Formal reasoning and automated theorem proving constitute a challenging subfield of machine learning, in which machines are tasked with proving mathematical theorems using formal languages like Lean. A formal verification system can check whether a formal proof is correct or not almost instantaneously, but generating a completely correct formal proof with large language models (LLMs) remains a formidable task. The usual approach in the literature is to prompt the LLM many times (up to several thousands) until one of the generated proofs passes the verification system. In this work, we present APOLLO (Automated PrOof repair via LLM and Lean cOllaboration), a modular, model-agnostic agentic framework that combines the strengths of the Lean compiler with an LLM's reasoning abilities to achieve better proof-generation results at a low token and sampling budgets. Apollo directs a fully automated process in which the LLM generates proofs for theorems, a set of agents analyze the proofs, fix the syntax errors, identify the mistakes in the proofs using Lean, isolate failing sub-lemmas, utilize automated solvers, and invoke an LLM on each remaining goal with a low top-K budget. The repaired sub-proofs are recombined and reverified, iterating up to a user-controlled maximum number of attempts. On the miniF2F benchmark, we establish a new state-of-the-art accuracy of 84.9% among sub 8B-parameter models (as of August 2025) while keeping the sampling budget below one hundred. Moreover, Apollo raises the state-of-the-art accuracy for Goedel-Prover-SFT to 65.6% while cutting sample complexity from 25,600 to a few hundred.

Formal reasoning and automated theorem proving constitute a challenging subfield of machine learning, in which machines are tasked with proving mathematical theorems using formal languages like Lean. A formal verification system can check whether a formal proof is correct or not almost instantaneously, but generating a completely correct formal proof with large language models (LLMs) remains a formidable task. The usual approach in the literature is to prompt the LLM many times (up to several thousands) until one of the generated proofs passes the verification system.

Having won only one of their past six Premier League games and drawn 2-2 at Tottenham after being 2-0 up, can second-placed Manchester City get back on track at Liverpool on Sunday? I wouldn't rule City out of anything at the moment said BBC Sport football expert Chris Sutton. But the way they folded in the second half against Tottenham was a real worry. Sutton is making predictions for all 380 Premier League games this season, against AI, BBC Sport readers and a variety of guests. His guest for week 25 is Gladiators star Apollo, real name Alex Gray, who supports Newcastle . Before becoming a Gladiator, the 6ft 6in Gray played Premiership rugby for three teams and also American Football for NFL side Atlanta Falcons.

Training large language models (LLMs) typically relies on adaptive optimizers such as Adam, which introduce extra operations and require significant more memory to maintain first- and second-order moments than SGD. While recent works such as GaLore, Fira and APOLLO have proposed state-compressed variants to reduce memory consumption, a fundamental question remains: What are the minimum modifications to plain SGD needed to match state-of-the-art pretraining performance? We systematically investigate this question using a bottom-up approach, and identify two simple yet highly (memory- and compute-) efficient techniques: (1) column-wise gradient normalization (normalizing the gradient along the output dimension), which boosts SGD performance without momentum; and (2) applying first-order momentum only to the output layer, where gradient variance is highest. Combining these two techniques lead to SCALE (Stochastic Column-normAlized Last-layer momEntum), a simple optimizer for memory efficient pretraining. Across multiple LLaMA models (60M-1B), SCALE matches or exceeds the performance of Adam while using only 35-45% of the total memory. It also consistently outperforms memory-efficient optimizers such as GaLore, Fira and APOLLO, making it a strong candidate for large-scale pretraining under memory constraints. For LLaMA 7B model, SCALE outperforms the state-of-the-art memory-efficient methods APOLLO and Muon, in terms of both perplexity and memory consumption.

Large language models (LLMs) have demonstrated remarkable performance due to their large parameter counts and extensive training data. However, their scale leads to significant memory bottlenecks during training, especially when using memory-intensive optimizers like Adam. Existing memory-efficient approaches often rely on techniques such as singular value decomposition (SVD), projections, or weight freezing, which can introduce substantial computational overhead, require additional memory for projections, or degrade model performance. In this paper, we propose Folded Optimizer with Approximate Moment (FOAM), a method that compresses optimizer states by computing block-wise gradient means and incorporates a residual correction to recover lost information. Theoretically, FOAM achieves convergence rates equivalent to vanilla Adam under standard non-convex optimization settings. Empirically, FOAM reduces total training memory by approximately 50\%, eliminates up to 90\% of optimizer state memory overhead, and accelerates convergence. Furthermore, FOAM is compatible with other memory-efficient optimizers, delivering performance and throughput that match or surpass both full-rank and existing memory-efficient baselines.

