GPU Mathematics for Machine Learning

Master the mathematical foundations, scaling strategies, and optimization formulas that power modern AI from single GPUs to massive distributed clusters

16×
Performance Improvement
1.2
PFLOPS Peak
95%
Matrix Operations
3.35
TB/s Memory

Real-Time Performance Dashboard

Current GPU Efficiency
75.2%
Memory Utilization
68.4%
Training Speed
342 tok/s
Cost Efficiency
$0.045
Scale Factor
6.8×
Overall Score
8.4/10

GPU Architectural Foundations

Tensor Core Performance Calculator

Operations Count
137.4 GFLOPS
Execution Time
0.034 ms
Memory Required
201.3 MB
Arithmetic Intensity
682.7 ops/byte
GEMM Formula: C = αAB + βC
Operations = 2 × N³ (for N×N matrices)
Memory = 3 × N² × bytes_per_element
Arithmetic Intensity = Operations / Memory_transferred

Architecture Comparison

Architecture Parallelization Memory Pattern GPU Utilization Tensor Core Benefit
Transformers Excellent Large matrix ops 70-85% 8× speedup
CNNs Excellent Regular, coalesced 80-95% 4× speedup
RNNs/LSTMs Limited Sequential dependencies 10-40% Minimal
GNNs Poor Irregular, sparse 10-30% None

GPU Performance Benchmark

GPU Model Memory FP32 TFLOPS FP16 TFLOPS Memory BW Power Price Performance Score
H100 SXM 80GB HBM3 67 1,979 3.35 TB/s 700W $30,000 9.8/10
A100 SXM 80GB HBM2e 19.5 312 2.0 TB/s 400W $15,000 8.5/10
V100 SXM 32GB HBM2 15.7 125 900 GB/s 300W $8,000 6.8/10
RTX 4090 24GB GDDR6X 35 165 1.0 TB/s 450W $1,600 7.2/10

Memory Calculations

Training Memory Calculator

Model Parameters
700 GB
Gradient Memory
700 GB
Optimizer Memory
1400 GB
Activation Memory
2.4 TB
Total Memory Required
5.2 TB
GPUs Required (80GB each)
65 GPUs
Training Memory Formula
Total = Parameters + Gradients + Optimizer + Activations
Activations = B × S × H × L × (16 + 2/p) bytes
Mixed Precision: 10 bytes/param vs FP32: 16 bytes/param

Parameter Counting for Transformers

Embedding Parameters
39.0M
Attention Parameters
84.9M
MLP Parameters
56.6M
Total Parameters
124.4M
Transformer Parameter Formula
C = E(V + P) + L(12E² + 13E) + 2E
E=embedding, V=vocab, P=position, L=layers
12E² = attention weights, 8E² ≈ MLP (4E hidden dim)

Distributed Training Mathematics

Scaling Efficiency Calculator

Communication Volume
612.5 GB
Communication Time
12.25 s
Scaling Efficiency
0.8%
Effective Speedup
0.06x
Recommended Strategy
ZeRO-3 + Pipeline
Distributed Training Formulas
Efficiency = Compute_time / (Compute_time + Communication_time)
All-reduce: 2×(N-1)/N × model_size
Pipeline bubble: (stages-1) / microbatches

ZeRO Optimizer Analysis

ZeRO Stage 1

4×
Optimizer State Partitioning

ZeRO Stage 2

8×
+ Gradient Partitioning

ZeRO Stage 3

Linear
+ Parameter Partitioning

FLOPS Analysis & Performance

Training FLOPS Calculator

FLOPS per Token
1.05 TFLOPS
Total Training FLOPS
3.15 × 10²³
Model FLOPS Utilization
60.6%
Training Time (days)
3.0 days
Performance Rating
Excellent
OpenAI Scaling Law
Training FLOPS ≈ 6N per token (N = non-embedding parameters)
Forward: 2N FLOPS, Backward: 4N FLOPS
MFU = Model_FLOPS_per_second / Hardware_peak_FLOPS

Inference FLOPS & KV Cache

Prefill FLOPS
35 TFLOPS
Generation FLOPS (total)
17.5 TFLOPS
KV Cache Memory
4.5 GB
Total Inference FLOPS
52.5 TFLOPS
Tokens per Second (est.)
4.5 tok/s
Inference FLOPS Formulas
Autoregressive: T × 2N FLOPS for T output tokens
KV Cache: B × S × H × L × 2 × bytes_per_param
Prefill scales with sequence length squared

Cost Mathematics & Economics

Total Cost of Ownership Calculator

Hardware Cost
$240,000
Annual Power Cost
$36,792
5-Year On-Premises TCO
$423,960
5-Year Cloud Cost
$2,102,400
Savings with On-Premises
$1,678,440
Breakeven Point
6.8 months
TCO Analysis Formula
On-Premises = Hardware + (Power × Hours × Days × Years)
Cloud = Hourly_rate × Hours × Days × Years
Power includes 30-50% cooling overhead

Inference Cost per Token

Effective Tokens/Hour
288,000
Cost per 1K Tokens
$0.083
Daily Revenue Potential
$5,760
Optimization Opportunity
Medium
$0.001
Optimized Cost
80%+ utilization
$0.005
Average Cost
40-60% utilization
$0.01+
Poor Utilization
<30% utilization

Latest Optimization Strategies

FlashAttention Performance Impact

Memory Usage
2.1 GB
Speed Improvement
2.8×
H100 Utilization
75%
Memory Reduction
65%

FlashAttention-3 Improvements:

  • • Warp-specialization for H100 architecture
  • • Native FP8 support for maximum throughput
  • • 75% of theoretical H100 maximum utilization
  • • 1.5-2× speedup over FlashAttention-2

Gradient Checkpointing Trade-offs

Memory Reduction
√L ≈ 75%
Compute Overhead
+20%
Effective Memory
125 GB
Batch Size Increase
4×
Gradient Checkpointing Formula
Memory = O(√L) instead of O(L)
Compute overhead ≈ 20% additional time
Net benefit: 2-4× larger batch sizes possible

5D Parallelization Strategy

Data Parallel

Batch splitting

Tensor Parallel

Weight distribution

Pipeline Parallel

Layer splitting

Context Parallel

Sequence splitting

Expert Parallel

MoE experts
Total GPUs Required
128
Communication Complexity
High

Meta's Llama 3.1 405B Configuration:

  • • 16,000+ H100 GPUs across multiple datacenters
  • • 5D parallelization with advanced scheduling
  • • Optimal efficiency achieved at 64-512 GPU clusters
  • • Sophisticated fault tolerance and checkpointing

GPU Mathematics Mastery

Understanding GPU mathematics is essential for building efficient, scalable ML infrastructure. From memory optimization through distributed communication complexity, these mathematical principles determine the success of modern AI projects.

10×
Cost Reduction Possible
75%
Max H100 Utilization
3.5×
Tensor Core Speedup
37%
Mixed Precision Savings