Perplexity

The perplexity example can be used to calculate the so-called perplexity value of a language model over a given text corpus. Perplexity measures how well the model can predict the next token with lower values being better. Note that perplexity is not directly comparable between models, especially if they use different tokenizers. Also note that finetunes typically result in a higher perplexity value even though the human-rated quality of outputs increases.

Within llama.cpp the perplexity of base models is used primarily to judge the quality loss from e.g. quantized models vs. FP16. The convention among contributors is to use the Wikitext-2 test set for testing unless noted otherwise (can be obtained with scripts/get-wikitext-2.sh). When numbers are listed all command line arguments and compilation options are left at their defaults unless noted otherwise. llama.cpp numbers are not directly comparable to those of other projects because the exact values depend strongly on the implementation details.

By default only the mean perplexity value and the corresponding uncertainty is calculated. The uncertainty is determined empirically by assuming a Gaussian distribution of the "correct" logits per and then applying error propagation.

More statistics can be obtained by recording the logits from the FP16 version of a model. To do this, supply perplexity with --kl-divergence-base path/to/logit/binary/file.kld. The program will then record all logits and save them to the provided path in binary format. The logit file will be very large, 11 GiB for LLaMA 2 or 37 GiB for LLaMA 3 when using the Wikitext-2 test set. Once you have the file, supply perplexity with the quantized model, the logits file via --kl-divergence-base, and finally the --kl-divergence argument to indicate that the program should calculate the so-called Kullback-Leibler divergence. This is a measure of how similar the FP16 and the quantized logit distributions are with a value of 0 indicating that the distribution are the same. The uncertainty on the mean KL divergence is calculated by assuming the KL divergence per token follows a Gaussian distribution.

In addition to the KL divergence the following statistics are calculated with --kl-divergence:

• Ratio of mean FP16 PPL and quantized PPL. Uncertainty is estimated on logits, then propagated. The logarithm of this metric is also calculated and printed, it is 0 if the logit distributions are the same.
• Difference of mean FP16 PPL and quantized PPL. Uncertainty is estimated on logits, then propagated.
• Mean change in "correct" token probability. Positive values mean the model gets better at prediction, negative values mean it gets worse.
• Pearson correlation coefficient of the "correct" token probabilites between models.
• Percentiles of change in "correct" token probability. Positive values mean the model gets better at prediction, negative values mean it gets worse. Can be used to judge noise vs. quality loss from quantization. If the percentiles are symmetric then the quantization is essentially just adding noise. If the negative values are significantly larger than the positive values then this indicates that the model is actually becoming worse from the quantization.
• The root mean square of the change in token probabilities. If you were to assume that the quantization simply causes Gaussian noise on the token probabilities then this would be the standard deviation of said noise. The uncertainty on the value is calculated that the change in token probabilities follows a Gaussian distribution. Related discussion: https://github.com/ggerganov/llama.cpp/discussions/2875 .
• Same top p: Percentage of how often the token was assigned the highest probabilites by both models. The uncertainty is calculated from the Gaussian approximation of the binomial distribution.

LLaMA 3 8b Scoreboard

Revision	f364eb6f
Backend	CUDA
CPU	AMD Epyc 7742
GPU	1x NVIDIA RTX 4090

Results were generated using the CUDA backend and are sorted by Kullback-Leibler divergence relative to FP16. The "WT" importance matrices were created using varying numbers of Wikitext tokens and can be found here. Note: the FP16 logits used for the calculation of all metrics other than perplexity are stored in a binary file between runs. In order to save space this file does not contain the exact same FP32 logits but instead casts them to 16 bit unsigned integers (with some scaling). So the "f16" results are to be understood as the difference resulting only from this downcast.