Formal reasoning and automated theorem proving constitute a challenging subfield of machine learning, in which machines are tasked with proving mathematical theorems using formal languages like Lean. A formal verification system can check whether a formal proof is correct or not almost instantaneously, but generating a completely correct formal proof with large language models (LLMs) remains a formidable task. The usual approach in the literature is to prompt the LLM many times (up to several thousands) until one of the generated proofs passes the verification system. In this work, we present APOLLO (Automated PrOof repair viaLLM and Lean cOllaboration), a modular, model-agnostic agentic framework that combines the strengths of the Lean compiler with an LLM's reasoning abilities to achieve better proof-generation results at a low token and sampling budgets. Apollo directs a fully automated process in which the LLM generates proofs for theorems, a set of agents analyze the proofs, fix the syntax errors, identify the mistakes in the proofs using Lean, isolate failing sub-lemmas, utilize automated solvers, and invoke an LLM on each remaining goal with a low top-K budget. The repaired sub-proofs are recombined and reverified, iterating up to a user-controlled maximum number of attempts. On the miniF2F benchmark, we establish a new state-of-the-art accuracy of 84.9% among sub 8B-parameter models (as of August 2025) while keeping the sampling budget below one hundred. Moreover, Apollo raises the state-of-the-art accuracy for Goedel-Prover-SFT to 65.6% while cutting sample complexity from 25,600 to a few hundred. General-purpose models (o3-mini, o4-mini) jump from 3-7% to over 40% accuracy. Our results demonstrate that targeted, compiler-guided repair of LLM outputs yields dramatic gains in both efficiency and correctness, suggesting a general paradigm for scalable automated theorem proving.

Machine Unlearning (MU) aims to update Machine Learning (ML) models following requests to remove training samples and their influences on a trained model efficiently without retraining the original ML model from scratch. While MU itself has been employed to provide privacy protection and regulatory compliance, it can also increase the attack surface of the model. Existing privacy inference attacks towards MU that aim to infer properties of the unlearned set rely on the weaker threat model that assumes the attacker has access to both the unlearned model and the original model, limiting their feasibility toward real-life scenarios. We propose a novel privacy attack, A Posteriori Label-Only Membership Inference Attack towards MU, Apollo, that infers whether a data sample has been unlearned, following a strict threat model where an adversary has access to the label-output of the unlearned model only. We demonstrate that our proposed attack, while requiring less access to the target model compared to previous attacks, can achieve relatively high precision on the membership status of the unlearned samples.

Despite continuous advancements in the capabilities of large language models (LLMs), numerical reasoning remains a challenging area. Techniques like chain-of-thought prompting, tree-of-thought prompting, and program-of-thought prompting guide LLMs through intermediate reasoning steps. Although in-context learning with few-shot prompting has improved performance, LLMs still lag behind state-of-the-art models on financial numerical reasoning datasets such as FinQA and ConvFinQA. In this work, we introduce FINDER, a novel two-step framework, to enhance LLMs' capabilities in financial numerical reasoning. The first step utilizes a generative retriever to extract relevant facts from unstructured data, including both text and tables. This is followed by context-aware Program of Thought prompting with dynamic selection of in-context examples. Our model FINDER achieves a new state-of-the-art performance on both the FinQA and ConvFinQA datasets, surpassing previous benchmarks with execution accuracy improvements of 5.98% and 4.05%, respectively.

Virtual testing has emerged as an effective approach to accelerate the deployment of automated driving systems. Nevertheless, existing simulation toolchains encounter difficulties in integrating rapid, automated scenario generation with simulation environments supporting advanced automated driving capabilities. To address this limitation, a full-stack toolchain is presented, enabling automatic scenario generation from real-world datasets and efficient validation through a co-simulation platform based on CarMaker, ROS, and Apollo. The simulation results demonstrate the effectiveness of the proposed toolchain. A demonstration video showcasing the toolchain is available at the provided link: https://youtu.be/taJw_-CmSiY.

The safety and reliability of Automated Driving Systems (ADS) are paramount, necessitating rigorous testing methodologies to uncover potential failures before deployment. Traditional testing approaches often prioritize either natural scenario sampling or safety-critical scenario generation, resulting in overly simplistic or unrealistic hazardous tests. In practice, the demand for natural scenarios (e.g., when evaluating the ADS's reliability in real-world conditions), critical scenarios (e.g., when evaluating safety in critical situations), or somewhere in between (e.g., when testing the ADS in regions with less civilized drivers) varies depending on the testing objectives. To address this issue, we propose the On-demand Scenario Generation (OSG) Framework, which generates diverse scenarios with varying risk levels. Achieving the goal of OSG is challenging due to the complexity of quantifying the criticalness and naturalness stemming from intricate vehicle-environment interactions, as well as the need to maintain scenario diversity across various risk levels. OSG learns from real-world traffic datasets and employs a Risk Intensity Regulator to quantitatively control the risk level. It also leverages an improved heuristic search method to ensure scenario diversity. We evaluate OSG on the Carla simulators using various ADSs. We verify OSG's ability to generate scenarios with different risk levels and demonstrate its necessity by comparing accident types across risk levels. With the help of OSG, we are now able to systematically and objectively compare the performance of different ADSs based on different risk levels.