingual Contexts

Maksim Kostritsya, Kseniia Kuvshinova, Rauf Parchiev, Konstantin Polev

Large language models (LLMs) have shown significant adaptability in tackling various human issues; however, their efficacy in resolving mathematical problems remains inadequate. Recent research has identified steering vectors — hidden attributes that can guide the actions and outputs of LLMs. Nonetheless, the exploration of universal vectors that can consistently affect model responses across different languages is still limited. This project aims to confront two primary challenges in contemporary LLM research by utilizing the Goodfire API to examine whether common latent features can improve mathematical problem-solving capabilities, regardless of the language employed.

Reviewer's Comments

Reviewer's Comments

Arrow
Arrow
Arrow

Mateusz Dziemian

Quite surprising results! Wasn’t expecting the model to have features which it doesn’t activate enough during inference that could boost performance. The results are very intriguing and warrant a further study to get a better understanding on the trade-offs for such features and if they just improve maths performance or other domains too.

Simon Lermen

It seems strange to use Google Translate or Mistral for translation when there are much better options available. This is especially problematic for challenging math problems. There’s also an inconsistency in Figure 1, where the numbers add up to 101 despite stating that there are 100 problems. French translations worked much worse, and I’m afraid the translations might not have been accurate. The text didn’t clarify it fully, but it seems they used Mistral for French and Google Translate for Russian. In general, the finding is interesting: they ran a math benchmark, used a contrastive method to find differences between correct and incorrect math answers, and checked if these differences transferred between languages—and they do. This provides evidence that the vector truly captures some sense of mathematical accuracy. That being said, they only evaluated the steering vector on samples where it was incorrect without steering. While this fixes some outcomes, it’s possible this also breaks previously correct responses. In French, it only corrected 2 out of 21, and I’d guess that some false answers might arise just from resampling. They also applied it to only 21 out of 77 false samples. I would also find it more interesting to take the correctness vector for a language in which the model performed better. Using a control feature and validating on all math problems could be beneficial too. You’d expect better results if you resampled on false outputs, so it would be interesting to see what happens if you steer for an unrelated feature.

Alana Xiang

This is creative paper which finds a new domain on which SAE features generalize well (across languages in grade school math).

The surprising finding that the French steering vectors had a larger impact on English and Russian performance than French performance warrants further inquiry. I think this paper could've significantly improved on novelty if it pursued this direction.

Given more time, I would also love to see the authors inspect whether the features they found generalize beyond GSM8K.

Good work!

Cite this work

@misc {

title={

Math Speaks All Languages: Enhancing LLM Problem-Solving Across Multilingual Contexts

},

author={

Maksim Kostritsya, Kseniia Kuvshinova, Rauf Parchiev, Konstantin Polev

},

date={

11/25/24

},

organization={Apart Research},

note={Research submission to the research sprint hosted by Apart.},

howpublished={https://apartresearch.com}

}

Jul 28, 2025

Local Learning Coefficients Predict Developmental Milestones During Group Relative Policy Optimization

In this work, we investigate the emergence of capabilities in reinforcement learning (RL) by framing them as developmental phase transitions. We propose that the individual components of the reward function can serve as direct observables for these transitions, avoiding the need for complex, derived metrics. To test this, we trained a language model on an arithmetic task using Group Relative Policy Optimization (GRPO) and analyzed its learning trajectory with the Local Learning Coefficient (LLC) from Singular Learning Theory. Our findings show a strong qualitative correlation between spikes in the LLC—indicating a phase transition—and significant shifts in the model's behavior, as reflected by changes in specific reward components for correctness and conciseness. This demonstrates a more direct and scalable method for monitoring capability acquisition, offering a valuable proof-of-concept for developmental interpretability and AI safety. To facilitate reproducibility, we make our code available at \url{github.com/ilijalichkovski/apart-physics}.

Read More

Jul 28, 2025

AI agentic system epidemiology

As AI systems scale into decentralized, multi-agent deployments, emergent vulnerabilities challenge our ability to evaluate and manage systemic risks.

In this work, we adapt classical epidemiological modeling (specifically SEIR compartment models) to model adversarial behavior propagation in AI agents.