Quantization	imatrix	Model size [GiB]	PPL	ΔPPL	KLD	Mean Δp	RMS Δp
f16	None	14.97	6.233160 ± 0.037828	0.001524 ± 0.000755	0.000551 ± 0.000002	0.001 ± 0.002 %	0.787 ± 0.004 %
q8_0	None	7.96	6.234284 ± 0.037878	0.002650 ± 0.001006	0.001355 ± 0.000006	-0.019 ± 0.003 %	1.198 ± 0.007 %
q6_K	None	6.14	6.253382 ± 0.038078	0.021748 ± 0.001852	0.005452 ± 0.000035	-0.007 ± 0.006 %	2.295 ± 0.019 %
q5_K_M	None	5.33	6.288607 ± 0.038338	0.056974 ± 0.002598	0.010762 ± 0.000079	-0.114 ± 0.008 %	3.160 ± 0.031 %
q5_K_S	None	5.21	6.336598 ± 0.038755	0.104964 ± 0.003331	0.016595 ± 0.000122	-0.223 ± 0.010 %	3.918 ± 0.036 %
q5_1	None	5.65	6.337857 ± 0.038677	0.106223 ± 0.003476	0.018045 ± 0.000139	-0.287 ± 0.011 %	4.123 ± 0.039 %
q5_0	None	5.21	6.363224 ± 0.038861	0.131591 ± 0.003894	0.022239 ± 0.000166	-0.416 ± 0.012 %	4.634 ± 0.043 %
q4_K_M	WT 10m	4.58	6.382937 ± 0.039055	0.151303 ± 0.004429	0.028152 ± 0.000240	-0.389 ± 0.014 %	5.251 ± 0.049 %
q4_K_M	None	4.58	6.407115 ± 0.039119	0.175482 ± 0.004620	0.031273 ± 0.000238	-0.596 ± 0.014 %	5.519 ± 0.050 %
q4_K_S	WT 10m	4.37	6.409697 ± 0.039189	0.178064 ± 0.004744	0.031951 ± 0.000259	-0.531 ± 0.015 %	5.645 ± 0.051 %
iq4_NL	WT 10m	4.35	6.455593 ± 0.039630	0.223959 ± 0.005201	0.035742 ± 0.000288	-0.590 ± 0.016 %	5.998 ± 0.054 %
iq4_XS	WT 10m	4.14	6.459705 ± 0.039595	0.228071 ± 0.005207	0.036334 ± 0.000284	-0.668 ± 0.016 %	6.044 ± 0.054 %
q4_K_S	None	4.37	6.500529 ± 0.039778	0.268895 ± 0.005638	0.043136 ± 0.000314	-0.927 ± 0.017 %	6.562 ± 0.055 %
q4_1	None	4.78	6.682737 ± 0.041285	0.451103 ± 0.008030	0.071683 ± 0.000505	-0.927 ± 0.017 %	8.512 ± 0.063 %
q4_0	None	4.34	6.700147 ± 0.041226	0.468514 ± 0.007951	0.071940 ± 0.000491	-1.588 ± 0.022 %	8.434 ± 0.061 %
q3_K_L	WT 10m	4.03	6.671223 ± 0.041427	0.439590 ± 0.008154	0.073077 ± 0.000529	-0.940 ± 0.023 %	8.662 ± 0.064 %
q3_K_M	WT 10m	3.74	6.734255 ± 0.041838	0.502622 ± 0.008901	0.084358 ± 0.000588	-1.198 ± 0.024 %	9.292 ± 0.065 %
q3_K_L	None	4.03	6.787876 ± 0.042104	0.556242 ± 0.009171	0.087176 ± 0.000614	-1.532 ± 0.025 %	9.432 ± 0.067 %
q3_K_M	None	3.74	6.888498 ± 0.042669	0.656864 ± 0.010071	0.101913 ± 0.000677	-1.990 ± 0.026 %	10.203 ± 0.068 %
iq3_M	WT 10m	3.53	6.898327 ± 0.041643	0.666694 ± 0.009449	0.102534 ± 0.000663	-3.178 ± 0.026 %	10.513 ± 0.066 %
iq3_S	WT 10m	3.42	6.965501 ± 0.042406	0.733867 ± 0.010245	0.111278 ± 0.000710	-3.066 ± 0.027 %	10.845 ± 0.068 %
iq3_XS	WT 10m	3.28	7.163043 ± 0.043772	0.931409 ± 0.012084	0.138693 ± 0.000857	-3.667 ± 0.031 %	12.148 ± 0.070 %
iq3_XXS	WT 10m	3.05	7.458436 ± 0.046404	1.226803 ± 0.015234	0.183625 ± 0.001042	-3.918 ± 0.035 %	13.836 ± 0.074 %
