| Data: Layer i to k with: Momentum Mi,Weight Wi, binary Maski prune rate pi, density d 1 fori←O to k do |
| Wi ← xavierInit(Wi) |
| 2 3 | Maski ← createMaskForWeight(Wi,d) |
| 4 | applyMask(Wi,Maski) |
| 5 end | |
|
| 6 for epoch ← O to numEpochs do |
| 7 | for j←O to numBatches do |
| 8 | batch ← getBatch(j) E |
| 9 | W = computeGradients(W, batch) |
| 10 | UpdateMomentum( 器） |
| 11 | UpdateWeights(M) |
| 12 | fori←O to k do |
| 13 | applyMask(Wi,Maski) end |
| 14 | end |
| 15 | |
| 16 | totalMomentum ← getTotalMomentum(M) |
| 17 | totalPruned ← getTotalPrunedWeights(W, p) |
| 18 | fori←O to k do |
| 19 | mi ← getMomentumContribution(Mi,Maski,totalMomentum) |
| 20 | magnitudePruneWeight(Wi,Maski, Pi) |
| 21 | regrowWeights(Wi,Maski,mi · totalPruned) |
| 22 | Pi←decayPrunerate(pi) |
| 23 | applyMask(Wi,Maski) |
| 24 | end |
| 25 end |
+
+Before training, we initialize the network with a certain sparsity $s$ : we initialize the network as usual and then remove a fraction of $s$ weights for each layer. We train the network normally and mask the weights after each gradient update to enforce sparsity. We apply sparse momentum after each epoch. We can break the sparse momentum into three major parts: (a) redistribution of weights, (b) pruning weights, (c) regrowing weights. In step (a), we we take the mean of the element-wise momentum momentum $m _ { i }$ agnitude of all layers . The resulting pr $i$ ortion is the momentum magnitude $\scriptstyle \sum _ { i = 0 } ^ { k } m _ { i }$
+removed weights multiplied by each layers momentum contribution: $\mathrm { R e g r o w } _ { i } =$ Total Removed · $m _ { i }$ . In step (b), we prune a proportion of $p$ (prune rate) of the weights with the lowest magnitude for each layer. In step (c), we regrow weights by enabling the gradient flow of zero-valued (missing) weights which have the largest momentum magnitude.
+
+Additionally, there are two edge-cases which we did not include in Algorithm 1 for clarity. (1) If we allocate more weights to be regrown than is possible for a specific layer, for example regrowing 100 weights for a layer of maximum 10 weights, we redistribute the excess number of weights equally among all other layers. (2) For some layers, our algorithm will converge in that the average weight in layer $i$ has much larger momentum magnitude than weights in other layers, but at the same time, this layer is dense and cannot grow further. We do not want to prune weights from such important layers. Thus, for these layers, we reduce the prune rate $p _ { i }$ proportional to the sparsity: $p _ { i } = \mathrm { { m i n } } ( p , \mathrm { { s p a r s i t y } } _ { i } )$ .
+
+After each epoch, we decay the prune rate in Algorithm 1 in the same way learning rates are decayed. We use a cosine decay schedule that anneals the prune rate to zero on the last epoch. See Appendix A.1 for an analysis on how decay schedule and starting prune rate affects training.
+
+# 4 EXPERIMENTAL SETUP
+
+For comparison, we follow three different experimental settings, one from Lee et al. (2019) and two settings follow Mostafa and Wang (2019): For MNIST (LeCun, 1998), we use a batch size of 100, decay the learning rate by a factor of 0.1 every 25000 mini-batches. For CIFAR-10 (Krizhevsky and Hinton, 2009), we use standard data augmentations (horizontal flip, and random crop with reflective padding), a batch size of 128, and decay the learning rate every 30000 mini-batches. We train for 100 and 250 epochs on MNIST and CIFAR-10, use a learning rate of 0.1, stochastic gradient descent with Nesterov momentum of $\alpha = 0 . 9$ , and we use a weight decay of 0.0005. We use a fixed $10 \%$ of the training data as the validation set and train on the remaining $90 \%$ . We evaluate the test set performance of our models on the last epoch. For all experiments on MNIST and CIFAR-10, we report the standard errors. Our sample size is generally between 10 and 12 experiments per method/architecture/sparsity level with different random seeds for each experiment.
+
+We use the modified network architectures of AlexNet, VGG16, and LeNet-5 as introduced by Lee et al. (2019). We consider two different variations of the experimental setup of Mostafa and Wang (2019) for ImageNet and CIFAR-10. The first follows their procedure closely, in that we run the networks in a partially dense setting where the first convolutional layer and downsampling convolutional layers are dense. Additionally, for CIFAR-10 the last fully connected layer is dense. In the second setting, we compare in a fully sparse setting – no layer is dense at the beginning of training. For the fully sparse setting we increase overall number of weights according to the extra parameters in the dense layers and distribute them equally among the network. The parameters in the dense layers make up $5 . 6 3 \%$ weights of the ResNet-50 network. We refer to these two settings as the partially dense and fully sparse settings.
+
+On ImageNet (Deng et al., 2009), we use ResNet-50 (He et al., 2016) with a stride of 2 for the $3 \mathrm { x } 3$ convolution in the bottleneck layers. We use a batch size of 256, input size of 224, momentum of $\alpha = 0 . 9$ , and weight decay of $1 \dot { 0 } ^ { - 4 }$ . We train for 100 epochs and report validation set performance after the last epoch. We report results for the fully sparse and the partially dense setting.
+
+For all experiments, we keep biases and batch normalization weights dense. We tuned the prune rate $p$ and momentum rate $\alpha$ searching the parameter space $\{ 0 . 2 , 0 . 3 , \bar { 0 } . 4 , 0 . 5 , 0 . 6 , 0 . 7 \}$ and $\{ 0 . 5 , 0 . 6 , 0 . 7$ $0 . 8 , 0 . 9 , 0 . 9 5 , 0 . 9 9 \}$ on MNIST and CIFAR-10 and found that $p = 0 . 2$ and $\alpha = 0 . 9$ work well for most architectures. We use this prune and momentum rate throughout all experiments.
+
+ImageNet experiments were run on $4 \mathbf { x }$ RTX 2080 Ti and all other experiments on individual GPUs.
+
+Our software builds on PyTorch (Paszke et al., 2017) and is a wrapper for PyTorch neural networks with a modular architecture for growth, redistribution, and pruning algorithms. Currently, no GPUaccelerated libraries that utilize sparse tensors exist, and as such we use masked weights to simulate sparse neural networks. Using our software, any PyTorch neural network can be adapted to be a sparse momentum network with less than 10 lines of code. We will open-source our software along with trained models and individual experimental results.1
+
+# 5 RESULTS
+
+Results in Figure 2 and Table 1 show a comparison with model compression methods. On MNIST, sparse momentum is the only method that provides consistent strong performance across both LeNet 300-100 and LeNet-5 Caffe models. Soft-weight sharing (Ullrich et al., 2017) and Layer-wise Brain Damage (Dong et al., 2017) are competitive with sparse momentum for one model, but underperforms for the other model. For $1 \%$ of weights, variational dropout is more effective – but this method also uses dropout for further regularization while we only use weight decay. We can see that sparse momentum achieves equal performance to the LeNet-5 Caffe dense baseline with $8 \%$ weights.
+
+On CIFAR-10 in Table 1, we can see that sparse momentum outperforms Single-shot Network Pruning (SNIP) for all models and can achieve the same performance level as a dense model for VGG16-D with just $5 \%$ of weights.
+
+
+Figure 2: Comparisons against compression methods on MNIST with $9 5 \%$ confidence intervals.
+
+Figure 3 and Table 2 show comparisons of sparse learning methods on MNIST and CIFAR that follows the experimental procedure of Mostafa and Wang (2019) where some selected layers are dense. For LeNet 300-100 on MNIST, we can see that sparse momentum outperforms all other methods. For CIFAR-10, sparse momentum is better than dynamic sparse in 4 out of 5 cases. However, in general, the confidence intervals for most methods overlap – this particular setup for CIFAR-10 with specifically selected dense layers seems to be too easy to determine difference in performance between methods and we do not recommend this setup for future work. Table 2 shows that sparse momentum outperforms all other methods on ImageNet (ILSVRC2012) for the Top-1 accuracy measure. Dynamic sparse is better for the Top-5 accuracy with $20 \%$ weights. In the fully sparse setting, sparse momentum remains competitive and seems to find a weight distribution which works equally well for the $10 \%$ weights case. For $20 \%$ weights, the performance decreases slightly.
+
+
+Figure 3: Test set accuracy with $9 5 \%$ confidence intervals on MNIST and CIFAR at varying sparsity levels for LeNet 300-100 and WRN 28-2.
+
+# 5.1 SPEEDUPS AND WEIGHTS NEEDED FOR DENSE PERFORMANCE LEVELS
+
+We analyzed how many weights are needed to achieve dense performance for our networks on CIFAR-10 and how much faster would we able to train such a sparse network compared to a dense one. We do this analysis by increasing the number of weights by $5 \%$ until the sparse network trained with sparse momentum reaches a performance level that overlaps with a $9 5 \%$ confidence interval of the dense performance. We then measure the speedup of the model. For each network-density combination we perform ten training runs with different random seeds to calculate the mean test error and its standard error.
+
+To estimated the speedups that could be obtained using sparse momentum for these dense networks we follow two approaches: Theoretical speedups for sparse convolution algorithms which are proportional to reductions in FLOPS and practical speedups using dense convolutional algorithms which are proportional to empty convolutional channels. For our sparse convolution estimates, we calculate the FLOPS saved for each convolution operation throughout training as well as the runtime for each convolution. To receive the maximum speedups for sparse convolution, we then scale the runtime for each convolution operation by the FLOPS saved. While a fast sparse convolution algorithm for coarse block structures exist for GPUs (Gray et al., 2017), optimal sparse convolution algorithms for fine-grained patterns do not and need to be developed to enable these speedups.
+
+Table 1: CIFAR-10 test set error ( $\pm$ standard error) for dense baselines, Sparse Momentum and SNIP.
+
+| Data: Layer i to k with: Momentum Mi,Weight Wi, binary Maski prune rate pi, density d 1 fori←O to k do |
| Wi ← xavierInit(Wi) |
| 2 3 | Maski ← createMaskForWeight(Wi,d) |
| 4 | applyMask(Wi,Maski) |
| 5 end | |
|
| 6 for epoch ← O to numEpochs do |
| 7 | for j←O to numBatches do |
| 8 | batch ← getBatch(j) E |
| 9 | W = computeGradients(W, batch) |
| 10 | UpdateMomentum( 器） |
| 11 | UpdateWeights(M) |
| 12 | fori←O to k do |
| 13 | applyMask(Wi,Maski) end |
| 14 | end |
| 15 | |
| 16 | totalMomentum ← getTotalMomentum(M) |
| 17 | totalPruned ← getTotalPrunedWeights(W, p) |
| 18 | fori←O to k do |
| 19 | mi ← getMomentumContribution(Mi,Maski,totalMomentum) |
| 20 | magnitudePruneWeight(Wi,Maski, Pi) |
| 21 | regrowWeights(Wi,Maski,mi · totalPruned) |
| 22 | Pi←decayPrunerate(pi) |
| 23 | applyMask(Wi,Maski) |
| 24 | end |
| 25 end |
",
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+ "type": "text",
+ "text": "4 EXPERIMENTAL SETUP ",
+ "text_level": 1,
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+ },
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+ "text": "For comparison, we follow three different experimental settings, one from Lee et al. (2019) and two settings follow Mostafa and Wang (2019): For MNIST (LeCun, 1998), we use a batch size of 100, decay the learning rate by a factor of 0.1 every 25000 mini-batches. For CIFAR-10 (Krizhevsky and Hinton, 2009), we use standard data augmentations (horizontal flip, and random crop with reflective padding), a batch size of 128, and decay the learning rate every 30000 mini-batches. We train for 100 and 250 epochs on MNIST and CIFAR-10, use a learning rate of 0.1, stochastic gradient descent with Nesterov momentum of $\\alpha = 0 . 9$ , and we use a weight decay of 0.0005. We use a fixed $10 \\%$ of the training data as the validation set and train on the remaining $90 \\%$ . We evaluate the test set performance of our models on the last epoch. For all experiments on MNIST and CIFAR-10, we report the standard errors. Our sample size is generally between 10 and 12 experiments per method/architecture/sparsity level with different random seeds for each experiment. ",
+ "bbox": [
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+ "text": "We use the modified network architectures of AlexNet, VGG16, and LeNet-5 as introduced by Lee et al. (2019). We consider two different variations of the experimental setup of Mostafa and Wang (2019) for ImageNet and CIFAR-10. The first follows their procedure closely, in that we run the networks in a partially dense setting where the first convolutional layer and downsampling convolutional layers are dense. Additionally, for CIFAR-10 the last fully connected layer is dense. In the second setting, we compare in a fully sparse setting – no layer is dense at the beginning of training. For the fully sparse setting we increase overall number of weights according to the extra parameters in the dense layers and distribute them equally among the network. The parameters in the dense layers make up $5 . 6 3 \\%$ weights of the ResNet-50 network. We refer to these two settings as the partially dense and fully sparse settings. ",
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+ "text": "On ImageNet (Deng et al., 2009), we use ResNet-50 (He et al., 2016) with a stride of 2 for the $3 \\mathrm { x } 3$ convolution in the bottleneck layers. We use a batch size of 256, input size of 224, momentum of $\\alpha = 0 . 9$ , and weight decay of $1 \\dot { 0 } ^ { - 4 }$ . We train for 100 epochs and report validation set performance after the last epoch. We report results for the fully sparse and the partially dense setting. ",
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+ "type": "text",
+ "text": "For all experiments, we keep biases and batch normalization weights dense. We tuned the prune rate $p$ and momentum rate $\\alpha$ searching the parameter space $\\{ 0 . 2 , 0 . 3 , \\bar { 0 } . 4 , 0 . 5 , 0 . 6 , 0 . 7 \\}$ and $\\{ 0 . 5 , 0 . 6 , 0 . 7$ $0 . 8 , 0 . 9 , 0 . 9 5 , 0 . 9 9 \\}$ on MNIST and CIFAR-10 and found that $p = 0 . 2$ and $\\alpha = 0 . 9$ work well for most architectures. We use this prune and momentum rate throughout all experiments. ",
+ "bbox": [
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+ "page_idx": 4
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+ {
+ "type": "text",
+ "text": "ImageNet experiments were run on $4 \\mathbf { x }$ RTX 2080 Ti and all other experiments on individual GPUs. ",
+ "bbox": [
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+ "page_idx": 4
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+ "type": "text",
+ "text": "Our software builds on PyTorch (Paszke et al., 2017) and is a wrapper for PyTorch neural networks with a modular architecture for growth, redistribution, and pruning algorithms. Currently, no GPUaccelerated libraries that utilize sparse tensors exist, and as such we use masked weights to simulate sparse neural networks. Using our software, any PyTorch neural network can be adapted to be a sparse momentum network with less than 10 lines of code. We will open-source our software along with trained models and individual experimental results.1 ",
+ "bbox": [
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+ },
+ {
+ "type": "text",
+ "text": "5 RESULTS ",
+ "text_level": 1,
+ "bbox": [
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+ ],
+ "page_idx": 4
+ },
+ {
+ "type": "text",
+ "text": "Results in Figure 2 and Table 1 show a comparison with model compression methods. On MNIST, sparse momentum is the only method that provides consistent strong performance across both LeNet 300-100 and LeNet-5 Caffe models. Soft-weight sharing (Ullrich et al., 2017) and Layer-wise Brain Damage (Dong et al., 2017) are competitive with sparse momentum for one model, but underperforms for the other model. For $1 \\%$ of weights, variational dropout is more effective – but this method also uses dropout for further regularization while we only use weight decay. We can see that sparse momentum achieves equal performance to the LeNet-5 Caffe dense baseline with $8 \\%$ weights. ",
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+ "page_idx": 4
+ },
+ {
+ "type": "text",
+ "text": "On CIFAR-10 in Table 1, we can see that sparse momentum outperforms Single-shot Network Pruning (SNIP) for all models and can achieve the same performance level as a dense model for VGG16-D with just $5 \\%$ of weights. ",
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+ "img_path": "images/9e005205b0e6994a85124aca20160b59043af33e15e75edf770b4775ce008edb.jpg",
+ "image_caption": [
+ "Figure 2: Comparisons against compression methods on MNIST with $9 5 \\%$ confidence intervals. "
+ ],
+ "image_footnote": [],
+ "bbox": [
+ 173,
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+ {
+ "type": "text",
+ "text": "Figure 3 and Table 2 show comparisons of sparse learning methods on MNIST and CIFAR that follows the experimental procedure of Mostafa and Wang (2019) where some selected layers are dense. For LeNet 300-100 on MNIST, we can see that sparse momentum outperforms all other methods. For CIFAR-10, sparse momentum is better than dynamic sparse in 4 out of 5 cases. However, in general, the confidence intervals for most methods overlap – this particular setup for CIFAR-10 with specifically selected dense layers seems to be too easy to determine difference in performance between methods and we do not recommend this setup for future work. Table 2 shows that sparse momentum outperforms all other methods on ImageNet (ILSVRC2012) for the Top-1 accuracy measure. Dynamic sparse is better for the Top-5 accuracy with $20 \\%$ weights. In the fully sparse setting, sparse momentum remains competitive and seems to find a weight distribution which works equally well for the $10 \\%$ weights case. For $20 \\%$ weights, the performance decreases slightly. ",
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+ "img_path": "images/9dd88284b32d232481d4be4d6a060b725bbcbf77df1fa6d28d3b3fe0b839a50c.jpg",
+ "image_caption": [
+ "Figure 3: Test set accuracy with $9 5 \\%$ confidence intervals on MNIST and CIFAR at varying sparsity levels for LeNet 300-100 and WRN 28-2. "
+ ],
+ "image_footnote": [],
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+ {
+ "type": "text",
+ "text": "5.1 SPEEDUPS AND WEIGHTS NEEDED FOR DENSE PERFORMANCE LEVELS ",
+ "text_level": 1,
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+ "type": "text",
+ "text": "We analyzed how many weights are needed to achieve dense performance for our networks on CIFAR-10 and how much faster would we able to train such a sparse network compared to a dense one. We do this analysis by increasing the number of weights by $5 \\%$ until the sparse network trained with sparse momentum reaches a performance level that overlaps with a $9 5 \\%$ confidence interval of the dense performance. We then measure the speedup of the model. For each network-density combination we perform ten training runs with different random seeds to calculate the mean test error and its standard error. ",
+ "bbox": [
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+ ],
+ "page_idx": 5
+ },
+ {
+ "type": "text",
+ "text": "To estimated the speedups that could be obtained using sparse momentum for these dense networks we follow two approaches: Theoretical speedups for sparse convolution algorithms which are proportional to reductions in FLOPS and practical speedups using dense convolutional algorithms which are proportional to empty convolutional channels. For our sparse convolution estimates, we calculate the FLOPS saved for each convolution operation throughout training as well as the runtime for each convolution. To receive the maximum speedups for sparse convolution, we then scale the runtime for each convolution operation by the FLOPS saved. While a fast sparse convolution algorithm for coarse block structures exist for GPUs (Gray et al., 2017), optimal sparse convolution algorithms for fine-grained patterns do not and need to be developed to enable these speedups. ",
+ "bbox": [
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+ {
+ "type": "table",
+ "img_path": "images/b7d4320ae9a6727fc4baf1e048755e25eb23692cb1889a49a1d9cc5ed4c96675.jpg",
+ "table_caption": [
+ "Table 1: CIFAR-10 test set error ( $\\pm$ standard error) for dense baselines, Sparse Momentum and SNIP. "
+ ],
+ "table_footnote": [
+ "\\* $9 5 \\%$ confidence intervals overlap with dense model. Table 2: Results for ResNet-50 on ImageNet. "
+ ],
+ "table_body": "| Network Recovery | AUC | Prec@500 | Community | Density | Silhouette |
| 1 | 5 | 10 | 1 | 5 | 10 | Detection | 5 | 25 | 50 5 | 25 | 50 |
| SEGEN(PS4) | 0.879 (1) | 0.881(1) | | | 0.881 (1) 0.914(3) 0.638(3) 0.370 (3) | | SEGEN(PS5) | 0.980 (2) | 0.845 (3) | 0.770 (3) | ）0.566(4) | 0.353(3) | 0.341(2) |
| SEGEN(PS1) | 0.814 (4) | | | | 0.813 (4) 0.814(4)0.606(4) 0.194 (4) 0.102 (4) | | SEGEN(PS1) | 0.786 (7) | 0.751(4) | | | 0.753(4) 0.481(10) 0.328(4) | 0.318(4) |
| SEGEN-HS(PS4) | 0.862 (2) | | | | | | ）0.863(2).6(2).594(5)(-)0(-)N-(96(6)(1)(.45)(1)( | | | | | | |
| SEENS(P4 | | | | | | | | | | | | | |
| EGENDS | | | | | | | | | | | | | |
| SEGEN-NS(PS)(57(6()8()E57986( | | | | | | | | | | | | | |
| SEGENES(PS | | | | | | | | | | | | | |
| LINE | | | | | 0.254 (11)0()0(1)0()18(7)(7) | | LINE | | | | | | 0.524 (11) 0.324(10) 0.251(9) 0.465 (11) -0.012(9) -0.012 (7) |
| DEEPWALK | | | | | 0.533(9)0.531(9)0.532(9)0.524(9) 0.146(6)0.070 (6) | | DEEPWALK | | | | | | 0.545(10) 0.542(7)0.503(7)0.492(9)0.173(8）0.150(6) |
| NODE2VEC | | | | | 0.704(7)0.703(7) 0.704(7）0.528 (8) 0.012(8) 0.000(-)| | | NODE2VEC | | | | | | 0.697(8) 0.693(5) 0.694(5) 0.530 (8) -0.020 (10） -0.015 (8) |
| HPE | | | | | 0.593(8)0.595 (8)0.594 (8)0.534(7) 0.186(5) 0.094 (5) | | HPE | | | | | | 0.579(9)0.579(6)0.579(6) 0.544(6)0.208(7)0.187(5) |
",
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+ {
+ "type": "text",
+ "text": "• SEGEN: Model SEGEN proposed in this paper is based on the genetic algorithm and ensemble learning, which effectively combines the learned sub-network representation feature vectors from the unit models to generate the feature vectors of the whole network. • LINE: The LINE model is a scalable network embedding model proposed in Tang et al. (2015), which optimizes an objective function that preserves both the local and global network structures. LINE uses a edge-sampling algorithm to addresses the limitation of the classical stochastic gradient descent. DEEPWALK: The DEEPWALK model Perozzi et al. (2014) extends the word2vec model Mikolov et al. (2013) to the network embedding scenario. DEEPWALK uses local information obtained from truncated random walks to learn latent representations. \n• NODE2VEC: The NODE2VEC model Grover & Leskovec (2016) introduces a flexible notion of a node’s network neighborhood and design a biased random walk procedure to sample the neighbors for node representation learning. \n• HPE: The HPE model Chen et al. (2016) is originally proposed for learning user preference in recommendation problems, which can effectively project the information from heterogeneous networks to a low-dimensional space. ",
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+ },
+ {
+ "type": "text",
+ "text": "5.1.3 EVALUATION TASKS AND METRICS ",
+ "text_level": 1,
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+ "text": "The network representation learning results can hardly be evaluated directly, whose evaluations are usually based on certain application tasks. In this paper, we propose to use application tasks, network recovery and clustering, to evaluate the learned representation features from the comparison methods. Furthermore, the network recovery results are evaluated by metrics, like AUC and Precision $@ 5 0 0$ . Meanwhile the clustering results are evaluated by Density and Silhouette. Without specific remarks, the default parameter setting for SEGEN in the experiments will be Parameter Setting 1 (PS1): sub-network size: 10, pool size: 200, batch size: 10, generation unit model number: 10, generation number: 30. ",
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+ "text": "5.2 SOCIAL NETWORK DATASET EXPERIMENTAL RESULTS ",
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+ "text": "The model training convergence analysis, and detailed analysis about the pool sampling and model learning parameters is available in the Appendix in Section 7.1. Besides these analysis results, we also provide the performance analysis of SEGEN and baseline methods in Tables 1-2, where the parameter settings are specified next to the method name. We provide the rank of method performance among all the methods, which are denoted by the numbers in blue font, and the top 5 results are in a bolded font. As shown in the Tables, we have the network recovery and community detection results on the left and right sections respectively. For the network recovery task, we change the ratio of negative links compared with positive links with values $\\{ 1 , 5 , 1 0 \\}$ , which are evaluated by the metrics AUC and Prec $@ 5 0 0$ . For the community detection task, we change the number of clusters with values $\\{ 5 , 2 5 , 5 0 \\}$ , and the results are evaluated by the metrics Density and Silhouette. ",
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+ "text": "Besides PS1 introduced at the beginning of Section 5.1, we have 4 other parameter settings selected based on the parameter analysis introduced before. PS2 for network recovery on Foursquare: sub-network size 50, pool size 600, batch size 5, generation size 50. PS3 for community detection on Foursquare: sub-network size 25, pool size 300, batch size 35, generation size 5. PS4 for network recovery on Twitter: sub-network size 50, pool size 700, batch size 10, generation size 5. PS5 for community detection on Twitter: sub-network size 45, pool size 500, batch size 50, generation size 5. ",
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+ "text": "According to the results shown in Table 1, method SEGEN with PS2 can obtain very good performance for both the network recovery task and the community detection task. For instance, for the network recovery task, method SEGEN with PS2 achieves 0.909 AUC score, which ranks the second and only lose to SEGEN-HS with PS2; meanwhile, SEGEN with PS2 also achieves the second highest Prec $@ 5 0 0$ score (i.e., 0.872 for np-ratio $= 1$ ) and the third highest Prec $@ 5 0 0$ score (i.e., 0.642 and 0.530 for np-ratios 5 and 10) among the comparison methods. On the other hand, for the community detection task, SEGEN with PS3 can generally rank the second/third among the comparison methods for both density and silhouette evaluation metrics. For instance, with the cluster number is 5, the density obtained by SEGEN ranks the second among the methods, which loses to SEGEN-LS only. Similar results can be observed for the Twitter network as shown in Figure 2. ",
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+ "Table 3: Experiments on MNIST Dataset. "
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+ "table_body": "| Network Recovery | AUC | Prec@500 | Community | Density | Silhouette |
| 1 | 5 | 10 | 1 | 5 | 10 | Detection | 5 | 25 | 50 5 | 25 | 50 |
| SEGEN(PS4) | 0.879 (1) | 0.881(1) | | | 0.881 (1) 0.914(3) 0.638(3) 0.370 (3) | | SEGEN(PS5) | 0.980 (2) | 0.845 (3) | 0.770 (3) | ）0.566(4) | 0.353(3) | 0.341(2) |
| SEGEN(PS1) | 0.814 (4) | | | | 0.813 (4) 0.814(4)0.606(4) 0.194 (4) 0.102 (4) | | SEGEN(PS1) | 0.786 (7) | 0.751(4) | | | 0.753(4) 0.481(10) 0.328(4) | 0.318(4) |
| SEGEN-HS(PS4) | 0.862 (2) | | | | | | ）0.863(2).6(2).594(5)(-)0(-)N-(96(6)(1)(.45)(1)( | | | | | | |
| SEENS(P4 | | | | | | | | | | | | | |
| EGENDS | | | | | | | | | | | | | |
| SEGEN-NS(PS)(57(6()8()E57986( | | | | | | | | | | | | | |
| SEGENES(PS | | | | | | | | | | | | | |
| LINE | | | | | 0.254 (11)0()0(1)0()18(7)(7) | | LINE | | | | | | 0.524 (11) 0.324(10) 0.251(9) 0.465 (11) -0.012(9) -0.012 (7) |
| DEEPWALK | | | | | 0.533(9)0.531(9)0.532(9)0.524(9) 0.146(6)0.070 (6) | | DEEPWALK | | | | | | 0.545(10) 0.542(7)0.503(7)0.492(9)0.173(8）0.150(6) |
| NODE2VEC | | | | | 0.704(7)0.703(7) 0.704(7）0.528 (8) 0.012(8) 0.000(-)| | | NODE2VEC | | | | | | 0.697(8) 0.693(5) 0.694(5) 0.530 (8) -0.020 (10） -0.015 (8) |
| HPE | | | | | 0.593(8)0.595 (8)0.594 (8)0.534(7) 0.186(5) 0.094 (5) | | HPE | | | | | | 0.579(9)0.579(6)0.579(6) 0.544(6)0.208(7)0.187(5) |
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+ "html": "| Input: input x,confidence γ,threshold T,noise parameters O1 ando2,total number of teachers M |
| 1:Ask the student to provide prediction scores p(x) 2:if maxj{nj(x)-Mpj(x)}+N(0,o²)≥Tthen >Student does not agree with teachers |
|
| 3: return argmaxj{nj(x) +N(0,0²)} |
| 4: else if max{pi(x)}> γ then > Student agrees with teachers and is confident |
| 5: return arg maxj Pj(x) |
| Reinforce student's prediction 6: else return⊥ No output given for this label |
+
+student’s current predictions for its public training data, we design an Interactive Aggregator that discards queries where the student already confidently predicts the same label as the teachers.
+
+Given a set of queries, the Interactive Aggregator (Algorithm 2) selects those answered by comparing student predictions to teacher votes for each class. Similar to Step 1 in the Confident Aggregator, queries where the plurality of these noised differences crosses a threshold are answered with GNMax. This noisy threshold suffices to enforce privacy of the first step because student predictions can be considered public information (the student is trained in a differentially private manner).
+
+For queries that fail this check, the mechanism reinforces the predicted student label if the student is confident enough and does this without looking at teacher votes again. This limited form of supervision comes at a small privacy cost. Moreover, the order of the checks ensures that a student falsely confident in its predictions on a query is not accidentally reinforced if it disagrees with the teacher consensus. The privacy accounting is identical to the Confident Aggregator except in considering the difference between teachers and the student instead of only the teachers votes.
+
+In practice, the Confident Aggregator can be used to start training a student when it can make no meaningful predictions and training can be finished off with the Interactive Aggregator after the student gains some proficiency.
+
+# 5 EXPERIMENTAL EVALUATION
+
+Our goal is first to show that the improved aggregators introduced in Section 4 enable the application of PATE to uncurated data, thus departing from previous results on tasks with balanced and wellseparated classes. We experiment with the Glyph dataset described below to address two aspects left open by Papernot et al. (2017): (a) the performance of PATE on a task with a larger number of classes (the framework was only evaluated on datasets with at most 10 classes) and (b) the privacy-utility tradeoffs offered by PATE on data that is class imbalanced and partly mislabeled. In Section 5.2, we evaluate the improvements given by the GNMax aggregator over its Laplace counterpart (LNMax) and demonstrate the necessity of the Gaussian mechanism for uncurated tasks.
+
+In Section 5.3, we then evaluate the performance of PATE with both the Confident and Interactive Aggregators on all datasets used to benchmark the original PATE framework, in addition to Glyph. With the right teacher and student training, the two mechanisms from Section 4 achieve high accuracy with very tight privacy bounds. Not answering queries for which teacher consensus is too low (Confident-GNMax) or the student’s predictions already agree with teacher votes (InteractiveGNMax) better aligns utility and privacy: queries are answered at a significantly reduced cost.
+
+# 5.1 EXPERIMENTAL SETUP
+
+MNIST, SVHN, and the UCI Adult databases. We evaluate with two computer vision tasks (MNIST and Street View House Numbers (Netzer et al., 2011)) and census data from the UCI Adult dataset (Kohavi, 1996). This enables a comparative analysis of the utility-privacy tradeoff achieved with our Confident-GNMax aggregator and the LNMax originally used in PATE. We replicate the experimental setup and results found in Papernot et al. (2017) with code and teacher votes made available online. The source code for the privacy analysis in this paper as well as supporting data required to run this analysis is available on Github.1
+
+A detailed description of the experimental setup can be found in Papernot et al. (2017); we provide here only a brief overview. For MNIST and SVHN, teachers are convolutional networks trained on partitions of the training set. For UCI Adult, each teacher is a random forest. The test set is split in two halves: the first is used as unlabeled inputs to simulate the student’s public data and the second is used as a hold out to evaluate test performance. The MNIST and SVHN students are convolutional networks trained using semi-supervised learning with GANs à la Salimans et al. (2016). The student for the Adult dataset are fully supervised random forests.
+
+Glyph. This optical character recognition task has an order of magnitude more classes than all previous applications of PATE. The Glyph dataset also possesses many characteristics shared by real-world tasks: e.g., it is imbalanced and some inputs are mislabeled. Each input is a $2 8 \times 2 8$ grayscale image containing a single glyph generated synthetically from a collection of over 500K computer fonts.2 Samples representative of the difficulties raised by the data are depicted in Figure 3. The task is to classify inputs as one of the 150 Unicode symbols used to generate them.
+
+This set of 150 classes results from pre-processing efforts. We discarded additional classes that had few samples; some classes had at least 50 times fewer inputs than the most popular classes, and these were almost exclusively incorrectly labeled inputs. We also merged classes that were too ambiguous for even a human to differentiate them. Nevertheless, a manual inspection of samples grouped by classes—favorably to the human observer—led to the conservative estimate that some classes remain 5 times more frequent, and mislabeled inputs represent at least $1 0 \%$ of the data.
+
+To simulate the availability of private and public data (see Section 3.1), we split data originally marked as the training set (about 65M points) into partitions given to the teachers. Each teacher is a ResNet (He et al., 2016) made of 32 leaky ReLU layers. We train on batches of 100 inputs for 40K steps using SGD with momentum. The learning rate, initially set to 0.1, is decayed after 10K steps to 0.01 and again after 20K steps to 0.001. These parameters were found with a grid search.
+
+We split holdout data in two subsets of 100K and 400K samples: the first acts as public data to train the student and the second as its testing data. The student architecture is a convolutional network learnt in a semi-supervised fashion with virtual adversarial training (VAT) from Miyato et al. (2017). Using unlabeled data, we show how VAT can regularize the student by making predictions constant in adversarial3 directions. Indeed, we found that GANs did not yield as much utility for Glyph as for MNIST or SVHN. We train with Adam for 400 epochs and a learning rate of $6 \cdot \mathrm { i 0 ^ { - 5 } }$ .
+
+# 5.2 COMPARING THE LNMAX AND GNMAX MECHANISMS
+
+Section 4.1 introduces the GNMax mechanism and the accompanying privacy analysis. With a Gaussian distribution, whose tail diminishes more rapidly than the Laplace distribution, we expect better utility when using the new mechanism (albeit with a more involved privacy analysis).
+
+To study the tradeoff between privacy and accuracy with the two mechanisms, we run experiments training several ensembles of $M$ teachers for $M \in \{ 1 0 0 , 5 0 0 , 1 0 0 0 , 5 0 0 0 \}$ on the Glyph data. Recall that 65 million training inputs are partitioned and distributed among the $M$ teachers with each teacher receiving between 650K and 13K inputs for the values of $M$ above. The test data is used to query the teacher ensemble and the resulting labels (after the LNMax and GNMax mechanisms) are compared with the ground truth labels provided in the dataset. This predictive performance of the teachers is essential to good student training with accurate labels and is a useful proxy for utility.
+
+For each mechanism, we compute $( \varepsilon , \delta )$ -differential privacy guarantees. As is common in literature, for a dataset on the order of $\bar { 1 0 } ^ { 8 }$ samples, we choose $\delta = \bar { 1 0 } ^ { - 8 }$ and denote the corresponding $\varepsilon$ as the privacy cost. The total $\varepsilon$ is calculated on a subset of 4,000 queries, which is representative of the number of labels needed by a student for accurate training (see Section 5.3). We visualize in Figure 4 the effect of the noise distribution (left) and the number of teachers (right) on the tradeoff between privacy costs and label accuracy.
+
+Observations. On the left of Figure 1, we compare our GNMax aggregator to the LNMax aggregator used by the original PATE proposal, on an ensemble of 1000 teachers and for varying noise scales $\sigma$ . At fixed test accuracy, the GNMax algorithm consistently outperforms the LNMax mechanism in terms of privacy cost. To explain this improved performance, recall notation from Section 4.1. For both mechanisms, the data dependent privacy cost scales linearly with $\tilde { q }$ —the likelihood of an answer other than the true plurality. The value of $\tilde { q }$ falls of as $\exp ( - x ^ { 2 } )$ for GNMax and $\exp ( - x )$ for LNMax, where $x$ is the ratio $( \dot { n _ { i ^ { * } } } - n _ { i } ) / \sigma$ . Thus, when $n _ { i ^ { * } } - n _ { i }$ is (say) $4 \sigma$ , LNMax would have $\tilde { q } \approx e ^ { - 4 } = 0 . 0 1 8 . . .$ , whereas GNMax would have $\tilde { q } \approx e ^ { - 1 6 } \approx 1 0 ^ { - 7 }$ , thereby leading to a much higher likelihood of returning the true plurality. Moreover, this reduced $\tilde { q }$ translates to a smaller privacy cost for a given $\sigma$ leading to a better utility-privacy tradeoff.
+
+As long as each teacher has sufficient data to learn a good-enough model, increasing the number $M$ of teachers improves the tradeoff—as illustrated on the right of Figure 4 with GNMax. The larger ensembles lower the privacy cost of answering queries by tolerating larger $\sigma$ ’s. Combining the two observations made in this Figure, for a fixed label accuracy, we lower privacy costs by switching to the GNMax aggregator and training a larger number $M$ of teachers.
+
+# 5.3 STUDENT TRAINING WITH THE GNMAX AGGREGATION MECHANISMS
+
+As outlined in Section 3, we train a student on public data labeled by the aggregation mechanisms. We take advantage of PATE’s flexibility and apply the technique that performs best on each dataset: semi-supervised learning with Generative Adversarial Networks (Salimans et al., 2016) for MNIST and SVHN, Virtual Adversarial Training (Miyato et al., 2017) for Glyph, and fully-supervised random forests for UCI Adult. In addition to evaluating the total privacy cost associated with training the student model, we compare its utility to a non-private baseline obtained by training on the sensitive data (used to train teachers in PATE): we use the baselines of $9 9 . 2 \%$ , $9 2 . 8 \%$ , and $8 \bar { 5 } . 0 \%$ reported by Papernot et al. (2017) respectively for MNIST, SVHN, and UCI Adult, and we measure a baseline of $8 2 . 2 \%$ for Glyph. We compute $( \varepsilon , \delta )$ -privacy bounds and denote the privacy cost as the $\varepsilon$ value at a value of $\delta$ set accordingly to number of training samples.
+
+Confident-GNMax Aggregator. Given a pool of 500 to 12,000 samples to learn from (depending on the dataset), the student submits queries to the teacher ensemble running the Confident-GNMax aggregator from Section 4.2. A grid search over a range of plausible values for parameters $T$ , $\sigma _ { 1 }$ and $\sigma _ { 2 }$ yielded the values reported in Table 1, illustrating the tradeoff between utility and privacy achieved. We additionally measure the number of queries selected by the teachers to be answered and compare student utility to a non-private baseline.
