# BitDelta: Your Fine-Tune May Only Be Worth One Bit

The *pretrain-finetune* paradigm has revolutionized machine learning; Through fine-tuning, LLMs are adeptly equipped to align with distinct user preferences or specialized task requirements, showcasing an unprecedented level of adaptability. Thus, the prospect of serving *millions of uniquely fine-tuned models*, each tailored to individual tasks and user needs, presents a promising vision for the future of machine learning.

However, multi-tenant serving is challenging due to two key reasons: 1) **Expensive Storage.** Each new fine-tuned model is large, even if we have relatively few base models, making them expensive to store and challenging to manage on disk. 2) **Expensive Serving.** Distinct fine-tuned models each demand significant GPU memory, making it difficult and expensive to concurrently serve such models without noticeable downtime.

**Insight: Information Disparity in Pre-training vs. Fine-tuning**

Given the higher computational demand of pre-training, it makes sense to assume that fine-tuning adds less new information to the model. This implies that fine-tuned models that are derived from the same base model may share a significant amount of redundant information. Can we exploit this to address the above storage and serving challenges?

Quantization results for Vicuna-7B v1.5 with base model Llama 2-7B. Adjusted average is over ARC, BBH, HellaSwag, Winogrande. We highlight TruthfulQA, GSM8K, MT-Bench as the base model struggles on these tasks, showing that BitDelta effectively retains fine-tune information.

**It turns out that we can!** We introduce BitDelta, which decomposes the weights of fine-tuned models into their pre-trained components and an additional delta: \(W_\text{fine} = W_\text{base} + \Delta\) Drawing from this insight, we find that we can quantize this delta, which encodes the fine-tuning information, down to **1 bit** without compromising performance. We conduct experiments over 17 popular fine-tuned models across the Llama-2 and Mistral families, and show that BitDelta is quite general. BitDelta is fast (compression takes minutes), works for models across a wide range of sizes (we test models between 7B and 70B parameters), and can retain all sorts of fine-tuning information (we test SFT, RLHF, DPO, and RoPE based context extension).

**Aside: What is quantization?**

In mathematics, quantization is the process of mapping infinite continuous values to a smaller set of discrete finite values. In efficient deep learning, we use quantization as a way to lossily compress models, reducing their memory footprint. This is especially important today, as LLMs have grown to the point where they have become difficult to store and serve with current hardware. For example, four A6000 GPUs are required to run Llama 2-70B with 16-bit weights. With a 4-bit weight quantization method like GPTQ or AWQ, we can run this on one A6000 GPU instead, reducing the memory footprint by 4x.

We conduct experiments over 17 popular fine-tuned models across the Llama-2 and Mistral families, and show that BitDelta is quite general. BitDelta is fast (compression takes minutes), works for models across a wide range of sizes (we test models between 7B and 70B parameters), and can retain all sorts of fine-tuning information (we test SFT, RLHF, DPO, and RoPE based context extension).

By representing multiple fine-tuned models as a single high-precision base model accompanied by multiple 1-bit deltas, we can drastically reduce GPU memory requirements. This addresses the **storage challenge**. We also note that LLM inference is **memory bound** - i.e., it takes more time to transfer weight matrices to GPU cache than it does to perform matrix multiplication. Taking this into account, we can also translate this memory reduction into **faster inference** (2x for now) in multi-tenant settings, using an efficient 1-bit matrix multiplication kernel! This addresses the **serving challenge**.

Past work (GPT-Zip, DeltaZip) has also explored quantization of the weight delta, achieving quantization levels as low as 2-bits by applying methods introduced by GPTQ. We find that the weight delta is extremely compressible, and are able to achieve **1-bit** quantization with minimal performance degradation using a simpler methodology.

**BitDelta Overview**

**1-bit quantization**

Let \(W_\text{base}, W_\text{fine} \in \mathbb{R}^{n \times m}\) be weight matrices from the base model and fine-tuned model, respectively. We define the weight delta as \(\Delta = W_\text{fine} - W_\text{base}\), representing the modification in weights post-fine-tuning. For efficient representation, we aim to obtain a binarized estimator of this weight delta, denoted as \(\hat{\Delta}\), by encoding its sign bits:

$$\hat{\Delta} = \alpha \odot \text{Sign}(\Delta)$$

where

$$\text{Sign}(W_{ij}) =

\begin{cases}

+1, & \text{if } W_{ij} > 0, \\

-1, & \text{if } W_{ij} \leq 0,

\end{cases}$$

and \(\alpha\) is a high-precision scaling factor for the entire matrix. To minimize the approximation error in \(L_2\) norm:

$$||\Delta - \hat{\Delta}||_2^2 = \sum_{ij}(|W_{ij}|-\alpha)^2$$

we take

$$\alpha = \frac{1}{nm} \sum_{ij} |W_{ij}|$$

Surprisingly, we find that the above quantization approach already does quite well and retains most of the fine-tuned models' performance.

