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+# data2vec: A General Framework for Self-supervised Learning in Speech, Vision and Language
+
+Alexei Baevski<sup>1</sup> Wei-Ning Hsu<sup>1</sup> Qiantong Xu<sup>2</sup> Arun Babu<sup>1</sup> Jiatao Gu<sup>1</sup> Michael Auli<sup>1</sup>
+
+# Abstract
+
+While the general idea of self-supervised learning is identical across modalities, the actual algorithms and objectives differ widely because they were developed with a single modality in mind. To get us closer to general self-supervised learning, we present data2vec, a framework that uses the same learning method for either speech, NLP or computer vision. The core idea is to predict latent representations of the full input data based on a masked view of the input in a self-distillation setup using a standard Transformer architecture. Instead of predicting modality-specific targets such as words, visual tokens or units of human speech which are local in nature, data2vec predicts contextualized latent representations that contain information from the entire input. Experiments on the major benchmarks of speech recognition, image classification, and natural language understanding demonstrate a new state of the art or competitive performance to predominant approaches. Models and code are available at www.github.com/pytorch/fairseq/tree/master/examples/data2vec.
+
+# 1. Introduction
+
+Self-supervised learning builds representations of data without human annotated labels which led to significant advances in natural language processing (NLP; Peters et al. 2018; Radford et al. 2018; Devlin et al. 2019; Brown et al. 2020), speech processing (van den Oord et al., 2018; Schneider et al., 2019; Baevski et al., 2020b) as well as computer vision (Chen et al., 2020; 2021b; Caron et al., 2021; Bao et al., 2021; He et al., 2021). Self-supervised representations have even enabled completely unsupervised learning
+
+$^{1}$ Meta AI  $^{2}$ SambaNova, work done while at Meta AI. Correspondence to: Alexei Baevski <abaevski@fb.com>, Michael Auli <michaelauli@fb.com>.
+
+Proceedings of the  $39^{th}$  International Conference on Machine Learning, Baltimore, Maryland, USA, PMLR 162, 2022. Copyright 2022 by the author(s).
+
+in tasks such as machine translation (Lample et al., 2018) and speech recognition (Baevski et al., 2021).
+
+Research in self-supervised algorithms has focused on individual modalities which results in specific designs and learning biases. For example, in speech processing, there is no vocabulary of speech units over which we can define a self-supervised learning task such as words in  $\mathrm{NLP}^1$  and therefore several prominent models are equipped with mechanisms to learn an inventory of speech units (Baevski et al., 2020b; Hsu et al., 2021). A similar problem exists for computer vision, where researchers either learn discrete visual tokens (Radford et al., 2021a; Bao et al., 2021), regress the input (He et al., 2021) or learn representations invariant to data augmentation (Chen et al., 2020; Grill et al., 2020; Caron et al., 2021).
+
+While learning biases are certainly helpful, it is often unclear whether they will generalize to other modalities. Moreover, leading theories on the biology of learning (Friston & Kiebel, 2009; Friston, 2010) imply that humans likely use similar learning processes to understand the visual world as they do for language. Relatedly, general neural network architectures have been shown to perform very well compared to modality-specific counterparts (Jaegle et al., 2021b).
+
+In an effort to get closer to machines that learn in general ways about the environment, we designed data2vec, a framework for general self-supervised learning that works for images, speech and text where the learning objective is identical in each modality. The present work unifies the learning algorithm but still learns representations individually for each modality. We hope that a single algorithm will make future multi-modal learning simpler, more effective and lead to models that understand the world better through multiple modalities.
+
+Our method combines masked prediction (Devlin et al., 2019; Baevski et al., 2020b; Bao et al., 2021) with the learning of latent target representations (Grill et al., 2020; Caron et al., 2021) but generalizes the latter by using multiple network layers as targets and shows that this approach works across several modalities. Specifically, we train an off-the
+
+![](images/99c60efe9a98cd4dce5d6e274ce6d1f57d929f4d927ee4f96b81ab70faa8e162.jpg)  
+Figure 1. Illustration of how data2vec follows the same learning process for different modalities. The model first produces representations of the original input example (teacher mode) which are then regressed by the same model based on a masked version of the input. The teacher parameters are an exponentially moving average of the student weights. The student predicts the average of  $K$  network layers of the teacher (shaded in blue).