+
+The Confident-GNMax aggregator outperforms LNMax for the four datasets considered in the original PATE proposal: it reduces the privacy cost $\varepsilon$ , increases student accuracy, or both simultaneously. On the uncurated Glyph data, despite the imbalance of classes and mislabeled data (as evidenced by the $8 2 . 2 \%$ baseline), the Confident Aggregator achieves $7 3 . 5 \%$ accuracy with a privacy cost of just $\varepsilon = 1 . 0 2$ . Roughly 1,300 out of 12,000 queries made are not answered, indicating that several expensive queries were successfully avoided. This selectivity is analyzed in more details in Section 5.4.
+
+Interactive-GNMax Aggregator. On Glyph, we evaluate the utility and privacy of an interactive training routine that proceeds in two rounds. Round one runs student training with a Confident
+
+
+Figure 3: Some example inputs from the Glyph dataset along with the class they are labeled as. Note the ambiguity (between the comma and apostrophe) and the mislabeled input.
+
+
+Figure 4: Tradeoff between utility and privacy for the LNMax and GNMax aggregators on Glyph: effect of the noise distribution (left) and size of the teacher ensemble (right). The LNMax aggregator uses a Laplace distribution and GNMax a Gaussian. Smaller values of the privacy cost $\varepsilon$ (often obtained by increasing the noise scale $\sigma$ —see Section 4) and higher accuracy are better.
+
+Table 1: Utility and privacy of the students. The Confident- and Interactive-GNMax aggregators introduced in Section 4 offer better tradeoffs between privacy (characterized by the value of the bound $\varepsilon$ ) and utility (the accuracy of the student compared to a non-private baseline) than the LNMax aggregator used by the original PATE proposal on all datasets we evaluated with. For MNIST, Adult, and SVHN, we use the labels of ensembles of 250 teachers published by Papernot et al. (2017) and set $\delta = 1 0 ^ { - 5 }$ to compute values of $\varepsilon$ (to the exception of SVHN where $\dot { \delta _ { \mathrm { } } } = 1 \dot { 0 } ^ { - 6 }$ ). All Glyph results use an ensemble of 5000 teachers and $\varepsilon$ is computed for $\delta = 1 0 ^ { - 8 }$ .
+
+| Dataset | Aggregator | Queriesanswered | Privacybound ε | Accuracy Student Baseline |
| MNIST | LNMax (Papernot et al., 2017) | 100 | 2.04 | 98.0% | 99.2% |
| LNMax (Papernot et al., 2017) | 1,000 | 8.03 | 98.1% |
| Confident-GNMax (T=200,σ1=150,σ2=40) | 286 | 1.97 | 98.5% |
| SVHN | LNMax (Papernot et al., 2017) | 500 | 5.04 | 82.7% | 92.8% |
| LNMax (Papernot et al., 2017) | 1,000 | 8.19 | 90.7% |
| Confident-GNMax (T=300,σ1=200,σ2=40) | 3,098 | 4.96 | 91.6% |
| Adult | LNMax (Papernot et al., 2017) | 500 | 2.66 | 83.0% | 85.0% |
| Confident-GNMax (T=300,σ1=200,σ2=40) | 524 | 1.90 | 83.7% |
| Glyph | LNMax | 4,000 | 4.3 | 72.4% | 82.2% |
| Confident-GNMax (T=1000,σ1=500,σ2=100) | 10,762 | 2.03 | 75.5% |
| Interactive-GNMax, two rounds | 4,341 | 0.837 | 73.2% |
+
+
+Figure 5: Effects of the noisy threshold checking: Left: The number of queries answered by LNMax, Confident-GNMax moderate $\scriptstyle { T = 3 5 0 0 }$ , $\sigma _ { 1 } { = } 1 5 0 0$ ), and Confident-GNMax aggressive ( ${ T } \mathrm { { = } } 5 0 0 0$ , $\sigma _ { 1 } { = } 1 5 0 0$ ). The black dots and the right axis (in log scale) show the expected cost of answering a single query in each bin (via GNMax, $\sigma _ { 2 } { = } 1 0 0$ ). Right: Privacy cost of answering all (LNMax) vs only inexpensive queries (GNMax) for a given number of answered queries. The very dark area under the curve is the cost of selecting queries; the rest is the cost of answering them.
+
+Aggregator. A grid search targeting the best privacy for roughly 3,400 answered queries (out of 6,000)—sufficient to bootstrap a student—led us to setting ( $\scriptstyle { T = 3 5 0 0 }$ , $\sigma _ { 1 } { = } 1 5 0 0$ , $\sigma _ { 2 } { = } 1 0 0 $ ) and a privacy cost of $\varepsilon \approx 0 . 5 9$ .
+
+In round two, this student was then trained with 10,000 more queries made with the InteractiveGNMax Aggregator $\scriptstyle { T = 3 5 0 0 }$ , $\sigma _ { 1 } { = } 2 0 0 0$ , $\sigma _ { 2 } { = } 2 0 0$ ). We computed the resulting (total) privacy cost and utility at an exemplar data point through another grid search of plausible parameter values. The result appears in the last row of Table 1. With just over 10,422 answered queries in total at a privacy cost of $\varepsilon = 0 . 8 4$ , the trained student was able to achieve $7 3 . 2 \%$ accuracy. Note that this students required fewer answered queries compared to the Confident Aggregator. The best overall cost of student training occurred when the privacy costs for the first and second rounds of training were roughly the same. (The total $\varepsilon$ is less than $0 . 5 9 \times 2 = 1 . 1 8$ due to better composition—via Theorems 4 and 5.)
+
+Comparison with Baseline. Note that the Glyph student’s accuracy remains seven percentage points below the non-private model’s accuracy achieved by training on the 65M sensitive inputs. We hypothesize that this is due to the uncurated nature of the data considered. Indeed, the class imbalance naturally requires more queries to return labels from the less represented classes. For instance, a model trained on 200K queries is only $7 7 \%$ accurate on test data. In addition, the large fraction of mislabeled inputs are likely to have a large privacy cost: these inputs are sensitive because they are outliers of the distribution, which is reflected by the weak consensus among teachers on these inputs.
+
+# 5.4 NOISY THRESHOLD CHECKS AND PRIVACY COSTS
+
+Sections 4.1 and 4.2 motivated the need for a noisy threshold checking step before having the teachers answer queries: it prevents most of the privacy budget being consumed by few queries that are expensive and also likely to be incorrectly answered. In Figure 5, we compare the privacy cost $\varepsilon$ of answering all queries to only answering confident queries for a fixed number of queries.
+
+We run additional experiments to support the evaluation from Section 5.3. With the votes of 5,000 teachers on the Glyph dataset, we plot in Figure 5 the histogram of the plurality vote counts $( n _ { i ^ { * } }$ in the notation of Section 4.1) across 25,000 student queries. We compare these values to the vote counts of queries that passed the noisy threshold check for two sets of parameters $T$ and $\sigma _ { 1 }$ in Algorithm 1. Smaller values imply weaker teacher agreements and consequently more expensive queries.
+
+When $\scriptstyle { T = 3 5 0 0 }$ , $\sigma _ { 1 } { = } 1 5 0 0$ ) we capture a significant fraction of queries where teachers have a strong consensus (roughly $> 4 0 0 0$ votes) while managing to filter out many queries with poor consensus. This moderate check ensures that although many queries with plurality votes between 2,500 and 3,500 are answered (i.e., only $5 0 \mathrm { - } 7 0 \%$ of teachers agree on a label) the expensive ones are most likely discarded. For ( $\scriptstyle { T = 5 0 0 0 }$ , $\sigma _ { 1 } { = } 1 5 0 0$ ), queries with poor consensus are completely culled out. This selectivity comes at the expense of a noticeable drop for queries that might have had a strong consensus and little-to-no privacy cost. Thus, this aggressive check answer fewer queries with very strong privacy guarantees. We reiterate that this threshold checking step itself is done in a private manner. Empirically, in our Interactive Aggregator experiments, we expend about a third to a half of our privacy budget on this step, which still yields a very small cost per query across 6,000 queries.
+
+# 6 CONCLUSIONS
+
+The key insight motivating the addition of a noisy thresholding step to the two aggregation mechanisms proposed in our work is that there is a form of synergy between the privacy and accuracy of labels output by the aggregation: labels that come at a small privacy cost also happen to be more likely to be correct. As a consequence, we are able to provide more quality supervision to the student by choosing not to output labels when the consensus among teachers is too low to provide an aggregated prediction at a small cost in privacy. This observation was further confirmed in some of our experiments where we observed that if we trained the student on either private or non-private labels, the former almost always gave better performance than the latter—for a fixed number of labels.
+
+Complementary with these aggregation mechanisms is the use of a Gaussian (rather than Laplace) distribution to perturb teacher votes. In our experiments with Glyph data, these changes proved essential to preserve the accuracy of the aggregated labels—because of the large number of classes. The analysis presented in Section 4 details the delicate but necessary adaptation of analogous results for the Laplace NoisyMax.
+
+As was the case for the original PATE proposal, semi-supervised learning was instrumental to ensure the student achieves strong utility given a limited set of labels from the aggregation mechanism. However, we found that virtual adversarial training outperforms the approach from Salimans et al. (2016) in our experiments with Glyph data. These results establish lower bounds on the performance that a student can achieve when supervised with our aggregation mechanisms; future work may continue to investigate virtual adversarial training, semi-supervised generative adversarial networks and other techniques for learning the student in these particular settings with restricted supervision.
+
+# ACKNOWLEDGMENTS
+
+We are grateful to Martín Abadi, Vincent Vanhoucke, and Daniel Levy for their useful inputs and discussions towards this paper.
+
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+# A APPENDIX: PRIVACY ANALYSIS
+
+In this appendix, we provide the proofs of Theorem 6 and Proposition 7. Moreover, we present Proposition 10, which provides optimal values of $\mu _ { 1 }$ and $\mu _ { 2 }$ to apply towards Theorem 6 for the GNMax mechanism. We start off with a statement about the Rényi differential privacy guarantee of the GNMax.
+
+Proposition 8. The GNMax aggregator $\mathcal { M } _ { \sigma }$ guarantees $\left( \lambda , \lambda / \sigma ^ { 2 } \right)$ -RDP for all $\lambda \geq 1$
+
+Proof. The result follows from observing that $\mathcal { M } _ { \sigma }$ can be decomposed into applying the argmax operator to a noisy histogram resulted from adding Gaussian noise to each dimension of the original histogram. The Gaussian mechanism satisfies $( { \bar { \lambda } } , \lambda / 2 \sigma ^ { 2 } )$ -RDP (Mironov, 2017), and since each teacher may change two counts (incrementing one and decrementing the other), the overall RDP guarantee is as claimed. ■
+
+Proposition 7. For a GNMax aggregator $\mathcal { M } _ { \sigma }$ , the teachers’ votes histogram $\bar { n } = ( n _ { 1 } , \ldots , n _ { m } ) $ , and for any $i ^ { * } \in [ m ]$ , we have
+
+where
+
+$$
+\begin{array} { c l c r } { { \displaystyle { \bf P r } \left[ \mathcal { M } _ { \sigma } ( D ) \neq i ^ { * } \right] \le q ( \bar { n } ) , } } \\ { { \displaystyle } } \\ { { q ( \bar { n } ) \triangleq \frac { 1 } { 2 } \sum _ { i \neq i ^ { * } } \mathrm { e r f c } \left( \frac { n _ { i ^ { * } } - n _ { i } } { 2 \sigma } \right) . } } \end{array}
+$$
+
+Proof. Recall that $\mathcal { M } _ { \sigma } ( D ) = \operatorname { a r g m a x } ( n _ { i } + Z _ { i } )$ , where $Z _ { i }$ are distributed as ${ \mathcal { N } } ( 0 , \sigma ^ { 2 } )$ . Then for any $i ^ { * } \in [ m ]$ , we have
+
+$$
+\begin{array} { r } { \mathbf { P r } [ \mathcal { M } _ { \sigma } ( D ) \neq i ^ { * } ] = \mathbf { P r } \left[ \exists i , n _ { i } + Z _ { i } > n _ { i ^ { * } } + Z _ { i ^ { * } } \right] \leq \displaystyle \sum _ { i \neq i ^ { * } } \mathbf { P r } \left[ n _ { i } + Z _ { i } > n _ { i ^ { * } } + Z _ { i ^ { * } } \right] } \\ { = \displaystyle \sum _ { i \neq i ^ { * } } \mathbf { P r } \left[ Z _ { i } - Z _ { i ^ { * } } > n _ { i ^ { * } } - n _ { i } \right] } \\ { = \displaystyle \sum _ { i \neq i ^ { * } } \frac { 1 } { 2 } \left( 1 - \mathrm { e r f } \left( \frac { n _ { i ^ { * } } - n _ { i } } { 2 \sigma } \right) \right) . } \end{array}
+$$
+
+where the last equality follows from the fact that $Z _ { i } - Z _ { j }$ is a Gaussian random variable with mean zero and variance $2 \sigma ^ { 2 }$ .
+
+We now present a precise statement of Theorem 6.
+
+Theorem 6. Let $\mathcal { M }$ be a randomized algorithm with $( \mu _ { 1 } , \varepsilon _ { 1 } )$ -RDP and $\left( \mu _ { 2 } , \varepsilon _ { 2 } \right)$ -RDP guarantees and suppose that there exists a likely outcome $i ^ { * }$ given a dataset $D$ and a bound $\tilde { q } \leq 1$ such that $\tilde { q } \geq \mathbf { P r } \left[ \mathcal { M } ( D ) \neq i ^ { * } \right]$ . Additionally suppose that $\lambda \le \mu _ { 1 }$ and $\begin{array} { r } { \tilde { q } \le e ^ { ( \mu _ { 2 } - 1 ) \varepsilon _ { 2 } } / \biggl ( \frac { \mu _ { 1 } } { \mu _ { 1 } - 1 } \cdot \frac { \mu _ { 2 } } { \mu _ { 2 } - 1 } \biggr ) ^ { \mu _ { 2 } } } \end{array}$ Then, for any neighboring dataset $D ^ { \prime }$ of $D$ , we have:
+
+$$
+D _ { \lambda } ( \mathcal { M } ( D ) \| \mathcal { M } ( D ^ { \prime } ) ) \le \frac { 1 } { \lambda - 1 } \log \left( ( 1 - \tilde { q } ) \cdot A ( \tilde { q } , \mu _ { 2 } , \varepsilon _ { 2 } ) ^ { \lambda - 1 } + \tilde { q } \cdot B ( \tilde { q } , \mu _ { 1 } , \varepsilon _ { 1 } ) ^ { \lambda - 1 } \right)
+$$
+
+$$
+\begin{array} { r } { \pmb { A } ( \tilde { q } , \mu _ { 2 } , \varepsilon _ { 2 } ) \triangleq ( 1 - \tilde { q } ) / \bigg ( 1 - \big ( \tilde { q } e ^ { \varepsilon _ { 2 } } \big ) ^ { \frac { \mu _ { 2 } - 1 } { \mu _ { 2 } } } \bigg ) a n d \pmb { B } ( \tilde { q } , \mu _ { 1 } , \varepsilon _ { 1 } ) \triangleq e ^ { \varepsilon _ { 1 } } / \tilde { q } ^ { \frac { 1 } { \mu _ { 1 } - 1 } } . } \end{array}
+$$
+
+Proof. Before we proceed to the proof, we introduce some simplifying notation. For a randomized mechanism $\mathcal { M }$ and neighboring datasets $D$ and $D ^ { \prime }$ , we define
+
+$$
+\begin{array} { l } { \displaystyle \beta _ { \mathcal { M } } ( \lambda ; D , D ^ { \prime } ) \triangleq D _ { \lambda } ( \mathcal { M } ( D ) \| \mathcal { M } ( D ^ { \prime } ) ) } \\ { \displaystyle = \frac { 1 } { \lambda - 1 } \log \mathbb { E } _ { x \sim \mathcal { M } ( D ) } \left[ \left( \frac { \mathbf { P r } \left[ \mathcal { M } ( D ) = x \right] } { \mathbf { P r } \left[ \mathcal { M } ( D ^ { \prime } ) = x \right] } \right) ^ { \lambda - 1 } \right] . } \end{array}
+$$
+
+As the proof involves working with the RDP bounds in the exponent, we set $\zeta _ { 1 } \triangleq e ^ { \varepsilon _ { 1 } ( \mu _ { 1 } - 1 ) }$ and ζ2 , eε2(µ2−1).
+
+Finally, we define the following shortcuts:
+
+$$
+\begin{array} { l } { q _ { i } \triangleq \mathbf { P r } \left[ \mathcal { M } ( D ) = i \right] \mathrm { ~ a n d ~ } q \triangleq \displaystyle \sum _ { i \neq i ^ { * } } q _ { i } = \mathbf { P r } \left[ \mathcal { M } ( D ) \neq i ^ { * } \right] , } \\ { p _ { i } \triangleq \mathbf { P r } \left[ \mathcal { M } ( D ^ { \prime } ) = i \right] \mathrm { ~ a n d ~ } p \triangleq \displaystyle \sum _ { i \neq i ^ { * } } p _ { i } = \mathbf { P r } \left[ \mathcal { M } ( D ^ { \prime } ) \neq i ^ { * } \right] , } \end{array}
+$$
+
+and note that $q \leq \tilde { q }$
+
+From the definition of Rényi differential privacy, $( \mu _ { 1 } , \varepsilon _ { 1 } )$ -RDP implies:
+
+$$
+\begin{array} { r l } & { \exp \left( \beta _ { \mathcal M } ( \mu _ { 1 } ; D , D ^ { \prime } ) \right) = \displaystyle \left( \frac { ( 1 - q ) ^ { \mu _ { 1 } } } { ( 1 - p ) ^ { \mu _ { 1 } - 1 } } + \sum _ { i \neq i * } \frac { q _ { i } ^ { \mu _ { 1 } } } { p _ { i } ^ { \mu _ { 1 } - 1 } } \right) ^ { 1 / ( \mu _ { 1 } - 1 ) } \leq \exp ( \varepsilon _ { 1 } ) } \\ & { \qquad \implies \displaystyle \sum _ { i > 1 } \frac { q _ { i } ^ { \mu _ { 1 } } } { p _ { i } ^ { \mu _ { 1 } - 1 } } = \sum _ { i > 1 } q _ { i } \left( \frac { q _ { i } } { p _ { i } } \right) ^ { \mu _ { 1 } - 1 } \leq \zeta _ { 1 } . } \end{array}
+$$
+
+Since $\mu _ { 1 } \geq \lambda , f ( x ) \triangleq x ^ { \frac { \mu _ { 1 } - 1 } { \lambda - 1 } }$ is convex. Applying Jensen’s Inequality we have the following:
+
+$$
+\begin{array} { r l } & { \left( \frac { \sum _ { i \neq j : i } \cdot q _ { i } \left( \frac { q _ { i } } { p _ { i } } \right) ^ { \lambda - 1 } } { q } \right) ^ { \frac { n _ { 1 } - 1 } { \lambda - 1 } } \leq \frac { \sum _ { i \neq i } \cdot q _ { i } \left( \frac { q _ { i } } { p _ { i } } \right) ^ { \mu _ { 1 } - 1 } } { q } } \\ & { \qquad \implies \sum _ { i \neq i ^ { * } } q _ { i } \left( \frac { q _ { i } } { p _ { i } } \right) ^ { \lambda - 1 } \leq q \left( \frac { \sum _ { i \neq i } \cdot q _ { i } \left( \frac { q _ { i } } { p _ { i } } \right) ^ { \mu _ { 1 } - 1 } } { q } \right) ^ { \frac { \lambda - 1 } { \mu _ { 1 } - 1 } } } \\ & { \qquad \overset { ( 3 ) } { \Longrightarrow } \sum _ { i \neq i ^ { * } } q _ { i } \left( \frac { q _ { i } } { p _ { i } } \right) ^ { \lambda - 1 } \leq \zeta _ { 1 } ^ { \frac { \lambda - 1 } { \mu _ { 1 } - 1 } } \cdot q ^ { 1 - \frac { \lambda - 1 } { \mu _ { 1 } - 1 } } . } \end{array}
+$$
+
+Next, by the bound at order $\mu _ { 2 }$ , we have:
+
+$$
+\begin{array} { c } { { \displaystyle \exp \left( \beta _ { { \mathcal M } } ( \mu _ { 2 } ; D ^ { \prime } , D ) \right) = \left( \frac { ( 1 - p ) ^ { \mu _ { 2 } } } { ( 1 - q ) ^ { \mu _ { 2 } - 1 } } + \sum _ { i \neq i ^ { * } } \frac { p _ { i } ^ { \mu _ { 2 } } } { q _ { i } ^ { \mu _ { 2 } - 1 } } \right) ^ { 1 / ( \mu _ { 2 } - 1 ) } \leq \exp ( \varepsilon _ { 2 } ) } } \\ { { \displaystyle \implies \frac { ( 1 - p ) ^ { \mu _ { 2 } } } { ( 1 - q ) ^ { \mu _ { 2 } - 1 } } + \sum _ { i \neq i ^ { * } } \frac { p _ { i } ^ { \mu _ { 2 } } } { q _ { i } ^ { \mu _ { 2 } - 1 } } \leq \zeta _ { 2 } . } } \end{array}
+$$
+
+By the data processing inequality of Rényi divergence, we have
+
+$$
+\frac { ( 1 - p ) ^ { \mu _ { 2 } } } { ( 1 - q ) ^ { \mu _ { 2 } - 1 } } + \frac { p ^ { \mu _ { 2 } } } { q ^ { \mu _ { 2 } - 1 } } \leq \zeta _ { 2 } ,
+$$
+
+which implies pµ2qµ2−1 ≤ ζ2 and thus
+
+$$
+p \leq \left( q ^ { \mu _ { 2 } - 1 } \zeta _ { 2 } \right) ^ { \frac { 1 } { \mu _ { 2 } } } .
+$$
+
+Combining (4) and (5), we can derive a bound at $\lambda$ .
+
+$$
+\begin{array} { r l } & { \exp \left( \beta _ { \mathcal M } ( \lambda , D , D ^ { \prime } ) \right) = \left( \frac { ( 1 - q ) ^ { \lambda } } { ( 1 - p ) ^ { \lambda - 1 } } + \displaystyle \sum _ { i \neq i ^ { * } } \frac { q _ { i } ^ { \lambda } } { p _ { i } ^ { \lambda - 1 } } \right) ^ { 1 / ( \lambda - 1 ) } } \\ & { \quad \le \left( \frac { ( 1 - q ) ^ { \lambda } } { \left( 1 - ( q ^ { \mu _ { 2 } - 1 } \zeta _ { 2 } ) ^ { \frac { 1 } { \mu _ { 2 } } } \right) ^ { \lambda - 1 } } + \zeta _ { 1 } ^ { \frac { \lambda - 1 } { \mu _ { 1 } - 1 } } \cdot q ^ { 1 - \frac { \lambda - 1 } { \mu _ { 1 } - 1 } } \right) ^ { 1 / ( \lambda - 1 ) } . } \end{array}
+$$
+
+Although Equation (6) is very close to the corresponding statement in the theorem’s claim, one subtlety remains. The bound (6) applies to the exact probability $q = \mathbf { P r } \left[ \mathcal { M } ( D ) \neq i ^ { * } \right]$ . In the theorem statement, and in practice, we can only derive an upper bound $\tilde { q }$ on $\mathbf { P r } \left[ \mathcal { M } ( D ) \neq i ^ { * } \right]$ . The last step of the proof requires showing that the expression in Equation (6) is monotone in the range of values of $q$ that we care about.
+
+Lemma 9 (Monotonicity of the bound). Let the functions $f _ { 1 } ( \cdot )$ and $f _ { 2 } ( \cdot )$ be
+
+$$
+\begin{array} { l } { { f _ { 1 } ( x ) \triangleq \displaystyle \frac { ( 1 - x ) ^ { \lambda } } { \Big ( 1 - ( x ^ { \mu _ { 2 } - 1 } \zeta _ { 2 } ) ^ { \frac { 1 } { \mu _ { 2 } } } \Big ) ^ { \lambda - 1 } } \qquad \ a n d \qquad f _ { 2 } ( x ) \triangleq \zeta _ { 1 } ^ { \frac { \lambda - 1 } { \mu _ { 1 } - 1 } } \cdot x ^ { 1 - \frac { \lambda - 1 } { \mu _ { 1 } - 1 } } , } } \\ { { \mathrm { } } } \\ { { \mathrm { } } } \\ { { \mathrm { } } { \mathrm { } } { } \\ { \mathrm { } { } \mathrm { } + f _ { 2 } ( x ) \ i s \ i n c r e a s i n g \ i n \Big [ 0 , \operatorname* { m i n } \Big ( 1 , \zeta _ { 2 } \Big / \Big ( \frac { \mu _ { 1 } } { \mu _ { 1 } - 1 } \cdot \frac { \mu _ { 2 } } { \mu _ { 2 } - 1 } \Big ) ^ { \mu _ { 2 } } \Big ) \Big ] . } } \end{array}
+$$
+
+Proof. Taking the derivative of $f _ { 1 } ( x )$ , we have:
+
+$$
+f _ { 1 } ^ { \prime } ( x ) = \frac { - \lambda ( 1 - x ) ^ { \lambda - 1 } ( 1 - ( x ^ { \mu _ { 2 } - 1 } \zeta _ { 2 } ) ^ { \frac { 1 } { \mu _ { 2 } } } ) ^ { \lambda - 1 } } { ( 1 - ( x ^ { \mu _ { 2 } - 1 } \zeta _ { 2 } ) ^ { \frac { 1 } { \mu _ { 2 } } } ) ^ { 2 \lambda - 2 } } \\ & { \qquad + \frac { ( 1 - x ) ^ { \lambda } ( \lambda - 1 ) ( 1 - ( x ^ { \mu _ { 2 } - 1 } \zeta _ { 2 } ) ^ { \frac { 1 } { \mu _ { 2 } } } ) ^ { \lambda - 2 } \zeta _ { 2 } ^ { \frac { 1 } { \mu _ { 2 } } } \cdot \frac { \mu _ { 2 } - 1 } { \mu _ { 2 } } \cdot x ^ { - \frac { 1 } { \mu _ { 2 } } } } { ( 1 - ( x ^ { \mu _ { 2 } - 1 } \zeta _ { 2 } ) ^ { \frac { 1 } { \mu _ { 2 } } } ) ^ { 2 \lambda - 2 } } } \\ & = \frac { ( 1 - x ) ^ { \lambda - 1 } } { ( 1 - ( x ^ { \mu _ { 2 } - 1 } \zeta _ { 2 } ) ^ { \frac { 1 } { \mu _ { 2 } } } ) ^ { \lambda - 1 } } \left( - \lambda + ( \lambda - 1 ) \left( 1 - \frac { 1 } { \mu _ { 2 } } \right) \frac { 1 - x } { 1 - ( x ^ { \mu _ { 2 } - 1 } \zeta _ { 2 } ) ^ { \frac { 1 } { \mu _ { 2 } } } } \left( \frac { \zeta _ { 2 } } { x } \right) ^ { \frac { 1 } { \mu _ { 2 } } } \right) .
+$$
+
+We intend to show that:
+
+$$
+f _ { 1 } ^ { \prime } ( x ) \geq - \lambda + ( \lambda - 1 ) \left( 1 - \frac { 1 } { \mu _ { 2 } } \right) \left( \frac { \zeta _ { 2 } } { x } \right) ^ { \frac { 1 } { \mu _ { 2 } } } .
+$$
+
+For $\begin{array} { r } { x \in \left[ 0 , \zeta _ { 2 } / \left( \frac { \mu _ { 1 } } { \mu _ { 1 } - 1 } \cdot \frac { \mu _ { 2 } } { \mu _ { 2 } - 1 } \right) ^ { \mu _ { 2 } } \right] } \end{array}$ and $y \in [ 1 , \infty )$ , define $g ( x , y )$ as:
+
+$$
+g ( x , y ) \triangleq - \lambda \cdot y ^ { \lambda - 1 } + ( \lambda - 1 ) \left( 1 - { \frac { 1 } { \mu _ { 2 } } } \right) \left( { \frac { \zeta _ { 2 } } { x } } \right) ^ { \frac { 1 } { \mu _ { 2 } } } y ^ { \lambda } .
+$$
+
+We claim that $g ( x , y )$ is increasing in $y$ and therefore $g ( x , y ) \geq g ( x , 1 )$ , and prove it by showing the partial derivative of $g ( x , y )$ with respect to $y$ is non-negative. Take a derivative with respect to $y$ as:
+
+$$
+\begin{array} { l } { { g _ { y } ^ { \prime } ( x , y ) = - \lambda ( \lambda - 1 ) y ^ { \lambda - 2 } + \lambda ( \lambda - 1 ) \left( 1 - { \frac { 1 } { \mu _ { 2 } } } \right) \left( { \frac { \zeta _ { 2 } } { x } } \right) ^ { \frac { 1 } { \mu _ { 2 } } } y ^ { \lambda - 1 } } } \\ { { \ = \lambda ( \lambda - 1 ) y ^ { \lambda - 2 } \left( - 1 + \left( 1 - { \frac { 1 } { \mu _ { 2 } } } \right) \left( { \frac { \zeta _ { 2 } } { x } } \right) ^ { \frac { 1 } { \mu _ { 2 } } } y \right) . } } \end{array}
+$$
+
+To see why $g _ { y } ^ { \prime } ( x , y )$ is non-negative in the respective ranges of $x$ and $y$ , note that:
+
+$$
+\begin{array} { r l } { x \leq \zeta _ { 2 } \Big / \Big ( \frac { \mu _ { 1 } } { \mu _ { 1 } - 1 } \cdot \frac { \mu _ { 2 } } { \mu _ { 2 } - 1 } \Big ) ^ { \mu _ { 2 } } \implies x \leq \zeta _ { 2 } \Big / \Big ( \frac { \mu _ { 2 } } { \mu _ { 2 } - 1 } \Big ) ^ { \mu _ { 2 } } } \\ & { \implies 1 \leq \frac { \zeta _ { 2 } } { x } \cdot \Big ( \frac { \mu _ { 2 } - 1 } { \mu _ { 2 } } \Big ) ^ { \mu _ { 2 } } } \\ & { \implies 1 \leq \frac { \mu _ { 2 } - 1 } { \mu _ { 2 } } \Big ( \frac { \zeta _ { 2 } } { x } \Big ) ^ { \frac { 1 } { \mu _ { 2 } } } } \\ & { \implies 1 \leq \frac { \mu _ { 2 } - 1 } { \mu _ { 2 } } \Big ( \frac { \zeta _ { 2 } } { x } \Big ) ^ { \frac { 1 } { \mu _ { 2 } } } g } \\ & { \implies 0 \leq - 1 + \frac { \mu _ { 2 } - 1 } { \mu _ { 2 } } \Big ( \frac { \zeta _ { 2 } } { x } \Big ) ^ { \frac { 1 } { \mu _ { 2 } } } g } \\ & { \implies 0 \leq - 1 + \frac { \mu _ { 2 } - 1 } { \mu _ { 2 } } \Big ( \frac { \zeta _ { 2 } } { x } \Big ) ^ { \frac { 1 } { \mu _ { 2 } } } g } \\ & { \implies 0 \leq g _ { g } ^ { \prime } ( x , y ) . } \end{array}
+$$
+
+Consider $\frac { 1 - x } { 1 - ( x ^ { \mu _ { 2 } - 1 } \zeta _ { 2 } ) ^ { 1 / \mu _ { 2 } } }$ . Since $\zeta _ { 2 } \geq 1$ and $x \leq 1$ , we have $x \leq \zeta _ { 2 }$ and hence
+
+$$
+{ \frac { 1 - x } { 1 - ( x ^ { \mu _ { 2 } - 1 } \zeta _ { 2 } ) ^ { \frac { 1 } { \mu _ { 2 } } } } } \geq { \frac { 1 - x } { 1 - ( x ^ { \mu _ { 2 } - 1 } x ) ^ { \frac { 1 } { \mu _ { 2 } } } } } = 1 .
+$$
+
+Therefore we can set $\begin{array} { r } { y = \frac { 1 - x } { 1 - ( x ^ { \mu _ { 2 } - 1 } \zeta _ { 2 } ) ^ { 1 / \mu _ { 2 } } } } \end{array}$ and apply the fact that $g ( x , y ) \geq g ( x , 1 )$ for all $y \geq 1$ to get
+
+$$
+f _ { 1 } ^ { \prime } ( x ) \geq - \lambda + ( \lambda - 1 ) \left( 1 - \frac { 1 } { \mu _ { 2 } } \right) \left( \frac { \zeta _ { 2 } } { x } \right) ^ { \frac { 1 } { \mu _ { 2 } } } ,
+$$
+
+as required by (7).
+
+Taking the derivative of $f _ { 2 } ( x )$ , we have:
+
+$$
+f _ { 2 } ^ { \prime } ( x ) = \zeta _ { 1 } ^ { \frac { \lambda - 1 } { \mu _ { 1 } - 1 } } \cdot \left( 1 - { \frac { \lambda - 1 } { \mu _ { 1 } - 1 } } \right) x ^ { - { \frac { \lambda - 1 } { \mu _ { 1 } - 1 } } } = \left( { \frac { \zeta _ { 1 } } { x } } \right) ^ { { \frac { \lambda - 1 } { \mu _ { 1 } - 1 } } } \left( 1 - { \frac { \lambda - 1 } { \mu _ { 1 } - 1 } } \right) \geq 1 - { \frac { \lambda - 1 } { \mu _ { 1 } - 1 } } .
+$$
+
+Combining the two terms together, we have:
+
+$$
+\begin{array} { c } { { f ^ { \prime } ( x ) \geq - \lambda + ( \lambda - 1 ) \left( 1 - { \frac { 1 } { \mu _ { 2 } } } \right) \left( { \frac { \zeta _ { 2 } } { x } } \right) ^ { \frac { 1 } { \mu _ { 2 } } } + 1 - { \frac { \lambda - 1 } { \mu _ { 1 } - 1 } } } } \\ { { = ( \lambda - 1 ) \left( - { \frac { \mu _ { 1 } } { \mu _ { 1 } - 1 } } + { \frac { \mu _ { 2 } - 1 } { \mu _ { 2 } } } \left( { \frac { \zeta _ { 2 } } { x } } \right) ^ { \frac { 1 } { \mu _ { 2 } } } \right) . } } \end{array}
+$$
+
+For $f ^ { \prime } ( x )$ to be non-negative we need:
+
+$$
+\begin{array} { r l } & { - \frac { \mu _ { 1 } } { \mu _ { 1 } - 1 } + \frac { \mu _ { 2 } - 1 } { \mu _ { 2 } } \left( \frac { \zeta _ { 2 } } { x } \right) ^ { \frac { 1 } { \mu _ { 2 } } } \geq 0 } \\ & { \iff \left( \frac { \mu _ { 1 } } { \mu _ { 1 } - 1 } \cdot \frac { \mu _ { 2 } } { \mu _ { 2 } - 1 } \right) ^ { \mu _ { 2 } } \leq \frac { \zeta _ { 2 } } { x } . } \end{array}
+$$
+
+So $f ( x )$ is increasing for $\begin{array} { r c l } { x } & { \in } & { \left[ 0 , \zeta _ { 2 } / \left( { \frac { \mu _ { 1 } } { \mu _ { 1 } - 1 } } \cdot { \frac { \mu _ { 2 } } { \mu _ { 2 } - 1 } } \right) ^ { \mu _ { 2 } } \right] } \end{array}$ . This means for $q \leq \tilde { q } \leq$ $\zeta _ { 2 } / { \left( \frac { \mu _ { 1 } } { \mu _ { 1 } - 1 } \cdot \frac { \mu _ { 2 } } { \mu _ { 2 } - 1 } \right) } ^ { \mu _ { 2 } }$ , we have $f ( q ) \leq f ( \tilde { q } )$ . This completes the proof of the lemma and that of the theorem. ■
+
+Theorem 6 yields data-dependent Rényi differential privacy bounds for any value of $\mu _ { 1 }$ and $\mu _ { 2 }$ larger than $\lambda$ . The following proposition simplifies this search by calculating optimal higher moments $\mu _ { 1 }$ and $\mu _ { 2 }$ for the GNMax mechanism with variance $\sigma ^ { 2 }$ .
+
+Proposition 10. When applying Theorem 6 and Proposition 8 for GNMax with Gaussian of variance $\sigma ^ { 2 }$ , the right-hand side of (2) is minimized at
+
+$$
+\mu _ { 2 } = \sigma \cdot \sqrt { \log ( 1 / \tilde { q } ) } , a n d \mu _ { 1 } = \mu _ { 2 } + 1 .
+$$
+
+Proof. We can minimize both terms in (2) independently. To minimize the first term in (6), we minimize $\left( \tilde { q } e ^ { \varepsilon _ { 2 } } \right) ^ { 1 - 1 / \mu _ { 2 } }$ by considering logarithms:
+
+$$
+\begin{array} { r l } { \log \left\{ \big ( \tilde { q } e ^ { \varepsilon _ { 2 } } \big ) ^ { 1 - 1 / \mu _ { 2 } } \right\} = \log \Bigg \{ \tilde { q } ^ { 1 - \frac { 1 } { \mu _ { 2 } } } \exp \left( \frac { \mu _ { 2 } - 1 } { \sigma ^ { 2 } } \right) \Bigg \} } & { } \\ { \quad \quad \quad \quad \quad = \left( 1 - \frac { 1 } { \mu _ { 2 } } \right) \cdot \log \tilde { q } + \frac { \mu _ { 2 } - 1 } { \sigma ^ { 2 } } } & { } \\ { \quad \quad \quad \quad = \frac { 1 } { \mu _ { 2 } } \log \frac { 1 } { \tilde { q } } + \frac { \mu _ { 2 } } { \sigma ^ { 2 } } - \frac { 1 } { \sigma ^ { 2 } } - \log \frac { 1 } { \tilde { q } } , } \end{array}
+$$
+
+which is minimized at $\mu _ { 2 } = \sigma \cdot \sqrt { \log ( 1 / \tilde { q } ) }$ .
+
+To minimize the second term in (6), we minimize $e ^ { \varepsilon _ { 1 } } / \tilde { q } ^ { 1 / ( \mu _ { 1 } - 1 ) }$ as follows:
+
+$$
+\begin{array} { r l } { \log \left\{ \displaystyle \frac { e ^ { \varepsilon _ { 1 } } } { \tilde { q } ^ { 1 / ( \mu _ { 1 } - 1 ) } } \right\} = \log \left\{ \tilde { q } ^ { - 1 / ( \mu _ { 1 } - 1 ) } \exp \left( \frac { \mu _ { 1 } } { \sigma ^ { 2 } } \right) \right\} } & { { } } \\ { = \displaystyle \frac { \mu _ { 1 } } { \sigma ^ { 2 } } + \frac { 1 } { \mu _ { 1 } - 1 } \log \frac { 1 } { \tilde { q } } } & { { } } \\ { = \displaystyle \frac { 1 } { \sigma ^ { 2 } } + \frac { \mu _ { 1 } - 1 } { \sigma ^ { 2 } } + \frac { 1 } { \mu _ { 1 } - 1 } \log \frac { 1 } { \tilde { q } } , } \end{array}
+$$
+
+which is minimized at $\mu _ { 1 } = 1 + \sigma \cdot \sqrt { \log ( 1 / \tilde { q } ) }$ completing the proof.
