ABSTRACT

Large neural networks spend most computation on floating point tensor multiplications. In this work, we find that a floating point multiplier can be approximated by one integer adder with high precision. We propose the linear-complexity multiplication (L-Mul) algorithm that approximates floating point number multiplication with integer addition operations. The new algorithm costs significantly less computation resource than 8-bit floating point multiplication but achieves higher precision. Compared to 8-bit floating point multiplications, the proposed method achieves higher precision but consumes significantly less bit-level computation. Since multiplying floating point numbers requires substantially higher energy compared to integer addition operations, applying the L-Mul operation in tensor processing hardware can potentially reduce 95% energy cost by elementwise floating point tensor multiplications and 80% energy cost of dot products. We calculated the theoretical error expectation of L-Mul, and evaluated the algorithm on a wide range of textual, visual, and symbolic tasks, including natural language understanding, structural reasoning, mathematics, and commonsense question answering. Our numerical analysis experiments agree with the theoretical error estimation, which indicates that L-Mul with 4-bit mantissa achieves comparable precision as float8 e4m3 multiplications, and L-Mul with 3-bit mantissa outperforms float8 e5m2. Evaluation results on popular benchmarks show that directly applying L-Mul to the attention mechanism is almost lossless. We further show that replacing all floating point multiplications with 3-bit mantissa L-Mul in a transformer model achieves equivalent precision as using float8 e4m3 as accumulation precision in both fine-tuning and inference.

I tried implementing this for AVX512 with tinyBLAS in llamafile.

    inline __m512 lmul512(__m512 x, __m512 y) {
        __m512i sign_mask = _mm512_set1_epi32(0x80000000);
        __m512i exp_mask = _mm512_set1_epi32(0x7F800000);
        __m512i mant_mask = _mm512_set1_epi32(0x007FFFFF);
        __m512i exp_bias = _mm512_set1_epi32(127);
        __m512i x_bits = _mm512_castps_si512(x);
        __m512i y_bits = _mm512_castps_si512(y);
        __m512i sign_x = _mm512_and_si512(x_bits, sign_mask);
        __m512i sign_y = _mm512_and_si512(y_bits, sign_mask);
        __m512i exp_x = _mm512_srli_epi32(_mm512_and_si512(x_bits, exp_mask), 23);
        __m512i exp_y = _mm512_srli_epi32(_mm512_and_si512(y_bits, exp_mask), 23);
        __m512i mant_x = _mm512_and_si512(x_bits, mant_mask);
        __m512i mant_y = _mm512_and_si512(y_bits, mant_mask);
        __m512i sign_result = _mm512_xor_si512(sign_x, sign_y);
        __m512i exp_result = _mm512_sub_epi32(_mm512_add_epi32(exp_x, exp_y), exp_bias);
        __m512i mant_result = _mm512_srli_epi32(_mm512_add_epi32(mant_x, mant_y), 1);
        __m512i result_bits = _mm512_or_si512(
            _mm512_or_si512(sign_result, _mm512_slli_epi32(exp_result, 23)), mant_result);
        return _mm512_castsi512_ps(result_bits);
    }

> In this section, we show that L-Mul is more precise than fp8 e4m3 multiplications

> To be concise, we do not consider the rounding to nearest even mode in both error analysis and complexity estimation for both Mul and L-Mul

These two statements together are non-sensical. Sure, if you analyze accuracy while ignoring the part of the algorithm that gives you accuracy in the baseline you can derive whatever cherry-picked result you want.

The multiplication of two floating point values if you round to nearest even will be the correctly rounded result of multiplying the original values at infinite precision, this is how floating point rounding usually works and what IEEE 754 mandates for fundamental operations if you choose to follow those guidelines (e.g., multiplication here). But not rounding to nearest even will result in a lot more quantization noise, and biased noise at that too.

> applying the L-Mul operation in tensor processing hardware can potentially reduce 95% energy cost by elementwise floating point tensor multiplications and 80% energy cost of dot products

A good chunk of the energy cost is simply moving data between memories (especially external DRAM/HBM/whatever) and along wires, buffering values in SRAMs and flip-flops and the like. Combinational logic cost is usually not a big deal. While having a ton of fixed-function matrix multipliers does raise the cost of combinational logic quite a bit, at most what they have will probably cut the power of an overall accelerator by 10-20% or so.

> In this section, we demonstrate that L-Mul can replace tensor multiplications in the attention mechanism without any loss of performance, whereas using fp8 multiplications for the same purpose degrades inference accuracy

I may have missed it in the paper, but they have provided no details on (re)scaling and/or using higher precision accumulation for intermediate results as one would experience on an H100 for instance. Without this information, I don't trust these evaluation results either.

https://news.ycombinator.com/item?id=41816598

This has been done for decades in digital circuits, FPGA’s, Digital Signal Processing, etc. Floating point is both resource and power intensive and using FP without the use of dedicated FP processing hardware is something that has been avoided and done without for decades unless absolutely necessary.

Why not publish some actual benchmarks that prove your claim in even a few special cases?

In other sensationalized words: "AI engineers can claim new algorithm allows them to fit GPT-5 in an RTX5090 running at 600 watts."

https://arxiv.org/abs/2307.01415

https://github.com/microsoft/BitNet

I had assumed the latency etc were based on what was desirable for the use case and hardware, rather than power consumption.

Maybe even generate a table of the approximate results and use that, in various stages? Like the way sin/cos was done 30y ago before FP coprocessors arrived

For he sake of the climate and environment it would be nice to be true.

Bad news for Nvidia. “Sell your stock” bad.

Does it come with a demonstration?

Surely even the OpenAI devs must have done this like the minute they got done training that model, right? I wonder if they'd even admit it was an AI that came up with the solution rather than just publishing it, and taking credit. haha.