+
+shelf Transformer network (Vaswani et al., 2017) which we use either in teacher or student mode (Illustration in Figure 1): we first build representations of the full input data whose purpose is to serve as targets in the learning task (teacher mode). Next, we encode a masked version of the input sample with which we predict the full data representations (student mode). The weights of the teacher are an exponentially decaying average of the student (He et al., 2019; Grill et al., 2020; Caron et al., 2021). Since different modalities have vastly different inputs, e.g., pixels vs. words, we use modality-specific feature encoders and masking strategies from the literature.
+
+Since our method works with the latent network representations of the learner itself, it can be seen as a simplification of many modality-specific designs such as learning a fixed set of visual tokens (Radford et al., 2021a; van den Oord et al., 2017), or normalization of the input to create suitable targets (He et al., 2021), or the learning of a vocabulary of discrete speech units (Baevski et al., 2020b; Hsu et al., 2021). Moreover, our target representations are continuous and contextualized, through the use of self-attention, which makes them richer than a fixed set of targets and/or targets based on local context such as used in most prior work.
+
+Experimental results show data2vec to be effective in all three modalities, setting a new state of the art for ViT-B with single models and ViT-L on ImageNet-1K, improving over the best prior work in speech processing on speech recognition (Baevski et al., 2020b; Hsu et al., 2021) and outperforming a like for like RoBERTa baseline on the GLUE natural language understanding benchmark (Liu et al., 2019).
+
+# 2. Related Work
+
+Self-supervised Learning in Computer Vision. Unsupervised pre-training for computer vision has been a very active area of research with methods contrasting representa
+
+tions of augmentations of the same image, entirely different images (Chen et al., 2020; Grill et al., 2020; Caron et al., 2021; Chen et al., 2021b) as well as online clustering (Caron et al., 2020). Similar to our work, both BYOL (Grill et al., 2020) and DINO (Caron et al., 2021) regress neural network representations of a momentum encoder, but our work differs in that it uses a masked prediction task and we regress multiple neural network layer representations instead of just the top layer which we find to be more effective. Moreover, data2vec works for multiple modalities.
+
+The most recent work focuses on training vision Transformers (Dosovitskiy et al., 2020) with masked prediction objectives (Bao et al., 2021; He et al., 2021; Xie et al., 2021) whose performance surpasses supervised-only training on ImageNet-1K. Several of these methods predict visual tokens (Bao et al., 2021; He et al., 2021; Dong et al., 2022) learned in a separate step before pre-training (van den Oord et al., 2017; Ramesh et al., 2021), during pretraining (Zhou et al., 2021), and others directly predict the input pixels (He et al., 2021; Xie et al., 2021).
+
+Instead, data2vec predicts the latent representations of the input data. Another difference to this body of work is that the latent target representations are contextualized, incorporating relevant features from the entire image instead of targets which contain information isolated to the current patch, such as visual tokens or pixels.
+
+Self-supervised Learning in NLP. Pre-training has been very successful in advancing natural language understanding (McCann et al., 2017; Peters et al., 2018; Radford et al., 2018; Baevski et al., 2019; Devlin et al., 2019; Yang et al., 2019; Brown et al., 2020). The most prominent model is BERT (Devlin et al., 2019) which solves a masked prediction task where some of the input tokens are blanked out in order to be predicted given the remaining input. For many languages it is easy to determine word boundaries and most
+
+methods therefore predict word or sub-word units for pretraining. There is also work on knowledge distillation to obtain smaller BERT-style models, both for pre-training and fine-tuning (Jiao et al., 2020).
+
+Compared to prior NLP algorithms, data2vec does not predict discrete linguistic tokens such as words, sub-words or bytes but rather a continuous and contextualized representation. This has two advantages: first, the targets themselves are not predefined, nor is their number limited. This enables the model to adapt to a particular input example. Second, targets are contextualized, taking context information into account. This is unlike BERT-style models which learn a single embedding for each target which needs to fit all instances of a particular target in the data.