By solving systems of ODEs describing the systems with physics-informed neural networks (PINNs), we analyze stable and unstable equilibria, bifurcation points, and the effectiveness of interventions.

We estimate parameters from real-world data (e.g., adversarial success rates, detection latency, patching delays) and simulate attack propagation scenarios across 8 sectors (enterprise, retail, trading, development, customer service, academia, medical, and critical infrastructure AI tools).

Our results demonstrate how agent population dynamics interact with architectural and policy design interventions to stabilize the system.

This framework bridges concepts from dynamical systems and cybersecurity to offer a proactive, quantitative toolbox on AI safety.

We argue that epidemic-style monitoring and tools grounded in interpretable, physics-aligned dynamics can serve as early warning systems for cascading AI agentic failures.

Read More

Jul 28, 2025

Momentum–Point-Perplexity Mechanics in Large Language Models

This work analyzes the hidden states of twenty different open-source transformer language models, ranging from small to medium size and covering five major architectures. The key discovery is that these models show signs of "energy conservation" during inference—meaning a certain measure combining changes in hidden states and token unpredictability stays almost constant as the model processes text.

The authors developed a new framework inspired by physics to jointly analyze how hidden states and prediction confidence evolve over time. They propose that transformers' behavior can be understood as following certain mechanical principles, much like how physical systems follow rules like conservation of energy.

Their experiments show that this conserved quantity varies very little between tokens, especially in untrained (random-weight) models, where it's extremely stable. In pre-trained models, the average energy drops more due to training, but there are larger relative fluctuations from token to token.

They also introduce a new method, based on this framework, for controlling transformer outputs by "steering" the hidden states. This method achieves good results—producing completions rated as higher in semantic quality, while still maintaining the same kind of energy stability.

Overall, the findings suggest that viewing transformer models through the lens of physical mechanics gives new, principled ways to interpret and control their behavior. It also highlights a key difference: random models behave more like balanced systems, while trained models make quicker, more decisive state changes at the cost of less precise energy conservation.

Read More

Jul 28, 2025

Local Learning Coefficients Predict Developmental Milestones During Group Relative Policy Optimization

In this work, we investigate the emergence of capabilities in reinforcement learning (RL) by framing them as developmental phase transitions. We propose that the individual components of the reward function can serve as direct observables for these transitions, avoiding the need for complex, derived metrics. To test this, we trained a language model on an arithmetic task using Group Relative Policy Optimization (GRPO) and analyzed its learning trajectory with the Local Learning Coefficient (LLC) from Singular Learning Theory. Our findings show a strong qualitative correlation between spikes in the LLC—indicating a phase transition—and significant shifts in the model's behavior, as reflected by changes in specific reward components for correctness and conciseness. This demonstrates a more direct and scalable method for monitoring capability acquisition, offering a valuable proof-of-concept for developmental interpretability and AI safety. To facilitate reproducibility, we make our code available at \url{github.com/ilijalichkovski/apart-physics}.

Read More

Jul 28, 2025

AI agentic system epidemiology

As AI systems scale into decentralized, multi-agent deployments, emergent vulnerabilities challenge our ability to evaluate and manage systemic risks.

In this work, we adapt classical epidemiological modeling (specifically SEIR compartment models) to model adversarial behavior propagation in AI agents.

By solving systems of ODEs describing the systems with physics-informed neural networks (PINNs), we analyze stable and unstable equilibria, bifurcation points, and the effectiveness of interventions.

We estimate parameters from real-world data (e.g., adversarial success rates, detection latency, patching delays) and simulate attack propagation scenarios across 8 sectors (enterprise, retail, trading, development, customer service, academia, medical, and critical infrastructure AI tools).

Our results demonstrate how agent population dynamics interact with architectural and policy design interventions to stabilize the system.

This framework bridges concepts from dynamical systems and cybersecurity to offer a proactive, quantitative toolbox on AI safety.

We argue that epidemic-style monitoring and tools grounded in interpretable, physics-aligned dynamics can serve as early warning systems for cascading AI agentic failures.