+
+Putting this together, we apply the following steps to calculate RDP of order $\lambda$ for GNMax with variance $\sigma ^ { 2 }$ on a given dataset $D$ . First, we compute a bound $q$ according to Proposition 7. Then we use the smaller of two bounds: a data-dependent (Theorem 6) and a data-independent one (Proposition 8) :
+
+$$
+\beta _ { \sigma } ( q ) \triangleq \operatorname* { m i n } \left\{ \frac { 1 } { \lambda - 1 } \log \left\{ ( 1 - q ) \cdot A ( q , \mu _ { 2 } , \varepsilon _ { 2 } ) ^ { \lambda - 1 } + q \cdot B ( q , \mu _ { 1 } , \varepsilon _ { 1 } ) ^ { \lambda - 1 } \right\} , \lambda / \sigma ^ { 2 } \right\} ,
+$$
+
+where $\pmb { A }$ and $\textbf { { B } }$ are defined as in the statement of Theorem 6, the parameters $\mu _ { 1 }$ and $\mu _ { 2 }$ are selected according to Proposition 10, and $\varepsilon _ { 1 } \triangleq \mu _ { 1 } / \sigma ^ { 2 }$ and $\varepsilon _ { 2 } \triangleq \mu _ { 2 } / \sigma ^ { 2 }$ (Proposition 8). Importantly, the first expression is evaluated only when $q < 1 , \mu _ { 1 } \geq \lambda , \mu _ { 2 } > 1$ , and $q \leq e ^ { ( \mu _ { 2 } - 1 ) \varepsilon _ { 2 } } / \left( \frac { \mu _ { 1 } } { \mu _ { 1 } - 1 } \cdot \frac { \mu _ { 2 } } { \mu _ { 2 } - 1 } \right) ^ { \mu _ { 2 } }$ . These conditions can either be checked for each application of the aggregation mechanism, or a critical value of $q _ { 0 }$ that separates the range of applicability of the data-dependent and data-independent bounds can be computed for given $\sigma$ and $\lambda$ . In our implementation we pursue the second approach.
+
+The following corollary offers a simple asymptotic expression of the privacy of GNMax for the case when there are large (relative to $\sigma$ ) gaps between the highest three vote counts.
+
+Corollary 11. If the top three vote counts are $n _ { 1 } ~ > ~ n _ { 2 } ~ > ~ n _ { 3 }$ and $n _ { 1 } \mathrm { ~ - ~ } n _ { 2 }$ , $n _ { 2 } \mathrm { ~ - ~ } n _ { 3 } \mathrm { ~ } \gg \sigma$ , then the mechanism GNMax with Gaussian of variance $\sigma ^ { 2 }$ satisfies $( \lambda , \exp ( - 2 \lambda / \sigma ^ { 2 } ) / \lambda )$ -RDP for $\lambda = ( n _ { 1 } - n _ { 2 } ) / 4$ .
+
+Proof. Denote the noisy counts as $\tilde { n } _ { i } = n _ { i } + \mathcal { N } ( 0 , \sigma ^ { 2 } )$ . Ignoring outputs other than those with the highest and the second highest counts, we bound $q = \mathbf { \bar { P r } } \left[ \mathcal { \bar { M } } ( D ) \right] \neq 1 ]$ as $\mathbf { P r } [ \tilde { n } _ { 1 } ~ < ~ \tilde { n } _ { 2 } ] ~ =$ ${ \bf P r } [ N ( 0 , 2 \sigma ^ { 2 } ) > n _ { 1 } - n _ { 2 } ] < \exp { \left( - ( n _ { 1 } - n _ { 2 } ) ^ { 2 } / 4 \sigma ^ { 2 } \right) }$ , which we use as $\tilde { q }$ . Plugging $\tilde { q }$ in Proposition 10, we have $\mu _ { 1 } - 1 = \mu _ { 2 } = ( n _ { 1 } - n _ { 2 } ) / 2$ , limiting the range of applicability of Theorem 6 to $\lambda < ( n _ { 1 } - n _ { 2 } ) / 2$ .
+
+Choosing $\lambda = ( n _ { 1 } - n _ { 2 } ) / 4$ ensures $A ( \tilde { q } , \mu _ { 2 } , \varepsilon _ { 2 } ) \approx 1$ , which allows approximating the bound (2) as $\tilde { q } \cdot B ( \tilde { q } , \mu _ { 1 } , \varepsilon _ { 1 } ) ^ { \lambda - 1 } / ( \lambda - 1 )$ . The proof follows by straightforward calculation.
+
+# B SMOOTH SENSITIVITY AND PUBLISHING THE PRIVACY PARAMETER
+
+The privacy guarantees obtained for the mechanisms in this paper via Theorem 6 take as input $\tilde { q }$ , an upper bound on the probability that the aggregate mechanism returns the true plurality. This means that the resulting privacy parameters computed depend on teacher votes and hence the underlying data. To avoid potential privacy breaches from simply publishing the data-dependent parameter, we need to publish a sanitized version of the privacy loss. This is done by adding noise to the computed privacy loss estimates using the smooth sensitivity algorithm proposed by Nissim et al. (2007).
+
+This section has the following structure. First we recall the notion of smooth sensitivity and introduce an algorithm for computing the smooth sensitivity of the privacy loss function of the GNMax mechanism. In the rest of the section we prove correctness of these algorithms by stating several conditions on the mechanism, proving that these conditions are sufficient for correctness of the algorithm, and finally demonstrating that GNMax satisfies these conditions.
+
+# B.1 COMPUTING SMOOTH SENSITIVITY
+
+Any dataset $D$ defines a histogram $\bar { n } = ( n _ { 1 } , \ldots , n _ { m } ) \in \mathbb { N } ^ { m }$ of the teachers’ votes. We have a natural notion of the distance between two histograms $\operatorname { d i s t } ( \bar { n } , \bar { n } ^ { \prime } )$ and a function $q \colon \mathbb { N } ^ { m } [ 0 , 1 ]$ on these histograms computing the bound according to Proposition 7. The value $q ( { \bar { n } } )$ can be used as $\tilde { q }$ in the application of Theorem 6. Additionally we have $\boldsymbol { n } ^ { ( i ) }$ denote the $i$ -th highest bar in the histogram.
+
+We aim at calculating a smooth sensitivity of $\beta \left( q ( \bar { n } ) \right)$ whose definition we recall now.
+
+Definition 12 (Smooth Sensitivity). Given the smoothness parameter $\beta$ , a $\beta$ -smooth sensitivity of $f ( n )$ is defined as
+
+$$
+\mathrm { S S } _ { \beta } ( \bar { n } ) \triangleq \operatorname* { m a x } _ { d \geq 0 } e ^ { - \beta d } \cdot \operatorname* { m a x } _ { \bar { n } ^ { \prime } : \mathrm { d i s t } ( \bar { n } , \bar { n } ^ { \prime } ) \leq d } \tilde { \mathrm { L S } } ( \bar { n } ^ { \prime } ) ,
+$$
+
+where
+
+$$
+\tilde { \mathrm { L S } } ( \bar { n } ) \geq \operatorname* { m a x } _ { \bar { n } ^ { \prime } : \mathrm { d i s t } ( \bar { n } , \bar { n } ^ { \prime } ) = 1 } | f ( n ) - f ( n ^ { \prime } ) |
+$$
+
+is an upper bound on the local sensitivity.
+
+We now describe Algorithms 3–5 computing a smooth sensitivity of $\beta \left( q ( \cdot ) \right)$ . The algorithms assume the existence of efficiently computable functions $q \colon \mathbb { N } ^ { m } [ 0 , 1 ]$ , $\mathrm { B _ { L } } , \mathrm { B _ { U } } \colon [ 0 , 1 ] \to [ 0 , 1 ]$ , and a constant $q _ { 0 }$ .
+
+Informally, the functions $\mathrm { B _ { U } }$ and $\mathrm { B _ { L } }$ respectively upper and lower bound the value of $q$ evaluated at any neighbor of $\bar { n }$ given $q ( { \bar { n } } )$ , and $[ 0 , q _ { 0 } )$ limits the range of applicability of data-dependent analysis.
+
+The functions $\mathrm { B _ { L } }$ and $\mathrm { B _ { U } }$ are defined as follows. Their derivation appears in Section B.4.
+
+$$
+\begin{array} { l } { { \displaystyle { \mathrm { B } } _ { \mathrm { U } } ( q ) \triangleq \operatorname* { m i n } \left\{ \frac { m - 1 } { 2 } \mathrm { e r f c } \left( \mathrm { e r f c } ^ { - 1 } \left( \frac { 2 q } { m - 1 } \right) - \frac { 1 } { \sigma } \right) , 1 \right\} , } } \\ { { \displaystyle { \mathrm { B } } _ { \mathrm { L } } ( q ) \triangleq \frac { m - 1 } { 2 } \mathrm { e r f c } \left( \mathrm { e r f c } ^ { - 1 } \left( \frac { 2 q } { m - 1 } \right) + \frac { 1 } { \sigma } \right) , } } \end{array}
+$$
+
+Algorithm 3 – Local Sensitivity: use the functions $\mathrm { B _ { U } }$ and $\mathrm { B _ { L } }$ to compute (an upper bound) of the local sensitivity at a given $q$ value by looking at the difference of $\beta \left( \cdot \right)$ evaluated on the bounds.
+
+| case analysis that finds the neighbor at distance d with the maximum sensitivity. |
| 1: procedure ATDISTANCED(n, d) |
| 2: q←q(n) | > q is in the flat region. |
| 3: 4: | if q1 ≤q≤ go then |
| return LS(q), STOP | |
| 5: end if | |
| 6: | if q<qithen Need to increase q. |
| 7: | if n(1)-n(2）< 2d then n(i) is the ith largest element. |
| 8: | return LS(qi), STOP |
| 9: | else |
| 10: | n'← SORT(n)+[-d,d,O,...,0] |
| 11: | q'←q(n') |
| 12: | if q' >qi then |
| 13: | return LS(qo), STOP |
| 14: | else |
| 15: | return LS(q'),CONTINUE |
| 16: | end if |
| 17: | end if |
| 18: | else Need to decrease q. |
| 19: | |
| 20: | n↑[n,0,...,0] |
| 21: | q'←q(n'） |
| 22: | return LS(q'), STOP |
| 23: | else |
| 24: | n' ← SORT(n)+[d,0,...,0] |
| 25: | for d' =1,...,d do |
| 26: | n'(2) ←n(2)-1 The index of n'(2) may change. |
| 27: | end for |
| 28: | q'←q(n'） |
| 29: | if q'<qo then |
| 30: | return LS(qo), STOP |
| 31: | else |
| 32: | return LS(q'), CONTINUE |
| 33: | end if |
| |
| 34: | end if |
| 35: end if 36: end procedure |
+
+# Algorithm 5 – Smooth Sensitivity: Compute the $\beta$ smooth sensitivity of $\beta \left( \cdot \right)$ via Definition 12 by looking at sensitivities at various distances and returning the maximum weighted by $e ^ { - \beta d }$ .
+
+1: procedure SMOOTHSENSITIVITY $( \bar { n } , \beta )$
+2: S ← 0
+3: d ← 0
+4: repeat
+5: $c$ , StoppingCondition ATDISTANCED $( \bar { n } , d )$
+6: S ← max{S, c · e−βd}
+7: d ← d + 1
+8: until StoppingCondition $=$ STOP
+9: end procedure
+
+• Define the distance between two histograms $\bar { n } = ( n _ { 1 } , \dots , n _ { m } )$ and $\hat { n } ^ { \prime } = ( n _ { 1 } ^ { \prime } , \dots , n _ { m } ^ { \prime } )$ as
+
+$$
+d ( \bar { n } , \bar { n } ^ { \prime } ) \triangleq \operatorname* { m a x } \left\{ \sum _ { \substack { i : n _ { i } > n _ { i } ^ { \prime } } } \lceil n _ { i } - n _ { i } ^ { \prime } \rceil , \quad \sum _ { \substack { i : n _ { i } < n _ { i } ^ { \prime } } } \lceil n _ { i } ^ { \prime } - n _ { i } \rceil \right\} ,
+$$
+
+which is equal to the smallest number of “moves” needed to make the two histograms identical. We use the ceiling function since a single step can increase/decrease one bar by at most 1.
+
+We say that two histograms are neighbors if their distance $d$ is 1.
+
+Notice that analyses of Rényi differential privacy for LNMax, GNMax and the exponential mechanism are still applicable when the neighboring datasets are defined in this manner.
+
+• Given a randomized aggregator $\mathcal { M } : \mathbb { R } _ { \ge 0 } ^ { m } [ m ]$ , let $q \colon \mathbb { R } _ { \geq 0 } ^ { m } [ 0 , 1 ]$ be so that
+
+$$
+q ( \bar { n } ) \geq \mathbf { P r } [ \mathcal { M } ( \bar { n } ) \neq \operatorname { a r g m a x } ( \bar { n } ) ] .
+$$
+
+When the context is clear, we use $q$ to denote a specific value of the function, which, in particular, can be used as $\tilde { q }$ in applications of Theorem 6.
+
+• Let $\beta \colon [ 0 , 1 ] \mathbb { R }$ be the function that maps a $q$ value to the value of the Rényi accountant.
+
+Conditions. Throughout this section we will be referring to the list of conditions on $q ( \cdot )$ and $\beta \left( \cdot \right)$
+
+C1. The function $q ( \cdot )$ is continuous in each argument $n _ { i }$ .
+C2. There exist functions $\mathrm { B _ { U } , B _ { L } \colon [ 0 , 1 ] \to [ 0 , 1 ] }$ such that for any neighbor $\bar { n } ^ { \prime }$ of $\bar { n }$ , we have $\mathrm { B } _ { \mathrm { L } } ( q ( \bar { n } ) ) \leq q ( \bar { n } ^ { \prime } ) \leq \mathrm { B } _ { \mathrm { U } } ( q ( \bar { n } ) )$ , i.e., $\mathrm { B _ { U } }$ and $\mathrm { B _ { L } }$ provide upper and lower bounds on the $q$ value of any neighbor of $\bar { n }$ .
+C3. $\operatorname { B _ { L } } ( q )$ is increasing in $q$ .
+C4. $\mathrm { B _ { U } }$ and $\mathrm { B _ { L } }$ are functional inverses of each other in part of the range, i.e., $q = \mathrm { B _ { L } } ( \mathrm { B _ { U } } ( q ) )$ for all $q \in [ 0 , q _ { 0 } ]$ , where $q _ { 0 }$ is defined below. Additionally $\mathrm { B } _ { \mathrm { L } } ( \bar { q } ) \leq q \bar { \leq } \mathrm { B } _ { \mathrm { U } } ( q )$ for all $q \in [ 0 , 1 ]$ .
+C5. $\beta \left( \cdot \right)$ has the following shape: there exist constants $\beta ^ { * }$ and $q _ { 0 } \leq 0 . 5$ , such that $\beta \left( q \right)$ nondecreasing in $[ 0 , q _ { 0 } ]$ and $\bar { \beta ( q ) } = \beta ^ { * } \ge \beta ( q _ { 0 } )$ for $q > q _ { 0 }$ . The constant $\beta ^ { * }$ corresponds to a data-independent bound.
+C6. $\Delta \beta ( q ) \triangleq \beta ( \mathrm { B } _ { \mathrm { U } } ( q ) ) - \beta ( q )$ is non-decreasing in $[ 0 , \mathrm { B _ { L } } ( q _ { 0 } ) ]$ , i.e., when $\mathrm { B _ { U } } ( q ) \le q _ { 0 }$ .
+C7. Recall that $\boldsymbol { n } ^ { ( i ) }$ is the $i$ -th largest coordinate of a histogram $\bar { n }$ . Then, if $q ( \bar { n } ) \leq \mathrm { B _ { U } } ( q _ { 0 } )$ , then $q ( { \bar { n } } )$ is differentiable in all coordinates and
+
+$$
+\forall i > j \ge 2 \quad \frac { \partial q } { \partial n ^ { ( j ) } } ( \bar { n } ) \ge \frac { \partial q } { \partial n ^ { ( i ) } } ( \bar { n } ) \ge 0 .
+$$
+
+C8. The function $q ( { \bar { n } } )$ is invariant under addition of a constant, i.e., and $q ( { \bar { n } } )$ is invariant under permutation of $\bar { n }$ , i.e.,
+
+Finally, we require that if $n ^ { ( 1 ) } = n ^ { ( 2 ) }$ , then $q ( { \bar { n } } ) \geq q _ { 0 }$
+
+We may additionally assume that $q _ { 0 } \ \ge \ q ( [ n , 0 , \ldots , 0 ] )$ . Indeed, if this condition is not satisfied, then the data-dependent analysis is not going to be used anywhere. The most extreme histogram— $[ n , 0 , \ldots , 0 ]$ —is the most advantageous setting for applying data-dependent bounds. If we cannot use the data-dependent bound even in that case, we would be using the data-independent bound everywhere and do not need to compute smooth sensitivity anyway. Yet this condition is not automatically satisfied. For example, if $m$ (the number of classes) is large compared to $n$ (the number of teachers), we might have large $q ( [ n , 0 , \ldots , 0 ] )$ . So we need to check this condition in the code before doing smooth sensitivity calculation.
+
+# B.3 CORRECTNESS OF ALGORITHMS 3–5
+
+Recall that local sensitivity of a deterministic function $f$ is defined as max $f ( D ) - f ( D ^ { \prime } )$ , where $D$ and $D ^ { \prime }$ are neighbors.
+
+Proposition 13. Under conditions C2–C6, Algorithm $^ 3$ computes an upper bound on local sensitivity of $\beta ( q ( \bar { n } ) )$ .
+
+Proof. Since $\beta \left( \cdot \right)$ is non-decreasing everywhere (by C5), and for any neighbors $\bar { n }$ and $\bar { n } ^ { \prime }$ it holds that $\mathrm { B } _ { \mathrm { L } } ( q ( \bar { n } ) ) \leq q ( \bar { n } ^ { \prime } ) \leq \mathrm { B } _ { \mathrm { U } } ( q ( \bar { n } ) )$ (by C2), we have the following
+
+$$
+\begin{array} { r l } & { | \beta \left( q ( \bar { n } ) \right) - \beta \left( q ( \bar { n } ^ { \prime } ) \right) | \leq \operatorname* { m a x } \bigg \{ \beta \Big ( \mathrm { B } _ { \mathrm { U } } { \left( q ( \bar { n } ) \right) } \Big ) - \beta \Big ( q ( \bar { n } ) \Big ) , \beta \Big ( q ( \bar { n } ) \Big ) - \beta \Big ( \mathrm { B } _ { \mathrm { L } } { \left( q ( \bar { n } ) \right) } \Big ) \bigg \} } \\ & { \quad \quad \quad \quad = \operatorname* { m a x } \bigg \{ \Delta \beta \Big ( q ( \bar { n } ) \Big ) , \Delta \beta \Big ( \mathrm { B } _ { \mathrm { L } } { \left( q ( \bar { n } ) \right) } \Big ) \bigg \} } \end{array}
+$$
+
+as an upper bound on the local sensitivity of $\beta \left( q ( \cdot ) \right)$ at input $\bar { n }$ .
+
+The function computed by Algorithm 3 differs from above when $q ( \bar { n } ) \in ( \mathrm { B _ { L } } ( q _ { 0 } ) , q _ { 0 } )$ . To complete the proof we need to argue that the local sensitivity is upper bounded by $\Delta \beta \left( \mathrm { B _ { L } } ( q _ { 0 } ) \right)$ for $q ( { \bar { n } } )$ in this interval. The bound follows from the following three observations.
+
+First, $\Delta \beta \left( q \right)$ is non-increasing in the range $( \mathrm { B _ { L } } ( q _ { 0 } ) , 1 ]$ , since $\beta \left( \mathrm { B } _ { \mathrm { U } } ( q ) \right)$ is constant (by $\operatorname { B _ { U } } ( q ) \geq$ $\mathrm { B _ { U } } ( \mathrm { B _ { L } } ( q _ { 0 } ) ) = q _ { 0 }$ and C5) and $\beta \left( q \right)$ is non-decreasing in the range (by C5). In particular,
+
+$$
+\Delta \beta ( q ) \leq \Delta \beta ( \mathrm { B } _ { \mathrm { L } } ( q _ { 0 } ) ) { \mathrm { ~ i f ~ } } q \geq \mathrm { B } _ { \mathrm { L } } ( q _ { 0 } ) .
+$$
+
+Second, $\Delta \beta \left( \mathrm { B } _ { \mathrm { L } } ( q ) \right)$ is non-decreasing in the range $[ 0 , q _ { 0 } ]$ since $\operatorname { B _ { L } } ( q )$ is increasing (by C3 and C6). This implies that
+
+$$
+\Delta \beta ( \mathrm { B _ { L } } ( q ) ) \leq \Delta \beta ( \mathrm { B _ { L } } ( q _ { 0 } ) ) { \mathrm { ~ i f ~ } } q \leq q _ { 0 } .
+$$
+
+By (8) and (9) applied to the intersection of the two ranges, it holds that
+
+$$
+\begin{array} { r } { \operatorname* { m a x } \bigg \{ \Delta \beta \Big ( q ( \bar { n } ) \Big ) , ~ \Delta \beta \Big ( \mathrm { B } _ { \mathrm { L } } ( q ( \bar { n } ) ) \Big ) \bigg \} \leq \Delta \beta \left( \mathrm { B } _ { \mathrm { L } } ( q _ { 0 } ) \right) \mathrm { i f } ~ \mathrm { B } _ { \mathrm { L } } ( q _ { 0 } ) \leq q \leq q _ { 0 } , } \end{array}
+$$
+
+as needed.
+
+We thus established that the function computed by Algorithm 3, which we call $\tilde { \mathrm { L S } } ( q )$ from now on, is an upper bound on the local sensitivity. Formally,
+
+$$
+\begin{array} { r } { \tilde { \mathrm { L S } } ( q ) \triangleq \left\{ \begin{array} { l l } { \Delta \beta \left( \mathrm { B } _ { \mathrm { L } } ( q _ { 0 } ) \right) } & { \mathrm { i f ~ } q \in ( \mathrm { B } _ { \mathrm { L } } ( q _ { 0 } ) , q _ { 0 } ) , } \\ { \operatorname* { m a x } \left\{ \Delta \beta \left( q \right) , \Delta \beta \left( \mathrm { B } _ { \mathrm { L } } ( q ) \right) \right\} } & { \mathrm { o t h e r w i s e } . } \end{array} \right. } \end{array}
+$$
+
+The following proposition characterizes the growth of $\tilde { \mathrm { L S } } ( q )$ .
+
+Proposition 14. Assuming conditions $C 2 { - } C 6 ,$ , the function $\tilde { \mathrm { L S } } ( q )$ is non-decreasing in $[ 0 , \mathrm { B _ { L } } ( q _ { 0 } ) ]$ , constant in $[ \mathrm { B _ { L } } ( q _ { 0 } ) , q _ { 0 } ]$ , and non-increasing in $[ q _ { 0 } , 1 ]$ .
+
+Proof. Consider separately three intervals.
+
+• By construction, $\tilde { \mathrm { L S } }$ is constant in $[ \mathrm { B _ { L } } ( q _ { 0 } ) , q _ { 0 } ]$ .
+• Since both functions $\Delta \beta \left( \cdot \right)$ and $\Delta \beta \left( \mathrm { B _ { L } } ( \cdot ) \right)$ are each non-decreasing in $[ 0 , \mathrm { B _ { L } } ( q _ { 0 } ) )$ , so is their max.
+• In the interval $( q _ { 0 } , 1 ] , \beta ( q )$ is constant. Hence $\Delta \beta ( q ) = 0$ and $\Delta \beta ( \mathrm { B } _ { \mathrm { L } } ( q ) ) = \beta ( q ) -$ $\beta \left( \mathrm { B _ { L } } ( q ) \right)$ is non-decreasing. Their maximum value $\Delta \beta \left( \mathrm { B _ { L } } ( q ) \right)$ is non-decreasing.
+
+The claim follows.
+
+We next prove correctness of Algorithm 4, which computes the maximal sensitivity of $\beta$ at a fixed distance.
+
+The proof relies on the following notion of a partial order between histograms.
+
+Definition 15. Prefix sums $S _ { i } ( \bar { n } )$ are defined as follows:
+
+$$
+S _ { i } ( \bar { n } ) \triangleq \sum _ { j = 1 } ^ { i } ( n ^ { ( 1 ) } - n ^ { ( j ) } ) .
+$$
+
+We say that a histogram $\bar { n }$ dominates $\bar { n } ^ { \prime }$ , denoted as $\bar { n } \succeq \bar { n } ^ { \prime }$ , iff:
+
+The function $q ( \cdot )$ is monotone under this notion of dominance (assuming certain conditions hold): Proposition 16. If $q ( \cdot )$ satisfies C1, C2, C7, and $C 8 _ { \mathrm { : } }$ , and $q ( \bar { n } ) < \mathrm { B _ { U } } ( q _ { 0 } )$ , then
+
+$$
+\bar { n } \succeq \bar { n } ^ { \prime } \Rightarrow q ( \bar { n } ) \leq q ( \bar { n } ^ { \prime } ) .
+$$
+
+Proof. We may assume that $n ^ { ( 1 ) } = n ^ { \prime ( 1 ) }$ . Indeed, if this does not hold, add $| n ^ { ( 1 ) } - n ^ { \prime ( 1 ) } |$ to all coordinates of the histogram with the smaller of the two values. This transform does not change the $q$ value (by C8) and it preserves the $\succeq$ relationship as all prefix sums $S _ { i } ( \cdot )$ remain unchanged.
+
+We make a simple observation that will be helpful later:
+
+$$
+\forall i \in [ m ] \mathrm { i t } \mathrm { h o l d s ~ t h a t } \sum _ { j = 1 } ^ { i } ( n ^ { ( 1 ) } - n _ { j } ) \geq S _ { i } ( \bar { n } ) .
+$$
+
+The inequality holds because the prefix sum accumulates the gaps between the largest value of $\bar { n }$ and all other values in the non-decreasing order. Any deviation from this order may only increase the prefix sums.
+
+The following lemma constructs a monotone chain (in the partial order of dominance) of histograms connecting $\bar { n }$ and $\bar { n } ^ { \prime }$ via a sequence of intermediate steps that either do not change the value of $q$ or touch at most two coordinates at a time.
+
+Lemma 17. There exists a chain ${ \bar { n } } = { \bar { n } } _ { 0 } \succeq { \bar { n } } _ { 1 } \succeq \cdot \cdot \cdot \succeq { \bar { n } } _ { d } = { \bar { n } } ^ { \prime }$ , such that for all $i \in [ d ]$ either $d ( \bar { n } _ { i - 1 } , \bar { n } _ { i } ) = 1$ or $\bar { n } _ { i - 1 } = \pi ( \bar { n } _ { i } )$ for some permutation $\pi$ on $[ m ]$ . Additionally, = · · · = n(1)d .
+
+Proof (Lemma). Wlog we assume that $\bar { n }$ and $\bar { n } ^ { \prime }$ are each sorted in the descending order. The proof is by induction on $\begin{array} { r } { \ell ( \bar { n } , \bar { n } ^ { \prime } ) \stackrel { \Delta } { = } \sum _ { i } \lceil n _ { i } - n _ { i } ^ { \prime } \rceil \leq 2 d ( \bar { n } , \bar { n } ^ { \prime } ) } \end{array}$ , which, by construction, only assumes non-negative integer values.
+
+If the distance is 0, the statement is immediate. Otherwise, find the smallest $i$ so that $S _ { i } ( \bar { n } ) > S _ { i } ( \bar { n } ^ { \prime } )$ (if all prefix sums are equal and $n ^ { ( 1 ) } = n ^ { \prime ( 1 ) }$ , it would imply that $\bar { n } = \bar { n } ^ { \prime }$ ). In particular, it means that $n _ { j } = n _ { j } ^ { \prime }$ for $j < i$ and $n _ { i } < n _ { i } ^ { \prime } \le n _ { i - 1 } = n _ { i - 1 } ^ { \prime }$ . Let $x \triangleq \operatorname* { m i n } ( n _ { i } ^ { \prime } - n _ { i } , 1 )$ . Define $\bar { n } ^ { \prime \prime }$ as identical to $\bar { n } ^ { \prime }$ except that $n _ { i } ^ { \prime \prime } = n _ { i } ^ { \prime } - x$ . The new value is guaranteed to be non-negative, since $x \leq n _ { i } ^ { \prime } - n _ { i }$ and $n _ { i } \geq 0$ . Note that $\bar { n } ^ { \prime \prime }$ is not necessarily sorted anymore. Consider two possibilities.
+
+Case I: $\bar { n } \succeq \bar { n } ^ { \prime \prime }$ . Since $\bar { n } ^ { \prime \prime } \succeq \bar { n } ^ { \prime }$ , $\ell ( \bar { n } , \bar { n } ^ { \prime \prime } ) < \ell ( \bar { n } , \bar { n } ^ { \prime } )$ , and $d ( \bar { n } ^ { \prime \prime } , \bar { n } ^ { \prime } ) = 1$ , we may apply the induction hypothesis to the pair $\bar { n } , \bar { n } ^ { \prime \prime }$ .
+
+Case II: $\bar { n } \not \subset \bar { n } ^ { \prime \prime }$ . This may happen because the prefix sums of $\bar { n } ^ { \prime \prime }$ increase compared to $S _ { j } ( \bar { n } ^ { \prime } )$ for $j \geq i$ . Find the smallest such $i ^ { \prime }$ so that $\textstyle \sum _ { j = 1 } ^ { i ^ { \prime } } ( n _ { 1 } ^ { \prime \prime } - n _ { j } ^ { \prime \prime } ) > S _ { i ^ { \prime } } ( \bar { n } )$ . (Since $\bar { n } ^ { \prime \prime }$ is not sorted, we fix the order in which prefix sums are accumulated to be the same as in $\bar { n }$ ; by (10) $i ^ { \prime }$ is well defined). Next we let $\bar { n } ^ { \prime \prime \prime }$ be identical to $\bar { n } ^ { \prime \prime }$ except that $n _ { i ^ { \prime } } ^ { \prime \prime \prime } = n _ { i ^ { \prime } } ^ { \prime \prime } + x$ . In other words, $\bar { n } ^ { \prime \prime \prime }$ differs from $\bar { n } ^ { \prime }$ by shifting $x$ from coordinate $i$ to coordinate $i ^ { \prime }$ .
+
+We argue that incrementing $n _ { i ^ { \prime } } ^ { \prime \prime }$ by $x$ does not change the maximal value of $\bar { n } ^ { \prime \prime }$ , i.e., $n _ { 1 } ^ { \prime \prime \prime } > n _ { i \prime } ^ { \prime \prime \prime }$ . Our choice of $i ^ { \prime }$ , which is the smallest index so that the prefix sum over $\bar { n } ^ { \prime \prime }$ overtakes that over $\bar { n }$ , implies that $n _ { 1 } ^ { \prime \prime } - n _ { i ^ { \prime } } ^ { \prime \prime } > n _ { 1 } - n _ { i ^ { \prime } }$ . Since $n _ { 1 } ^ { \prime \prime } = n _ { 1 }$ , it means that $n _ { i ^ { \prime } } > n _ { i ^ { \prime } } ^ { \prime \prime }$ (and by adding $x$ we move $n _ { i ^ { \prime } } ^ { \prime \prime }$ towards $n _ { i ^ { \prime } }$ ). Furthermore,
+
+$$
+n _ { i ^ { \prime } } ^ { \prime \prime \prime } = n _ { i ^ { \prime } } ^ { \prime \prime } + x \leq n _ { i ^ { \prime } } + ( n _ { i } ^ { \prime } - n _ { i } ) = n _ { i } ^ { \prime } + ( n _ { i ^ { \prime } } - n _ { i } ) \leq n _ { i } ^ { \prime } \leq n _ { 1 } ^ { \prime } = n _ { 1 } ^ { \prime \prime \prime } ,
+$$
+
+(We use $n _ { i ^ { \prime } } \leq n _ { i }$ , which is implied by $i ^ { \prime } > i$ .)
+
+We claim that $\begin{array} { r } { \sum _ { j = 1 } ^ { t } ( n _ { 1 } ^ { \prime \prime \prime } - n _ { j } ^ { \prime \prime \prime } ) \le S _ { t } ( \bar { n } ) } \end{array}$ for all $t$ , and thus, via (10), $\bar { n } \succeq \bar { n } ^ { \prime \prime \prime }$ . The choice of $i ^ { \prime }$ makes the statement trivial for $t < i ^ { \prime }$ . For $t \geq i ^ { \prime }$ the following holds:
+
+$$
+\sum _ { j = 1 } ^ { t } ( n _ { 1 } ^ { \prime \prime \prime } - n _ { j } ^ { \prime \prime \prime } ) = \sum _ { j = 1 } ^ { t } ( n _ { 1 } ^ { \prime \prime } - n _ { j } ^ { \prime \prime } ) - x \leq \left( \sum _ { j = 1 } ^ { t } ( n _ { 1 } ^ { \prime } - n _ { j } ^ { \prime } ) + x \right) - x = S _ { t } ( n ^ { \prime } ) \leq S _ { t } ( \bar { n } ) .
+$$
+
+By construction $d ( \bar { n } ^ { \prime } , \bar { n } ^ { \prime \prime \prime } ) = 1$ (the two histograms differ in two locations, in positive and negative directions, by $x \leq 1$ in each). For the same reasons $\bar { n } ^ { \prime \prime \prime } \succeq \bar { n } ^ { \prime }$ . To show that $\bar { \ell } ( \bar { n } , \bar { n } ^ { \prime } ) > \ell ( \bar { n } , \bar { n } ^ { \prime \prime \prime } )$ ,
+
+compare $\lceil n _ { j } - n _ { j } ^ { \prime } \rceil$ and $\lceil n _ { j } - n _ { j } ^ { \prime \prime \prime } \rceil$ for $j = i , i ^ { \prime }$ . At $j = i$ the first term is strictly larger than the second. At $j = i ^ { \prime }$ , the inequality holds too but it may be not strict.
+
+We may again apply the induction hypothesis to the pair $\bar { n }$ and $\bar { n } ^ { \prime \prime \prime }$ , thus completing the proof of the lemma.
+
+To complete the proof of the proposition, we need to argue that the values of $q$ are also monotone in the chain constructed by the previous lemma. Concretely, we put forth
+
+Proof. The fact that $d ( \bar { n } , \bar { n } ^ { \prime } ) = 1$ and $\bar { n } \succeq \bar { n } ^ { \prime }$ means that there is either a single index $i$ so that $n _ { i } ^ { \prime } < n _ { i }$ , or there exist two indices $i$ and $j$ so that $n _ { i } ^ { \prime } < n _ { i }$ and $n _ { j } ^ { \prime } > n _ { j }$ . The first case is immediate, since $q$ is non-decreasing in all inputs except for the largest (by C7).
+
+Let $n _ { i } ^ { \prime } = n _ { i } - x$ and $n _ { j } ^ { \prime } = n _ { j } + y$ , where $x , y > 0$ . Since $\bar { n } \succeq \bar { n } ^ { \prime }$ , it follows that $n _ { i } \geq n _ { j }$ and $x > y$ Consider two cases.
+
+Case I: $n _ { i } ^ { \prime } \geq n _ { j } ^ { \prime }$ , i.e., removing $x$ from $n _ { i }$ and adding $y$ to $n _ { j }$ does not change their ordering. Let
+
+$$
+\bar { n } ( t ) \triangleq ( 1 - t ) \bar { n } + t \cdot \bar { n } ^ { \prime } = [ n _ { 1 } , \ldots , n _ { i } - t \cdot x , \ldots , n _ { j } + t \cdot y , \ldots , n _ { m } ] .
+$$
+
+Then,
+
+$$
+\begin{array} { l l } { \displaystyle q ( \bar { n } ^ { \prime } ) - q ( \bar { n } ) = q ( \bar { n } ( 1 ) ) - q ( \bar { n } ( 0 ) ) = \int _ { t = 0 } ^ { 1 } ( q \circ \bar { n } ) ^ { \prime } ( t ) \mathrm { d } t } \\ { \displaystyle \qquad = \int _ { t = 0 } ^ { 1 } \left\{ - x \frac { \partial q } { \partial n _ { i } } \bar { n } ( t ) + y \frac { \partial q } { \partial n _ { j } } \bar { n } ( t ) \right\} \mathrm { d } t } \\ { \displaystyle \qquad \leq 0 . } \end{array}
+$$
+
+The last inequality follows from C7 and the facts that $x > y > 0$ and $n _ { i } ( t ) > n _ { j } ( t )$ . (The condition that $q ( \bar { n } ( t ) ) \overset { \cdot } { \leq } \dot { \mathrm { B } _ { \mathrm { U } } } ( q _ { 0 } )$ follows from C2 and the fact that $d ( \bar { n } ^ { \prime } , \bar { n } ( t ) ) \leq 1 .$ )
+
+Case II: $n _ { i } ^ { \prime } \leq n _ { j } ^ { \prime }$ . In this case we swap the $i$ th and $j$ th indices in $\bar { n } ^ { \prime }$ by defining $\bar { n } ^ { \prime \prime }$ which differs from it in $\bar { n } _ { i } ^ { \prime \prime } = \bar { n } _ { j } ^ { \prime }$ and $\bar { n } _ { j \_ } ^ { \prime \prime } = \bar { n } _ { i } ^ { \prime }$ . By C8, $q ( \bar { n } ^ { \prime \prime } ) = q ( \bar { n } ^ { \prime } )$ and, of course, $\bar { n } ^ { \prime \prime } \succeq \bar { n }$ since the prefix sums remain unchanged. The benefit of doing this transformation is that we are back in Case I, where the relative order of coordinates that change between $\bar { n }$ and $\bar { n } ^ { \prime \prime }$ remains the same.
+
+This concludes the proof of the lemma.
+
+Applying Lemma 17 we construct a chain of histograms between $\bar { n }$ and $\bar { n } ^ { \prime }$ , which, by Lemma 18, is non-increasing in $q ( \cdot )$ . Together this implies that $q ( \bar { n } ) \leq q ( \bar { n } ^ { \prime } )$ , as claimed.
+
+We apply the notion of dominance in proving the following proposition, which is used later in arguing correctness of Algorithm 4.
+
+Proposition 19. Let n¯ be an integer-valued histogram and d be a positive integer. And $q ( \cdot )$ satisfies C1, C7, and C8. The following holds:
+
+1. Assuming $n ^ { ( 1 ) } - n ^ { ( 2 ) } \geq 2 d ,$ , let $\bar { n } ^ { * }$ be obtained from $\bar { n }$ by decrementing ${ \boldsymbol n } ^ { ( 1 ) }$ by $d$ and incrementing $n ^ { ( 2 ) }$ by $d .$ . Then
+
+2. Assuming repeatedly $\begin{array} { r } { \sum _ { i = 2 } ^ { m } n ^ { ( i ) } \geq d } \end{array}$ , let e hi $\bar { n } ^ { * * }$ be obtained from am’s current secon $\bar { n }$ by incrementing highest value by o $n _ { 1 }$ y ti $d$ , and byes. Then $d$
+
+Proof. Towards proving the claims, we argue that $\bar { n } ^ { * }$ and $\bar { n } ^ { * * }$ are, respectively, the minimal and the maximal elements in the histogram dominance order (Definition 15) in the set of histograms at distance $d$ from $\bar { n }$ . By Proposition 16 the claims follow.