q3_K_S	WT 10m	3.41	7.602878 ± 0.046848	1.371244 ± 0.015688	0.199821 ± 0.001008	-5.046 ± 0.037 %	14.980 ± 0.070 %
q3_K_S	None	3.41	7.863786 ± 0.048885	1.632152 ± 0.017733	0.228217 ± 0.001079	-5.604 ± 0.038 %	15.541 ± 0.070 %
iq2_M	WT 10m	2.74	8.600799 ± 0.055124	2.369166 ± 0.025244	0.325989 ± 0.00160	-6.463 ± 0.046 %	18.519 ± 0.080 %
q2_K	WT 10k	2.96	8.652290 ± 0.055572	2.420657 ± 0.025587	0.331393 ± 0.001562	-6.606 ± 0.046 %	18.790 ± 0.078 %
q2_K	WT 100k	2.96	8.641993 ± 0.055406	2.410359 ± 0.025495	0.331672 ± 0.001569	-6.628 ± 0.047 %	18.856 ± 0.078 %
q2_K	WT 10m	2.96	8.647825 ± 0.055610	2.416191 ± 0.025683	0.332223 ± 0.001572	-6.500 ± 0.047 %	18.881 ± 0.078 %
q2_K	WT 1m	2.96	8.674365 ± 0.055743	2.442732 ± 0.025843	0.335308 ± 0.001576	-6.634 ± 0.047 %	19.009 ± 0.079 %
q2_K	WT 1k	2.96	8.682605 ± 0.055916	2.450972 ± 0.026069	0.337093 ± 0.001596	-6.596 ± 0.047 %	18.977 ± 0.079 %
q2_K_S	WT 10m	2.96	9.323778 ± 0.061551	3.092145 ± 0.031914	0.403360 ± 0.001787	-7.131 ± 0.049 %	20.050 ± 0.081 %
q2_K_S	WT 1m	2.96	9.329321 ± 0.061378	3.097688 ± 0.031816	0.403590 ± 0.001797	-7.289 ± 0.049 %	20.123 ± 0.081 %
q2_K_S	WT 100k	2.96	9.362973 ± 0.061740	3.131339 ± 0.032169	0.408367 ± 0.001802	-7.198 ± 0.050 %	20.132 ± 0.081 %
q2_K_S	WT 10k	2.96	9.376479 ± 0.062045	3.144846 ± 0.032464	0.408662 ± 0.001819	-7.141 ± 0.050 %	20.120 ± 0.081 %
q2_K_S	WT 1k	2.96	9.415200 ± 0.062475	3.183567 ± 0.032993	0.415865 ± 0.001846	-7.153 ± 0.050 %	20.311 ± 0.082 %
iq2_S	WT 10m	2.56	9.650781 ± 0.063209	3.419148 ± 0.034017	0.439197 ± 0.001976	-8.319 ± 0.052 %	21.491 ± 0.083 %
q2_K	None	2.96	9.751568 ± 0.063312	3.519934 ± 0.033863	0.445132 ± 0.001835	-9.123 ± 0.051 %	21.421 ± 0.079 %
iq2_XS	WT 10m	2.43	10.761424 ± 0.071056	4.529791 ± 0.042229	0.546290 ± 0.002133	-10.576 ± 0.056 %	23.872 ± 0.082 %
iq2_XXS	WT 10m	2.24	14.091782 ± 0.098396	7.860148 ± 0.070752	0.812022 ± 0.002741	-14.363 ± 0.065 %	28.576 ± 0.084 %
iq1_M	WT 10m	2.01	25.493722 ± 0.177903	19.262089 ± 0.152396	1.393084 ± 0.003529	-24.672 ± 0.077 %	38.287 ± 0.084 %
iq1_S	WT 1m	1.88	58.097760 ± 0.438604	51.866126 ± 0.416604	2.211278 ± 0.004688	-32.471 ± 0.087 %	46.418 ± 0.085 %
iq1_S	WT 1k	1.88	58.267851 ± 0.446208	52.036218 ± 0.424373	2.214858 ± 0.004778	-31.880 ± 0.089 %	46.330 ± 0.086 %
iq1_S	WT 100k	1.88	58.581498 ± 0.453145	52.349864 ± 0.431360	2.220834 ± 0.004818	-32.261 ± 0.089 %	46.002 ± 0.086 %
iq1_S	WT 10m	1.88	60.694593 ± 0.471290	54.462959 ± 0.449644	2.254554 ± 0.004868	-31.973 ± 0.088 %	46.271 ± 0.086 %
iq1_S	WT 10k	1.88	63.221324 ± 0.493077	56.989691 ± 0.471423	2.293527 ± 0.004885	-32.261 ± 0.089 %	46.562 ± 0.086 %