Read More

Jul 28, 2025

Local Learning Coefficients Predict Developmental Milestones During Group Relative Policy Optimization

In this work, we investigate the emergence of capabilities in reinforcement learning (RL) by framing them as developmental phase transitions. We propose that the individual components of the reward function can serve as direct observables for these transitions, avoiding the need for complex, derived metrics. To test this, we trained a language model on an arithmetic task using Group Relative Policy Optimization (GRPO) and analyzed its learning trajectory with the Local Learning Coefficient (LLC) from Singular Learning Theory. Our findings show a strong qualitative correlation between spikes in the LLC—indicating a phase transition—and significant shifts in the model's behavior, as reflected by changes in specific reward components for correctness and conciseness. This demonstrates a more direct and scalable method for monitoring capability acquisition, offering a valuable proof-of-concept for developmental interpretability and AI safety. To facilitate reproducibility, we make our code available at \url{github.com/ilijalichkovski/apart-physics}.

Read More

Jul 28, 2025

AI agentic system epidemiology

As AI systems scale into decentralized, multi-agent deployments, emergent vulnerabilities challenge our ability to evaluate and manage systemic risks.

In this work, we adapt classical epidemiological modeling (specifically SEIR compartment models) to model adversarial behavior propagation in AI agents.

By solving systems of ODEs describing the systems with physics-informed neural networks (PINNs), we analyze stable and unstable equilibria, bifurcation points, and the effectiveness of interventions.

We estimate parameters from real-world data (e.g., adversarial success rates, detection latency, patching delays) and simulate attack propagation scenarios across 8 sectors (enterprise, retail, trading, development, customer service, academia, medical, and critical infrastructure AI tools).

Our results demonstrate how agent population dynamics interact with architectural and policy design interventions to stabilize the system.

This framework bridges concepts from dynamical systems and cybersecurity to offer a proactive, quantitative toolbox on AI safety.

We argue that epidemic-style monitoring and tools grounded in interpretable, physics-aligned dynamics can serve as early warning systems for cascading AI agentic failures.

Read More

Jul 28, 2025

Local Learning Coefficients Predict Developmental Milestones During Group Relative Policy Optimization

In this work, we investigate the emergence of capabilities in reinforcement learning (RL) by framing them as developmental phase transitions. We propose that the individual components of the reward function can serve as direct observables for these transitions, avoiding the need for complex, derived metrics. To test this, we trained a language model on an arithmetic task using Group Relative Policy Optimization (GRPO) and analyzed its learning trajectory with the Local Learning Coefficient (LLC) from Singular Learning Theory. Our findings show a strong qualitative correlation between spikes in the LLC—indicating a phase transition—and significant shifts in the model's behavior, as reflected by changes in specific reward components for correctness and conciseness. This demonstrates a more direct and scalable method for monitoring capability acquisition, offering a valuable proof-of-concept for developmental interpretability and AI safety. To facilitate reproducibility, we make our code available at \url{github.com/ilijalichkovski/apart-physics}.

Read More

Jul 28, 2025

AI agentic system epidemiology

As AI systems scale into decentralized, multi-agent deployments, emergent vulnerabilities challenge our ability to evaluate and manage systemic risks.

In this work, we adapt classical epidemiological modeling (specifically SEIR compartment models) to model adversarial behavior propagation in AI agents.

By solving systems of ODEs describing the systems with physics-informed neural networks (PINNs), we analyze stable and unstable equilibria, bifurcation points, and the effectiveness of interventions.

We estimate parameters from real-world data (e.g., adversarial success rates, detection latency, patching delays) and simulate attack propagation scenarios across 8 sectors (enterprise, retail, trading, development, customer service, academia, medical, and critical infrastructure AI tools).

Our results demonstrate how agent population dynamics interact with architectural and policy design interventions to stabilize the system.

This framework bridges concepts from dynamical systems and cybersecurity to offer a proactive, quantitative toolbox on AI safety.

We argue that epidemic-style monitoring and tools grounded in interpretable, physics-aligned dynamics can serve as early warning systems for cascading AI agentic failures.

Read More

This work was done during one weekend by research workshop participants and does not represent the work of Apart Research.
This work was done during one weekend by research workshop participants and does not represent the work of Apart Research.