+
+1. Take any histogram $\bar { n } ^ { \prime }$ at distance $d$ from $\bar { n }$ . Our goal is to prove that $\bar { n } ^ { \prime } \succeq$ $\bar { n } ^ { * }$ . Recall the definition of the distance $d ( \cdot , \cdot )$ between two histograms $d ( \bar { n } , \bar { n } ^ { \prime } ) ~ =$ max $\begin{array} { r l } { \bigg \{ \sum _ { i : n _ { i } > n _ { i } ^ { \prime } } \lceil n _ { i } - n _ { i } ^ { \prime } \rceil , \sum _ { i : n _ { i } < n _ { i } ^ { \prime } } \lceil n _ { i } ^ { \prime } - n _ { i } \rceil \bigg \} } & { { } } \end{array}$ . If the distance is bounded by $d$ , it means, in particular, that
+
+$$
+\sum _ { j = 2 } ^ { i } n ^ { \prime ( j ) } \leq \sum _ { j = 2 } ^ { i } n ^ { ( j ) } + d \quad { \mathrm { ~ f o r ~ a l l ~ } } i > 2 .
+$$
+
+That lets us bound the prefix sums of $\bar { n } ^ { \prime }$ as follows:
+
+$$
+\begin{array} { l } { S _ { i } ( \bar { n } ^ { \prime } ) = \displaystyle \sum _ { j = 2 } ^ { i } ( n ^ { \prime ( 1 ) } - n ^ { \prime ( j ) } ) = ( i - 1 ) \cdot n ^ { \prime ( 1 ) } - \displaystyle \sum _ { j = 2 } ^ { i } n ^ { \prime ( j ) } } \\ { \geq ( i - 1 ) \cdot ( n ^ { ( 1 ) } - d ) - \displaystyle \left( \sum _ { j = 2 } ^ { i } n ^ { ( j ) } + d \right) = S _ { i } ( \bar { n } ^ { * } ) . } \end{array}
+$$
+
+We demonstrated that $\bar { n } ^ { \prime } \succeq \bar { n } ^ { * }$ , which, by Proposition 16, implies that $q ( \bar { n } ^ { \prime } ) \leq q ( \bar { n } ^ { * } )$ Together with the immediate $d ( \bar { n } , \bar { n } ^ { * } ) = d$ we prove the claim.
+
+2. Assume wlog that $\bar { n }$ is sorted in the descending order. Define the following value that depends on $\bar { n }$ and $d$ :
+
+$$
+u \triangleq \operatorname* { m i n } \left\{ x \in { \mathbb { N } } \colon \sum _ { i : i > 1 , n _ { i } \geq x } n _ { i } - x \leq d \right\} .
+$$
+
+The constant $u$ is the smallest such $x$ so that the total mass that can be shaved from elements of $\bar { n }$ above $x$ (excluding $n _ { 1 }$ ) is at most $d$ .
+
+We give the following equivalent definition of $\bar { n } ^ { * * }$ :
+
+$$
+n _ { i } ^ { * * } = \left\{ { \begin{array} { l l } { n _ { 1 } + d } & { { \mathrm { i f ~ } } i = 1 , } \\ { u } & { { \mathrm { i f ~ } } i > 1 { \mathrm { ~ a n d ~ } } n _ { i } \geq u , } \\ { n _ { i } } & { { \mathrm { o t h e r w i s e } } . } \end{array} } \right.
+$$
+
+Fix any $i \in [ m ]$ and any histogram $\bar { n } ^ { \prime }$ at distance $d$ from $\bar { n }$ . Our goal is to prove that $S _ { i } ( \bar { n } ^ { * * } ) \geq S _ { i } ^ { \bar { ( n ^ { \prime } ) } }$ and thus $\bar { n } ^ { * * } \succeq \bar { n } ^ { \prime }$ . Assume the opposite and take largest $i$ such that $S _ { i } ( { \bar { n } } ^ { * * } ) < S _ { i } ( { \bar { n } } ^ { \prime } )$ .
+
+We may assume that $n ^ { \prime ( 1 ) } = n _ { 1 } ^ { * * } = n _ { 1 } + d$ . Consider the following cases.
+
+Case I. If $n ^ { * * ( i ) } < u$ , the contradiction follows from
+
+$$
+\begin{array} { r } { S _ { i } ( \bar { n } ^ { \prime } ) = \displaystyle \sum _ { j = 2 } ^ { i } ( n ^ { \prime ( 1 ) } - n ^ { \prime ( j ) } ) = \displaystyle \sum _ { j = 2 } ^ { i } \Big ( ( n ^ { \prime ( 1 ) } - n ^ { ( 1 ) } ) + ( n ^ { ( 1 ) } - n ^ { ( j ) } ) + ( n ^ { ( j ) } - n ^ { \prime ( j ) } ) \Big ) } \\ { \le ( i - 1 ) d + S _ { i } ( \bar { n } ) + d = S _ { i } ( \bar { n } ^ { * * } ) . } \end{array}
+$$
+
+The last equality is due to the fact that all differences between $\bar { n }$ and $\bar { n } ^ { * * }$ are confined to the indices that are less than $i$ .
+
+Case $\mathbf { I I }$ . If $n ^ { * * ( i ) } = u$ and $n ^ { \prime { ( i ) } } \geq u$ , the contradiction with $S _ { i } ( \bar { n } ^ { * * } ) < S _ { i } ( \bar { n } ^ { \prime } )$ follows immediately from
+
+$$
+S _ { i } ( \bar { n } ^ { \prime } ) = \sum _ { j = 2 } ^ { i } ( n ^ { \prime ( 1 ) } - n ^ { \prime ( j ) } ) \le ( i - 1 ) ( n ^ { \prime ( 1 ) } - u ) = S _ { i } ( \bar { n } ^ { * * } ) .
+$$
+
+Case III. Finally, consider the case when $n ^ { * * ( i ) } = u$ and $v \ \triangleq n ^ { \prime ( i ) } < u$ . Since $i$ is the largest such that $S _ { i } ( { \bar { n } } ^ { * * } ) < S _ { i } ( { \bar { n } } ^ { \prime } )$ , it means that $n ^ { * * ( i + 1 ) } < n ^ { \prime ( i + 1 ) } \leq v < u = n ^ { * * ( i ) }$
+
+and thus $n ^ { * * ( i ) } - n ^ { * * ( i + 1 ) } \geq 2$ (we rely on the fact that the histograms are integer-valued). It implies that all differences between $\bar { n }$ and $\bar { n } ^ { * * }$ are confined to the indices in $[ 1 , i ]$ . Then,
+
+$$
+\begin{array} { l } { { S _ { i } ( \bar { n } ^ { * * } ) - S _ { i } ( \bar { n } ^ { \prime } ) \geq \displaystyle \sum _ { j = 2 } ^ { i } ( n _ { 1 } ^ { * * } - n _ { j } ^ { * * } ) - \displaystyle \sum _ { j = 2 } ^ { i } ( n _ { 1 } ^ { \prime } - n _ { j } ^ { \prime } ) } } \\ { { \mathrm { ~ } = \displaystyle \sum _ { j = 2 } ^ { i } ( ( n _ { j } - n _ { j } ^ { * * } ) + ( n _ { j } ^ { \prime } - n _ { j } ) ) } } \\ { { \mathrm { ~ } \geq d - d ( \bar { n } , \bar { n } ^ { \prime } ) } } \\ { { \mathrm { ~ } \geq 0 , } } \end{array}
+$$
+
+which contradicts the assumption that $S _ { i } ( { \bar { n } } ^ { * * } ) < S _ { i } ( { \bar { n } } ^ { \prime } )$ .
+
+We may now state and prove the main result of this section.
+
+Theorem 20. Assume that $q ( \cdot )$ satisfies conditions $C l { - } C 8$ and $\bar { n }$ is an integer-valued histogram. Then the following two claims are true:
+
+1. Algorithm 4 computes $\begin{array} { r } { \operatorname* { m a x } _ { \bar { n } ^ { \prime } : \mathrm { d i s t } ( \bar { n } , \bar { n } ^ { \prime } ) \leq d } \tilde { \mathrm { L S } } ( \bar { n } ^ { \prime } ) . } \end{array}$
+
+2. Algorithm 5 computes $\mathrm { S S } _ { \beta } ( \bar { n } )$ , which is a $\beta$ -smooth upper bound on smooth sensitivity of $\beta \left( q ( \cdot ) \right)$ .
+
+Proof. Claim 1. Recall that $q _ { 1 } = \mathrm { B _ { L } } ( q _ { 0 } )$ , and therefore, by Proposition 14 the function $\tilde { \mathrm { L S } } ( q )$ is non-decreasing in $[ 0 , q _ { 1 } ]$ , constant in $[ q _ { 1 } , q _ { 0 } ]$ , and non-increasing in $[ q _ { 0 } , 1 ]$ . It means, in particular, that to maximize $\tilde { \mathrm { L S } } ( q ( \bar { n } ^ { \prime } ) )$ over histograms satisfying $d ( \bar { n } , \bar { n } ^ { \prime } ) = d$ , it suffices to consider the following cases.
+
+If $\tilde { \mathrm { L S } } ( q ( \bar { n } ) ) < q _ { 1 }$ , then higher values of $\mathrm { \tilde { L S } ( \cdot ) }$ may be attained only by histograms with higher values of $q$ . Proposition 19 enables us to efficiently find a histogram $\bar { n } ^ { * }$ with the highest $q$ at distance $d$ , or conclude that we may reach the plateau by making the two highest histogram entries be equal.
+
+$q _ { 1 } \leq \tilde { \mathrm { L S } } ( q ( \bar { n } ) ) \leq q _ { 0 }$ , it means that $\tilde { \mathrm { L S } } ( q ( n ) )$ is already as high as it can be.
+
+If $q _ { 0 } < \tilde { \mathrm { L S } } ( q ( \bar { n } ) )$ , then, according to Proposition 14, higher values of $\mathrm { \tilde { L S } ( \cdot ) }$ can be achieved by histograms with smaller values of $q$ , which we explore using the procedure outlined by Proposition 19. The stopping condition—when the plateau is reached—happens when $q$ becomes smaller than $q _ { 0 }$ .
+
+Claim 2. The second claim follows from the specification of Algorithm 5 and the first claim.
+
+# B.4 GNMAX SATISFIES CONDITIONS C1–C8
+
+The previous sections laid down a framework for computing smooth sensitivity of a randomized aggregator mechanism: defining functions $q ( \cdot ) , \mathrm { B _ { U } ( \cdot ) , B _ { L } ( \cdot ) }$ , verifying that they satisfy conditions C1–C8, and applying Theorem 20, which asserts correctness of Algorithm 5. In this section we instantiate this framework for the GNMax mechanism.
+
+# B.4.1 CONDITIONS C1–C4, C7 AND C8
+
+Defining $q$ and conditions C1, C7, and C8. Following Proposition 7, we define $q \colon \mathbb { R } _ { \geq 0 } ^ { m } [ 0 , 1 ]$ for a GNMax mechanism parameterized with $\sigma$ as:
+
+$$
+\begin{array} { r l } & { q ( \bar { n } ) \triangleq \operatorname* { m i n } \left\{ \underset { i \neq i ^ { * } } { \sum } \mathbf { P r } ( Z _ { i } - Z _ { i ^ { * } } \geq n _ { i ^ { * } } - n _ { i } ) , 1 \right\} } \\ & { \quad = \operatorname* { m i n } \left\{ \underset { i \neq i ^ { * } } { \sum } \frac { 1 } { 2 } \left( 1 - \mathrm { e r f } \left( \frac { n _ { i ^ { * } } - n _ { i } } { 2 \sigma } \right) \right) , 1 \right\} } \\ & { \quad = \operatorname* { m i n } \left\{ \underset { i \neq i ^ { * } } { \sum } \frac { 1 } { 2 } \mathrm { e r f c } \left( \frac { n _ { i ^ { * } } - n _ { i } } { 2 \sigma } \right) , 1 \right\} , } \end{array}
+$$
+
+where $i ^ { * }$ is the histogram $\bar { n }$ ’s highest coordinate, i.e., $n _ { i ^ { * } } \geq n _ { i }$ for all $i$ (if there are multiple highest, let $i ^ { * }$ be any of them). Recall that erf is the error function, and erfc is the complement error function.
+
+Proposition 7 demonstrates that $q ( { \bar { n } } )$ bounds from above the probability that GNMax outputs anything but the highest coordinate of the histogram.
+
+Conditions C1, C7, and C8 follow by simple calculus $( q _ { 0 }$ , defined below, is at most 0.5).
+
+Functions $\mathrm { B } _ { \mathrm { L } } , \mathrm { B } _ { \mathrm { U } }$ , and conditions C2–C4. Recall that the functions $\mathrm { B _ { L } }$ and $\mathrm { B _ { U } }$ are defined in Appendix $\mathbf { B }$ as follows:
+
+$$
+\begin{array} { l } { { \displaystyle { \mathrm { B } } _ { \mathrm { U } } ( q ) \triangleq \operatorname* { m i n } \left\{ \frac { m - 1 } { 2 } \mathrm { e r f c } \left( \mathrm { e r f c } ^ { - 1 } \left( \frac { 2 q } { m - 1 } \right) - \frac { 1 } { \sigma } \right) , 1 \right\} , } } \\ { { \displaystyle { \mathrm { B } } _ { \mathrm { L } } ( q ) \triangleq \frac { m - 1 } { 2 } \mathrm { e r f c } \left( \mathrm { e r f c } ^ { - 1 } \left( \frac { 2 q } { m - 1 } \right) + \frac { 1 } { \sigma } \right) , } } \end{array}
+$$
+
+Proposition 21 (Condition C2). For any neighbor $\bar { n } ^ { \prime }$ of $\bar { n }$ , i.e., $d ( \bar { n } ^ { \prime } , \bar { n } ) = 1$ , the following bounds hold:
+
+$$
+\mathrm { B } _ { \mathrm { L } } ( q ( \bar { n } ) ) \leq q ( \bar { n } ^ { \prime } ) \leq \mathrm { B } _ { \mathrm { U } } ( q ( \bar { n } ) ) .
+$$
+
+Proof. Assume wlog that $i ^ { * } = 1$ . Let $x _ { i } \triangleq n _ { 1 } - n _ { i }$ and $q _ { i } \triangleq \mathrm { e r f c } ( x _ { i } / 2 \sigma ) / 2$ , and similarly define $\boldsymbol { x } _ { i } ^ { \prime }$ for $\bar { n } ^ { \prime }$ . Observe that $| x _ { i } - x _ { i } ^ { \prime } | \leq 2$ , which, by monotonicity of erfc, implies that
+
+$$
+{ \frac { 1 } { 2 } } \operatorname { e r f c } \left( { \frac { x _ { i } + 2 } { 2 \sigma } } \right) \leq q _ { i } ( { \bar { n } } ^ { \prime } ) \leq { \frac { 1 } { 2 } } \operatorname { e r f c } \left( { \frac { x _ { i } - 2 } { 2 \sigma } } \right) .
+$$
+
+Thus
+
+$$
+\frac { 1 } { 2 } \sum _ { i > 1 } \mathrm { e r f c } \left( \frac { x _ { i } + 2 } { 2 \sigma } \right) \leq q ( \bar { n } ^ { \prime } ) \leq \frac { 1 } { 2 } \sum _ { i > 1 } \mathrm { e r f c } \left( \frac { x _ { i } - 2 } { 2 \sigma } \right) .
+$$
+
+(Although $i ^ { * }$ may change between $\bar { n }$ and $\bar { n } ^ { \prime }$ , the bounds still hold.)
+
+Our first goal is to upper bound $q ( \bar { n } ^ { \prime } )$ for a given value of $q ( { \bar { n } } )$ . To this end we set up the following maximization problem
+
+$$
+\operatorname* { m a x } _ { \{ x _ { i } \} } { \frac { 1 } { 2 } } \sum _ { i > 1 } { \mathrm { e r f c } } \left( { \frac { x _ { i } - 2 } { 2 \sigma } } \right) { \mathrm { ~ s u c h ~ t h a t ~ } } { \frac { 1 } { 2 } } \sum _ { i > 1 } { \mathrm { e r f c } } \left( { \frac { x _ { i } } { 2 \sigma } } \right) = q { \mathrm { ~ a n d ~ } } x _ { i } \geq 0 .
+$$
+
+We may temporarily ignore the non-negative constraints, which end up being satisfied by our solution. Consider using the method of Lagrange multipliers and take a derivative in $x _ { i } { } ^ { \ ' }$ :
+
+$$
+\begin{array} { c } { \displaystyle - \exp \left( - \left( \frac { x _ { i } - 2 } { 2 \sigma } \right) ^ { 2 } \right) + \lambda \exp \left( - \left( \frac { x _ { i } } { 2 \sigma } \right) ^ { 2 } \right) = 0 } \\ { \displaystyle \Leftrightarrow \lambda = \exp \left( \frac { x _ { i } - 1 } { \sigma ^ { 2 } } \right) . } \end{array}
+$$
+
+Since the expression is symmetric in $i > 1$ , it means that the local optima are attained at $x _ { 2 } =$ $\cdots = x _ { m }$ (the second derivative confirms that these are local maxima). After solving for $( m -$ $1 ) \operatorname { e r f c } ( x / 2 \sigma ) = 2 q$ we have
+
+$$
+q ( \bar { n } ^ { \prime } ) \leq \frac { m - 1 } { 2 } \mathrm { e r f c } \left( \mathrm { e r f c ^ { - 1 } } \left( \frac { 2 q } { m - 1 } \right) - \frac { 1 } { \sigma } \right) .
+$$
+
+where $m$ is the number of classes. Similarly,
+
+$$
+q ( \bar { n } ^ { \prime } ) \geq \frac { m - 1 } { 2 } \mathrm { e r f c } \left( \mathrm { e r f c ^ { - 1 } } \left( \frac { 2 q } { m - 1 } \right) + \frac { 1 } { \sigma } \right) .
+$$
+
+Conditions C3, i.e., $\operatorname { B _ { L } } ( q )$ is monotonically increasing in $q$ , and $\mathrm { C 4 }$ , i.e., $\mathrm { B _ { L } }$ and $\mathrm { B _ { U } }$ are functional inverses of each other in $[ 0 , q _ { 0 } ]$ and $\mathrm { B } _ { \mathrm { L } } ( { \dot { q } } ) \leq q \leq \mathbf { \bar { B } } _ { \mathrm { U } } ( { \dot { q } } )$ for all $q \in [ 0 , 1 ]$ , follow from basic properties of erfc. The restriction that $q \in [ 0 , q _ { 0 } ]$ ensures that $\operatorname { B _ { U } } ( q )$ is strictly less than one, and the minimum in the definition of $\operatorname { B _ { U } } ( \cdot )$ simplifies to its first argument in this range.
+
+# B.4.2 CONDITIONS C5 AND C6
+
+Conditions C5 and C6 stipulate that the function $\beta ( q ) \triangleq \beta _ { \sigma } ( q )$ (defined in Appendix A) exhibits a specific growth pattern. Concretely, C5 states that $\beta \left( q \right)$ is monotonically increasing for $0 \leq q \leq q _ { 0 }$ , and constant for $q _ { 0 } < q \leq 1$ . (Additionally, we require that $\mathrm { B _ { U } } ( q _ { 0 } ) < 1 )$ . Condition C6 requires that $\Delta \beta ( q ) = \beta ( \mathrm { B } _ { \mathrm { U } } ( q ) ) - \beta ( q )$ is non-decreasing in $[ 0 , \mathrm { B _ { L } } ( q _ { 0 } ) ]$ .
+
+Rather than proving these statements analytically, we check these assumptions for any fixed $\sigma$ and $\lambda$ via a combination of symbolic and numeric analyses.
+
+More concretely, we construct symbolic expressions for $\beta \left( \cdot \right)$ and $\Delta \beta \left( \cdot \right)$ and (symbolically) differentiate them. We then minimize (numerically) the resulting expressions over $[ 0 , q _ { 0 } ]$ and $[ 0 , \mathrm { B _ { L } } ( q _ { 0 } ) ]$ , and verify that their minimal values are indeed non-negative.
+
+# B.5 RÉNYI DIFFERENTIAL PRIVACY AND SMOOTH SENSITIVITY
+
+Although the procedure for computing a smooth sensitivity bound may be quite involved (such as Algorithms 3–5), its use in a differentially private data release is straightforward. Following Nissim et al. (2007), we define an additive Gaussian mechanism where the noise distribution is scaled by $\sigma$ and a smooth sensitivity bound:
+
+Definition 22. Given a real-valued function $f \colon { \mathcal { D } } \mathbb { R }$ and a $\beta$ -smooth sensitivity bound $\operatorname { S S } ( \cdot )$ , let $( \beta , \sigma )$ -GNSS mechanism be
+
+$$
+{ \mathcal { F } } _ { \sigma } ( D ) \triangleq f ( D ) + \operatorname { S S } _ { \beta } ( D ) \cdot { \mathcal { N } } ( 0 , \sigma ^ { 2 } ) .
+$$
+
+We claim that this mechanism satisfies Rényi differential privacy for finite orders from a certain range.
+
+Theorem 23. The $( \beta , \sigma )$ -GNSS mechanism ${ \mathcal { F } } _ { \sigma }$ is $( \lambda , \varepsilon )$ -RDP, where
+
+$$
+\varepsilon \triangleq { \frac { \lambda \cdot e ^ { 2 \beta } } { \sigma ^ { 2 } } } + { \frac { \beta \lambda - 0 . 5 \ln ( 1 - 2 \lambda \beta ) } { \lambda - 1 } }
+$$
+
+for all $1 < \lambda < 1 / ( 2 \beta )$ .
+
+Proof. Consider two neighboring datasets $D$ and $D ^ { \prime }$ . The output distributions of the $( \beta , \sigma )$ -GNSS mechanism on $D$ and $D ^ { \prime }$ are, respectively,
+
+$$
+P \triangleq f ( D ) + \mathrm { S S } _ { \beta } ( D ) \cdot { \mathcal { N } } ( 0 , \sigma ^ { 2 } ) = { \mathcal { N } } ( f ( D ) , ( \mathrm { S S } _ { \beta } ( D ) \sigma ) ^ { 2 } ) { \mathrm { ~ a n d ~ } } Q \triangleq { \mathcal { N } } ( f ( D ^ { \prime } ) , ( \mathrm { S S } _ { \beta } ( D ^ { \prime } ) \sigma ) ^ { 2 } ) .
+$$
+
+The Rényi divergence between two normal distributions can be computed in closed form (van Erven & Harremoës, 2014):
+
+$$
+D _ { \lambda } ( P \| Q ) = \lambda { \frac { ( f ( D ) - f ( D ^ { \prime } ) ) ^ { 2 } } { 2 \sigma ^ { 2 } s ^ { 2 } } } + { \frac { 1 } { 1 - \lambda } } \ln { \frac { s } { { \mathrm { S S } } _ { \beta } ( D ) ^ { 1 - \lambda } \cdot { \mathrm { S S } } _ { \beta } ( D ^ { \prime } ) ^ { \lambda } } } ,
+$$
+
+provided $s ^ { 2 } \triangleq ( 1 - \lambda ) \cdot \mathrm { S S } _ { \beta } ( D ) ^ { 2 } + \lambda \cdot \mathrm { S S } _ { \beta } ( D ^ { \prime } ) ^ { 2 } > 0 .$
+
+According to the definition of smooth sensitivity (Definition 12)
+
+$$
+e ^ { - \beta } \cdot \mathrm { S S } _ { \beta } ( D ) \leq \mathrm { S S } _ { \beta } ( D ^ { \prime } ) \leq e ^ { \beta } \cdot \mathrm { S S } _ { \beta } ( D ) ,
+$$
+
+and
+
+$$
+| f ( D ) - f ( D ^ { \prime } ) | \leq e ^ { \beta } \cdot \operatorname * { m i n } ( \mathrm { S S } _ { \beta } ( D ) , \mathrm { S S } _ { \beta } ( D ^ { \prime } ) ) .
+$$
+
+Bound (12) together with the condition that $\lambda \le 1 / ( 2 \beta )$ implies that
+
+$$
+\begin{array} { r l } & { s ^ { 2 } = ( 1 - \lambda ) \cdot \operatorname { S S } _ { \beta } ( D ) ^ { 2 } + \lambda \cdot \operatorname { S S } _ { \beta } ( D ^ { \prime } ) ^ { 2 } = \operatorname { S S } _ { \beta } ( D ) ^ { 2 } + \lambda ( \operatorname { S S } _ { \beta } ( D ^ { \prime } ) ^ { 2 } - \operatorname { S S } _ { \beta } ( D ) ^ { 2 } ) } \\ & { \qquad \geq \operatorname { S S } _ { \beta } ( D ) ^ { 2 } ( 1 + \lambda ( e ^ { - 2 \beta } - 1 ) ) \geq \operatorname { S S } _ { \beta } ( D ) ^ { 2 } ( 1 - 2 \lambda \beta ) > 0 . } \end{array}
+$$
+
+The above lower bound ensures that $s ^ { 2 }$ is well-defined, i.e., non-negative, as required for application of (11).
+
+Combining bounds (12)– (14), we have that
+
+$$
+D _ { \lambda } ( P \| Q ) \le \frac { \lambda \cdot e ^ { 2 \beta } } { \sigma ^ { 2 } } + \frac { 1 } { 1 - \lambda } \ln \left\{ \frac { s } { \mathrm { S S } _ { \beta } ( D ) } e ^ { - \lambda \beta } \right\} \le \frac { \lambda \cdot e ^ { 2 \beta } } { \sigma ^ { 2 } } + \frac { \beta \lambda - 0 . 5 \ln ( 1 - 2 \lambda \beta ) } { \lambda - 1 }
+$$
+
+as claimed.
+
+Note that if $\lambda \gg 1$ , $\sigma \ll \lambda$ , and $\beta \ll 1 / ( 2 \lambda )$ , then $( \beta , \sigma )$ -GNSS satisfies $( \lambda , ( \lambda { + } 1 ) / \sigma ^ { 2 } )$ -RDP. Compare this with RDP analysis of the standard additive Gaussian mechanism, which satisfies $( \lambda , \lambda / \sigma ^ { 2 } )$ - RDP. The difference is that GNSS scales noise in proportion to smooth sensitivity, which is no larger and can be much smaller than global sensitivity.
+
+# B.6 PUTTING IT ALL TOGETHER: APPLYING SMOOTH SENSITIVITY
+
+Recall our initial motivation for the smooth sensitivity analysis: enabling privacy-preserving release of data-dependent privacy guarantees. Indeed, these guarantees vary greatly between queries (see Figure 5) and are typically much smaller than data-independent privacy bounds. Since datadependent bounds may leak information about underlying data, publishing the bounds themselves requires a differentially private mechanism. As we explain shortly, smooth sensitivity analysis is a natural fit for this task.
+
+We first consider the standard additive noise mechanism where the noise (such as Laplace or Gaussian) is calibrated to the global sensitivity of the function we would like to make differentially private. We know that Rényi differential privacy is additive for any fixed order $\lambda$ , and thus the cumulative RDP cost is the sum of RDP costs of individual queries each upper bounded by a dataindependent bound. Thus, it might be tempting to use the standard additive noise mechanism for sanitizing the total, but that would be a mistake.
+
+To see why, consider a sequence of queries $\bar { n } _ { 1 } , \ldots , \bar { n } _ { \ell }$ answered by the aggregator. Their total (unsanitized) RDP cost of order $\lambda$ is $\begin{array} { r } { B _ { \sigma } = \sum _ { i = 1 } ^ { \ell } \beta _ { \sigma } ( q ( \bar { n } _ { i } ) ) } \end{array}$ . Even though $\beta _ { \sigma } ( q ( \bar { n } _ { i } ) ) \leq \lambda / \sigma ^ { 2 }$ (the data-independent bound, Proposition 8), the sensitivity of their sum is not $\lambda / \sigma ^ { 2 }$ . The reason is that the (global) sensitivity is defined as the maximal difference in the function’s output between two neighboring datasets $D$ and $D ^ { \prime }$ . Transitioning from $D$ to $D ^ { \prime }$ may change one teacher’s output on all student queries.
+
+In contrast with the global sensitivity of $B _ { \sigma }$ that may be quite high—particularly for the second step of the Confident GNMax aggregator—its smooth sensitivity can be extremely small. Towards computing a smooth sensitivity bound on $B _ { \sigma }$ , we prove the following theorem which defines a smooth sensitivity of the sum in terms of local sensitivities of its parts.
+
+Theorem 24. Let $f _ { i } { : } \mathcal { D } \mathbb { R }$ for $1 \leq i \leq \ell _ { \mathrm { { i } } }$ , $\begin{array} { r } { F ( D ) \triangleq \sum _ { i = 1 } ^ { \ell } f _ { i } ( D ) } \end{array}$ and $\beta > 0$ . Then
+
+$$
+\mathrm { S S } ( D ) \triangleq \operatorname* { m a x } _ { d \geq 0 } e ^ { - \beta d } \cdot \sum _ { i = 1 } ^ { \ell } \operatorname* { m a x } _ { D ^ { \prime } : \mathrm { d i s t } ( D , D ^ { \prime } ) \leq d } \tilde { \mathrm { L S } } _ { f _ { i } } ( D ^ { \prime } ) ,
+$$
+
+is a $\beta$ -smooth bound on $F ( \cdot ) i f \tilde { \mathrm { L S } } _ { f _ { i } } ( D ^ { \prime } )$ are upper bounds on the local sensitivity of $f _ { i } ( D ^ { \prime } )$ .
+
+Proof. We need to argue that $\operatorname { S S } ( { \mathord { \cdot } } )$ is $\beta$ -smooth, i.e., $\mathrm { S S } ( D _ { 1 } ) \le e ^ { \beta } \cdot \mathrm { S S } ( D _ { 2 } )$ for any neighboring $D _ { 1 } , D _ { 2 } \ \in \ { \mathcal { D } }$ , and it is an upper bound on the local sensitivity of $F ( D _ { 1 } )$ , i.e., $\mathrm { S S } ( D _ { 1 } ) \ \geq$ $| F ( D _ { 1 } ) - F ( D _ { 2 } ) |$ .
+
+Smoothness follows from the observation that
+
+$$
+\operatorname* { m a x } _ { D : \mathrm { d i s t } ( D _ { 1 } , D ) \leq d } \tilde { \mathrm { L S } } _ { f _ { i } } ( D ) \leq \operatorname* { m a x } _ { D : \mathrm { d i s t } ( D _ { 2 } , D ) \leq d + 1 } \tilde { \mathrm { L S } } _ { f _ { i } } ( D )
+$$
+
+for all neighboring datasets $D _ { 1 }$ and $D _ { 2 }$ (by the triangle inequality over distances). Then
+
+$$
+\begin{array} { r l } & { \displaystyle \mathrm { S S } ( D _ { 1 } ) = \operatorname* { m a x } _ { d \geq 0 } e ^ { - \beta d } \cdot \sum _ { i = 1 } ^ { \ell } \operatorname* { m a x } _ { D : \mathrm { d i s t } ( D _ { 1 } , D ) \leq d } \mathrm { I } \tilde { \mathrm { S } S } _ { f _ { i } } ( D ) , } \\ & { \qquad \leq \displaystyle \operatorname* { m a x } _ { d \geq 0 } e ^ { - \beta d } \cdot \sum _ { i = 1 } ^ { \ell } \operatorname* { m a x } _ { D : \mathrm { d i s t } ( D _ { 2 } , D ) \leq d + 1 } \mathrm { I } \tilde { \mathrm { S } S } _ { f _ { i } } ( D ) } \\ & { \qquad = \displaystyle \operatorname* { m a x } _ { d \geq 1 } e ^ { - \beta ( d ^ { \prime } - 1 ) } \cdot \sum _ { i = 1 } ^ { \ell } \operatorname* { m a x } _ { D : \mathrm { d i s t } ( D _ { 2 } , D ) \leq d ^ { \prime } } \mathrm { I } \tilde { \mathrm { S } S } _ { f _ { i } } ( D ) } \\ & { \qquad \leq e ^ { \beta } \cdot \mathrm { S S } ( D _ { 2 } ) } \end{array}
+$$
+
+as needed for $\beta$ -smoothness.
+
+The fact that $\operatorname { S S } ( { \mathord { \cdot } } )$ is an upper bound on the local sensitivity of $F ( \cdot )$ is implied by the following:
+
+$$
+\begin{array} { r l } { \displaystyle | F ( D _ { 1 } ) - F ( D _ { 2 } ) | = \left| \displaystyle \sum _ { i = 1 } ^ { \ell } f _ { i } ( D _ { 1 } ) - \displaystyle \sum _ { i = 1 } ^ { \ell } f _ { i } ( D _ { 2 } ) \right| } & { } \\ { \displaystyle } & { \le \displaystyle \sum _ { i = 1 } ^ { \ell } | f _ { i } ( D _ { 1 } ) - f _ { i } ( D _ { 2 } ) | } \\ { \displaystyle } & { \le \displaystyle \sum _ { i = 1 } ^ { \ell } \mathrm { I } \tilde { \mathrm { S } } f _ { i } ( D _ { 1 } ) } \\ { \displaystyle } & { \le \mathrm { S S } ( D _ { 1 } ) , } \end{array}
+$$
+
+which concludes the proof.
+
+Applying Theorem 24 allows us to compute a smooth sensitivity of the sum more efficiently than summing up smooth sensitivities of its parts. Results below rely on this strategy.
+
+Empirical results. Table 2 revisits the privacy bounds in Table 1. For all data-dependent privacy claims of the Confident GNMax aggregator we report parameters for their smooth sensitivity analysis and results of applying the GNSS mechanism for their release.
+
+Consider the first row of the table. The MNIST dataset was partitioned among 250 teachers, each getting 200 training examples. After the teachers were individually trained, the student selected at random 640 unlabeled examples, and submitted them to the Confident GNMax aggregator with the threshold of 200, and noise parameters $\sigma _ { 1 } = 1 5 0$ and $\sigma _ { 2 } = 4 0 $ . The expected number of answered examples (those that passed the first step of Algorithm 1) is 283, and the expected Rényi differential privacy is $\varepsilon = 1 . 1 8$ at order $\lambda = 1 4$ . This translates (via Theorem 5) to $( 2 . 0 0 , 1 0 ^ { - 5 } )$ -differential privacy, where 2.00 is the expectation of the privacy parameter $\varepsilon$ .
+
+These costs are data-dependent and they cannot be released without further sanitization, which we handle by adding Gaussian noise scaled by the smooth sensitivity of $\varepsilon$ (the GNSS mechanism, Definition 22). At $\beta = 0 . 0 3 2 9$ the expected value of smooth sensitivity is 0.0618. We choose $\sigma _ { \mathrm { S S } } = 6 . 2 3$ , which incurs, according to Theorem 23, an additional (data-independent) (14, 0.52)- RDP cost. Applying $( \beta , \sigma _ { \mathrm { S S } } )$ -GNSS where $\sigma _ { \mathrm { S S } } ~ = ~ 6 . 2 3$ , we may publish differentially private estimate of the total privacy cost that consists of a fixed part—the cost of applying Confident GNMax and GNSS—and random noise. The fixed part is $2 . { \bar { 5 } } 2 = 1 . 1 8 + 0 . 5 2 { \stackrel { - } { - } } \mathrm { l n } ( 1 { \bar { 0 } } ^ { - 5 } ) / 1 4$ , and the noise is normally distributed with mean 0 and standard deviation $\sigma _ { \mathrm { S S } } \cdot 0 . 0 6 1 8 = 0 . 3 8 5$ . We note that, in contrast with the standard additive noise, one cannot publish its standard deviation without going through additional privacy analysis.
+
+Table 2: Privacy-preserving reporting of privacy costs. The table augments Table 1 by including smooth sensitivity analysis of the total privacy cost. The expectations are taken over the student’s queries and outcomes of the first step of the Confident GNMax aggregator. Order $\lambda$ , smooth sensitivity parameter $\beta$ , $\sigma _ { \mathrm { S S } }$ are parameters of the GNSS mechanism (Section B.5). The final column sums up the data-dependent cost $\varepsilon$ , the cost of applying GNSS (Theorem 23), and the standard deviation of Gaussian noise calibrated to smooth sensitivity (the product of $\mathbb { E }$ $[ \mathrm { S S } _ { \beta } ]$ and $\sigma _ { \mathrm { S S } }$ ).