### Scale distillation

Intuitively, the scaling factor \alpha plays a more significant role in the low-bit regime, so we further optimize these scales by performing model distillation to align the output logits of the quantized model to that of the original fine-tuned model. More concretely, we freeze the model weights and optimize for the following objective:

$$\boldsymbol{\alpha}^* = \arg\min_{\boldsymbol{\alpha}} \mathbb{E}_{x \sim \mathbf{X}}\left[ \left\| \mathbf{Z}_{\text{fine}}(x) - \mathbf{Z}_{\text{bin}}(x; \boldsymbol{\alpha}) \right\|^2 \right]$$

where \(\mathbf{X}\) is a calibration dataset, and \(\mathbf{Z}(\cdot)\) are the logits of the respective models. We find that scale distillation is fairly insensitive to choice \(\mathbf{X}\), as 1) the process is extremely parameter efficient, and 2) the crucial aspect of the process is to logit match with the fine-tuned model, regardless of the actual text content. We denote the method without scale distillation as BitDelta-Initial, and the method with scale distillation as BitDelta. As seen in the table above, scale distillation is effective in further recovering fine-tune performance.

**Inference speedup**

Since LLM inference follows the memory-bound computation pattern where generation latency is proportional to the GPU memory used by the model weights, this reduced memory consumption also suggests the opportunity to improve the serving latency. For example, Punica and S-LoRA exploit LoRA's structure and memory saving by developing a CUDA kernel that can efficiently calculate the batched delta-activation product for low rank deltas. Similarly, we decompose the forward pass of each linear layer as follows:

$$X'_i = W_{\text{fine}, i}X_i \approx W_{\text{base}}X_i + \underbrace{ \hat{\Delta}_iX_i}_\textbf{Kernel} \label{eqn:kernel_decomp}$$

where \(X_i\) and \(X_i’\) represent input and output features to the \(i-th\) fine-tuned model, and the base model weight and the delta are computed separately. For a batch of requests,

\(W_{\text{base}}X_i\) can be computed with the classic batched GEMM kernel. We implement a fused binary GEMM kernel in Triton that allows us to calculate

\(\hat{\Delta}_iX\) in a batched setting while keeping the 1-bit deltas quantized until they are transferred to the GPU cache. This kernel fuses the dequantization operation with the GEMM calculation, reducing the data moving overhead by a large factor!

To illustrate the speedup, we benchmark the decoding latency of our kernel, a batched linear operation over multiple deltas with a single base model, as in the decomposed forward pass, and compare against naively computing the forward pass separately for each model. We ablate across the batch size and hidden size dimensions and find that our kernel consistently achieves a ~2x speedup.

Decoding latency of a linear layer with and without BitDelta. Blue: Naive forward pass with B distinct fine-tuned models. Yellow: Batched forward pass with BitDelta, corresponding to one base model and B 1-bit deltas, utilizing a Triton kernel.

**Ablation Studies**

**Quantized base models**

We apply BitDelta to Llama 2-7B Chat, and find it holds up when the underlying base model is quantized at various levels. Because 8-bit RTN and GPTQ work with 16-bit activations, we can keep the fine-tune weights \(W_\text{fine}\) and scaling factors \(\alpha\) in high precision, only quantizing the base weights \(W_\text{base}\).

\(FP16\) + \(\Delta\) outperforms GPTQ across the board. In the performance engineering context of multi-tenancy serving, we would rather store a single high precision base model with many 1-bit deltas than store many quantized fine-tuned models. This interesting result implies that the above also holds true in the *model quality* context of multi-tenancy serving.

We try using Llama 2-7B Chat as both the base model and fine-tune model, quantizing the base model using GPTQ, and find that we're able to outperform baseline GPTQ on many evaluations. We hypothesize this is because we can diffuse 16-bit information into the model through high precision scaling factors, at the cost of including a 1-bit delta.

**Varying fidelity of \(\Delta\)**

By successively applying BitDelta, treating the compressed model from the previous iteration as our base model, we can vary the granularity over the delta, associating it with multiple 1-bit masks. One advantage of doing this is the ability to assign arbitrary scale factors to each 1-bit mask. In contrast, when just increasing the bit size, scale factors are implicitly fixed with respect to each other. The figure shows how the TruthfulQA of Llama 2-7B plus an increasingly granular delta approaches that of Vicuna-7B v1.5.

**Future Work**

There are many exciting directions for future work. On the model quality side, we can incorporate saliency aware quantization in the weight deltas, similar to AWQ (Ji et. al.). On the compression side, we can investigate sub 1-bit quantization methods that maintain hardware-friendliness. On the serving side, we can further optimize the Triton kernel; it is actually fairly slow compared to the theoretical upper bound, considering small memory footprint of weight deltas. With further optimization, it should be possible to achieve a ~4-8x speedup. Finally, the idea of calibrating certain scale factors through distillation may be applied more generally to PTQ methods, which we hope will make low-bit quantized LLMs more robust.

- Lower

Cost20% - faster

training4x - network

compression117x

## Q: Should I use the RedPajama-V2 Dataset out of the box?

RedPajama-V2 is conceptualized as a pool of data that serves as a foundation for creating high quality datasets. The dataset is thus not intended to be used out of the box and, depending on the application, data should be filtered out using the quality signals that accompany the data. With this dataset, we take the view that the optimal filtering of data is dependent on the intended use. Our goal is to provide all the signals and tooling that enables this.

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