There seems to be no consistent improvement from using more Wikitext tokens for the importance matrix. K-quants score better on mean Δp than the legacy quants than e.g. KL divergence would suggest.

LLaMA 2 vs. LLaMA 3 Quantization comparison

Revision	f364eb6f
Backend	CUDA
CPU	AMD Epyc 7742
GPU	1x NVIDIA RTX 4090

Metric	L2 7b q2_K	L3 8b q2_K	L2 7b q4_K_M	L3 8b q4_K_M	L2 7b q6_K	L3 8b q6_K	L2 7b q8_0	L3 8b q8_0
Mean PPL	5.794552 ± 0.032298	9.751568 ± 0.063312	5.877078 ± 0.032781	6.407115 ± 0.039119	5.808494 ± 0.032425	6.253382 ± 0.038078	5.798542 ± 0.032366	6.234284 ± 0.037878
Mean PPL ratio	1.107955 ± 0.001427	1.564849 ± 0.004525	1.014242 ± 0.000432	1.028160 ± 0.000723	1.002406 ± 0.000191	1.003490 ± 0.000296	1.000689 ± 0.000107	1.000425 ± 0.000161
Mean ΔPPL	0.625552 ± 0.008725	3.519934 ± 0.033863	0.082526 ± 0.002530	0.175482 ± 0.004620	0.013941 ± 0.001110	0.021748 ± 0.001852	0.003990 ± 0.000624	0.002650 ± 0.001006
PPL correlation	97.36%	89.62%	99.71%	99.34%	99.94%	99.88%	99.98%	99.96%
Mean KLD	0.108903 ± 0.000645	0.445132 ± 0.001835	0.012686 ± 0.000079	0.031273 ± 0.000238	0.002098 ± 0.000014	0.005452 ± 0.000035	0.000369 ± 0.000007	0.001355 ± 0.000006
Mean Δp	-2.710 ± 0.023 %	-9.123 ± 0.051 %	-0.416 ± 0.008 %	-0.596 ± 0.014 %	-0.035 ± 0.003 %	-0.007 ± 0.006 %	-0.005 ± 0.002 %	-0.019 ± 0.003 %
Maximum Δp	85.136%	94.268%	45.209%	95.054%	23.593%	53.601%	43.925%	28.734%
99.9% Δp	37.184%	50.003%	17.461%	27.084%	7.798%	13.613%	3.387%	6.402%
99.0% Δp	18.131%	25.875%	7.798%	12.084%	3.838%	6.407%	1.867%	3.544%
Median Δp	-0.391%	-2.476%	-0.026%	-0.024%	-0.001%	0.000%	-0.000%	-0.000%
1.0% Δp	-39.762%	-87.173%	-11.433%	-19.567%	-4.222%	-6.767%	-1.862%	-3.698%
0.1% Δp	-79.002%	-98.897%	-26.433%	-56.054%	-9.091%	-16.584%	-3.252%	-6.579%
Minimum Δp	-99.915%	-99.965%	-83.383%	-98.699%	-43.142%	-68.487%	-9.343%	-24.301%
RMS Δp	9.762 ± 0.053 %	21.421 ± 0.079 %	3.252 ± 0.024 %	5.519 ± 0.050 %	1.339 ± 0.010 %	2.295 ± 0.019 %	0.618 ± 0.011 %	1.198 ± 0.007 %
Same top p	85.584 ± 0.086 %	71.138 ± 0.119 %	94.665 ± 0.055 %	91.901 ± 0.072 %	97.520 ± 0.038 %	96.031 ± 0.051 %	98.846 ± 0.026 %	97.674 ± 0.040 %