+
+| Input: input x,confidence γ,threshold T,noise parameters O1 ando2,total number of teachers M |
| 1:Ask the student to provide prediction scores p(x) 2:if maxj{nj(x)-Mpj(x)}+N(0,o²)≥Tthen >Student does not agree with teachers |
|
| 3: return argmaxj{nj(x) +N(0,0²)} |
| 4: else if max{pi(x)}> γ then > Student agrees with teachers and is confident |
| 5: return arg maxj Pj(x) |
| Reinforce student's prediction 6: else return⊥ No output given for this label |
",
+ "bbox": [
+ 178,
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+ ],
+ "page_idx": 7
+ },
+ {
+ "type": "text",
+ "text": "student’s current predictions for its public training data, we design an Interactive Aggregator that discards queries where the student already confidently predicts the same label as the teachers. ",
+ "bbox": [
+ 176,
+ 449,
+ 821,
+ 477
+ ],
+ "page_idx": 7
+ },
+ {
+ "type": "text",
+ "text": "Given a set of queries, the Interactive Aggregator (Algorithm 2) selects those answered by comparing student predictions to teacher votes for each class. Similar to Step 1 in the Confident Aggregator, queries where the plurality of these noised differences crosses a threshold are answered with GNMax. This noisy threshold suffices to enforce privacy of the first step because student predictions can be considered public information (the student is trained in a differentially private manner). ",
+ "bbox": [
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+ ],
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+ },
+ {
+ "type": "text",
+ "text": "For queries that fail this check, the mechanism reinforces the predicted student label if the student is confident enough and does this without looking at teacher votes again. This limited form of supervision comes at a small privacy cost. Moreover, the order of the checks ensures that a student falsely confident in its predictions on a query is not accidentally reinforced if it disagrees with the teacher consensus. The privacy accounting is identical to the Confident Aggregator except in considering the difference between teachers and the student instead of only the teachers votes. ",
+ "bbox": [
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+ ],
+ "page_idx": 7
+ },
+ {
+ "type": "text",
+ "text": "In practice, the Confident Aggregator can be used to start training a student when it can make no meaningful predictions and training can be finished off with the Interactive Aggregator after the student gains some proficiency. ",
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+ },
+ {
+ "type": "text",
+ "text": "5 EXPERIMENTAL EVALUATION ",
+ "text_level": 1,
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+ "type": "text",
+ "text": "Our goal is first to show that the improved aggregators introduced in Section 4 enable the application of PATE to uncurated data, thus departing from previous results on tasks with balanced and wellseparated classes. We experiment with the Glyph dataset described below to address two aspects left open by Papernot et al. (2017): (a) the performance of PATE on a task with a larger number of classes (the framework was only evaluated on datasets with at most 10 classes) and (b) the privacy-utility tradeoffs offered by PATE on data that is class imbalanced and partly mislabeled. In Section 5.2, we evaluate the improvements given by the GNMax aggregator over its Laplace counterpart (LNMax) and demonstrate the necessity of the Gaussian mechanism for uncurated tasks. ",
+ "bbox": [
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+ },
+ {
+ "type": "text",
+ "text": "In Section 5.3, we then evaluate the performance of PATE with both the Confident and Interactive Aggregators on all datasets used to benchmark the original PATE framework, in addition to Glyph. With the right teacher and student training, the two mechanisms from Section 4 achieve high accuracy with very tight privacy bounds. Not answering queries for which teacher consensus is too low (Confident-GNMax) or the student’s predictions already agree with teacher votes (InteractiveGNMax) better aligns utility and privacy: queries are answered at a significantly reduced cost. ",
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+ "text": "5.1 EXPERIMENTAL SETUP ",
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+ "text": "MNIST, SVHN, and the UCI Adult databases. We evaluate with two computer vision tasks (MNIST and Street View House Numbers (Netzer et al., 2011)) and census data from the UCI Adult dataset (Kohavi, 1996). This enables a comparative analysis of the utility-privacy tradeoff achieved with our Confident-GNMax aggregator and the LNMax originally used in PATE. We replicate the experimental setup and results found in Papernot et al. (2017) with code and teacher votes made available online. The source code for the privacy analysis in this paper as well as supporting data required to run this analysis is available on Github.1 ",
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+ "text": "A detailed description of the experimental setup can be found in Papernot et al. (2017); we provide here only a brief overview. For MNIST and SVHN, teachers are convolutional networks trained on partitions of the training set. For UCI Adult, each teacher is a random forest. The test set is split in two halves: the first is used as unlabeled inputs to simulate the student’s public data and the second is used as a hold out to evaluate test performance. The MNIST and SVHN students are convolutional networks trained using semi-supervised learning with GANs à la Salimans et al. (2016). The student for the Adult dataset are fully supervised random forests. ",
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+ "text": "Glyph. This optical character recognition task has an order of magnitude more classes than all previous applications of PATE. The Glyph dataset also possesses many characteristics shared by real-world tasks: e.g., it is imbalanced and some inputs are mislabeled. Each input is a $2 8 \\times 2 8$ grayscale image containing a single glyph generated synthetically from a collection of over 500K computer fonts.2 Samples representative of the difficulties raised by the data are depicted in Figure 3. The task is to classify inputs as one of the 150 Unicode symbols used to generate them. ",
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+ "text": "This set of 150 classes results from pre-processing efforts. We discarded additional classes that had few samples; some classes had at least 50 times fewer inputs than the most popular classes, and these were almost exclusively incorrectly labeled inputs. We also merged classes that were too ambiguous for even a human to differentiate them. Nevertheless, a manual inspection of samples grouped by classes—favorably to the human observer—led to the conservative estimate that some classes remain 5 times more frequent, and mislabeled inputs represent at least $1 0 \\%$ of the data. ",
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+ "text": "To simulate the availability of private and public data (see Section 3.1), we split data originally marked as the training set (about 65M points) into partitions given to the teachers. Each teacher is a ResNet (He et al., 2016) made of 32 leaky ReLU layers. We train on batches of 100 inputs for 40K steps using SGD with momentum. The learning rate, initially set to 0.1, is decayed after 10K steps to 0.01 and again after 20K steps to 0.001. These parameters were found with a grid search. ",
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+ "text": "We split holdout data in two subsets of 100K and 400K samples: the first acts as public data to train the student and the second as its testing data. The student architecture is a convolutional network learnt in a semi-supervised fashion with virtual adversarial training (VAT) from Miyato et al. (2017). Using unlabeled data, we show how VAT can regularize the student by making predictions constant in adversarial3 directions. Indeed, we found that GANs did not yield as much utility for Glyph as for MNIST or SVHN. We train with Adam for 400 epochs and a learning rate of $6 \\cdot \\mathrm { i 0 ^ { - 5 } }$ . ",
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+ "text": "5.2 COMPARING THE LNMAX AND GNMAX MECHANISMS ",
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+ "text": "Section 4.1 introduces the GNMax mechanism and the accompanying privacy analysis. With a Gaussian distribution, whose tail diminishes more rapidly than the Laplace distribution, we expect better utility when using the new mechanism (albeit with a more involved privacy analysis). ",
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+ "text": "To study the tradeoff between privacy and accuracy with the two mechanisms, we run experiments training several ensembles of $M$ teachers for $M \\in \\{ 1 0 0 , 5 0 0 , 1 0 0 0 , 5 0 0 0 \\}$ on the Glyph data. Recall that 65 million training inputs are partitioned and distributed among the $M$ teachers with each teacher receiving between 650K and 13K inputs for the values of $M$ above. The test data is used to query the teacher ensemble and the resulting labels (after the LNMax and GNMax mechanisms) are compared with the ground truth labels provided in the dataset. This predictive performance of the teachers is essential to good student training with accurate labels and is a useful proxy for utility. ",
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+ "text": "For each mechanism, we compute $( \\varepsilon , \\delta )$ -differential privacy guarantees. As is common in literature, for a dataset on the order of $\\bar { 1 0 } ^ { 8 }$ samples, we choose $\\delta = \\bar { 1 0 } ^ { - 8 }$ and denote the corresponding $\\varepsilon$ as the privacy cost. The total $\\varepsilon$ is calculated on a subset of 4,000 queries, which is representative of the number of labels needed by a student for accurate training (see Section 5.3). We visualize in Figure 4 the effect of the noise distribution (left) and the number of teachers (right) on the tradeoff between privacy costs and label accuracy. ",
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+ "text": "Observations. On the left of Figure 1, we compare our GNMax aggregator to the LNMax aggregator used by the original PATE proposal, on an ensemble of 1000 teachers and for varying noise scales $\\sigma$ . At fixed test accuracy, the GNMax algorithm consistently outperforms the LNMax mechanism in terms of privacy cost. To explain this improved performance, recall notation from Section 4.1. For both mechanisms, the data dependent privacy cost scales linearly with $\\tilde { q }$ —the likelihood of an answer other than the true plurality. The value of $\\tilde { q }$ falls of as $\\exp ( - x ^ { 2 } )$ for GNMax and $\\exp ( - x )$ for LNMax, where $x$ is the ratio $( \\dot { n _ { i ^ { * } } } - n _ { i } ) / \\sigma$ . Thus, when $n _ { i ^ { * } } - n _ { i }$ is (say) $4 \\sigma$ , LNMax would have $\\tilde { q } \\approx e ^ { - 4 } = 0 . 0 1 8 . . .$ , whereas GNMax would have $\\tilde { q } \\approx e ^ { - 1 6 } \\approx 1 0 ^ { - 7 }$ , thereby leading to a much higher likelihood of returning the true plurality. Moreover, this reduced $\\tilde { q }$ translates to a smaller privacy cost for a given $\\sigma$ leading to a better utility-privacy tradeoff. ",
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+ "text": "As long as each teacher has sufficient data to learn a good-enough model, increasing the number $M$ of teachers improves the tradeoff—as illustrated on the right of Figure 4 with GNMax. The larger ensembles lower the privacy cost of answering queries by tolerating larger $\\sigma$ ’s. Combining the two observations made in this Figure, for a fixed label accuracy, we lower privacy costs by switching to the GNMax aggregator and training a larger number $M$ of teachers. ",
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+ "text": "5.3 STUDENT TRAINING WITH THE GNMAX AGGREGATION MECHANISMS ",
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+ "text": "As outlined in Section 3, we train a student on public data labeled by the aggregation mechanisms. We take advantage of PATE’s flexibility and apply the technique that performs best on each dataset: semi-supervised learning with Generative Adversarial Networks (Salimans et al., 2016) for MNIST and SVHN, Virtual Adversarial Training (Miyato et al., 2017) for Glyph, and fully-supervised random forests for UCI Adult. In addition to evaluating the total privacy cost associated with training the student model, we compare its utility to a non-private baseline obtained by training on the sensitive data (used to train teachers in PATE): we use the baselines of $9 9 . 2 \\%$ , $9 2 . 8 \\%$ , and $8 \\bar { 5 } . 0 \\%$ reported by Papernot et al. (2017) respectively for MNIST, SVHN, and UCI Adult, and we measure a baseline of $8 2 . 2 \\%$ for Glyph. We compute $( \\varepsilon , \\delta )$ -privacy bounds and denote the privacy cost as the $\\varepsilon$ value at a value of $\\delta$ set accordingly to number of training samples. ",
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+ "text": "Confident-GNMax Aggregator. Given a pool of 500 to 12,000 samples to learn from (depending on the dataset), the student submits queries to the teacher ensemble running the Confident-GNMax aggregator from Section 4.2. A grid search over a range of plausible values for parameters $T$ , $\\sigma _ { 1 }$ and $\\sigma _ { 2 }$ yielded the values reported in Table 1, illustrating the tradeoff between utility and privacy achieved. We additionally measure the number of queries selected by the teachers to be answered and compare student utility to a non-private baseline. ",
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+ "text": "The Confident-GNMax aggregator outperforms LNMax for the four datasets considered in the original PATE proposal: it reduces the privacy cost $\\varepsilon$ , increases student accuracy, or both simultaneously. On the uncurated Glyph data, despite the imbalance of classes and mislabeled data (as evidenced by the $8 2 . 2 \\%$ baseline), the Confident Aggregator achieves $7 3 . 5 \\%$ accuracy with a privacy cost of just $\\varepsilon = 1 . 0 2$ . Roughly 1,300 out of 12,000 queries made are not answered, indicating that several expensive queries were successfully avoided. This selectivity is analyzed in more details in Section 5.4. ",
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+ "text": "Interactive-GNMax Aggregator. On Glyph, we evaluate the utility and privacy of an interactive training routine that proceeds in two rounds. Round one runs student training with a Confident ",
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+ "Figure 3: Some example inputs from the Glyph dataset along with the class they are labeled as. Note the ambiguity (between the comma and apostrophe) and the mislabeled input. "
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+ "Figure 4: Tradeoff between utility and privacy for the LNMax and GNMax aggregators on Glyph: effect of the noise distribution (left) and size of the teacher ensemble (right). The LNMax aggregator uses a Laplace distribution and GNMax a Gaussian. Smaller values of the privacy cost $\\varepsilon$ (often obtained by increasing the noise scale $\\sigma$ —see Section 4) and higher accuracy are better. "
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+ "Table 1: Utility and privacy of the students. The Confident- and Interactive-GNMax aggregators introduced in Section 4 offer better tradeoffs between privacy (characterized by the value of the bound $\\varepsilon$ ) and utility (the accuracy of the student compared to a non-private baseline) than the LNMax aggregator used by the original PATE proposal on all datasets we evaluated with. For MNIST, Adult, and SVHN, we use the labels of ensembles of 250 teachers published by Papernot et al. (2017) and set $\\delta = 1 0 ^ { - 5 }$ to compute values of $\\varepsilon$ (to the exception of SVHN where $\\dot { \\delta _ { \\mathrm { } } } = 1 \\dot { 0 } ^ { - 6 }$ ). All Glyph results use an ensemble of 5000 teachers and $\\varepsilon$ is computed for $\\delta = 1 0 ^ { - 8 }$ . "
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+ "table_body": "| Dataset | Aggregator | Queriesanswered | Privacybound ε | Accuracy Student Baseline |
| MNIST | LNMax (Papernot et al., 2017) | 100 | 2.04 | 98.0% | 99.2% |
| LNMax (Papernot et al., 2017) | 1,000 | 8.03 | 98.1% |
| Confident-GNMax (T=200,σ1=150,σ2=40) | 286 | 1.97 | 98.5% |
| SVHN | LNMax (Papernot et al., 2017) | 500 | 5.04 | 82.7% | 92.8% |
| LNMax (Papernot et al., 2017) | 1,000 | 8.19 | 90.7% |
| Confident-GNMax (T=300,σ1=200,σ2=40) | 3,098 | 4.96 | 91.6% |
| Adult | LNMax (Papernot et al., 2017) | 500 | 2.66 | 83.0% | 85.0% |
| Confident-GNMax (T=300,σ1=200,σ2=40) | 524 | 1.90 | 83.7% |
| Glyph | LNMax | 4,000 | 4.3 | 72.4% | 82.2% |
| Confident-GNMax (T=1000,σ1=500,σ2=100) | 10,762 | 2.03 | 75.5% |
| Interactive-GNMax, two rounds | 4,341 | 0.837 | 73.2% |
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+ "Figure 5: Effects of the noisy threshold checking: Left: The number of queries answered by LNMax, Confident-GNMax moderate $\\scriptstyle { T = 3 5 0 0 }$ , $\\sigma _ { 1 } { = } 1 5 0 0$ ), and Confident-GNMax aggressive ( ${ T } \\mathrm { { = } } 5 0 0 0$ , $\\sigma _ { 1 } { = } 1 5 0 0$ ). The black dots and the right axis (in log scale) show the expected cost of answering a single query in each bin (via GNMax, $\\sigma _ { 2 } { = } 1 0 0$ ). Right: Privacy cost of answering all (LNMax) vs only inexpensive queries (GNMax) for a given number of answered queries. The very dark area under the curve is the cost of selecting queries; the rest is the cost of answering them. "
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+ "text": "Aggregator. A grid search targeting the best privacy for roughly 3,400 answered queries (out of 6,000)—sufficient to bootstrap a student—led us to setting ( $\\scriptstyle { T = 3 5 0 0 }$ , $\\sigma _ { 1 } { = } 1 5 0 0$ , $\\sigma _ { 2 } { = } 1 0 0 $ ) and a privacy cost of $\\varepsilon \\approx 0 . 5 9$ . ",
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+ "text": "In round two, this student was then trained with 10,000 more queries made with the InteractiveGNMax Aggregator $\\scriptstyle { T = 3 5 0 0 }$ , $\\sigma _ { 1 } { = } 2 0 0 0$ , $\\sigma _ { 2 } { = } 2 0 0$ ). We computed the resulting (total) privacy cost and utility at an exemplar data point through another grid search of plausible parameter values. The result appears in the last row of Table 1. With just over 10,422 answered queries in total at a privacy cost of $\\varepsilon = 0 . 8 4$ , the trained student was able to achieve $7 3 . 2 \\%$ accuracy. Note that this students required fewer answered queries compared to the Confident Aggregator. The best overall cost of student training occurred when the privacy costs for the first and second rounds of training were roughly the same. (The total $\\varepsilon$ is less than $0 . 5 9 \\times 2 = 1 . 1 8$ due to better composition—via Theorems 4 and 5.) ",
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+ "text": "Comparison with Baseline. Note that the Glyph student’s accuracy remains seven percentage points below the non-private model’s accuracy achieved by training on the 65M sensitive inputs. We hypothesize that this is due to the uncurated nature of the data considered. Indeed, the class imbalance naturally requires more queries to return labels from the less represented classes. For instance, a model trained on 200K queries is only $7 7 \\%$ accurate on test data. In addition, the large fraction of mislabeled inputs are likely to have a large privacy cost: these inputs are sensitive because they are outliers of the distribution, which is reflected by the weak consensus among teachers on these inputs. ",
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+ "text": "5.4 NOISY THRESHOLD CHECKS AND PRIVACY COSTS ",
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+ "text": "Sections 4.1 and 4.2 motivated the need for a noisy threshold checking step before having the teachers answer queries: it prevents most of the privacy budget being consumed by few queries that are expensive and also likely to be incorrectly answered. In Figure 5, we compare the privacy cost $\\varepsilon$ of answering all queries to only answering confident queries for a fixed number of queries. ",
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+ "text": "We run additional experiments to support the evaluation from Section 5.3. With the votes of 5,000 teachers on the Glyph dataset, we plot in Figure 5 the histogram of the plurality vote counts $( n _ { i ^ { * } }$ in the notation of Section 4.1) across 25,000 student queries. We compare these values to the vote counts of queries that passed the noisy threshold check for two sets of parameters $T$ and $\\sigma _ { 1 }$ in Algorithm 1. Smaller values imply weaker teacher agreements and consequently more expensive queries. ",
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+ "text": "When $\\scriptstyle { T = 3 5 0 0 }$ , $\\sigma _ { 1 } { = } 1 5 0 0$ ) we capture a significant fraction of queries where teachers have a strong consensus (roughly $> 4 0 0 0$ votes) while managing to filter out many queries with poor consensus. This moderate check ensures that although many queries with plurality votes between 2,500 and 3,500 are answered (i.e., only $5 0 \\mathrm { - } 7 0 \\%$ of teachers agree on a label) the expensive ones are most likely discarded. For ( $\\scriptstyle { T = 5 0 0 0 }$ , $\\sigma _ { 1 } { = } 1 5 0 0$ ), queries with poor consensus are completely culled out. This selectivity comes at the expense of a noticeable drop for queries that might have had a strong consensus and little-to-no privacy cost. Thus, this aggressive check answer fewer queries with very strong privacy guarantees. We reiterate that this threshold checking step itself is done in a private manner. Empirically, in our Interactive Aggregator experiments, we expend about a third to a half of our privacy budget on this step, which still yields a very small cost per query across 6,000 queries. ",
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+ "text": "6 CONCLUSIONS ",
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+ "text": "The key insight motivating the addition of a noisy thresholding step to the two aggregation mechanisms proposed in our work is that there is a form of synergy between the privacy and accuracy of labels output by the aggregation: labels that come at a small privacy cost also happen to be more likely to be correct. As a consequence, we are able to provide more quality supervision to the student by choosing not to output labels when the consensus among teachers is too low to provide an aggregated prediction at a small cost in privacy. This observation was further confirmed in some of our experiments where we observed that if we trained the student on either private or non-private labels, the former almost always gave better performance than the latter—for a fixed number of labels. ",
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+ "text": "Complementary with these aggregation mechanisms is the use of a Gaussian (rather than Laplace) distribution to perturb teacher votes. In our experiments with Glyph data, these changes proved essential to preserve the accuracy of the aggregated labels—because of the large number of classes. The analysis presented in Section 4 details the delicate but necessary adaptation of analogous results for the Laplace NoisyMax. ",
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+ "text": "As was the case for the original PATE proposal, semi-supervised learning was instrumental to ensure the student achieves strong utility given a limited set of labels from the aggregation mechanism. However, we found that virtual adversarial training outperforms the approach from Salimans et al. (2016) in our experiments with Glyph data. These results establish lower bounds on the performance that a student can achieve when supervised with our aggregation mechanisms; future work may continue to investigate virtual adversarial training, semi-supervised generative adversarial networks and other techniques for learning the student in these particular settings with restricted supervision. ",
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+ "text": "ACKNOWLEDGMENTS ",
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+ "text": "We are grateful to Martín Abadi, Vincent Vanhoucke, and Daniel Levy for their useful inputs and discussions towards this paper. ",
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+ "text": "REFERENCES ",
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+ "type": "text",
+ "text": "A APPENDIX: PRIVACY ANALYSIS ",
+ "text_level": 1,
+ "bbox": [
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+ "page_idx": 16
+ },
+ {
+ "type": "text",
+ "text": "In this appendix, we provide the proofs of Theorem 6 and Proposition 7. Moreover, we present Proposition 10, which provides optimal values of $\\mu _ { 1 }$ and $\\mu _ { 2 }$ to apply towards Theorem 6 for the GNMax mechanism. We start off with a statement about the Rényi differential privacy guarantee of the GNMax. ",
+ "bbox": [
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+ "page_idx": 16
+ },
+ {
+ "type": "text",
+ "text": "Proposition 8. The GNMax aggregator $\\mathcal { M } _ { \\sigma }$ guarantees $\\left( \\lambda , \\lambda / \\sigma ^ { 2 } \\right)$ -RDP for all $\\lambda \\geq 1$ ",
+ "bbox": [
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+ "page_idx": 16
+ },
+ {
+ "type": "text",
+ "text": "Proof. The result follows from observing that $\\mathcal { M } _ { \\sigma }$ can be decomposed into applying the argmax operator to a noisy histogram resulted from adding Gaussian noise to each dimension of the original histogram. The Gaussian mechanism satisfies $( { \\bar { \\lambda } } , \\lambda / 2 \\sigma ^ { 2 } )$ -RDP (Mironov, 2017), and since each teacher may change two counts (incrementing one and decrementing the other), the overall RDP guarantee is as claimed. ■ ",
+ "bbox": [
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+ "page_idx": 16
+ },
+ {
+ "type": "text",
+ "text": "Proposition 7. For a GNMax aggregator $\\mathcal { M } _ { \\sigma }$ , the teachers’ votes histogram $\\bar { n } = ( n _ { 1 } , \\ldots , n _ { m } ) $ , and for any $i ^ { * } \\in [ m ]$ , we have ",
+ "bbox": [
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+ "page_idx": 16
+ },
+ {
+ "type": "text",
+ "text": "where ",
+ "bbox": [
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+ "page_idx": 16
+ },
+ {
+ "type": "equation",
+ "img_path": "images/1c3c919e9ada1269e499b7a19c3c10b68c843e51f2e814f986ff624779ec1f1b.jpg",
+ "text": "$$\n\\begin{array} { c l c r } { { \\displaystyle { \\bf P r } \\left[ \\mathcal { M } _ { \\sigma } ( D ) \\neq i ^ { * } \\right] \\le q ( \\bar { n } ) , } } \\\\ { { \\displaystyle } } \\\\ { { q ( \\bar { n } ) \\triangleq \\frac { 1 } { 2 } \\sum _ { i \\neq i ^ { * } } \\mathrm { e r f c } \\left( \\frac { n _ { i ^ { * } } - n _ { i } } { 2 \\sigma } \\right) . } } \\end{array}\n$$",
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+ "bbox": [
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+ "page_idx": 16
+ },
+ {
+ "type": "text",
+ "text": "Proof. Recall that $\\mathcal { M } _ { \\sigma } ( D ) = \\operatorname { a r g m a x } ( n _ { i } + Z _ { i } )$ , where $Z _ { i }$ are distributed as ${ \\mathcal { N } } ( 0 , \\sigma ^ { 2 } )$ . Then for any $i ^ { * } \\in [ m ]$ , we have ",
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+ "page_idx": 16
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+ {
+ "type": "equation",
+ "img_path": "images/ee155f4d624a5fba3b4a2ca84d4518af207c20765435b50cf2e74aeab36294e5.jpg",
+ "text": "$$\n\\begin{array} { r } { \\mathbf { P r } [ \\mathcal { M } _ { \\sigma } ( D ) \\neq i ^ { * } ] = \\mathbf { P r } \\left[ \\exists i , n _ { i } + Z _ { i } > n _ { i ^ { * } } + Z _ { i ^ { * } } \\right] \\leq \\displaystyle \\sum _ { i \\neq i ^ { * } } \\mathbf { P r } \\left[ n _ { i } + Z _ { i } > n _ { i ^ { * } } + Z _ { i ^ { * } } \\right] } \\\\ { = \\displaystyle \\sum _ { i \\neq i ^ { * } } \\mathbf { P r } \\left[ Z _ { i } - Z _ { i ^ { * } } > n _ { i ^ { * } } - n _ { i } \\right] } \\\\ { = \\displaystyle \\sum _ { i \\neq i ^ { * } } \\frac { 1 } { 2 } \\left( 1 - \\mathrm { e r f } \\left( \\frac { n _ { i ^ { * } } - n _ { i } } { 2 \\sigma } \\right) \\right) . } \\end{array}\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ "type": "text",
+ "text": "where the last equality follows from the fact that $Z _ { i } - Z _ { j }$ is a Gaussian random variable with mean zero and variance $2 \\sigma ^ { 2 }$ . ",
+ "bbox": [
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+ },
+ {
+ "type": "text",
+ "text": "We now present a precise statement of Theorem 6. ",
+ "bbox": [
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+ ],
+ "page_idx": 16
+ },
+ {
+ "type": "text",
+ "text": "Theorem 6. Let $\\mathcal { M }$ be a randomized algorithm with $( \\mu _ { 1 } , \\varepsilon _ { 1 } )$ -RDP and $\\left( \\mu _ { 2 } , \\varepsilon _ { 2 } \\right)$ -RDP guarantees and suppose that there exists a likely outcome $i ^ { * }$ given a dataset $D$ and a bound $\\tilde { q } \\leq 1$ such that $\\tilde { q } \\geq \\mathbf { P r } \\left[ \\mathcal { M } ( D ) \\neq i ^ { * } \\right]$ . Additionally suppose that $\\lambda \\le \\mu _ { 1 }$ and $\\begin{array} { r } { \\tilde { q } \\le e ^ { ( \\mu _ { 2 } - 1 ) \\varepsilon _ { 2 } } / \\biggl ( \\frac { \\mu _ { 1 } } { \\mu _ { 1 } - 1 } \\cdot \\frac { \\mu _ { 2 } } { \\mu _ { 2 } - 1 } \\biggr ) ^ { \\mu _ { 2 } } } \\end{array}$ Then, for any neighboring dataset $D ^ { \\prime }$ of $D$ , we have: ",
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+ },
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+ "type": "equation",
+ "img_path": "images/2cfe6826ca433929ddcaaa86ed5317161487a749198ab6784470924c289e6375.jpg",
+ "text": "$$\nD _ { \\lambda } ( \\mathcal { M } ( D ) \\| \\mathcal { M } ( D ^ { \\prime } ) ) \\le \\frac { 1 } { \\lambda - 1 } \\log \\left( ( 1 - \\tilde { q } ) \\cdot A ( \\tilde { q } , \\mu _ { 2 } , \\varepsilon _ { 2 } ) ^ { \\lambda - 1 } + \\tilde { q } \\cdot B ( \\tilde { q } , \\mu _ { 1 } , \\varepsilon _ { 1 } ) ^ { \\lambda - 1 } \\right)\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ "type": "equation",
+ "img_path": "images/991081fd038e139ccf105c9cd1c06e19fbca27adbe3faddd70696b59044ec1be.jpg",
+ "text": "$$\n\\begin{array} { r } { \\pmb { A } ( \\tilde { q } , \\mu _ { 2 } , \\varepsilon _ { 2 } ) \\triangleq ( 1 - \\tilde { q } ) / \\bigg ( 1 - \\big ( \\tilde { q } e ^ { \\varepsilon _ { 2 } } \\big ) ^ { \\frac { \\mu _ { 2 } - 1 } { \\mu _ { 2 } } } \\bigg ) a n d \\pmb { B } ( \\tilde { q } , \\mu _ { 1 } , \\varepsilon _ { 1 } ) \\triangleq e ^ { \\varepsilon _ { 1 } } / \\tilde { q } ^ { \\frac { 1 } { \\mu _ { 1 } - 1 } } . } \\end{array}\n$$",
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+ "page_idx": 16
+ },
+ {
+ "type": "text",
+ "text": "Proof. Before we proceed to the proof, we introduce some simplifying notation. For a randomized mechanism $\\mathcal { M }$ and neighboring datasets $D$ and $D ^ { \\prime }$ , we define ",
+ "bbox": [
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+ "type": "equation",
+ "img_path": "images/9010c1e03a91497835f0833ac793ed0943e13cdcb39a08e58087746837876c47.jpg",
+ "text": "$$\n\\begin{array} { l } { \\displaystyle \\beta _ { \\mathcal { M } } ( \\lambda ; D , D ^ { \\prime } ) \\triangleq D _ { \\lambda } ( \\mathcal { M } ( D ) \\| \\mathcal { M } ( D ^ { \\prime } ) ) } \\\\ { \\displaystyle = \\frac { 1 } { \\lambda - 1 } \\log \\mathbb { E } _ { x \\sim \\mathcal { M } ( D ) } \\left[ \\left( \\frac { \\mathbf { P r } \\left[ \\mathcal { M } ( D ) = x \\right] } { \\mathbf { P r } \\left[ \\mathcal { M } ( D ^ { \\prime } ) = x \\right] } \\right) ^ { \\lambda - 1 } \\right] . } \\end{array}\n$$",
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+ "type": "text",
+ "text": "As the proof involves working with the RDP bounds in the exponent, we set $\\zeta _ { 1 } \\triangleq e ^ { \\varepsilon _ { 1 } ( \\mu _ { 1 } - 1 ) }$ and ζ2 , eε2(µ2−1). ",
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+ "type": "text",
+ "text": "Finally, we define the following shortcuts: ",
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+ },
+ {
+ "type": "equation",
+ "img_path": "images/17b8657bad918393c4884aa39ff03bb82e08df30f2f9150755913075c13bbe1f.jpg",
+ "text": "$$\n\\begin{array} { l } { q _ { i } \\triangleq \\mathbf { P r } \\left[ \\mathcal { M } ( D ) = i \\right] \\mathrm { ~ a n d ~ } q \\triangleq \\displaystyle \\sum _ { i \\neq i ^ { * } } q _ { i } = \\mathbf { P r } \\left[ \\mathcal { M } ( D ) \\neq i ^ { * } \\right] , } \\\\ { p _ { i } \\triangleq \\mathbf { P r } \\left[ \\mathcal { M } ( D ^ { \\prime } ) = i \\right] \\mathrm { ~ a n d ~ } p \\triangleq \\displaystyle \\sum _ { i \\neq i ^ { * } } p _ { i } = \\mathbf { P r } \\left[ \\mathcal { M } ( D ^ { \\prime } ) \\neq i ^ { * } \\right] , } \\end{array}\n$$",
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+ "bbox": [
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+ "page_idx": 17
+ },
+ {
+ "type": "text",
+ "text": "and note that $q \\leq \\tilde { q }$ ",
+ "bbox": [
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+ "page_idx": 17
+ },
+ {
+ "type": "text",
+ "text": "From the definition of Rényi differential privacy, $( \\mu _ { 1 } , \\varepsilon _ { 1 } )$ -RDP implies: ",
+ "bbox": [
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+ },
+ {
+ "type": "equation",
+ "img_path": "images/52a7ba2a7619a5928d6cb42aa47aeb1280cd5c3d764b6a645e4043f9301878c8.jpg",
+ "text": "$$\n\\begin{array} { r l } & { \\exp \\left( \\beta _ { \\mathcal M } ( \\mu _ { 1 } ; D , D ^ { \\prime } ) \\right) = \\displaystyle \\left( \\frac { ( 1 - q ) ^ { \\mu _ { 1 } } } { ( 1 - p ) ^ { \\mu _ { 1 } - 1 } } + \\sum _ { i \\neq i * } \\frac { q _ { i } ^ { \\mu _ { 1 } } } { p _ { i } ^ { \\mu _ { 1 } - 1 } } \\right) ^ { 1 / ( \\mu _ { 1 } - 1 ) } \\leq \\exp ( \\varepsilon _ { 1 } ) } \\\\ & { \\qquad \\implies \\displaystyle \\sum _ { i > 1 } \\frac { q _ { i } ^ { \\mu _ { 1 } } } { p _ { i } ^ { \\mu _ { 1 } - 1 } } = \\sum _ { i > 1 } q _ { i } \\left( \\frac { q _ { i } } { p _ { i } } \\right) ^ { \\mu _ { 1 } - 1 } \\leq \\zeta _ { 1 } . } \\end{array}\n$$",
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+ "type": "text",
+ "text": "Since $\\mu _ { 1 } \\geq \\lambda , f ( x ) \\triangleq x ^ { \\frac { \\mu _ { 1 } - 1 } { \\lambda - 1 } }$ is convex. Applying Jensen’s Inequality we have the following: ",
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+ "type": "equation",
