We argue that current definitions of machine unlearning are underspecified for second-order optimizers. We compare first-order and second-order learners for their ability to handle the data deletion task with varying degrees of eigendecomposition to mimic the loss model memory. While both first and second-order methods realign with the ideal counterfactul in terms of performance and gradient, the second-order optimizer shows significant volatility in the optimizer state. This indicates residual information, supposedly deleted, that isn't detectable by first-order analysis. Various eigendecay treatments show that stability and information loss is regained only under controlled state pertubation where geometric information (or memory) is erased.

Second-order optimizers, maintaining a matrix termed a preconditioner, are superior to first-order optimizers in both theory and practice.The states forming the preconditioner and its inverse root restrict the maximum size of models trained by second-order optimizers. To address this, compressing 32-bit optimizer states to lower bitwidths has shown promise in reducing memory usage.

Training large neural networks requires substantial hardware and energy resources.

While parameter-efficient fine-tuning (PEFT) methods aim to reduce the memory usage of the optimizer state during fine-tuning, the inherent size of pre-trained LLM weights continues to be a pressing concern. Even though quantization techniques are widely proposed to ease memory demands and accelerate LLM inference, most of these techniques are geared towards the deployment phase.To bridge this gap, this paper presents Parameter-Efficient and Quantization-aware Adaptation (PEQA) - a simple yet effective method that combines the advantages of PEFT with quantized LLMs. By updating solely the quantization scales, PEQA can be directly applied to quantized LLMs, ensuring seamless task transitions. Parallel to existing PEFT methods, PEQA significantly reduces the memory overhead associated with the optimizer state. Furthermore, it leverages the advantages of quantization to substantially reduce model sizes. Even after fine-tuning, the quantization structure of a PEQA-tuned LLM remains intact, allowing for accelerated inference on the deployment stage.We employ PEQA-tuning for task-specific adaptation on LLMs with up to $65$ billion parameters. To assess the logical reasoning and language comprehension of PEQA-tuned LLMs, we fine-tune low-bit quantized LLMs using a instruction dataset. Our results show that even when LLMs are quantized to below 4-bit precision, their capabilities in language modeling, few-shot in-context learning, and comprehension can be resiliently restored to (or even improved over) their full-precision original performances with PEQA.

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.

As large language models (LLMs) continue to grow in size and complexity, efficient checkpoint saving\&loading has become crucial for managing storage, memory usage, and fault tolerance in LLM training. The current works do not comprehensively take into account the optimization of these several aspects. This paper proposes a novel checkpoint sparsification and quantization method that adapts dynamically to different training stages and model architectures. We present a comprehensive analysis of existing lossy and lossless compression techniques, identify current limitations, and introduce our adaptive approach that balances compression ratio, speed, and precision impact throughout the training process. Experiments on different sizes of LLMs demonstrate that our bitmask-based sparsification method achieves 16x compression ratio without compromising model accuracy. Additionally, the cluster-based quantization method achieves 2x compression ratio with little precision loss.