LLaMA 3 BF16 vs. FP16 comparison

Revision	83330d8c
Backend	CPU
CPU	AMD Epyc 7742
GPU	N/A

Results were calculated with LLaMA 3 8b BF16 as --kl-divergence-base and LLaMA 3 8b FP16 as the --model for comparison.

Metric	Value
Mean PPL(Q)	6.227711 ± 0.037833
Mean PPL(base)	6.225194 ± 0.037771
Cor(ln(PPL(Q)), ln(PPL(base)))	99.990%
Mean ln(PPL(Q)/PPL(base))	0.000404 ± 0.000086
Mean PPL(Q)/PPL(base)	1.000404 ± 0.000086
Mean PPL(Q)-PPL(base)	0.002517 ± 0.000536
Mean KLD	0.00002515 ± 0.00000020
Maximum KLD	0.012206
99.9% KLD	0.000799
99.0% KLD	0.000222
99.0% KLD	0.000222
Median KLD	0.000013
10.0% KLD	-0.000002
5.0% KLD	-0.000008
1.0% KLD	-0.000023
Minimum KLD	-0.000059
Mean Δp	-0.0000745 ± 0.0003952 %
Maximum Δp	4.186%
99.9% Δp	1.049%
99.0% Δp	0.439%
95.0% Δp	0.207%
90.0% Δp	0.125%
75.0% Δp	0.029%
Median Δp	0.000%
25.0% Δp	-0.030%
10.0% Δp	-0.126%
5.0% Δp	-0.207%
1.0% Δp	-0.434%
0.1% Δp	-1.016%
Minimum Δp	-4.672%
RMS Δp	0.150 ± 0.001 %
Same top p	99.739 ± 0.013 %

Old Numbers

<details> <summary>Llama 2 70B Scoreboard</summary>

Quantization	Model size (GiB)	Perplexity	Delta to fp16
Q4_0	36.20	3.5550	3.61%
Q4_1	40.20	3.5125	2.37%
Q5_0	44.20	3.4744	1.26%
Q2_K	27.27	3.7339	8.82%
Q3_K_S	27.86	3.7019	7.89%
Q3_K_M	30.83	3.5932	4.72%
Q3_K_L	33.67	3.5617	3.80%
Q4_K_S	36.39	3.4852	1.57%
Q4_K_M	38.54	3.4725	1.20%
Q5_K_S	44.20	3.4483	0.50%
Q5_K_M	45.41	3.4451	0.40%
Q6_K	52.70	3.4367	0.16%
fp16	128.5	3.4313	-

</details>