+ "img_path": "images/ee95560a496090abec0a3e8a07eaaa1337df337b19f38e1f05f32c9200831389.jpg",
+ "text": "$$\n\\begin{array} { r l } & { \\left( \\frac { \\sum _ { i \\neq j : i } \\cdot q _ { i } \\left( \\frac { q _ { i } } { p _ { i } } \\right) ^ { \\lambda - 1 } } { q } \\right) ^ { \\frac { n _ { 1 } - 1 } { \\lambda - 1 } } \\leq \\frac { \\sum _ { i \\neq i } \\cdot q _ { i } \\left( \\frac { q _ { i } } { p _ { i } } \\right) ^ { \\mu _ { 1 } - 1 } } { q } } \\\\ & { \\qquad \\implies \\sum _ { i \\neq i ^ { * } } q _ { i } \\left( \\frac { q _ { i } } { p _ { i } } \\right) ^ { \\lambda - 1 } \\leq q \\left( \\frac { \\sum _ { i \\neq i } \\cdot q _ { i } \\left( \\frac { q _ { i } } { p _ { i } } \\right) ^ { \\mu _ { 1 } - 1 } } { q } \\right) ^ { \\frac { \\lambda - 1 } { \\mu _ { 1 } - 1 } } } \\\\ & { \\qquad \\overset { ( 3 ) } { \\Longrightarrow } \\sum _ { i \\neq i ^ { * } } q _ { i } \\left( \\frac { q _ { i } } { p _ { i } } \\right) ^ { \\lambda - 1 } \\leq \\zeta _ { 1 } ^ { \\frac { \\lambda - 1 } { \\mu _ { 1 } - 1 } } \\cdot q ^ { 1 - \\frac { \\lambda - 1 } { \\mu _ { 1 } - 1 } } . } \\end{array}\n$$",
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+ "type": "text",
+ "text": "Next, by the bound at order $\\mu _ { 2 }$ , we have: ",
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+ "type": "equation",
+ "img_path": "images/13ed527b73a3e9abc833e22c70cc45121f7db00bc6226f257b3bb446face8902.jpg",
+ "text": "$$\n\\begin{array} { c } { { \\displaystyle \\exp \\left( \\beta _ { { \\mathcal M } } ( \\mu _ { 2 } ; D ^ { \\prime } , D ) \\right) = \\left( \\frac { ( 1 - p ) ^ { \\mu _ { 2 } } } { ( 1 - q ) ^ { \\mu _ { 2 } - 1 } } + \\sum _ { i \\neq i ^ { * } } \\frac { p _ { i } ^ { \\mu _ { 2 } } } { q _ { i } ^ { \\mu _ { 2 } - 1 } } \\right) ^ { 1 / ( \\mu _ { 2 } - 1 ) } \\leq \\exp ( \\varepsilon _ { 2 } ) } } \\\\ { { \\displaystyle \\implies \\frac { ( 1 - p ) ^ { \\mu _ { 2 } } } { ( 1 - q ) ^ { \\mu _ { 2 } - 1 } } + \\sum _ { i \\neq i ^ { * } } \\frac { p _ { i } ^ { \\mu _ { 2 } } } { q _ { i } ^ { \\mu _ { 2 } - 1 } } \\leq \\zeta _ { 2 } . } } \\end{array}\n$$",
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+ "bbox": [
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+ "type": "text",
+ "text": "By the data processing inequality of Rényi divergence, we have ",
+ "bbox": [
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+ },
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+ "type": "equation",
+ "img_path": "images/e6b2fc5dffabde612584d3f9feb2aac087a0b0cb4e7aca4fbdb090f6fa564fe2.jpg",
+ "text": "$$\n\\frac { ( 1 - p ) ^ { \\mu _ { 2 } } } { ( 1 - q ) ^ { \\mu _ { 2 } - 1 } } + \\frac { p ^ { \\mu _ { 2 } } } { q ^ { \\mu _ { 2 } - 1 } } \\leq \\zeta _ { 2 } ,\n$$",
+ "text_format": "latex",
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+ {
+ "type": "text",
+ "text": "which implies pµ2qµ2−1 ≤ ζ2 and thus ",
+ "bbox": [
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+ "page_idx": 17
+ },
+ {
+ "type": "equation",
+ "img_path": "images/3f562dbbf1ca27002624f91f3397e84af94df77e7cd4be56bfdf095460f53a1e.jpg",
+ "text": "$$\np \\leq \\left( q ^ { \\mu _ { 2 } - 1 } \\zeta _ { 2 } \\right) ^ { \\frac { 1 } { \\mu _ { 2 } } } .\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ "type": "text",
+ "text": "Combining (4) and (5), we can derive a bound at $\\lambda$ . ",
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+ {
+ "type": "equation",
+ "img_path": "images/5a661797c3e6870246223e3f8843aceb0b11ca4a20b196bf4170f12278028295.jpg",
+ "text": "$$\n\\begin{array} { r l } & { \\exp \\left( \\beta _ { \\mathcal M } ( \\lambda , D , D ^ { \\prime } ) \\right) = \\left( \\frac { ( 1 - q ) ^ { \\lambda } } { ( 1 - p ) ^ { \\lambda - 1 } } + \\displaystyle \\sum _ { i \\neq i ^ { * } } \\frac { q _ { i } ^ { \\lambda } } { p _ { i } ^ { \\lambda - 1 } } \\right) ^ { 1 / ( \\lambda - 1 ) } } \\\\ & { \\quad \\le \\left( \\frac { ( 1 - q ) ^ { \\lambda } } { \\left( 1 - ( q ^ { \\mu _ { 2 } - 1 } \\zeta _ { 2 } ) ^ { \\frac { 1 } { \\mu _ { 2 } } } \\right) ^ { \\lambda - 1 } } + \\zeta _ { 1 } ^ { \\frac { \\lambda - 1 } { \\mu _ { 1 } - 1 } } \\cdot q ^ { 1 - \\frac { \\lambda - 1 } { \\mu _ { 1 } - 1 } } \\right) ^ { 1 / ( \\lambda - 1 ) } . } \\end{array}\n$$",
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+ "type": "text",
+ "text": "Although Equation (6) is very close to the corresponding statement in the theorem’s claim, one subtlety remains. The bound (6) applies to the exact probability $q = \\mathbf { P r } \\left[ \\mathcal { M } ( D ) \\neq i ^ { * } \\right]$ . In the theorem statement, and in practice, we can only derive an upper bound $\\tilde { q }$ on $\\mathbf { P r } \\left[ \\mathcal { M } ( D ) \\neq i ^ { * } \\right]$ . The last step of the proof requires showing that the expression in Equation (6) is monotone in the range of values of $q$ that we care about. ",
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+ "type": "text",
+ "text": "Lemma 9 (Monotonicity of the bound). Let the functions $f _ { 1 } ( \\cdot )$ and $f _ { 2 } ( \\cdot )$ be ",
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+ "type": "equation",
+ "img_path": "images/6c12f64042b39c6c2111babda04809a2511a69055071bc9aff6539d2e92de17c.jpg",
+ "text": "$$\n\\begin{array} { l } { { f _ { 1 } ( x ) \\triangleq \\displaystyle \\frac { ( 1 - x ) ^ { \\lambda } } { \\Big ( 1 - ( x ^ { \\mu _ { 2 } - 1 } \\zeta _ { 2 } ) ^ { \\frac { 1 } { \\mu _ { 2 } } } \\Big ) ^ { \\lambda - 1 } } \\qquad \\ a n d \\qquad f _ { 2 } ( x ) \\triangleq \\zeta _ { 1 } ^ { \\frac { \\lambda - 1 } { \\mu _ { 1 } - 1 } } \\cdot x ^ { 1 - \\frac { \\lambda - 1 } { \\mu _ { 1 } - 1 } } , } } \\\\ { { \\mathrm { } } } \\\\ { { \\mathrm { } } } \\\\ { { \\mathrm { } } { \\mathrm { } } { } \\\\ { \\mathrm { } { } \\mathrm { } + f _ { 2 } ( x ) \\ i s \\ i n c r e a s i n g \\ i n \\Big [ 0 , \\operatorname* { m i n } \\Big ( 1 , \\zeta _ { 2 } \\Big / \\Big ( \\frac { \\mu _ { 1 } } { \\mu _ { 1 } - 1 } \\cdot \\frac { \\mu _ { 2 } } { \\mu _ { 2 } - 1 } \\Big ) ^ { \\mu _ { 2 } } \\Big ) \\Big ] . } } \\end{array}\n$$",
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+ "type": "text",
+ "text": "Proof. Taking the derivative of $f _ { 1 } ( x )$ , we have: ",
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+ "img_path": "images/35b15c7ab1a2c4747cdca5fd542001f35c54d98e924dca0579f61a5a6e374196.jpg",
+ "text": "$$\nf _ { 1 } ^ { \\prime } ( x ) = \\frac { - \\lambda ( 1 - x ) ^ { \\lambda - 1 } ( 1 - ( x ^ { \\mu _ { 2 } - 1 } \\zeta _ { 2 } ) ^ { \\frac { 1 } { \\mu _ { 2 } } } ) ^ { \\lambda - 1 } } { ( 1 - ( x ^ { \\mu _ { 2 } - 1 } \\zeta _ { 2 } ) ^ { \\frac { 1 } { \\mu _ { 2 } } } ) ^ { 2 \\lambda - 2 } } \\\\ & { \\qquad + \\frac { ( 1 - x ) ^ { \\lambda } ( \\lambda - 1 ) ( 1 - ( x ^ { \\mu _ { 2 } - 1 } \\zeta _ { 2 } ) ^ { \\frac { 1 } { \\mu _ { 2 } } } ) ^ { \\lambda - 2 } \\zeta _ { 2 } ^ { \\frac { 1 } { \\mu _ { 2 } } } \\cdot \\frac { \\mu _ { 2 } - 1 } { \\mu _ { 2 } } \\cdot x ^ { - \\frac { 1 } { \\mu _ { 2 } } } } { ( 1 - ( x ^ { \\mu _ { 2 } - 1 } \\zeta _ { 2 } ) ^ { \\frac { 1 } { \\mu _ { 2 } } } ) ^ { 2 \\lambda - 2 } } } \\\\ & = \\frac { ( 1 - x ) ^ { \\lambda - 1 } } { ( 1 - ( x ^ { \\mu _ { 2 } - 1 } \\zeta _ { 2 } ) ^ { \\frac { 1 } { \\mu _ { 2 } } } ) ^ { \\lambda - 1 } } \\left( - \\lambda + ( \\lambda - 1 ) \\left( 1 - \\frac { 1 } { \\mu _ { 2 } } \\right) \\frac { 1 - x } { 1 - ( x ^ { \\mu _ { 2 } - 1 } \\zeta _ { 2 } ) ^ { \\frac { 1 } { \\mu _ { 2 } } } } \\left( \\frac { \\zeta _ { 2 } } { x } \\right) ^ { \\frac { 1 } { \\mu _ { 2 } } } \\right) .\n$$",
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+ "type": "text",
+ "text": "We intend to show that: ",
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+ "type": "equation",
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+ "text": "$$\nf _ { 1 } ^ { \\prime } ( x ) \\geq - \\lambda + ( \\lambda - 1 ) \\left( 1 - \\frac { 1 } { \\mu _ { 2 } } \\right) \\left( \\frac { \\zeta _ { 2 } } { x } \\right) ^ { \\frac { 1 } { \\mu _ { 2 } } } .\n$$",
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+ "type": "text",
+ "text": "For $\\begin{array} { r } { x \\in \\left[ 0 , \\zeta _ { 2 } / \\left( \\frac { \\mu _ { 1 } } { \\mu _ { 1 } - 1 } \\cdot \\frac { \\mu _ { 2 } } { \\mu _ { 2 } - 1 } \\right) ^ { \\mu _ { 2 } } \\right] } \\end{array}$ and $y \\in [ 1 , \\infty )$ , define $g ( x , y )$ as: ",
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+ "img_path": "images/2c7b796ca481daff12481695463e1d671fbf4c557174ac45a6fc23f691a8ad93.jpg",
+ "text": "$$\ng ( x , y ) \\triangleq - \\lambda \\cdot y ^ { \\lambda - 1 } + ( \\lambda - 1 ) \\left( 1 - { \\frac { 1 } { \\mu _ { 2 } } } \\right) \\left( { \\frac { \\zeta _ { 2 } } { x } } \\right) ^ { \\frac { 1 } { \\mu _ { 2 } } } y ^ { \\lambda } .\n$$",
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+ "type": "text",
+ "text": "We claim that $g ( x , y )$ is increasing in $y$ and therefore $g ( x , y ) \\geq g ( x , 1 )$ , and prove it by showing the partial derivative of $g ( x , y )$ with respect to $y$ is non-negative. Take a derivative with respect to $y$ as: ",
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+ "img_path": "images/38cdd5bb8155351f31061a3e2560a277a40179a17fad0039ec5ebe6035253653.jpg",
+ "text": "$$\n\\begin{array} { l } { { g _ { y } ^ { \\prime } ( x , y ) = - \\lambda ( \\lambda - 1 ) y ^ { \\lambda - 2 } + \\lambda ( \\lambda - 1 ) \\left( 1 - { \\frac { 1 } { \\mu _ { 2 } } } \\right) \\left( { \\frac { \\zeta _ { 2 } } { x } } \\right) ^ { \\frac { 1 } { \\mu _ { 2 } } } y ^ { \\lambda - 1 } } } \\\\ { { \\ = \\lambda ( \\lambda - 1 ) y ^ { \\lambda - 2 } \\left( - 1 + \\left( 1 - { \\frac { 1 } { \\mu _ { 2 } } } \\right) \\left( { \\frac { \\zeta _ { 2 } } { x } } \\right) ^ { \\frac { 1 } { \\mu _ { 2 } } } y \\right) . } } \\end{array}\n$$",
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+ "type": "text",
+ "text": "To see why $g _ { y } ^ { \\prime } ( x , y )$ is non-negative in the respective ranges of $x$ and $y$ , note that: ",
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+ "img_path": "images/d4bebd870642ce5ae5e53e436cfcf9373b4aff326f24e4b9cf297f1a2e793b07.jpg",
+ "text": "$$\n\\begin{array} { r l } { x \\leq \\zeta _ { 2 } \\Big / \\Big ( \\frac { \\mu _ { 1 } } { \\mu _ { 1 } - 1 } \\cdot \\frac { \\mu _ { 2 } } { \\mu _ { 2 } - 1 } \\Big ) ^ { \\mu _ { 2 } } \\implies x \\leq \\zeta _ { 2 } \\Big / \\Big ( \\frac { \\mu _ { 2 } } { \\mu _ { 2 } - 1 } \\Big ) ^ { \\mu _ { 2 } } } \\\\ & { \\implies 1 \\leq \\frac { \\zeta _ { 2 } } { x } \\cdot \\Big ( \\frac { \\mu _ { 2 } - 1 } { \\mu _ { 2 } } \\Big ) ^ { \\mu _ { 2 } } } \\\\ & { \\implies 1 \\leq \\frac { \\mu _ { 2 } - 1 } { \\mu _ { 2 } } \\Big ( \\frac { \\zeta _ { 2 } } { x } \\Big ) ^ { \\frac { 1 } { \\mu _ { 2 } } } } \\\\ & { \\implies 1 \\leq \\frac { \\mu _ { 2 } - 1 } { \\mu _ { 2 } } \\Big ( \\frac { \\zeta _ { 2 } } { x } \\Big ) ^ { \\frac { 1 } { \\mu _ { 2 } } } g } \\\\ & { \\implies 0 \\leq - 1 + \\frac { \\mu _ { 2 } - 1 } { \\mu _ { 2 } } \\Big ( \\frac { \\zeta _ { 2 } } { x } \\Big ) ^ { \\frac { 1 } { \\mu _ { 2 } } } g } \\\\ & { \\implies 0 \\leq - 1 + \\frac { \\mu _ { 2 } - 1 } { \\mu _ { 2 } } \\Big ( \\frac { \\zeta _ { 2 } } { x } \\Big ) ^ { \\frac { 1 } { \\mu _ { 2 } } } g } \\\\ & { \\implies 0 \\leq g _ { g } ^ { \\prime } ( x , y ) . } \\end{array}\n$$",
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+ "type": "text",
+ "text": "Consider $\\frac { 1 - x } { 1 - ( x ^ { \\mu _ { 2 } - 1 } \\zeta _ { 2 } ) ^ { 1 / \\mu _ { 2 } } }$ . Since $\\zeta _ { 2 } \\geq 1$ and $x \\leq 1$ , we have $x \\leq \\zeta _ { 2 }$ and hence ",
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+ "img_path": "images/e724b1d3dd791934b55c0a088cf927ac10524a6c57a269b37c82ec3763a694c2.jpg",
+ "text": "$$\n{ \\frac { 1 - x } { 1 - ( x ^ { \\mu _ { 2 } - 1 } \\zeta _ { 2 } ) ^ { \\frac { 1 } { \\mu _ { 2 } } } } } \\geq { \\frac { 1 - x } { 1 - ( x ^ { \\mu _ { 2 } - 1 } x ) ^ { \\frac { 1 } { \\mu _ { 2 } } } } } = 1 .\n$$",
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+ "type": "text",
+ "text": "Therefore we can set $\\begin{array} { r } { y = \\frac { 1 - x } { 1 - ( x ^ { \\mu _ { 2 } - 1 } \\zeta _ { 2 } ) ^ { 1 / \\mu _ { 2 } } } } \\end{array}$ and apply the fact that $g ( x , y ) \\geq g ( x , 1 )$ for all $y \\geq 1$ to get ",
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+ "type": "equation",
+ "img_path": "images/1beabd9acdabf6088d3fbe178aa8aebe94c6683d554c410e4e8ace6a00d13fbf.jpg",
+ "text": "$$\nf _ { 1 } ^ { \\prime } ( x ) \\geq - \\lambda + ( \\lambda - 1 ) \\left( 1 - \\frac { 1 } { \\mu _ { 2 } } \\right) \\left( \\frac { \\zeta _ { 2 } } { x } \\right) ^ { \\frac { 1 } { \\mu _ { 2 } } } ,\n$$",
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+ "type": "text",
+ "text": "as required by (7). ",
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+ },
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+ "type": "text",
+ "text": "Taking the derivative of $f _ { 2 } ( x )$ , we have: ",
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+ "type": "equation",
+ "img_path": "images/f94fde8eb40eb496e74481f048a47191098e0e671b1757a079026f2cc4b295c0.jpg",
+ "text": "$$\nf _ { 2 } ^ { \\prime } ( x ) = \\zeta _ { 1 } ^ { \\frac { \\lambda - 1 } { \\mu _ { 1 } - 1 } } \\cdot \\left( 1 - { \\frac { \\lambda - 1 } { \\mu _ { 1 } - 1 } } \\right) x ^ { - { \\frac { \\lambda - 1 } { \\mu _ { 1 } - 1 } } } = \\left( { \\frac { \\zeta _ { 1 } } { x } } \\right) ^ { { \\frac { \\lambda - 1 } { \\mu _ { 1 } - 1 } } } \\left( 1 - { \\frac { \\lambda - 1 } { \\mu _ { 1 } - 1 } } \\right) \\geq 1 - { \\frac { \\lambda - 1 } { \\mu _ { 1 } - 1 } } .\n$$",
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+ {
+ "type": "text",
+ "text": "Combining the two terms together, we have: ",
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+ "type": "equation",
+ "img_path": "images/8887c72367b365abe5de736be80daaf9b57ebe1a496c6fb4f009565260ebbfa2.jpg",
+ "text": "$$\n\\begin{array} { c } { { f ^ { \\prime } ( x ) \\geq - \\lambda + ( \\lambda - 1 ) \\left( 1 - { \\frac { 1 } { \\mu _ { 2 } } } \\right) \\left( { \\frac { \\zeta _ { 2 } } { x } } \\right) ^ { \\frac { 1 } { \\mu _ { 2 } } } + 1 - { \\frac { \\lambda - 1 } { \\mu _ { 1 } - 1 } } } } \\\\ { { = ( \\lambda - 1 ) \\left( - { \\frac { \\mu _ { 1 } } { \\mu _ { 1 } - 1 } } + { \\frac { \\mu _ { 2 } - 1 } { \\mu _ { 2 } } } \\left( { \\frac { \\zeta _ { 2 } } { x } } \\right) ^ { \\frac { 1 } { \\mu _ { 2 } } } \\right) . } } \\end{array}\n$$",
+ "text_format": "latex",
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+ },
+ {
+ "type": "text",
+ "text": "For $f ^ { \\prime } ( x )$ to be non-negative we need: ",
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+ },
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+ "type": "equation",
+ "img_path": "images/83b10b31b9cc7c088535979d7ffe5dbd44307a5a0c3b19ed6c5a8e091752b2b4.jpg",
+ "text": "$$\n\\begin{array} { r l } & { - \\frac { \\mu _ { 1 } } { \\mu _ { 1 } - 1 } + \\frac { \\mu _ { 2 } - 1 } { \\mu _ { 2 } } \\left( \\frac { \\zeta _ { 2 } } { x } \\right) ^ { \\frac { 1 } { \\mu _ { 2 } } } \\geq 0 } \\\\ & { \\iff \\left( \\frac { \\mu _ { 1 } } { \\mu _ { 1 } - 1 } \\cdot \\frac { \\mu _ { 2 } } { \\mu _ { 2 } - 1 } \\right) ^ { \\mu _ { 2 } } \\leq \\frac { \\zeta _ { 2 } } { x } . } \\end{array}\n$$",
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+ "type": "text",
+ "text": "So $f ( x )$ is increasing for $\\begin{array} { r c l } { x } & { \\in } & { \\left[ 0 , \\zeta _ { 2 } / \\left( { \\frac { \\mu _ { 1 } } { \\mu _ { 1 } - 1 } } \\cdot { \\frac { \\mu _ { 2 } } { \\mu _ { 2 } - 1 } } \\right) ^ { \\mu _ { 2 } } \\right] } \\end{array}$ . This means for $q \\leq \\tilde { q } \\leq$ $\\zeta _ { 2 } / { \\left( \\frac { \\mu _ { 1 } } { \\mu _ { 1 } - 1 } \\cdot \\frac { \\mu _ { 2 } } { \\mu _ { 2 } - 1 } \\right) } ^ { \\mu _ { 2 } }$ , we have $f ( q ) \\leq f ( \\tilde { q } )$ . This completes the proof of the lemma and that of the theorem. ■ ",
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+ {
+ "type": "text",
+ "text": "Theorem 6 yields data-dependent Rényi differential privacy bounds for any value of $\\mu _ { 1 }$ and $\\mu _ { 2 }$ larger than $\\lambda$ . The following proposition simplifies this search by calculating optimal higher moments $\\mu _ { 1 }$ and $\\mu _ { 2 }$ for the GNMax mechanism with variance $\\sigma ^ { 2 }$ . ",
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+ "text": "Proposition 10. When applying Theorem 6 and Proposition 8 for GNMax with Gaussian of variance $\\sigma ^ { 2 }$ , the right-hand side of (2) is minimized at ",
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+ "text": "$$\n\\mu _ { 2 } = \\sigma \\cdot \\sqrt { \\log ( 1 / \\tilde { q } ) } , a n d \\mu _ { 1 } = \\mu _ { 2 } + 1 .\n$$",
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+ "text": "Proof. We can minimize both terms in (2) independently. To minimize the first term in (6), we minimize $\\left( \\tilde { q } e ^ { \\varepsilon _ { 2 } } \\right) ^ { 1 - 1 / \\mu _ { 2 } }$ by considering logarithms: ",
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+ "text": "$$\n\\begin{array} { r l } { \\log \\left\\{ \\big ( \\tilde { q } e ^ { \\varepsilon _ { 2 } } \\big ) ^ { 1 - 1 / \\mu _ { 2 } } \\right\\} = \\log \\Bigg \\{ \\tilde { q } ^ { 1 - \\frac { 1 } { \\mu _ { 2 } } } \\exp \\left( \\frac { \\mu _ { 2 } - 1 } { \\sigma ^ { 2 } } \\right) \\Bigg \\} } & { } \\\\ { \\quad \\quad \\quad \\quad \\quad = \\left( 1 - \\frac { 1 } { \\mu _ { 2 } } \\right) \\cdot \\log \\tilde { q } + \\frac { \\mu _ { 2 } - 1 } { \\sigma ^ { 2 } } } & { } \\\\ { \\quad \\quad \\quad \\quad = \\frac { 1 } { \\mu _ { 2 } } \\log \\frac { 1 } { \\tilde { q } } + \\frac { \\mu _ { 2 } } { \\sigma ^ { 2 } } - \\frac { 1 } { \\sigma ^ { 2 } } - \\log \\frac { 1 } { \\tilde { q } } , } \\end{array}\n$$",
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+ "type": "text",
+ "text": "which is minimized at $\\mu _ { 2 } = \\sigma \\cdot \\sqrt { \\log ( 1 / \\tilde { q } ) }$ . ",
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+ "type": "text",
+ "text": "To minimize the second term in (6), we minimize $e ^ { \\varepsilon _ { 1 } } / \\tilde { q } ^ { 1 / ( \\mu _ { 1 } - 1 ) }$ as follows: ",
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+ "text": "$$\n\\begin{array} { r l } { \\log \\left\\{ \\displaystyle \\frac { e ^ { \\varepsilon _ { 1 } } } { \\tilde { q } ^ { 1 / ( \\mu _ { 1 } - 1 ) } } \\right\\} = \\log \\left\\{ \\tilde { q } ^ { - 1 / ( \\mu _ { 1 } - 1 ) } \\exp \\left( \\frac { \\mu _ { 1 } } { \\sigma ^ { 2 } } \\right) \\right\\} } & { { } } \\\\ { = \\displaystyle \\frac { \\mu _ { 1 } } { \\sigma ^ { 2 } } + \\frac { 1 } { \\mu _ { 1 } - 1 } \\log \\frac { 1 } { \\tilde { q } } } & { { } } \\\\ { = \\displaystyle \\frac { 1 } { \\sigma ^ { 2 } } + \\frac { \\mu _ { 1 } - 1 } { \\sigma ^ { 2 } } + \\frac { 1 } { \\mu _ { 1 } - 1 } \\log \\frac { 1 } { \\tilde { q } } , } \\end{array}\n$$",
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+ "type": "text",
+ "text": "which is minimized at $\\mu _ { 1 } = 1 + \\sigma \\cdot \\sqrt { \\log ( 1 / \\tilde { q } ) }$ completing the proof. ",
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+ "type": "text",
+ "text": "Putting this together, we apply the following steps to calculate RDP of order $\\lambda$ for GNMax with variance $\\sigma ^ { 2 }$ on a given dataset $D$ . First, we compute a bound $q$ according to Proposition 7. Then we use the smaller of two bounds: a data-dependent (Theorem 6) and a data-independent one (Proposition 8) : ",
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+ "text": "$$\n\\beta _ { \\sigma } ( q ) \\triangleq \\operatorname* { m i n } \\left\\{ \\frac { 1 } { \\lambda - 1 } \\log \\left\\{ ( 1 - q ) \\cdot A ( q , \\mu _ { 2 } , \\varepsilon _ { 2 } ) ^ { \\lambda - 1 } + q \\cdot B ( q , \\mu _ { 1 } , \\varepsilon _ { 1 } ) ^ { \\lambda - 1 } \\right\\} , \\lambda / \\sigma ^ { 2 } \\right\\} ,\n$$",
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+ "type": "text",
+ "text": "where $\\pmb { A }$ and $\\textbf { { B } }$ are defined as in the statement of Theorem 6, the parameters $\\mu _ { 1 }$ and $\\mu _ { 2 }$ are selected according to Proposition 10, and $\\varepsilon _ { 1 } \\triangleq \\mu _ { 1 } / \\sigma ^ { 2 }$ and $\\varepsilon _ { 2 } \\triangleq \\mu _ { 2 } / \\sigma ^ { 2 }$ (Proposition 8). Importantly, the first expression is evaluated only when $q < 1 , \\mu _ { 1 } \\geq \\lambda , \\mu _ { 2 } > 1$ , and $q \\leq e ^ { ( \\mu _ { 2 } - 1 ) \\varepsilon _ { 2 } } / \\left( \\frac { \\mu _ { 1 } } { \\mu _ { 1 } - 1 } \\cdot \\frac { \\mu _ { 2 } } { \\mu _ { 2 } - 1 } \\right) ^ { \\mu _ { 2 } }$ . These conditions can either be checked for each application of the aggregation mechanism, or a critical value of $q _ { 0 }$ that separates the range of applicability of the data-dependent and data-independent bounds can be computed for given $\\sigma$ and $\\lambda$ . In our implementation we pursue the second approach. ",
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+ "type": "text",
+ "text": "The following corollary offers a simple asymptotic expression of the privacy of GNMax for the case when there are large (relative to $\\sigma$ ) gaps between the highest three vote counts. ",
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+ "text": "Corollary 11. If the top three vote counts are $n _ { 1 } ~ > ~ n _ { 2 } ~ > ~ n _ { 3 }$ and $n _ { 1 } \\mathrm { ~ - ~ } n _ { 2 }$ , $n _ { 2 } \\mathrm { ~ - ~ } n _ { 3 } \\mathrm { ~ } \\gg \\sigma$ , then the mechanism GNMax with Gaussian of variance $\\sigma ^ { 2 }$ satisfies $( \\lambda , \\exp ( - 2 \\lambda / \\sigma ^ { 2 } ) / \\lambda )$ -RDP for $\\lambda = ( n _ { 1 } - n _ { 2 } ) / 4$ . ",
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+ "text": "Proof. Denote the noisy counts as $\\tilde { n } _ { i } = n _ { i } + \\mathcal { N } ( 0 , \\sigma ^ { 2 } )$ . Ignoring outputs other than those with the highest and the second highest counts, we bound $q = \\mathbf { \\bar { P r } } \\left[ \\mathcal { \\bar { M } } ( D ) \\right] \\neq 1 ]$ as $\\mathbf { P r } [ \\tilde { n } _ { 1 } ~ < ~ \\tilde { n } _ { 2 } ] ~ =$ ${ \\bf P r } [ N ( 0 , 2 \\sigma ^ { 2 } ) > n _ { 1 } - n _ { 2 } ] < \\exp { \\left( - ( n _ { 1 } - n _ { 2 } ) ^ { 2 } / 4 \\sigma ^ { 2 } \\right) }$ , which we use as $\\tilde { q }$ . Plugging $\\tilde { q }$ in Proposition 10, we have $\\mu _ { 1 } - 1 = \\mu _ { 2 } = ( n _ { 1 } - n _ { 2 } ) / 2$ , limiting the range of applicability of Theorem 6 to $\\lambda < ( n _ { 1 } - n _ { 2 } ) / 2$ . ",
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+ "type": "text",
+ "text": "Choosing $\\lambda = ( n _ { 1 } - n _ { 2 } ) / 4$ ensures $A ( \\tilde { q } , \\mu _ { 2 } , \\varepsilon _ { 2 } ) \\approx 1$ , which allows approximating the bound (2) as $\\tilde { q } \\cdot B ( \\tilde { q } , \\mu _ { 1 } , \\varepsilon _ { 1 } ) ^ { \\lambda - 1 } / ( \\lambda - 1 )$ . The proof follows by straightforward calculation. ",
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+ "type": "text",
+ "text": "B SMOOTH SENSITIVITY AND PUBLISHING THE PRIVACY PARAMETER ",
+ "text_level": 1,
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+ "type": "text",
+ "text": "The privacy guarantees obtained for the mechanisms in this paper via Theorem 6 take as input $\\tilde { q }$ , an upper bound on the probability that the aggregate mechanism returns the true plurality. This means that the resulting privacy parameters computed depend on teacher votes and hence the underlying data. To avoid potential privacy breaches from simply publishing the data-dependent parameter, we need to publish a sanitized version of the privacy loss. This is done by adding noise to the computed privacy loss estimates using the smooth sensitivity algorithm proposed by Nissim et al. (2007). ",
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+ "text": "This section has the following structure. First we recall the notion of smooth sensitivity and introduce an algorithm for computing the smooth sensitivity of the privacy loss function of the GNMax mechanism. In the rest of the section we prove correctness of these algorithms by stating several conditions on the mechanism, proving that these conditions are sufficient for correctness of the algorithm, and finally demonstrating that GNMax satisfies these conditions. ",
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+ "type": "text",
+ "text": "B.1 COMPUTING SMOOTH SENSITIVITY ",
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+ "text": "Any dataset $D$ defines a histogram $\\bar { n } = ( n _ { 1 } , \\ldots , n _ { m } ) \\in \\mathbb { N } ^ { m }$ of the teachers’ votes. We have a natural notion of the distance between two histograms $\\operatorname { d i s t } ( \\bar { n } , \\bar { n } ^ { \\prime } )$ and a function $q \\colon \\mathbb { N } ^ { m } [ 0 , 1 ]$ on these histograms computing the bound according to Proposition 7. The value $q ( { \\bar { n } } )$ can be used as $\\tilde { q }$ in the application of Theorem 6. Additionally we have $\\boldsymbol { n } ^ { ( i ) }$ denote the $i$ -th highest bar in the histogram. ",
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+ "type": "text",
+ "text": "We aim at calculating a smooth sensitivity of $\\beta \\left( q ( \\bar { n } ) \\right)$ whose definition we recall now. ",
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+ "text": "Definition 12 (Smooth Sensitivity). Given the smoothness parameter $\\beta$ , a $\\beta$ -smooth sensitivity of $f ( n )$ is defined as ",
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+ "img_path": "images/0e0f2fa2554b39340a31602d93847c48ad1d028bf20f0694b831ba5bf92dc15f.jpg",
+ "text": "$$\n\\mathrm { S S } _ { \\beta } ( \\bar { n } ) \\triangleq \\operatorname* { m a x } _ { d \\geq 0 } e ^ { - \\beta d } \\cdot \\operatorname* { m a x } _ { \\bar { n } ^ { \\prime } : \\mathrm { d i s t } ( \\bar { n } , \\bar { n } ^ { \\prime } ) \\leq d } \\tilde { \\mathrm { L S } } ( \\bar { n } ^ { \\prime } ) ,\n$$",
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+ "type": "text",
+ "text": "where ",
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+ "text": "$$\n\\tilde { \\mathrm { L S } } ( \\bar { n } ) \\geq \\operatorname* { m a x } _ { \\bar { n } ^ { \\prime } : \\mathrm { d i s t } ( \\bar { n } , \\bar { n } ^ { \\prime } ) = 1 } | f ( n ) - f ( n ^ { \\prime } ) |\n$$",
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+ "text": "is an upper bound on the local sensitivity. ",
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+ "type": "text",
+ "text": "We now describe Algorithms 3–5 computing a smooth sensitivity of $\\beta \\left( q ( \\cdot ) \\right)$ . The algorithms assume the existence of efficiently computable functions $q \\colon \\mathbb { N } ^ { m } [ 0 , 1 ]$ , $\\mathrm { B _ { L } } , \\mathrm { B _ { U } } \\colon [ 0 , 1 ] \\to [ 0 , 1 ]$ , and a constant $q _ { 0 }$ . ",
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+ "type": "text",
+ "text": "Informally, the functions $\\mathrm { B _ { U } }$ and $\\mathrm { B _ { L } }$ respectively upper and lower bound the value of $q$ evaluated at any neighbor of $\\bar { n }$ given $q ( { \\bar { n } } )$ , and $[ 0 , q _ { 0 } )$ limits the range of applicability of data-dependent analysis. ",
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+ "type": "text",
+ "text": "The functions $\\mathrm { B _ { L } }$ and $\\mathrm { B _ { U } }$ are defined as follows. Their derivation appears in Section B.4. ",
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+ "img_path": "images/4d98ba6b423631d450213ea11cc140d6b38b9f2a647cfb7427d6ff9b4724b3d0.jpg",
+ "text": "$$\n\\begin{array} { l } { { \\displaystyle { \\mathrm { B } } _ { \\mathrm { U } } ( q ) \\triangleq \\operatorname* { m i n } \\left\\{ \\frac { m - 1 } { 2 } \\mathrm { e r f c } \\left( \\mathrm { e r f c } ^ { - 1 } \\left( \\frac { 2 q } { m - 1 } \\right) - \\frac { 1 } { \\sigma } \\right) , 1 \\right\\} , } } \\\\ { { \\displaystyle { \\mathrm { B } } _ { \\mathrm { L } } ( q ) \\triangleq \\frac { m - 1 } { 2 } \\mathrm { e r f c } \\left( \\mathrm { e r f c } ^ { - 1 } \\left( \\frac { 2 q } { m - 1 } \\right) + \\frac { 1 } { \\sigma } \\right) , } } \\end{array}\n$$",
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+ "type": "text",
+ "text": "Algorithm 3 – Local Sensitivity: use the functions $\\mathrm { B _ { U } }$ and $\\mathrm { B _ { L } }$ to compute (an upper bound) of the local sensitivity at a given $q$ value by looking at the difference of $\\beta \\left( \\cdot \\right)$ evaluated on the bounds. ",
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+ {
+ "type": "table",
+ "img_path": "images/6a31d795155be45c1cb5e956266a989f2c78f2afa0fa20bb3ff6c48c0ba271f5.jpg",
+ "table_caption": [],
+ "table_footnote": [],
+ "table_body": "| case analysis that finds the neighbor at distance d with the maximum sensitivity. |
| 1: procedure ATDISTANCED(n, d) |
| 2: q←q(n) | > q is in the flat region. |
| 3: 4: | if q1 ≤q≤ go then |
| return LS(q), STOP | |
| 5: end if | |
| 6: | if q<qithen Need to increase q. |
| 7: | if n(1)-n(2）< 2d then n(i) is the ith largest element. |
| 8: | return LS(qi), STOP |
| 9: | else |
| 10: | n'← SORT(n)+[-d,d,O,...,0] |
| 11: | q'←q(n') |
| 12: | if q' >qi then |
| 13: | return LS(qo), STOP |
| 14: | else |
| 15: | return LS(q'),CONTINUE |
| 16: | end if |
| 17: | end if |
| 18: | else Need to decrease q. |
| 19: | |
| 20: | n↑[n,0,...,0] |
| 21: | q'←q(n'） |
| 22: | return LS(q'), STOP |
| 23: | else |
| 24: | n' ← SORT(n)+[d,0,...,0] |
| 25: | for d' =1,...,d do |
| 26: | n'(2) ←n(2)-1 The index of n'(2) may change. |
| 27: | end for |
| 28: | q'←q(n'） |
| 29: | if q'<qo then |
| 30: | return LS(qo), STOP |
| 31: | else |
| 32: | return LS(q'), CONTINUE |
| 33: | end if |
| |
| 34: | end if |
| 35: end if 36: end procedure |
",
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+ "type": "text",
+ "text": "Algorithm 5 – Smooth Sensitivity: Compute the $\\beta$ smooth sensitivity of $\\beta \\left( \\cdot \\right)$ via Definition 12 by looking at sensitivities at various distances and returning the maximum weighted by $e ^ { - \\beta d }$ . ",
+ "text_level": 1,
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+ "type": "text",
+ "text": "1: procedure SMOOTHSENSITIVITY $( \\bar { n } , \\beta )$ \n2: S ← 0 \n3: d ← 0 \n4: repeat \n5: $c$ , StoppingCondition ATDISTANCED $( \\bar { n } , d )$ \n6: S ← max{S, c · e−βd} \n7: d ← d + 1 \n8: until StoppingCondition $=$ STOP \n9: end procedure ",
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+ {
+ "type": "text",
+ "text": "• Define the distance between two histograms $\\bar { n } = ( n _ { 1 } , \\dots , n _ { m } )$ and $\\hat { n } ^ { \\prime } = ( n _ { 1 } ^ { \\prime } , \\dots , n _ { m } ^ { \\prime } )$ as ",
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+ {
+ "type": "equation",
+ "img_path": "images/d62dd9e9d87ca604e4a74552a88afbd4a6db6bcb558c27bd67e92c99f0b4692c.jpg",
+ "text": "$$\nd ( \\bar { n } , \\bar { n } ^ { \\prime } ) \\triangleq \\operatorname* { m a x } \\left\\{ \\sum _ { \\substack { i : n _ { i } > n _ { i } ^ { \\prime } } } \\lceil n _ { i } - n _ { i } ^ { \\prime } \\rceil , \\quad \\sum _ { \\substack { i : n _ { i } < n _ { i } ^ { \\prime } } } \\lceil n _ { i } ^ { \\prime } - n _ { i } \\rceil \\right\\} ,\n$$",
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+ "type": "text",
+ "text": "which is equal to the smallest number of “moves” needed to make the two histograms identical. We use the ceiling function since a single step can increase/decrease one bar by at most 1. ",
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+ "type": "text",
+ "text": "We say that two histograms are neighbors if their distance $d$ is 1. ",
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+ },
+ {
+ "type": "text",
+ "text": "Notice that analyses of Rényi differential privacy for LNMax, GNMax and the exponential mechanism are still applicable when the neighboring datasets are defined in this manner. ",
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+ },
+ {
+ "type": "text",
+ "text": "• Given a randomized aggregator $\\mathcal { M } : \\mathbb { R } _ { \\ge 0 } ^ { m } [ m ]$ , let $q \\colon \\mathbb { R } _ { \\geq 0 } ^ { m } [ 0 , 1 ]$ be so that ",
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+ "type": "equation",
+ "img_path": "images/8471e0e213718b3fe46ab230dfbfff17f68134b7f1e57f67aa3edc8cca914986.jpg",
+ "text": "$$\nq ( \\bar { n } ) \\geq \\mathbf { P r } [ \\mathcal { M } ( \\bar { n } ) \\neq \\operatorname { a r g m a x } ( \\bar { n } ) ] .\n$$",
+ "text_format": "latex",
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+ "type": "text",
+ "text": "When the context is clear, we use $q$ to denote a specific value of the function, which, in particular, can be used as $\\tilde { q }$ in applications of Theorem 6. ",
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+ "type": "text",
+ "text": "• Let $\\beta \\colon [ 0 , 1 ] \\mathbb { R }$ be the function that maps a $q$ value to the value of the Rényi accountant. ",
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+ "type": "text",
+ "text": "Conditions. Throughout this section we will be referring to the list of conditions on $q ( \\cdot )$ and $\\beta \\left( \\cdot \\right)$ ",
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+ "text": "C1. The function $q ( \\cdot )$ is continuous in each argument $n _ { i }$ . \nC2. There exist functions $\\mathrm { B _ { U } , B _ { L } \\colon [ 0 , 1 ] \\to [ 0 , 1 ] }$ such that for any neighbor $\\bar { n } ^ { \\prime }$ of $\\bar { n }$ , we have $\\mathrm { B } _ { \\mathrm { L } } ( q ( \\bar { n } ) ) \\leq q ( \\bar { n } ^ { \\prime } ) \\leq \\mathrm { B } _ { \\mathrm { U } } ( q ( \\bar { n } ) )$ , i.e., $\\mathrm { B _ { U } }$ and $\\mathrm { B _ { L } }$ provide upper and lower bounds on the $q$ value of any neighbor of $\\bar { n }$ . \nC3. $\\operatorname { B _ { L } } ( q )$ is increasing in $q$ . \nC4. $\\mathrm { B _ { U } }$ and $\\mathrm { B _ { L } }$ are functional inverses of each other in part of the range, i.e., $q = \\mathrm { B _ { L } } ( \\mathrm { B _ { U } } ( q ) )$ for all $q \\in [ 0 , q _ { 0 } ]$ , where $q _ { 0 }$ is defined below. Additionally $\\mathrm { B } _ { \\mathrm { L } } ( \\bar { q } ) \\leq q \\bar { \\leq } \\mathrm { B } _ { \\mathrm { U } } ( q )$ for all $q \\in [ 0 , 1 ]$ . \nC5. $\\beta \\left( \\cdot \\right)$ has the following shape: there exist constants $\\beta ^ { * }$ and $q _ { 0 } \\leq 0 . 5$ , such that $\\beta \\left( q \\right)$ nondecreasing in $[ 0 , q _ { 0 } ]$ and $\\bar { \\beta ( q ) } = \\beta ^ { * } \\ge \\beta ( q _ { 0 } )$ for $q > q _ { 0 }$ . The constant $\\beta ^ { * }$ corresponds to a data-independent bound. \nC6. $\\Delta \\beta ( q ) \\triangleq \\beta ( \\mathrm { B } _ { \\mathrm { U } } ( q ) ) - \\beta ( q )$ is non-decreasing in $[ 0 , \\mathrm { B _ { L } } ( q _ { 0 } ) ]$ , i.e., when $\\mathrm { B _ { U } } ( q ) \\le q _ { 0 }$ . \nC7. Recall that $\\boldsymbol { n } ^ { ( i ) }$ is the $i$ -th largest coordinate of a histogram $\\bar { n }$ . Then, if $q ( \\bar { n } ) \\leq \\mathrm { B _ { U } } ( q _ { 0 } )$ , then $q ( { \\bar { n } } )$ is differentiable in all coordinates and ",
