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CUDA Binary Cross-Entropy Loss Optimization Guide

December 25, 20259 minBy RightNow AI Team

Introduction

Binary Cross-Entropy (BCE) loss is used for binary classification and multi-label problems. Like softmax cross-entropy, naive implementation can overflow or underflow. Fusing sigmoid with BCE (BCEWithLogits) provides numerical stability.

Common Performance Issues

Overflow in sigmoid for large positive logits
Underflow in log(sigmoid) for large negative logits
Separate sigmoid and BCE wastes bandwidth
Class imbalance handling

Optimization Techniques

1. Stable Sigmoid-BCE Fusion

Use log-sum-exp formulation to avoid explicit sigmoid.

cuda

// BCE = -[y*log(sigmoid(x)) + (1-y)*log(1-sigmoid(x))]
// Stable: max(x,0) - x*y + log(1 + exp(-|x|))
__device__ float bce_stable(float logit, float target) {
    float max_val = fmaxf(logit, 0.0f);
    return max_val - logit * target + logf(1.0f + expf(-fabsf(logit)));
}

Implementation Comparison

Before (Naive Implementation)

Direct computation prone to overflow and log(0).

cuda

__global__ void bce_naive(float* logits, float* targets, float* loss, int n) {
    int idx = blockIdx.x * blockDim.x + threadIdx.x;
    if (idx < n) {
        float p = 1.0f / (1.0f + expf(-logits[idx]));  // Can overflow!
        float y = targets[idx];
        loss[idx] = -(y * logf(p) + (1-y) * logf(1-p));  // log(0)!
    }
}

After (Optimized Implementation)

Stable formulation with optional positive class weighting.

cuda

__global__ void bce_stable_fused(float* __restrict__ logits,
                                    float* __restrict__ targets,
                                    float* __restrict__ loss,
                                    float pos_weight, int n) {
    int idx = blockIdx.x * blockDim.x + threadIdx.x;
    int stride = blockDim.x * gridDim.x;

    for (int i = idx; i < n; i += stride) {
        float x = logits[i];
        float y = targets[i];

        // Stable BCE with pos_weight
        // loss = max(x,0) - x*y + log(1+exp(-|x|))
        // With pos_weight: multiply positive term by pos_weight
        float max_val = fmaxf(x, 0.0f);
        float log_term = logf(1.0f + expf(-fabsf(x)));

        if (pos_weight != 1.0f) {
            // Weighted: (1-y)*max(x,0) + pos_weight*y*max(-x,0) + ...
            loss[i] = (1.0f - y) * max_val + pos_weight * y * fmaxf(-x, 0.0f)
                    + (1.0f + (pos_weight - 1.0f) * y) * log_term;
        } else {
            loss[i] = max_val - x * y + log_term;
        }
    }
}

Performance Results

Metric	Naive	Optimized	Improvement
Throughput	180 GB/s	450 GB/s	2.5x faster
Numerical range	Fails \|x\|>20	Full float32 range	Robust

Frequently Asked Questions

When to use BCE vs Cross-Entropy?

BCE for binary/multi-label (independent outputs). Cross-entropy for multi-class (mutually exclusive). BCE allows multiple labels per sample.

How does pos_weight help class imbalance?

pos_weight scales the positive class loss. Set to neg_count/pos_count to balance. Equivalent to oversampling positives.

Cross-Entropy Loss

Multi-class version

→

Sigmoid

BCE uses sigmoid internally

→

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CUDA BCEbinary cross entropysigmoid lossmulti-label classificationlogistic lossbinary classification

Optimization Techniques

1. Stable Sigmoid-BCE Fusion

Use log-sum-exp formulation to avoid explicit sigmoid.

cuda

// BCE = -[y*log(sigmoid(x)) + (1-y)*log(1-sigmoid(x))]
// Stable: max(x,0) - x*y + log(1 + exp(-|x|))
__device__ float bce_stable(float logit, float target) {
    float max_val = fmaxf(logit, 0.0f);
    return max_val - logit * target + logf(1.0f + expf(-fabsf(logit)));
}

Implementation Comparison

Before (Naive Implementation)

Direct computation prone to overflow and log(0).

cuda

__global__ void bce_naive(float* logits, float* targets, float* loss, int n) {
    int idx = blockIdx.x * blockDim.x + threadIdx.x;
    if (idx < n) {
        float p = 1.0f / (1.0f + expf(-logits[idx]));  // Can overflow!
        float y = targets[idx];
        loss[idx] = -(y * logf(p) + (1-y) * logf(1-p));  // log(0)!
    }
}

After (Optimized Implementation)

Stable formulation with optional positive class weighting.

cuda

__global__ void bce_stable_fused(float* __restrict__ logits,
                                    float* __restrict__ targets,
                                    float* __restrict__ loss,
                                    float pos_weight, int n) {
    int idx = blockIdx.x * blockDim.x + threadIdx.x;
    int stride = blockDim.x * gridDim.x;

    for (int i = idx; i < n; i += stride) {
        float x = logits[i];
        float y = targets[i];

        // Stable BCE with pos_weight
        // loss = max(x,0) - x*y + log(1+exp(-|x|))
        // With pos_weight: multiply positive term by pos_weight
        float max_val = fmaxf(x, 0.0f);
        float log_term = logf(1.0f + expf(-fabsf(x)));

        if (pos_weight != 1.0f) {
            // Weighted: (1-y)*max(x,0) + pos_weight*y*max(-x,0) + ...
            loss[i] = (1.0f - y) * max_val + pos_weight * y * fmaxf(-x, 0.0f)
                    + (1.0f + (pos_weight - 1.0f) * y) * log_term;
        } else {
            loss[i] = max_val - x * y + log_term;
        }
    }
}

Metric

Naive

Optimized

Improvement

Throughput

180 GB/s

450 GB/s

2.5x faster

Numerical range

Fails |x|>20

Full float32 range

Robust

Frequently Asked Questions

When to use BCE vs Cross-Entropy?

BCE for binary/multi-label (independent outputs). Cross-entropy for multi-class (mutually exclusive). BCE allows multiple labels per sample.

How does pos_weight help class imbalance?

pos_weight scales the positive class loss. Set to neg_count/pos_count to balance. Equivalent to oversampling positives.

CUDA Binary Cross-Entropy Loss Optimization Guide

Introduction

Common Performance Issues

Optimization Techniques

1. Stable Sigmoid-BCE Fusion

Implementation Comparison

Before (Naive Implementation)

After (Optimized Implementation)

Performance Results

Frequently Asked Questions

When to use BCE vs Cross-Entropy?

How does pos_weight help class imbalance?

Related Guides

CUDA Binary Cross-Entropy Loss Optimization Guide

Introduction

Common Performance Issues

Optimization Techniques

1. Stable Sigmoid-BCE Fusion

Implementation Comparison

Before (Naive Implementation)

After (Optimized Implementation)

Performance Results

Frequently Asked Questions

When to use BCE vs Cross-Entropy?

How does pos_weight help class imbalance?

Related Guides