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+ "type": "equation",
+ "img_path": "images/0e5fbcd8bdd27e5d00a653ed39197c383ea484e313945c5681977cf3be7862fe.jpg",
+ "text": "$$\n\\forall i > j \\ge 2 \\quad \\frac { \\partial q } { \\partial n ^ { ( j ) } } ( \\bar { n } ) \\ge \\frac { \\partial q } { \\partial n ^ { ( i ) } } ( \\bar { n } ) \\ge 0 .\n$$",
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+ "type": "text",
+ "text": "C8. The function $q ( { \\bar { n } } )$ is invariant under addition of a constant, i.e., and $q ( { \\bar { n } } )$ is invariant under permutation of $\\bar { n }$ , i.e., ",
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+ "type": "text",
+ "text": "Finally, we require that if $n ^ { ( 1 ) } = n ^ { ( 2 ) }$ , then $q ( { \\bar { n } } ) \\geq q _ { 0 }$ ",
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+ "type": "text",
+ "text": "We may additionally assume that $q _ { 0 } \\ \\ge \\ q ( [ n , 0 , \\ldots , 0 ] )$ . Indeed, if this condition is not satisfied, then the data-dependent analysis is not going to be used anywhere. The most extreme histogram— $[ n , 0 , \\ldots , 0 ]$ —is the most advantageous setting for applying data-dependent bounds. If we cannot use the data-dependent bound even in that case, we would be using the data-independent bound everywhere and do not need to compute smooth sensitivity anyway. Yet this condition is not automatically satisfied. For example, if $m$ (the number of classes) is large compared to $n$ (the number of teachers), we might have large $q ( [ n , 0 , \\ldots , 0 ] )$ . So we need to check this condition in the code before doing smooth sensitivity calculation. ",
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+ "type": "text",
+ "text": "B.3 CORRECTNESS OF ALGORITHMS 3–5 ",
+ "text_level": 1,
+ "bbox": [
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+ "type": "text",
+ "text": "Recall that local sensitivity of a deterministic function $f$ is defined as max $f ( D ) - f ( D ^ { \\prime } )$ , where $D$ and $D ^ { \\prime }$ are neighbors. ",
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+ "text": "Proposition 13. Under conditions C2–C6, Algorithm $^ 3$ computes an upper bound on local sensitivity of $\\beta ( q ( \\bar { n } ) )$ . ",
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+ "text": "Proof. Since $\\beta \\left( \\cdot \\right)$ is non-decreasing everywhere (by C5), and for any neighbors $\\bar { n }$ and $\\bar { n } ^ { \\prime }$ it holds that $\\mathrm { B } _ { \\mathrm { L } } ( q ( \\bar { n } ) ) \\leq q ( \\bar { n } ^ { \\prime } ) \\leq \\mathrm { B } _ { \\mathrm { U } } ( q ( \\bar { n } ) )$ (by C2), we have the following ",
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+ "img_path": "images/5e37804c04f0b61b766c1c18b59af0847d6859b2e52026105130c8ec00cc52bc.jpg",
+ "text": "$$\n\\begin{array} { r l } & { | \\beta \\left( q ( \\bar { n } ) \\right) - \\beta \\left( q ( \\bar { n } ^ { \\prime } ) \\right) | \\leq \\operatorname* { m a x } \\bigg \\{ \\beta \\Big ( \\mathrm { B } _ { \\mathrm { U } } { \\left( q ( \\bar { n } ) \\right) } \\Big ) - \\beta \\Big ( q ( \\bar { n } ) \\Big ) , \\beta \\Big ( q ( \\bar { n } ) \\Big ) - \\beta \\Big ( \\mathrm { B } _ { \\mathrm { L } } { \\left( q ( \\bar { n } ) \\right) } \\Big ) \\bigg \\} } \\\\ & { \\quad \\quad \\quad \\quad = \\operatorname* { m a x } \\bigg \\{ \\Delta \\beta \\Big ( q ( \\bar { n } ) \\Big ) , \\Delta \\beta \\Big ( \\mathrm { B } _ { \\mathrm { L } } { \\left( q ( \\bar { n } ) \\right) } \\Big ) \\bigg \\} } \\end{array}\n$$",
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+ {
+ "type": "text",
+ "text": "as an upper bound on the local sensitivity of $\\beta \\left( q ( \\cdot ) \\right)$ at input $\\bar { n }$ . ",
+ "bbox": [
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+ "type": "text",
+ "text": "The function computed by Algorithm 3 differs from above when $q ( \\bar { n } ) \\in ( \\mathrm { B _ { L } } ( q _ { 0 } ) , q _ { 0 } )$ . To complete the proof we need to argue that the local sensitivity is upper bounded by $\\Delta \\beta \\left( \\mathrm { B _ { L } } ( q _ { 0 } ) \\right)$ for $q ( { \\bar { n } } )$ in this interval. The bound follows from the following three observations. ",
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+ "type": "text",
+ "text": "First, $\\Delta \\beta \\left( q \\right)$ is non-increasing in the range $( \\mathrm { B _ { L } } ( q _ { 0 } ) , 1 ]$ , since $\\beta \\left( \\mathrm { B } _ { \\mathrm { U } } ( q ) \\right)$ is constant (by $\\operatorname { B _ { U } } ( q ) \\geq$ $\\mathrm { B _ { U } } ( \\mathrm { B _ { L } } ( q _ { 0 } ) ) = q _ { 0 }$ and C5) and $\\beta \\left( q \\right)$ is non-decreasing in the range (by C5). In particular, ",
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+ "img_path": "images/e768b40261e07a278ed78af1d8eaacd26eb8efa83b039462e27d4aa6dbd3e341.jpg",
+ "text": "$$\n\\Delta \\beta ( q ) \\leq \\Delta \\beta ( \\mathrm { B } _ { \\mathrm { L } } ( q _ { 0 } ) ) { \\mathrm { ~ i f ~ } } q \\geq \\mathrm { B } _ { \\mathrm { L } } ( q _ { 0 } ) .\n$$",
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+ "type": "text",
+ "text": "Second, $\\Delta \\beta \\left( \\mathrm { B } _ { \\mathrm { L } } ( q ) \\right)$ is non-decreasing in the range $[ 0 , q _ { 0 } ]$ since $\\operatorname { B _ { L } } ( q )$ is increasing (by C3 and C6). This implies that ",
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+ "img_path": "images/87c019bbfca5244fdfab97ca5325e8b310a2a30fdd80c9a5c4621b63ac321022.jpg",
+ "text": "$$\n\\Delta \\beta ( \\mathrm { B _ { L } } ( q ) ) \\leq \\Delta \\beta ( \\mathrm { B _ { L } } ( q _ { 0 } ) ) { \\mathrm { ~ i f ~ } } q \\leq q _ { 0 } .\n$$",
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+ "type": "text",
+ "text": "By (8) and (9) applied to the intersection of the two ranges, it holds that ",
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+ "img_path": "images/a7c2a46609dee524b7163b3e6a3ecf2386ca376b44363b84544140aa39622b24.jpg",
+ "text": "$$\n\\begin{array} { r } { \\operatorname* { m a x } \\bigg \\{ \\Delta \\beta \\Big ( q ( \\bar { n } ) \\Big ) , ~ \\Delta \\beta \\Big ( \\mathrm { B } _ { \\mathrm { L } } ( q ( \\bar { n } ) ) \\Big ) \\bigg \\} \\leq \\Delta \\beta \\left( \\mathrm { B } _ { \\mathrm { L } } ( q _ { 0 } ) \\right) \\mathrm { i f } ~ \\mathrm { B } _ { \\mathrm { L } } ( q _ { 0 } ) \\leq q \\leq q _ { 0 } , } \\end{array}\n$$",
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+ {
+ "type": "text",
+ "text": "as needed. ",
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+ "type": "text",
+ "text": "We thus established that the function computed by Algorithm 3, which we call $\\tilde { \\mathrm { L S } } ( q )$ from now on, is an upper bound on the local sensitivity. Formally, ",
+ "bbox": [
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+ {
+ "type": "equation",
+ "img_path": "images/bf84673a8a9cc41fcf5b2ef17c4a9a8a0439b4835fdcdf431c81993a82464af2.jpg",
+ "text": "$$\n\\begin{array} { r } { \\tilde { \\mathrm { L S } } ( q ) \\triangleq \\left\\{ \\begin{array} { l l } { \\Delta \\beta \\left( \\mathrm { B } _ { \\mathrm { L } } ( q _ { 0 } ) \\right) } & { \\mathrm { i f ~ } q \\in ( \\mathrm { B } _ { \\mathrm { L } } ( q _ { 0 } ) , q _ { 0 } ) , } \\\\ { \\operatorname* { m a x } \\left\\{ \\Delta \\beta \\left( q \\right) , \\Delta \\beta \\left( \\mathrm { B } _ { \\mathrm { L } } ( q ) \\right) \\right\\} } & { \\mathrm { o t h e r w i s e } . } \\end{array} \\right. } \\end{array}\n$$",
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+ "type": "text",
+ "text": "The following proposition characterizes the growth of $\\tilde { \\mathrm { L S } } ( q )$ . ",
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+ "text": "Proposition 14. Assuming conditions $C 2 { - } C 6 ,$ , the function $\\tilde { \\mathrm { L S } } ( q )$ is non-decreasing in $[ 0 , \\mathrm { B _ { L } } ( q _ { 0 } ) ]$ , constant in $[ \\mathrm { B _ { L } } ( q _ { 0 } ) , q _ { 0 } ]$ , and non-increasing in $[ q _ { 0 } , 1 ]$ . ",
+ "bbox": [
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+ "type": "text",
+ "text": "Proof. Consider separately three intervals. ",
+ "bbox": [
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+ "type": "text",
+ "text": "• By construction, $\\tilde { \\mathrm { L S } }$ is constant in $[ \\mathrm { B _ { L } } ( q _ { 0 } ) , q _ { 0 } ]$ . \n• Since both functions $\\Delta \\beta \\left( \\cdot \\right)$ and $\\Delta \\beta \\left( \\mathrm { B _ { L } } ( \\cdot ) \\right)$ are each non-decreasing in $[ 0 , \\mathrm { B _ { L } } ( q _ { 0 } ) )$ , so is their max. \n• In the interval $( q _ { 0 } , 1 ] , \\beta ( q )$ is constant. Hence $\\Delta \\beta ( q ) = 0$ and $\\Delta \\beta ( \\mathrm { B } _ { \\mathrm { L } } ( q ) ) = \\beta ( q ) -$ $\\beta \\left( \\mathrm { B _ { L } } ( q ) \\right)$ is non-decreasing. Their maximum value $\\Delta \\beta \\left( \\mathrm { B _ { L } } ( q ) \\right)$ is non-decreasing. ",
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+ "type": "text",
+ "text": "The claim follows. ",
+ "bbox": [
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+ "type": "text",
+ "text": "We next prove correctness of Algorithm 4, which computes the maximal sensitivity of $\\beta$ at a fixed distance. ",
+ "bbox": [
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+ "type": "text",
+ "text": "The proof relies on the following notion of a partial order between histograms. ",
+ "bbox": [
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+ "type": "text",
+ "text": "Definition 15. Prefix sums $S _ { i } ( \\bar { n } )$ are defined as follows: ",
+ "bbox": [
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+ {
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+ "img_path": "images/61e36132164f56cbb5eac510ee6c00562799aab463400659a78d5be1f75c552e.jpg",
+ "text": "$$\nS _ { i } ( \\bar { n } ) \\triangleq \\sum _ { j = 1 } ^ { i } ( n ^ { ( 1 ) } - n ^ { ( j ) } ) .\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ "type": "text",
+ "text": "We say that a histogram $\\bar { n }$ dominates $\\bar { n } ^ { \\prime }$ , denoted as $\\bar { n } \\succeq \\bar { n } ^ { \\prime }$ , iff: ",
+ "bbox": [
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+ {
+ "type": "text",
+ "text": "The function $q ( \\cdot )$ is monotone under this notion of dominance (assuming certain conditions hold): Proposition 16. If $q ( \\cdot )$ satisfies C1, C2, C7, and $C 8 _ { \\mathrm { : } }$ , and $q ( \\bar { n } ) < \\mathrm { B _ { U } } ( q _ { 0 } )$ , then ",
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+ "page_idx": 25
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+ {
+ "type": "equation",
+ "img_path": "images/932364722b16b32734ed5f03e55e82f13f3610d021ef610e08875c1043083e30.jpg",
+ "text": "$$\n\\bar { n } \\succeq \\bar { n } ^ { \\prime } \\Rightarrow q ( \\bar { n } ) \\leq q ( \\bar { n } ^ { \\prime } ) .\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ "type": "text",
+ "text": "Proof. We may assume that $n ^ { ( 1 ) } = n ^ { \\prime ( 1 ) }$ . Indeed, if this does not hold, add $| n ^ { ( 1 ) } - n ^ { \\prime ( 1 ) } |$ to all coordinates of the histogram with the smaller of the two values. This transform does not change the $q$ value (by C8) and it preserves the $\\succeq$ relationship as all prefix sums $S _ { i } ( \\cdot )$ remain unchanged. ",
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+ "type": "text",
+ "text": "We make a simple observation that will be helpful later: ",
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+ "img_path": "images/87a290a91360b0afa6a80fe03af03f377df02790898ac4640ed1989f63ae2d4d.jpg",
+ "text": "$$\n\\forall i \\in [ m ] \\mathrm { i t } \\mathrm { h o l d s ~ t h a t } \\sum _ { j = 1 } ^ { i } ( n ^ { ( 1 ) } - n _ { j } ) \\geq S _ { i } ( \\bar { n } ) .\n$$",
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+ "type": "text",
+ "text": "The inequality holds because the prefix sum accumulates the gaps between the largest value of $\\bar { n }$ and all other values in the non-decreasing order. Any deviation from this order may only increase the prefix sums. ",
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+ "type": "text",
+ "text": "The following lemma constructs a monotone chain (in the partial order of dominance) of histograms connecting $\\bar { n }$ and $\\bar { n } ^ { \\prime }$ via a sequence of intermediate steps that either do not change the value of $q$ or touch at most two coordinates at a time. ",
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+ "type": "text",
+ "text": "Lemma 17. There exists a chain ${ \\bar { n } } = { \\bar { n } } _ { 0 } \\succeq { \\bar { n } } _ { 1 } \\succeq \\cdot \\cdot \\cdot \\succeq { \\bar { n } } _ { d } = { \\bar { n } } ^ { \\prime }$ , such that for all $i \\in [ d ]$ either $d ( \\bar { n } _ { i - 1 } , \\bar { n } _ { i } ) = 1$ or $\\bar { n } _ { i - 1 } = \\pi ( \\bar { n } _ { i } )$ for some permutation $\\pi$ on $[ m ]$ . Additionally, = · · · = n(1)d . ",
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+ "text": "Proof (Lemma). Wlog we assume that $\\bar { n }$ and $\\bar { n } ^ { \\prime }$ are each sorted in the descending order. The proof is by induction on $\\begin{array} { r } { \\ell ( \\bar { n } , \\bar { n } ^ { \\prime } ) \\stackrel { \\Delta } { = } \\sum _ { i } \\lceil n _ { i } - n _ { i } ^ { \\prime } \\rceil \\leq 2 d ( \\bar { n } , \\bar { n } ^ { \\prime } ) } \\end{array}$ , which, by construction, only assumes non-negative integer values. ",
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+ "type": "text",
+ "text": "If the distance is 0, the statement is immediate. Otherwise, find the smallest $i$ so that $S _ { i } ( \\bar { n } ) > S _ { i } ( \\bar { n } ^ { \\prime } )$ (if all prefix sums are equal and $n ^ { ( 1 ) } = n ^ { \\prime ( 1 ) }$ , it would imply that $\\bar { n } = \\bar { n } ^ { \\prime }$ ). In particular, it means that $n _ { j } = n _ { j } ^ { \\prime }$ for $j < i$ and $n _ { i } < n _ { i } ^ { \\prime } \\le n _ { i - 1 } = n _ { i - 1 } ^ { \\prime }$ . Let $x \\triangleq \\operatorname* { m i n } ( n _ { i } ^ { \\prime } - n _ { i } , 1 )$ . Define $\\bar { n } ^ { \\prime \\prime }$ as identical to $\\bar { n } ^ { \\prime }$ except that $n _ { i } ^ { \\prime \\prime } = n _ { i } ^ { \\prime } - x$ . The new value is guaranteed to be non-negative, since $x \\leq n _ { i } ^ { \\prime } - n _ { i }$ and $n _ { i } \\geq 0$ . Note that $\\bar { n } ^ { \\prime \\prime }$ is not necessarily sorted anymore. Consider two possibilities. ",
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+ "type": "text",
+ "text": "Case I: $\\bar { n } \\succeq \\bar { n } ^ { \\prime \\prime }$ . Since $\\bar { n } ^ { \\prime \\prime } \\succeq \\bar { n } ^ { \\prime }$ , $\\ell ( \\bar { n } , \\bar { n } ^ { \\prime \\prime } ) < \\ell ( \\bar { n } , \\bar { n } ^ { \\prime } )$ , and $d ( \\bar { n } ^ { \\prime \\prime } , \\bar { n } ^ { \\prime } ) = 1$ , we may apply the induction hypothesis to the pair $\\bar { n } , \\bar { n } ^ { \\prime \\prime }$ . ",
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+ "text": "Case II: $\\bar { n } \\not \\subset \\bar { n } ^ { \\prime \\prime }$ . This may happen because the prefix sums of $\\bar { n } ^ { \\prime \\prime }$ increase compared to $S _ { j } ( \\bar { n } ^ { \\prime } )$ for $j \\geq i$ . Find the smallest such $i ^ { \\prime }$ so that $\\textstyle \\sum _ { j = 1 } ^ { i ^ { \\prime } } ( n _ { 1 } ^ { \\prime \\prime } - n _ { j } ^ { \\prime \\prime } ) > S _ { i ^ { \\prime } } ( \\bar { n } )$ . (Since $\\bar { n } ^ { \\prime \\prime }$ is not sorted, we fix the order in which prefix sums are accumulated to be the same as in $\\bar { n }$ ; by (10) $i ^ { \\prime }$ is well defined). Next we let $\\bar { n } ^ { \\prime \\prime \\prime }$ be identical to $\\bar { n } ^ { \\prime \\prime }$ except that $n _ { i ^ { \\prime } } ^ { \\prime \\prime \\prime } = n _ { i ^ { \\prime } } ^ { \\prime \\prime } + x$ . In other words, $\\bar { n } ^ { \\prime \\prime \\prime }$ differs from $\\bar { n } ^ { \\prime }$ by shifting $x$ from coordinate $i$ to coordinate $i ^ { \\prime }$ . ",
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+ "text": "We argue that incrementing $n _ { i ^ { \\prime } } ^ { \\prime \\prime }$ by $x$ does not change the maximal value of $\\bar { n } ^ { \\prime \\prime }$ , i.e., $n _ { 1 } ^ { \\prime \\prime \\prime } > n _ { i \\prime } ^ { \\prime \\prime \\prime }$ . Our choice of $i ^ { \\prime }$ , which is the smallest index so that the prefix sum over $\\bar { n } ^ { \\prime \\prime }$ overtakes that over $\\bar { n }$ , implies that $n _ { 1 } ^ { \\prime \\prime } - n _ { i ^ { \\prime } } ^ { \\prime \\prime } > n _ { 1 } - n _ { i ^ { \\prime } }$ . Since $n _ { 1 } ^ { \\prime \\prime } = n _ { 1 }$ , it means that $n _ { i ^ { \\prime } } > n _ { i ^ { \\prime } } ^ { \\prime \\prime }$ (and by adding $x$ we move $n _ { i ^ { \\prime } } ^ { \\prime \\prime }$ towards $n _ { i ^ { \\prime } }$ ). Furthermore, ",
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+ "img_path": "images/de324607f5fe5bb5f2dc8482a81c533c6fbe3878cdc34a473edd73e26e83e3e3.jpg",
+ "text": "$$\nn _ { i ^ { \\prime } } ^ { \\prime \\prime \\prime } = n _ { i ^ { \\prime } } ^ { \\prime \\prime } + x \\leq n _ { i ^ { \\prime } } + ( n _ { i } ^ { \\prime } - n _ { i } ) = n _ { i } ^ { \\prime } + ( n _ { i ^ { \\prime } } - n _ { i } ) \\leq n _ { i } ^ { \\prime } \\leq n _ { 1 } ^ { \\prime } = n _ { 1 } ^ { \\prime \\prime \\prime } ,\n$$",
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+ "type": "text",
+ "text": "(We use $n _ { i ^ { \\prime } } \\leq n _ { i }$ , which is implied by $i ^ { \\prime } > i$ .) ",
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+ "type": "text",
+ "text": "We claim that $\\begin{array} { r } { \\sum _ { j = 1 } ^ { t } ( n _ { 1 } ^ { \\prime \\prime \\prime } - n _ { j } ^ { \\prime \\prime \\prime } ) \\le S _ { t } ( \\bar { n } ) } \\end{array}$ for all $t$ , and thus, via (10), $\\bar { n } \\succeq \\bar { n } ^ { \\prime \\prime \\prime }$ . The choice of $i ^ { \\prime }$ makes the statement trivial for $t < i ^ { \\prime }$ . For $t \\geq i ^ { \\prime }$ the following holds: ",
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+ "text": "$$\n\\sum _ { j = 1 } ^ { t } ( n _ { 1 } ^ { \\prime \\prime \\prime } - n _ { j } ^ { \\prime \\prime \\prime } ) = \\sum _ { j = 1 } ^ { t } ( n _ { 1 } ^ { \\prime \\prime } - n _ { j } ^ { \\prime \\prime } ) - x \\leq \\left( \\sum _ { j = 1 } ^ { t } ( n _ { 1 } ^ { \\prime } - n _ { j } ^ { \\prime } ) + x \\right) - x = S _ { t } ( n ^ { \\prime } ) \\leq S _ { t } ( \\bar { n } ) .\n$$",
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+ "type": "text",
+ "text": "By construction $d ( \\bar { n } ^ { \\prime } , \\bar { n } ^ { \\prime \\prime \\prime } ) = 1$ (the two histograms differ in two locations, in positive and negative directions, by $x \\leq 1$ in each). For the same reasons $\\bar { n } ^ { \\prime \\prime \\prime } \\succeq \\bar { n } ^ { \\prime }$ . To show that $\\bar { \\ell } ( \\bar { n } , \\bar { n } ^ { \\prime } ) > \\ell ( \\bar { n } , \\bar { n } ^ { \\prime \\prime \\prime } )$ , ",
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+ "type": "text",
+ "text": "compare $\\lceil n _ { j } - n _ { j } ^ { \\prime } \\rceil$ and $\\lceil n _ { j } - n _ { j } ^ { \\prime \\prime \\prime } \\rceil$ for $j = i , i ^ { \\prime }$ . At $j = i$ the first term is strictly larger than the second. At $j = i ^ { \\prime }$ , the inequality holds too but it may be not strict. ",
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+ "type": "text",
+ "text": "We may again apply the induction hypothesis to the pair $\\bar { n }$ and $\\bar { n } ^ { \\prime \\prime \\prime }$ , thus completing the proof of the lemma. ",
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+ "type": "text",
+ "text": "To complete the proof of the proposition, we need to argue that the values of $q$ are also monotone in the chain constructed by the previous lemma. Concretely, we put forth ",
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+ "type": "text",
+ "text": "Proof. The fact that $d ( \\bar { n } , \\bar { n } ^ { \\prime } ) = 1$ and $\\bar { n } \\succeq \\bar { n } ^ { \\prime }$ means that there is either a single index $i$ so that $n _ { i } ^ { \\prime } < n _ { i }$ , or there exist two indices $i$ and $j$ so that $n _ { i } ^ { \\prime } < n _ { i }$ and $n _ { j } ^ { \\prime } > n _ { j }$ . The first case is immediate, since $q$ is non-decreasing in all inputs except for the largest (by C7). ",
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+ "type": "text",
+ "text": "Let $n _ { i } ^ { \\prime } = n _ { i } - x$ and $n _ { j } ^ { \\prime } = n _ { j } + y$ , where $x , y > 0$ . Since $\\bar { n } \\succeq \\bar { n } ^ { \\prime }$ , it follows that $n _ { i } \\geq n _ { j }$ and $x > y$ Consider two cases. ",
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+ "type": "text",
+ "text": "Case I: $n _ { i } ^ { \\prime } \\geq n _ { j } ^ { \\prime }$ , i.e., removing $x$ from $n _ { i }$ and adding $y$ to $n _ { j }$ does not change their ordering. Let ",
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+ "img_path": "images/6494bd35497a509b8f7d73f77113409c8cd2b52a95a50f290b487553202cec9b.jpg",
+ "text": "$$\n\\bar { n } ( t ) \\triangleq ( 1 - t ) \\bar { n } + t \\cdot \\bar { n } ^ { \\prime } = [ n _ { 1 } , \\ldots , n _ { i } - t \\cdot x , \\ldots , n _ { j } + t \\cdot y , \\ldots , n _ { m } ] .\n$$",
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+ "type": "text",
+ "text": "Then, ",
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+ "img_path": "images/c3a7f188b7530edd23bdad41a77caecb8a059c7e5594abe8daa0b950f7b39059.jpg",
+ "text": "$$\n\\begin{array} { l l } { \\displaystyle q ( \\bar { n } ^ { \\prime } ) - q ( \\bar { n } ) = q ( \\bar { n } ( 1 ) ) - q ( \\bar { n } ( 0 ) ) = \\int _ { t = 0 } ^ { 1 } ( q \\circ \\bar { n } ) ^ { \\prime } ( t ) \\mathrm { d } t } \\\\ { \\displaystyle \\qquad = \\int _ { t = 0 } ^ { 1 } \\left\\{ - x \\frac { \\partial q } { \\partial n _ { i } } \\bar { n } ( t ) + y \\frac { \\partial q } { \\partial n _ { j } } \\bar { n } ( t ) \\right\\} \\mathrm { d } t } \\\\ { \\displaystyle \\qquad \\leq 0 . } \\end{array}\n$$",
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+ "type": "text",
+ "text": "The last inequality follows from C7 and the facts that $x > y > 0$ and $n _ { i } ( t ) > n _ { j } ( t )$ . (The condition that $q ( \\bar { n } ( t ) ) \\overset { \\cdot } { \\leq } \\dot { \\mathrm { B } _ { \\mathrm { U } } } ( q _ { 0 } )$ follows from C2 and the fact that $d ( \\bar { n } ^ { \\prime } , \\bar { n } ( t ) ) \\leq 1 .$ ) ",
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+ "type": "text",
+ "text": "Case II: $n _ { i } ^ { \\prime } \\leq n _ { j } ^ { \\prime }$ . In this case we swap the $i$ th and $j$ th indices in $\\bar { n } ^ { \\prime }$ by defining $\\bar { n } ^ { \\prime \\prime }$ which differs from it in $\\bar { n } _ { i } ^ { \\prime \\prime } = \\bar { n } _ { j } ^ { \\prime }$ and $\\bar { n } _ { j \\_ } ^ { \\prime \\prime } = \\bar { n } _ { i } ^ { \\prime }$ . By C8, $q ( \\bar { n } ^ { \\prime \\prime } ) = q ( \\bar { n } ^ { \\prime } )$ and, of course, $\\bar { n } ^ { \\prime \\prime } \\succeq \\bar { n }$ since the prefix sums remain unchanged. The benefit of doing this transformation is that we are back in Case I, where the relative order of coordinates that change between $\\bar { n }$ and $\\bar { n } ^ { \\prime \\prime }$ remains the same. ",
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+ "type": "text",
+ "text": "This concludes the proof of the lemma. ",
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+ "type": "text",
+ "text": "Applying Lemma 17 we construct a chain of histograms between $\\bar { n }$ and $\\bar { n } ^ { \\prime }$ , which, by Lemma 18, is non-increasing in $q ( \\cdot )$ . Together this implies that $q ( \\bar { n } ) \\leq q ( \\bar { n } ^ { \\prime } )$ , as claimed. ",
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+ "type": "text",
+ "text": "We apply the notion of dominance in proving the following proposition, which is used later in arguing correctness of Algorithm 4. ",
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+ "type": "text",
+ "text": "Proposition 19. Let n¯ be an integer-valued histogram and d be a positive integer. And $q ( \\cdot )$ satisfies C1, C7, and C8. The following holds: ",
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+ "type": "text",
+ "text": "1. Assuming $n ^ { ( 1 ) } - n ^ { ( 2 ) } \\geq 2 d ,$ , let $\\bar { n } ^ { * }$ be obtained from $\\bar { n }$ by decrementing ${ \\boldsymbol n } ^ { ( 1 ) }$ by $d$ and incrementing $n ^ { ( 2 ) }$ by $d .$ . Then ",
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+ "type": "text",
+ "text": "2. Assuming repeatedly $\\begin{array} { r } { \\sum _ { i = 2 } ^ { m } n ^ { ( i ) } \\geq d } \\end{array}$ , let e hi $\\bar { n } ^ { * * }$ be obtained from am’s current secon $\\bar { n }$ by incrementing highest value by o $n _ { 1 }$ y ti $d$ , and byes. Then $d$ ",
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+ "type": "text",
+ "text": "Proof. Towards proving the claims, we argue that $\\bar { n } ^ { * }$ and $\\bar { n } ^ { * * }$ are, respectively, the minimal and the maximal elements in the histogram dominance order (Definition 15) in the set of histograms at distance $d$ from $\\bar { n }$ . By Proposition 16 the claims follow. ",
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+ "type": "text",
+ "text": "1. Take any histogram $\\bar { n } ^ { \\prime }$ at distance $d$ from $\\bar { n }$ . Our goal is to prove that $\\bar { n } ^ { \\prime } \\succeq$ $\\bar { n } ^ { * }$ . Recall the definition of the distance $d ( \\cdot , \\cdot )$ between two histograms $d ( \\bar { n } , \\bar { n } ^ { \\prime } ) ~ =$ max $\\begin{array} { r l } { \\bigg \\{ \\sum _ { i : n _ { i } > n _ { i } ^ { \\prime } } \\lceil n _ { i } - n _ { i } ^ { \\prime } \\rceil , \\sum _ { i : n _ { i } < n _ { i } ^ { \\prime } } \\lceil n _ { i } ^ { \\prime } - n _ { i } \\rceil \\bigg \\} } & { { } } \\end{array}$ . If the distance is bounded by $d$ , it means, in particular, that ",
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+ "img_path": "images/e51cdc6036b3cf6ba1921706a666ce5da624dffafd279e30cb6d46ffbeb7ad42.jpg",
+ "text": "$$\n\\sum _ { j = 2 } ^ { i } n ^ { \\prime ( j ) } \\leq \\sum _ { j = 2 } ^ { i } n ^ { ( j ) } + d \\quad { \\mathrm { ~ f o r ~ a l l ~ } } i > 2 .\n$$",
+ "text_format": "latex",
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+ "type": "text",
+ "text": "That lets us bound the prefix sums of $\\bar { n } ^ { \\prime }$ as follows: ",
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+ "type": "equation",
+ "img_path": "images/bf48420f4900c7f8d7ff8206d29a0facecae5d732ed9e66b42c0d87828f6b09a.jpg",
+ "text": "$$\n\\begin{array} { l } { S _ { i } ( \\bar { n } ^ { \\prime } ) = \\displaystyle \\sum _ { j = 2 } ^ { i } ( n ^ { \\prime ( 1 ) } - n ^ { \\prime ( j ) } ) = ( i - 1 ) \\cdot n ^ { \\prime ( 1 ) } - \\displaystyle \\sum _ { j = 2 } ^ { i } n ^ { \\prime ( j ) } } \\\\ { \\geq ( i - 1 ) \\cdot ( n ^ { ( 1 ) } - d ) - \\displaystyle \\left( \\sum _ { j = 2 } ^ { i } n ^ { ( j ) } + d \\right) = S _ { i } ( \\bar { n } ^ { * } ) . } \\end{array}\n$$",
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+ "type": "text",
+ "text": "We demonstrated that $\\bar { n } ^ { \\prime } \\succeq \\bar { n } ^ { * }$ , which, by Proposition 16, implies that $q ( \\bar { n } ^ { \\prime } ) \\leq q ( \\bar { n } ^ { * } )$ Together with the immediate $d ( \\bar { n } , \\bar { n } ^ { * } ) = d$ we prove the claim. ",
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+ "type": "text",
+ "text": "2. Assume wlog that $\\bar { n }$ is sorted in the descending order. Define the following value that depends on $\\bar { n }$ and $d$ : ",
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+ "img_path": "images/2ae07e381ae3d22531b806739fd5ae3cf8a217c375f0f495fca76da961fa2406.jpg",
+ "text": "$$\nu \\triangleq \\operatorname* { m i n } \\left\\{ x \\in { \\mathbb { N } } \\colon \\sum _ { i : i > 1 , n _ { i } \\geq x } n _ { i } - x \\leq d \\right\\} .\n$$",
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+ "type": "text",
+ "text": "The constant $u$ is the smallest such $x$ so that the total mass that can be shaved from elements of $\\bar { n }$ above $x$ (excluding $n _ { 1 }$ ) is at most $d$ . ",
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+ "type": "text",
+ "text": "We give the following equivalent definition of $\\bar { n } ^ { * * }$ : ",
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+ "img_path": "images/813c8558474a39af939a72cbc535ee9c7e7b03c4a40cebcab34830ae3e0cc6f3.jpg",
+ "text": "$$\nn _ { i } ^ { * * } = \\left\\{ { \\begin{array} { l l } { n _ { 1 } + d } & { { \\mathrm { i f ~ } } i = 1 , } \\\\ { u } & { { \\mathrm { i f ~ } } i > 1 { \\mathrm { ~ a n d ~ } } n _ { i } \\geq u , } \\\\ { n _ { i } } & { { \\mathrm { o t h e r w i s e } } . } \\end{array} } \\right.\n$$",
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+ "type": "text",
+ "text": "Fix any $i \\in [ m ]$ and any histogram $\\bar { n } ^ { \\prime }$ at distance $d$ from $\\bar { n }$ . Our goal is to prove that $S _ { i } ( \\bar { n } ^ { * * } ) \\geq S _ { i } ^ { \\bar { ( n ^ { \\prime } ) } }$ and thus $\\bar { n } ^ { * * } \\succeq \\bar { n } ^ { \\prime }$ . Assume the opposite and take largest $i$ such that $S _ { i } ( { \\bar { n } } ^ { * * } ) < S _ { i } ( { \\bar { n } } ^ { \\prime } )$ . ",
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+ "type": "text",
+ "text": "We may assume that $n ^ { \\prime ( 1 ) } = n _ { 1 } ^ { * * } = n _ { 1 } + d$ . Consider the following cases. ",
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+ "type": "text",
+ "text": "Case I. If $n ^ { * * ( i ) } < u$ , the contradiction follows from ",
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+ "img_path": "images/20c6752be4a43e4789082067e1276bddffdc8903272c1a05f275d7d232fed5bf.jpg",
+ "text": "$$\n\\begin{array} { r } { S _ { i } ( \\bar { n } ^ { \\prime } ) = \\displaystyle \\sum _ { j = 2 } ^ { i } ( n ^ { \\prime ( 1 ) } - n ^ { \\prime ( j ) } ) = \\displaystyle \\sum _ { j = 2 } ^ { i } \\Big ( ( n ^ { \\prime ( 1 ) } - n ^ { ( 1 ) } ) + ( n ^ { ( 1 ) } - n ^ { ( j ) } ) + ( n ^ { ( j ) } - n ^ { \\prime ( j ) } ) \\Big ) } \\\\ { \\le ( i - 1 ) d + S _ { i } ( \\bar { n } ) + d = S _ { i } ( \\bar { n } ^ { * * } ) . } \\end{array}\n$$",
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+ "type": "text",
+ "text": "The last equality is due to the fact that all differences between $\\bar { n }$ and $\\bar { n } ^ { * * }$ are confined to the indices that are less than $i$ . ",
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+ "type": "text",
+ "text": "Case $\\mathbf { I I }$ . If $n ^ { * * ( i ) } = u$ and $n ^ { \\prime { ( i ) } } \\geq u$ , the contradiction with $S _ { i } ( \\bar { n } ^ { * * } ) < S _ { i } ( \\bar { n } ^ { \\prime } )$ follows immediately from ",
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+ "type": "equation",
+ "img_path": "images/d3122cb1b2334f94cde4f121fa76017d00abcd3307d0b0cbd2dab06d6ad68235.jpg",
+ "text": "$$\nS _ { i } ( \\bar { n } ^ { \\prime } ) = \\sum _ { j = 2 } ^ { i } ( n ^ { \\prime ( 1 ) } - n ^ { \\prime ( j ) } ) \\le ( i - 1 ) ( n ^ { \\prime ( 1 ) } - u ) = S _ { i } ( \\bar { n } ^ { * * } ) .\n$$",
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+ "type": "text",
+ "text": "Case III. Finally, consider the case when $n ^ { * * ( i ) } = u$ and $v \\ \\triangleq n ^ { \\prime ( i ) } < u$ . Since $i$ is the largest such that $S _ { i } ( { \\bar { n } } ^ { * * } ) < S _ { i } ( { \\bar { n } } ^ { \\prime } )$ , it means that $n ^ { * * ( i + 1 ) } < n ^ { \\prime ( i + 1 ) } \\leq v < u = n ^ { * * ( i ) }$ ",
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+ "type": "text",
+ "text": "and thus $n ^ { * * ( i ) } - n ^ { * * ( i + 1 ) } \\geq 2$ (we rely on the fact that the histograms are integer-valued). It implies that all differences between $\\bar { n }$ and $\\bar { n } ^ { * * }$ are confined to the indices in $[ 1 , i ]$ . Then, ",
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+ "text": "$$\n\\begin{array} { l } { { S _ { i } ( \\bar { n } ^ { * * } ) - S _ { i } ( \\bar { n } ^ { \\prime } ) \\geq \\displaystyle \\sum _ { j = 2 } ^ { i } ( n _ { 1 } ^ { * * } - n _ { j } ^ { * * } ) - \\displaystyle \\sum _ { j = 2 } ^ { i } ( n _ { 1 } ^ { \\prime } - n _ { j } ^ { \\prime } ) } } \\\\ { { \\mathrm { ~ } = \\displaystyle \\sum _ { j = 2 } ^ { i } ( ( n _ { j } - n _ { j } ^ { * * } ) + ( n _ { j } ^ { \\prime } - n _ { j } ) ) } } \\\\ { { \\mathrm { ~ } \\geq d - d ( \\bar { n } , \\bar { n } ^ { \\prime } ) } } \\\\ { { \\mathrm { ~ } \\geq 0 , } } \\end{array}\n$$",
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+ "type": "text",
+ "text": "which contradicts the assumption that $S _ { i } ( { \\bar { n } } ^ { * * } ) < S _ { i } ( { \\bar { n } } ^ { \\prime } )$ . ",
+ "bbox": [
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+ "page_idx": 28
+ },
+ {
+ "type": "text",
+ "text": "We may now state and prove the main result of this section. ",
+ "bbox": [
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+ "page_idx": 28
+ },
+ {
+ "type": "text",
+ "text": "Theorem 20. Assume that $q ( \\cdot )$ satisfies conditions $C l { - } C 8$ and $\\bar { n }$ is an integer-valued histogram. Then the following two claims are true: ",
+ "bbox": [
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+ "page_idx": 28
+ },
+ {
+ "type": "text",
+ "text": "1. Algorithm 4 computes $\\begin{array} { r } { \\operatorname* { m a x } _ { \\bar { n } ^ { \\prime } : \\mathrm { d i s t } ( \\bar { n } , \\bar { n } ^ { \\prime } ) \\leq d } \\tilde { \\mathrm { L S } } ( \\bar { n } ^ { \\prime } ) . } \\end{array}$ ",
+ "bbox": [
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+ "page_idx": 28
+ },
+ {
+ "type": "text",
+ "text": "2. Algorithm 5 computes $\\mathrm { S S } _ { \\beta } ( \\bar { n } )$ , which is a $\\beta$ -smooth upper bound on smooth sensitivity of $\\beta \\left( q ( \\cdot ) \\right)$ . ",
+ "bbox": [
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+ "page_idx": 28
+ },
+ {
+ "type": "text",
+ "text": "Proof. Claim 1. Recall that $q _ { 1 } = \\mathrm { B _ { L } } ( q _ { 0 } )$ , and therefore, by Proposition 14 the function $\\tilde { \\mathrm { L S } } ( q )$ is non-decreasing in $[ 0 , q _ { 1 } ]$ , constant in $[ q _ { 1 } , q _ { 0 } ]$ , and non-increasing in $[ q _ { 0 } , 1 ]$ . It means, in particular, that to maximize $\\tilde { \\mathrm { L S } } ( q ( \\bar { n } ^ { \\prime } ) )$ over histograms satisfying $d ( \\bar { n } , \\bar { n } ^ { \\prime } ) = d$ , it suffices to consider the following cases. ",
+ "bbox": [
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+ },
+ {
+ "type": "text",
+ "text": "If $\\tilde { \\mathrm { L S } } ( q ( \\bar { n } ) ) < q _ { 1 }$ , then higher values of $\\mathrm { \\tilde { L S } ( \\cdot ) }$ may be attained only by histograms with higher values of $q$ . Proposition 19 enables us to efficiently find a histogram $\\bar { n } ^ { * }$ with the highest $q$ at distance $d$ , or conclude that we may reach the plateau by making the two highest histogram entries be equal. ",
+ "bbox": [
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+ "page_idx": 28
+ },
+ {
+ "type": "text",
+ "text": "$q _ { 1 } \\leq \\tilde { \\mathrm { L S } } ( q ( \\bar { n } ) ) \\leq q _ { 0 }$ , it means that $\\tilde { \\mathrm { L S } } ( q ( n ) )$ is already as high as it can be. ",
+ "bbox": [
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+ "page_idx": 28
+ },
+ {
+ "type": "text",
+ "text": "If $q _ { 0 } < \\tilde { \\mathrm { L S } } ( q ( \\bar { n } ) )$ , then, according to Proposition 14, higher values of $\\mathrm { \\tilde { L S } ( \\cdot ) }$ can be achieved by histograms with smaller values of $q$ , which we explore using the procedure outlined by Proposition 19. The stopping condition—when the plateau is reached—happens when $q$ becomes smaller than $q _ { 0 }$ . ",
+ "bbox": [
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+ },
+ {
+ "type": "text",
+ "text": "Claim 2. The second claim follows from the specification of Algorithm 5 and the first claim. ",
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+ "page_idx": 28
+ },
+ {
+ "type": "text",
+ "text": "B.4 GNMAX SATISFIES CONDITIONS C1–C8 ",
+ "text_level": 1,
+ "bbox": [
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+ "page_idx": 28
+ },
+ {
+ "type": "text",
+ "text": "The previous sections laid down a framework for computing smooth sensitivity of a randomized aggregator mechanism: defining functions $q ( \\cdot ) , \\mathrm { B _ { U } ( \\cdot ) , B _ { L } ( \\cdot ) }$ , verifying that they satisfy conditions C1–C8, and applying Theorem 20, which asserts correctness of Algorithm 5. In this section we instantiate this framework for the GNMax mechanism. ",
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+ },
+ {
+ "type": "text",
+ "text": "B.4.1 CONDITIONS C1–C4, C7 AND C8 ",
+ "text_level": 1,
+ "bbox": [
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+ },
+ {
+ "type": "text",
+ "text": "Defining $q$ and conditions C1, C7, and C8. Following Proposition 7, we define $q \\colon \\mathbb { R } _ { \\geq 0 } ^ { m } [ 0 , 1 ]$ for a GNMax mechanism parameterized with $\\sigma$ as: ",
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+ {
+ "type": "equation",
+ "img_path": "images/b555ced3f1c79d37e679ec7a63759430adc04569f17e6b1a46ee55adbca4dbc6.jpg",
+ "text": "$$\n\\begin{array} { r l } & { q ( \\bar { n } ) \\triangleq \\operatorname* { m i n } \\left\\{ \\underset { i \\neq i ^ { * } } { \\sum } \\mathbf { P r } ( Z _ { i } - Z _ { i ^ { * } } \\geq n _ { i ^ { * } } - n _ { i } ) , 1 \\right\\} } \\\\ & { \\quad = \\operatorname* { m i n } \\left\\{ \\underset { i \\neq i ^ { * } } { \\sum } \\frac { 1 } { 2 } \\left( 1 - \\mathrm { e r f } \\left( \\frac { n _ { i ^ { * } } - n _ { i } } { 2 \\sigma } \\right) \\right) , 1 \\right\\} } \\\\ & { \\quad = \\operatorname* { m i n } \\left\\{ \\underset { i \\neq i ^ { * } } { \\sum } \\frac { 1 } { 2 } \\mathrm { e r f c } \\left( \\frac { n _ { i ^ { * } } - n _ { i } } { 2 \\sigma } \\right) , 1 \\right\\} , } \\end{array}\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 29
+ },
+ {
+ "type": "text",
+ "text": "where $i ^ { * }$ is the histogram $\\bar { n }$ ’s highest coordinate, i.e., $n _ { i ^ { * } } \\geq n _ { i }$ for all $i$ (if there are multiple highest, let $i ^ { * }$ be any of them). Recall that erf is the error function, and erfc is the complement error function. ",
+ "bbox": [
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+ "page_idx": 29
+ },
+ {
+ "type": "text",
+ "text": "Proposition 7 demonstrates that $q ( { \\bar { n } } )$ bounds from above the probability that GNMax outputs anything but the highest coordinate of the histogram. ",
+ "bbox": [
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+ "page_idx": 29
+ },
+ {
+ "type": "text",
+ "text": "Conditions C1, C7, and C8 follow by simple calculus $( q _ { 0 }$ , defined below, is at most 0.5). ",
+ "bbox": [
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+ "page_idx": 29
+ },
+ {
+ "type": "text",
+ "text": "Functions $\\mathrm { B } _ { \\mathrm { L } } , \\mathrm { B } _ { \\mathrm { U } }$ , and conditions C2–C4. Recall that the functions $\\mathrm { B _ { L } }$ and $\\mathrm { B _ { U } }$ are defined in Appendix $\\mathbf { B }$ as follows: ",
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+ },
+ {
+ "type": "equation",
+ "img_path": "images/2a46ffe3a3d74b3ca3f2abc703aaea0051f2a3bc248d7c5ef591287b7ffb861f.jpg",
+ "text": "$$\n\\begin{array} { l } { { \\displaystyle { \\mathrm { B } } _ { \\mathrm { U } } ( q ) \\triangleq \\operatorname* { m i n } \\left\\{ \\frac { m - 1 } { 2 } \\mathrm { e r f c } \\left( \\mathrm { e r f c } ^ { - 1 } \\left( \\frac { 2 q } { m - 1 } \\right) - \\frac { 1 } { \\sigma } \\right) , 1 \\right\\} , } } \\\\ { { \\displaystyle { \\mathrm { B } } _ { \\mathrm { L } } ( q ) \\triangleq \\frac { m - 1 } { 2 } \\mathrm { e r f c } \\left( \\mathrm { e r f c } ^ { - 1 } \\left( \\frac { 2 q } { m - 1 } \\right) + \\frac { 1 } { \\sigma } \\right) , } } \\end{array}\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 29
+ },
+ {
+ "type": "text",
+ "text": "Proposition 21 (Condition C2). For any neighbor $\\bar { n } ^ { \\prime }$ of $\\bar { n }$ , i.e., $d ( \\bar { n } ^ { \\prime } , \\bar { n } ) = 1$ , the following bounds hold: ",
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+ "page_idx": 29
+ },
+ {
+ "type": "equation",
+ "img_path": "images/3b3ca4537077ca14e32edae19eacc4db1a5b70ad1ad23fa159d24fac9a950114.jpg",
+ "text": "$$\n\\mathrm { B } _ { \\mathrm { L } } ( q ( \\bar { n } ) ) \\leq q ( \\bar { n } ^ { \\prime } ) \\leq \\mathrm { B } _ { \\mathrm { U } } ( q ( \\bar { n } ) ) .\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 29
+ },
+ {
+ "type": "text",
+ "text": "Proof. Assume wlog that $i ^ { * } = 1$ . Let $x _ { i } \\triangleq n _ { 1 } - n _ { i }$ and $q _ { i } \\triangleq \\mathrm { e r f c } ( x _ { i } / 2 \\sigma ) / 2$ , and similarly define $\\boldsymbol { x } _ { i } ^ { \\prime }$ for $\\bar { n } ^ { \\prime }$ . Observe that $| x _ { i } - x _ { i } ^ { \\prime } | \\leq 2$ , which, by monotonicity of erfc, implies that ",
+ "bbox": [
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+ "page_idx": 29
+ },
+ {
+ "type": "equation",
+ "img_path": "images/157fa8dfc2bd519ef8e7fe58aa7d2bf8c5cb507852a07d0280eb15058ef212a0.jpg",
+ "text": "$$\n{ \\frac { 1 } { 2 } } \\operatorname { e r f c } \\left( { \\frac { x _ { i } + 2 } { 2 \\sigma } } \\right) \\leq q _ { i } ( { \\bar { n } } ^ { \\prime } ) \\leq { \\frac { 1 } { 2 } } \\operatorname { e r f c } \\left( { \\frac { x _ { i } - 2 } { 2 \\sigma } } \\right) .\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 29
+ },
+ {
+ "type": "text",
+ "text": "Thus ",
+ "bbox": [
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+ ],
+ "page_idx": 29
+ },
+ {
+ "type": "equation",
+ "img_path": "images/8c90bdf88a16d5b9f2509ac9502a6cf98038c74118dd8f10283a298d6e8140cd.jpg",
+ "text": "$$\n\\frac { 1 } { 2 } \\sum _ { i > 1 } \\mathrm { e r f c } \\left( \\frac { x _ { i } + 2 } { 2 \\sigma } \\right) \\leq q ( \\bar { n } ^ { \\prime } ) \\leq \\frac { 1 } { 2 } \\sum _ { i > 1 } \\mathrm { e r f c } \\left( \\frac { x _ { i } - 2 } { 2 \\sigma } \\right) .\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 29
+ },
+ {
+ "type": "text",
+ "text": "(Although $i ^ { * }$ may change between $\\bar { n }$ and $\\bar { n } ^ { \\prime }$ , the bounds still hold.) ",
+ "bbox": [
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+ ],
+ "page_idx": 29
+ },
+ {
+ "type": "text",
+ "text": "Our first goal is to upper bound $q ( \\bar { n } ^ { \\prime } )$ for a given value of $q ( { \\bar { n } } )$ . To this end we set up the following maximization problem ",
+ "bbox": [
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+ "page_idx": 29
+ },
+ {
+ "type": "equation",
+ "img_path": "images/17bcdbb17a43550590c08be1e22c9d0c8696fc196a10de7c2e09ead1aa536605.jpg",
+ "text": "$$\n\\operatorname* { m a x } _ { \\{ x _ { i } \\} } { \\frac { 1 } { 2 } } \\sum _ { i > 1 } { \\mathrm { e r f c } } \\left( { \\frac { x _ { i } - 2 } { 2 \\sigma } } \\right) { \\mathrm { ~ s u c h ~ t h a t ~ } } { \\frac { 1 } { 2 } } \\sum _ { i > 1 } { \\mathrm { e r f c } } \\left( { \\frac { x _ { i } } { 2 \\sigma } } \\right) = q { \\mathrm { ~ a n d ~ } } x _ { i } \\geq 0 .\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 29
+ },
+ {
+ "type": "text",
+ "text": "We may temporarily ignore the non-negative constraints, which end up being satisfied by our solution. Consider using the method of Lagrange multipliers and take a derivative in $x _ { i } { } ^ { \\ ' }$ : ",
+ "bbox": [
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+ {
+ "type": "equation",
+ "img_path": "images/dfaa73f9d59389f801c72fec89860f0573218fd8733edba749cbaf91f7476414.jpg",
+ "text": "$$\n\\begin{array} { c } { \\displaystyle - \\exp \\left( - \\left( \\frac { x _ { i } - 2 } { 2 \\sigma } \\right) ^ { 2 } \\right) + \\lambda \\exp \\left( - \\left( \\frac { x _ { i } } { 2 \\sigma } \\right) ^ { 2 } \\right) = 0 } \\\\ { \\displaystyle \\Leftrightarrow \\lambda = \\exp \\left( \\frac { x _ { i } - 1 } { \\sigma ^ { 2 } } \\right) . } \\end{array}\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ },
+ {
+ "type": "text",
+ "text": "Since the expression is symmetric in $i > 1$ , it means that the local optima are attained at $x _ { 2 } =$ $\\cdots = x _ { m }$ (the second derivative confirms that these are local maxima). After solving for $( m -$ $1 ) \\operatorname { e r f c } ( x / 2 \\sigma ) = 2 q$ we have ",
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+ {
+ "type": "equation",
+ "img_path": "images/954a47d8f0473556818f5c73b15af652c16d18e39bd05fa295ee938074704cb1.jpg",
+ "text": "$$\nq ( \\bar { n } ^ { \\prime } ) \\leq \\frac { m - 1 } { 2 } \\mathrm { e r f c } \\left( \\mathrm { e r f c ^ { - 1 } } \\left( \\frac { 2 q } { m - 1 } \\right) - \\frac { 1 } { \\sigma } \\right) .\n$$",
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+ "bbox": [
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+ {
+ "type": "text",
+ "text": "where $m$ is the number of classes. Similarly, ",
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+ },
+ {
+ "type": "equation",
+ "img_path": "images/ed3718159ea1135242ea8c1435901f1ca18e0f871acb2bf82df56fb6363f23dc.jpg",
+ "text": "$$\nq ( \\bar { n } ^ { \\prime } ) \\geq \\frac { m - 1 } { 2 } \\mathrm { e r f c } \\left( \\mathrm { e r f c ^ { - 1 } } \\left( \\frac { 2 q } { m - 1 } \\right) + \\frac { 1 } { \\sigma } \\right) .\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 30
+ },
+ {
+ "type": "text",
+ "text": "Conditions C3, i.e., $\\operatorname { B _ { L } } ( q )$ is monotonically increasing in $q$ , and $\\mathrm { C 4 }$ , i.e., $\\mathrm { B _ { L } }$ and $\\mathrm { B _ { U } }$ are functional inverses of each other in $[ 0 , q _ { 0 } ]$ and $\\mathrm { B } _ { \\mathrm { L } } ( { \\dot { q } } ) \\leq q \\leq \\mathbf { \\bar { B } } _ { \\mathrm { U } } ( { \\dot { q } } )$ for all $q \\in [ 0 , 1 ]$ , follow from basic properties of erfc. The restriction that $q \\in [ 0 , q _ { 0 } ]$ ensures that $\\operatorname { B _ { U } } ( q )$ is strictly less than one, and the minimum in the definition of $\\operatorname { B _ { U } } ( \\cdot )$ simplifies to its first argument in this range. ",
+ "bbox": [
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+ },
+ {
+ "type": "text",
+ "text": "B.4.2 CONDITIONS C5 AND C6 ",
+ "text_level": 1,
+ "bbox": [
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+ ],
+ "page_idx": 30
+ },
+ {
+ "type": "text",
+ "text": "Conditions C5 and C6 stipulate that the function $\\beta ( q ) \\triangleq \\beta _ { \\sigma } ( q )$ (defined in Appendix A) exhibits a specific growth pattern. Concretely, C5 states that $\\beta \\left( q \\right)$ is monotonically increasing for $0 \\leq q \\leq q _ { 0 }$ , and constant for $q _ { 0 } < q \\leq 1$ . (Additionally, we require that $\\mathrm { B _ { U } } ( q _ { 0 } ) < 1 )$ . Condition C6 requires that $\\Delta \\beta ( q ) = \\beta ( \\mathrm { B } _ { \\mathrm { U } } ( q ) ) - \\beta ( q )$ is non-decreasing in $[ 0 , \\mathrm { B _ { L } } ( q _ { 0 } ) ]$ . ",
+ "bbox": [
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+ ],
+ "page_idx": 30
+ },
+ {
+ "type": "text",
+ "text": "Rather than proving these statements analytically, we check these assumptions for any fixed $\\sigma$ and $\\lambda$ via a combination of symbolic and numeric analyses. ",
+ "bbox": [
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+ ],
+ "page_idx": 30
+ },
+ {
+ "type": "text",
+ "text": "More concretely, we construct symbolic expressions for $\\beta \\left( \\cdot \\right)$ and $\\Delta \\beta \\left( \\cdot \\right)$ and (symbolically) differentiate them. We then minimize (numerically) the resulting expressions over $[ 0 , q _ { 0 } ]$ and $[ 0 , \\mathrm { B _ { L } } ( q _ { 0 } ) ]$ , and verify that their minimal values are indeed non-negative. ",
+ "bbox": [
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+ "page_idx": 30
+ },
+ {
+ "type": "text",
+ "text": "B.5 RÉNYI DIFFERENTIAL PRIVACY AND SMOOTH SENSITIVITY ",
+ "text_level": 1,
+ "bbox": [
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+ "page_idx": 30
+ },
+ {
+ "type": "text",
+ "text": "Although the procedure for computing a smooth sensitivity bound may be quite involved (such as Algorithms 3–5), its use in a differentially private data release is straightforward. Following Nissim et al. (2007), we define an additive Gaussian mechanism where the noise distribution is scaled by $\\sigma$ and a smooth sensitivity bound: ",
+ "bbox": [
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+ "page_idx": 30
+ },
+ {
+ "type": "text",
+ "text": "Definition 22. Given a real-valued function $f \\colon { \\mathcal { D } } \\mathbb { R }$ and a $\\beta$ -smooth sensitivity bound $\\operatorname { S S } ( \\cdot )$ , let $( \\beta , \\sigma )$ -GNSS mechanism be ",
+ "bbox": [
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+ "page_idx": 30
+ },
+ {
+ "type": "equation",
+ "img_path": "images/3e487bd6badf792115f3a5bc6a4ef192a039e9492014e1de8d341f550f516c69.jpg",
+ "text": "$$\n{ \\mathcal { F } } _ { \\sigma } ( D ) \\triangleq f ( D ) + \\operatorname { S S } _ { \\beta } ( D ) \\cdot { \\mathcal { N } } ( 0 , \\sigma ^ { 2 } ) .\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 30
+ },
+ {
+ "type": "text",
+ "text": "We claim that this mechanism satisfies Rényi differential privacy for finite orders from a certain range. ",
+ "bbox": [
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+ "page_idx": 30
+ },
+ {
+ "type": "text",
+ "text": "Theorem 23. The $( \\beta , \\sigma )$ -GNSS mechanism ${ \\mathcal { F } } _ { \\sigma }$ is $( \\lambda , \\varepsilon )$ -RDP, where ",
+ "bbox": [
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+ "page_idx": 30
+ },
+ {
+ "type": "equation",
+ "img_path": "images/e07df1ad04294ea2f19d0d403bc4bd2b2b7927f5a0ca379ecd42e2badb7c31a6.jpg",
+ "text": "$$\n\\varepsilon \\triangleq { \\frac { \\lambda \\cdot e ^ { 2 \\beta } } { \\sigma ^ { 2 } } } + { \\frac { \\beta \\lambda - 0 . 5 \\ln ( 1 - 2 \\lambda \\beta ) } { \\lambda - 1 } }\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 30
+ },
+ {
+ "type": "text",
+ "text": "for all $1 < \\lambda < 1 / ( 2 \\beta )$ . ",
+ "bbox": [
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+ "page_idx": 30
+ },
+ {
+ "type": "text",
+ "text": "Proof. Consider two neighboring datasets $D$ and $D ^ { \\prime }$ . The output distributions of the $( \\beta , \\sigma )$ -GNSS mechanism on $D$ and $D ^ { \\prime }$ are, respectively, ",
+ "bbox": [
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+ {
+ "type": "equation",
+ "img_path": "images/14eb515b6e73df43657653069dccefb52bff376888351b95d0656aea26ac1b45.jpg",
+ "text": "$$\nP \\triangleq f ( D ) + \\mathrm { S S } _ { \\beta } ( D ) \\cdot { \\mathcal { N } } ( 0 , \\sigma ^ { 2 } ) = { \\mathcal { N } } ( f ( D ) , ( \\mathrm { S S } _ { \\beta } ( D ) \\sigma ) ^ { 2 } ) { \\mathrm { ~ a n d ~ } } Q \\triangleq { \\mathcal { N } } ( f ( D ^ { \\prime } ) , ( \\mathrm { S S } _ { \\beta } ( D ^ { \\prime } ) \\sigma ) ^ { 2 } ) .\n$$",
+ "text_format": "latex",
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+ },
+ {
+ "type": "text",
+ "text": "The Rényi divergence between two normal distributions can be computed in closed form (van Erven & Harremoës, 2014): ",
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+ "text": "$$\nD _ { \\lambda } ( P \\| Q ) = \\lambda { \\frac { ( f ( D ) - f ( D ^ { \\prime } ) ) ^ { 2 } } { 2 \\sigma ^ { 2 } s ^ { 2 } } } + { \\frac { 1 } { 1 - \\lambda } } \\ln { \\frac { s } { { \\mathrm { S S } } _ { \\beta } ( D ) ^ { 1 - \\lambda } \\cdot { \\mathrm { S S } } _ { \\beta } ( D ^ { \\prime } ) ^ { \\lambda } } } ,\n$$",
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+ "text": "provided $s ^ { 2 } \\triangleq ( 1 - \\lambda ) \\cdot \\mathrm { S S } _ { \\beta } ( D ) ^ { 2 } + \\lambda \\cdot \\mathrm { S S } _ { \\beta } ( D ^ { \\prime } ) ^ { 2 } > 0 .$ ",
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+ "text": "According to the definition of smooth sensitivity (Definition 12) ",
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+ "text": "$$\ne ^ { - \\beta } \\cdot \\mathrm { S S } _ { \\beta } ( D ) \\leq \\mathrm { S S } _ { \\beta } ( D ^ { \\prime } ) \\leq e ^ { \\beta } \\cdot \\mathrm { S S } _ { \\beta } ( D ) ,\n$$",
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+ "text": "$$\n| f ( D ) - f ( D ^ { \\prime } ) | \\leq e ^ { \\beta } \\cdot \\operatorname * { m i n } ( \\mathrm { S S } _ { \\beta } ( D ) , \\mathrm { S S } _ { \\beta } ( D ^ { \\prime } ) ) .\n$$",
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+ "text": "Bound (12) together with the condition that $\\lambda \\le 1 / ( 2 \\beta )$ implies that ",
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+ "text": "$$\n\\begin{array} { r l } & { s ^ { 2 } = ( 1 - \\lambda ) \\cdot \\operatorname { S S } _ { \\beta } ( D ) ^ { 2 } + \\lambda \\cdot \\operatorname { S S } _ { \\beta } ( D ^ { \\prime } ) ^ { 2 } = \\operatorname { S S } _ { \\beta } ( D ) ^ { 2 } + \\lambda ( \\operatorname { S S } _ { \\beta } ( D ^ { \\prime } ) ^ { 2 } - \\operatorname { S S } _ { \\beta } ( D ) ^ { 2 } ) } \\\\ & { \\qquad \\geq \\operatorname { S S } _ { \\beta } ( D ) ^ { 2 } ( 1 + \\lambda ( e ^ { - 2 \\beta } - 1 ) ) \\geq \\operatorname { S S } _ { \\beta } ( D ) ^ { 2 } ( 1 - 2 \\lambda \\beta ) > 0 . } \\end{array}\n$$",
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+ "text": "The above lower bound ensures that $s ^ { 2 }$ is well-defined, i.e., non-negative, as required for application of (11). ",
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+ "text": "Combining bounds (12)– (14), we have that ",
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+ "text": "$$\nD _ { \\lambda } ( P \\| Q ) \\le \\frac { \\lambda \\cdot e ^ { 2 \\beta } } { \\sigma ^ { 2 } } + \\frac { 1 } { 1 - \\lambda } \\ln \\left\\{ \\frac { s } { \\mathrm { S S } _ { \\beta } ( D ) } e ^ { - \\lambda \\beta } \\right\\} \\le \\frac { \\lambda \\cdot e ^ { 2 \\beta } } { \\sigma ^ { 2 } } + \\frac { \\beta \\lambda - 0 . 5 \\ln ( 1 - 2 \\lambda \\beta ) } { \\lambda - 1 }\n$$",
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+ "text": "Note that if $\\lambda \\gg 1$ , $\\sigma \\ll \\lambda$ , and $\\beta \\ll 1 / ( 2 \\lambda )$ , then $( \\beta , \\sigma )$ -GNSS satisfies $( \\lambda , ( \\lambda { + } 1 ) / \\sigma ^ { 2 } )$ -RDP. Compare this with RDP analysis of the standard additive Gaussian mechanism, which satisfies $( \\lambda , \\lambda / \\sigma ^ { 2 } )$ - RDP. The difference is that GNSS scales noise in proportion to smooth sensitivity, which is no larger and can be much smaller than global sensitivity. ",
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+ "text": "B.6 PUTTING IT ALL TOGETHER: APPLYING SMOOTH SENSITIVITY ",
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+ "text": "Recall our initial motivation for the smooth sensitivity analysis: enabling privacy-preserving release of data-dependent privacy guarantees. Indeed, these guarantees vary greatly between queries (see Figure 5) and are typically much smaller than data-independent privacy bounds. Since datadependent bounds may leak information about underlying data, publishing the bounds themselves requires a differentially private mechanism. As we explain shortly, smooth sensitivity analysis is a natural fit for this task. ",
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+ "text": "We first consider the standard additive noise mechanism where the noise (such as Laplace or Gaussian) is calibrated to the global sensitivity of the function we would like to make differentially private. We know that Rényi differential privacy is additive for any fixed order $\\lambda$ , and thus the cumulative RDP cost is the sum of RDP costs of individual queries each upper bounded by a dataindependent bound. Thus, it might be tempting to use the standard additive noise mechanism for sanitizing the total, but that would be a mistake. ",
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+ "text": "To see why, consider a sequence of queries $\\bar { n } _ { 1 } , \\ldots , \\bar { n } _ { \\ell }$ answered by the aggregator. Their total (unsanitized) RDP cost of order $\\lambda$ is $\\begin{array} { r } { B _ { \\sigma } = \\sum _ { i = 1 } ^ { \\ell } \\beta _ { \\sigma } ( q ( \\bar { n } _ { i } ) ) } \\end{array}$ . Even though $\\beta _ { \\sigma } ( q ( \\bar { n } _ { i } ) ) \\leq \\lambda / \\sigma ^ { 2 }$ (the data-independent bound, Proposition 8), the sensitivity of their sum is not $\\lambda / \\sigma ^ { 2 }$ . The reason is that the (global) sensitivity is defined as the maximal difference in the function’s output between two neighboring datasets $D$ and $D ^ { \\prime }$ . Transitioning from $D$ to $D ^ { \\prime }$ may change one teacher’s output on all student queries. ",
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+ "text": "In contrast with the global sensitivity of $B _ { \\sigma }$ that may be quite high—particularly for the second step of the Confident GNMax aggregator—its smooth sensitivity can be extremely small. Towards computing a smooth sensitivity bound on $B _ { \\sigma }$ , we prove the following theorem which defines a smooth sensitivity of the sum in terms of local sensitivities of its parts. ",
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+ "text": "Theorem 24. Let $f _ { i } { : } \\mathcal { D } \\mathbb { R }$ for $1 \\leq i \\leq \\ell _ { \\mathrm { { i } } }$ , $\\begin{array} { r } { F ( D ) \\triangleq \\sum _ { i = 1 } ^ { \\ell } f _ { i } ( D ) } \\end{array}$ and $\\beta > 0$ . Then ",
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+ "text": "$$\n\\mathrm { S S } ( D ) \\triangleq \\operatorname* { m a x } _ { d \\geq 0 } e ^ { - \\beta d } \\cdot \\sum _ { i = 1 } ^ { \\ell } \\operatorname* { m a x } _ { D ^ { \\prime } : \\mathrm { d i s t } ( D , D ^ { \\prime } ) \\leq d } \\tilde { \\mathrm { L S } } _ { f _ { i } } ( D ^ { \\prime } ) ,\n$$",
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+ "text": "is a $\\beta$ -smooth bound on $F ( \\cdot ) i f \\tilde { \\mathrm { L S } } _ { f _ { i } } ( D ^ { \\prime } )$ are upper bounds on the local sensitivity of $f _ { i } ( D ^ { \\prime } )$ . ",
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+ "text": "Proof. We need to argue that $\\operatorname { S S } ( { \\mathord { \\cdot } } )$ is $\\beta$ -smooth, i.e., $\\mathrm { S S } ( D _ { 1 } ) \\le e ^ { \\beta } \\cdot \\mathrm { S S } ( D _ { 2 } )$ for any neighboring $D _ { 1 } , D _ { 2 } \\ \\in \\ { \\mathcal { D } }$ , and it is an upper bound on the local sensitivity of $F ( D _ { 1 } )$ , i.e., $\\mathrm { S S } ( D _ { 1 } ) \\ \\geq$ $| F ( D _ { 1 } ) - F ( D _ { 2 } ) |$ . ",
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+ "text": "$$\n\\operatorname* { m a x } _ { D : \\mathrm { d i s t } ( D _ { 1 } , D ) \\leq d } \\tilde { \\mathrm { L S } } _ { f _ { i } } ( D ) \\leq \\operatorname* { m a x } _ { D : \\mathrm { d i s t } ( D _ { 2 } , D ) \\leq d + 1 } \\tilde { \\mathrm { L S } } _ { f _ { i } } ( D )\n$$",
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+ "text": "for all neighboring datasets $D _ { 1 }$ and $D _ { 2 }$ (by the triangle inequality over distances). Then ",
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+ "text": "$$\n\\begin{array} { r l } & { \\displaystyle \\mathrm { S S } ( D _ { 1 } ) = \\operatorname* { m a x } _ { d \\geq 0 } e ^ { - \\beta d } \\cdot \\sum _ { i = 1 } ^ { \\ell } \\operatorname* { m a x } _ { D : \\mathrm { d i s t } ( D _ { 1 } , D ) \\leq d } \\mathrm { I } \\tilde { \\mathrm { S } S } _ { f _ { i } } ( D ) , } \\\\ & { \\qquad \\leq \\displaystyle \\operatorname* { m a x } _ { d \\geq 0 } e ^ { - \\beta d } \\cdot \\sum _ { i = 1 } ^ { \\ell } \\operatorname* { m a x } _ { D : \\mathrm { d i s t } ( D _ { 2 } , D ) \\leq d + 1 } \\mathrm { I } \\tilde { \\mathrm { S } S } _ { f _ { i } } ( D ) } \\\\ & { \\qquad = \\displaystyle \\operatorname* { m a x } _ { d \\geq 1 } e ^ { - \\beta ( d ^ { \\prime } - 1 ) } \\cdot \\sum _ { i = 1 } ^ { \\ell } \\operatorname* { m a x } _ { D : \\mathrm { d i s t } ( D _ { 2 } , D ) \\leq d ^ { \\prime } } \\mathrm { I } \\tilde { \\mathrm { S } S } _ { f _ { i } } ( D ) } \\\\ & { \\qquad \\leq e ^ { \\beta } \\cdot \\mathrm { S S } ( D _ { 2 } ) } \\end{array}\n$$",
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+ "text": "The fact that $\\operatorname { S S } ( { \\mathord { \\cdot } } )$ is an upper bound on the local sensitivity of $F ( \\cdot )$ is implied by the following: ",
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+ "text": "$$\n\\begin{array} { r l } { \\displaystyle | F ( D _ { 1 } ) - F ( D _ { 2 } ) | = \\left| \\displaystyle \\sum _ { i = 1 } ^ { \\ell } f _ { i } ( D _ { 1 } ) - \\displaystyle \\sum _ { i = 1 } ^ { \\ell } f _ { i } ( D _ { 2 } ) \\right| } & { } \\\\ { \\displaystyle } & { \\le \\displaystyle \\sum _ { i = 1 } ^ { \\ell } | f _ { i } ( D _ { 1 } ) - f _ { i } ( D _ { 2 } ) | } \\\\ { \\displaystyle } & { \\le \\displaystyle \\sum _ { i = 1 } ^ { \\ell } \\mathrm { I } \\tilde { \\mathrm { S } } f _ { i } ( D _ { 1 } ) } \\\\ { \\displaystyle } & { \\le \\mathrm { S S } ( D _ { 1 } ) , } \\end{array}\n$$",
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+ "text": "Applying Theorem 24 allows us to compute a smooth sensitivity of the sum more efficiently than summing up smooth sensitivities of its parts. Results below rely on this strategy. ",
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+ "text": "Empirical results. Table 2 revisits the privacy bounds in Table 1. For all data-dependent privacy claims of the Confident GNMax aggregator we report parameters for their smooth sensitivity analysis and results of applying the GNSS mechanism for their release. ",
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+ "text": "Consider the first row of the table. The MNIST dataset was partitioned among 250 teachers, each getting 200 training examples. After the teachers were individually trained, the student selected at random 640 unlabeled examples, and submitted them to the Confident GNMax aggregator with the threshold of 200, and noise parameters $\\sigma _ { 1 } = 1 5 0$ and $\\sigma _ { 2 } = 4 0 $ . The expected number of answered examples (those that passed the first step of Algorithm 1) is 283, and the expected Rényi differential privacy is $\\varepsilon = 1 . 1 8$ at order $\\lambda = 1 4$ . This translates (via Theorem 5) to $( 2 . 0 0 , 1 0 ^ { - 5 } )$ -differential privacy, where 2.00 is the expectation of the privacy parameter $\\varepsilon$ . ",
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+ "text": "These costs are data-dependent and they cannot be released without further sanitization, which we handle by adding Gaussian noise scaled by the smooth sensitivity of $\\varepsilon$ (the GNSS mechanism, Definition 22). At $\\beta = 0 . 0 3 2 9$ the expected value of smooth sensitivity is 0.0618. We choose $\\sigma _ { \\mathrm { S S } } = 6 . 2 3$ , which incurs, according to Theorem 23, an additional (data-independent) (14, 0.52)- RDP cost. Applying $( \\beta , \\sigma _ { \\mathrm { S S } } )$ -GNSS where $\\sigma _ { \\mathrm { S S } } ~ = ~ 6 . 2 3$ , we may publish differentially private estimate of the total privacy cost that consists of a fixed part—the cost of applying Confident GNMax and GNSS—and random noise. The fixed part is $2 . { \\bar { 5 } } 2 = 1 . 1 8 + 0 . 5 2 { \\stackrel { - } { - } } \\mathrm { l n } ( 1 { \\bar { 0 } } ^ { - 5 } ) / 1 4$ , and the noise is normally distributed with mean 0 and standard deviation $\\sigma _ { \\mathrm { S S } } \\cdot 0 . 0 6 1 8 = 0 . 3 8 5$ . We note that, in contrast with the standard additive noise, one cannot publish its standard deviation without going through additional privacy analysis. ",
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+ "Table 2: Privacy-preserving reporting of privacy costs. The table augments Table 1 by including smooth sensitivity analysis of the total privacy cost. The expectations are taken over the student’s queries and outcomes of the first step of the Confident GNMax aggregator. Order $\\lambda$ , smooth sensitivity parameter $\\beta$ , $\\sigma _ { \\mathrm { S S } }$ are parameters of the GNSS mechanism (Section B.5). The final column sums up the data-dependent cost $\\varepsilon$ , the cost of applying GNSS (Theorem 23), and the standard deviation of Gaussian noise calibrated to smooth sensitivity (the product of $\\mathbb { E }$ $[ \\mathrm { S S } _ { \\beta } ]$ and $\\sigma _ { \\mathrm { S S } }$ ). "
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+ "table_body": "| Input: input x,confidence γ,threshold T,noise parameters O1 ando2,total number of teachers M |
| 1:Ask the student to provide prediction scores p(x) 2:if maxj{nj(x)-Mpj(x)}+N(0,o²)≥Tthen >Student does not agree with teachers |
|
| 3: return argmaxj{nj(x) +N(0,0²)} |
| 4: else if max{pi(x)}> γ then > Student agrees with teachers and is confident |
| 5: return arg maxj Pj(x) |
| Reinforce student's prediction 6: else return⊥ No output given for this label |
",
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| 1: procedure ATDISTANCED(n, d) |
| 2: q←q(n) | > q is in the flat region. |
| 3: 4: | if q1 ≤q≤ go then |
| return LS(q), STOP | |
| 5: end if | |
| 6: | if q<qithen Need to increase q. |
| 7: | if n(1)-n(2）< 2d then n(i) is the ith largest element. |
| 8: | return LS(qi), STOP |
| 9: | else |
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| 12: | if q' >qi then |
| 13: | return LS(qo), STOP |
| 14: | else |
| 15: | return LS(q'),CONTINUE |
| 16: | end if |
| 17: | end if |
| 18: | else Need to decrease q. |
| 19: | |
| 20: | n↑[n,0,...,0] |
| 21: | q'←q(n'） |
| 22: | return LS(q'), STOP |
| 23: | else |
| 24: | n' ← SORT(n)+[d,0,...,0] |
| 25: | for d' =1,...,d do |
| 26: | n'(2) ←n(2)-1 The index of n'(2) may change. |
| 27: | end for |
| 28: | q'←q(n'） |
| 29: | if q'<qo then |
| 30: | return LS(qo), STOP |
| 31: | else |
| 32: | return LS(q'), CONTINUE |
| 33: | end if |
| |
| 34: | end if |
| 35: end if 36: end procedure |
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| 1: procedure ATDISTANCED(n, d) |
| 2: q←q(n) | > q is in the flat region. |
| 3: 4: | if q1 ≤q≤ go then |
| return LS(q), STOP | |
| 5: end if | |
| 6: | if q<qithen Need to increase q. |
| 7: | if n(1)-n(2）< 2d then n(i) is the ith largest element. |
| 8: | return LS(qi), STOP |
| 9: | else |
| 10: | n'← SORT(n)+[-d,d,O,...,0] |
| 11: | q'←q(n') |
| 12: | if q' >qi then |
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| 14: | else |
| 15: | return LS(q'),CONTINUE |
| 16: | end if |
| 17: | end if |
| 18: | else Need to decrease q. |
| 19: | |
| 20: | n↑[n,0,...,0] |
| 21: | q'←q(n'） |
| 22: | return LS(q'), STOP |
| 23: | else |
| 24: | n' ← SORT(n)+[d,0,...,0] |
| 25: | for d' =1,...,d do |
| 26: | n'(2) ←n(2)-1 The index of n'(2) may change. |
| 27: | end for |
| 28: | q'←q(n'） |
| 29: | if q'<qo then |
| 30: | return LS(qo), STOP |
| 31: | else |
| 32: | return LS(q'), CONTINUE |
| 33: | end if |
| |
| 34: | end if |
| 35: end if 36: end procedure |
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+ "html": "| case analysis that finds the neighbor at distance d with the maximum sensitivity. |
| 1: procedure ATDISTANCED(n, d) |
| 2: q←q(n) | > q is in the flat region. |
| 3: 4: | if q1 ≤q≤ go then |
| return LS(q), STOP | |
| 5: end if | |
| 6: | if q<qithen Need to increase q. |
| 7: | if n(1)-n(2）< 2d then n(i) is the ith largest element. |
| 8: | return LS(qi), STOP |
| 9: | else |
| 10: | n'← SORT(n)+[-d,d,O,...,0] |
| 11: | q'←q(n') |
| 12: | if q' >qi then |
| 13: | return LS(qo), STOP |
| 14: | else |
| 15: | return LS(q'),CONTINUE |
| 16: | end if |
| 17: | end if |
| 18: | else Need to decrease q. |
| 19: | |
| 20: | n↑[n,0,...,0] |
| 21: | q'←q(n'） |
| 22: | return LS(q'), STOP |
| 23: | else |
| 24: | n' ← SORT(n)+[d,0,...,0] |
| 25: | for d' =1,...,d do |
| 26: | n'(2) ←n(2)-1 The index of n'(2) may change. |
| 27: | end for |
| 28: | q'←q(n'） |
| 29: | if q'<qo then |
| 30: | return LS(qo), STOP |
| 31: | else |
| 32: | return LS(q'), CONTINUE |
| 33: | end if |
| |
| 34: | end if |
| 35: end if 36: end procedure |
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