+
+Self-supervised Learning in Speech. Work in self-supervised learning for speech includes autoregressive models (van den Oord et al., 2018; Schneider et al., 2019; Baevski et al., 2020a; Chung et al., 2019) as well as bidirectional models (Baevski et al., 2020b; Hsu et al., 2021; Ao et al., 2021; Chen et al., 2021a). Two prominent models, wav2vec 2.0 and HuBERT are based on predicting discrete units of speech, either learned jointly during pretraining (Baevski et al., 2020b), or in an iterative pipeline approach (Hsu et al., 2021) where pre-training and clustering alternate. Another line of work directly reconstructs the input features (Eloff et al., 2019; Liu et al., 2021).
+
+In comparison to wav2vec 2.0, data2vec directly predicts contextualized latent representations without quantization. HuBERT discretizes representations from different layers across iterations and predicts these discretized units whereas data2vec predicts the average over multiple layers. Similar to other modalities, there is work on distilling larger self-supervised models into smaller models but primarily for the purpose of efficiency (Chang et al., 2021).
+
+Multimodal Pre-training. There has been a considerable body of research on learning representations of multiple modalities simultaneously often using paired data (Aytar et al., 2017; Radford et al., 2021b; Wang et al., 2021; Singh et al., 2021) with the aim to produce cross-modal representations which can perform well on multi-modal tasks and with modalities benefiting from each other through joint training (Alayrac et al., 2020; Akbari et al., 2021) with recent methods exploring few-shot learning (Tsimpoukelli et al., 2021). Our work does not perform multimodal training but aims to unify the learning objective for self-supervised learning in different modalities. We hope that this will enable better multimodal representations in the future.
+
+2Quantization is optional for wav2vec 2.0 (Baevski et al., 2020b; Zhang et al., 2020) but helpful for noisy speech (Chung et al., 2021).
+
+# 3. Method
+
+data2vec is trained by predicting the model representations of the full input data given a partial view of the input (Figure 1). We first encode a masked version of the training sample (model in student mode) and then construct training targets by encoding the unmasked version of the input with the same model but when parameterized as an exponentially moving average of the model weights (model in teacher mode; Grill et al. 2020; Caron et al. 2021). The target representations encode all of the information in the training sample and the learning task is for the student to predict these representations given a partial view of the input.
+
+# 3.1. Model Architecture
+
+We use the standard Transformer architecture (Vaswani et al., 2017) with a modality-specific encoding of the input data borrowed from prior work: for computer vision, we use the ViT-strategy of encoding an image as a sequence of patches, each spanning 16x16 pixels, input to a linear transformation (Dosovitskiy et al., 2020; Bao et al., 2021). Speech data is encoded using a multi-layer 1-D convolutional neural network that maps  $16\mathrm{kHz}$  waveform to  $50\mathrm{Hz}$  representations (Baevski et al., 2020b). Text is pre-processed to obtain sub-word units (Sennrich et al., 2016; Devlin et al., 2019), which are then embedded in distributional space via learned embedding vectors. We detail these methods below ( $\S 4$ ).
+
+# 3.2. Masking
+
+After the input sample has been embedded as a sequence of tokens, we mask part of these units by replacing them with a learned MASK embedding token and feed the sequence to the Transformer network. For computer vision, we follow the block-wise masking strategy of Bao et al. (2021), for speech we mask spans of latent speech representations (Baevski et al., 2020b) and for language we mask tokens (Devlin et al., 2019); §4 details each strategy.
+
+# 3.3. Training Targets
+
+The model is trained to predict the model representations of the original unmasked training sample based on an encoding of the masked sample. We predict model representations only for time-steps which are masked. The representations we predict are contextualized representations, encoding the particular time-step but also other information from the sample due to the use of self-attention in the Transformer network. $^4$  This is an important difference to BERT (De
+
+vlin et al., 2019), wav2vec 2.0 (Baevski et al., 2020b) or BEiT, MAE, SimMIM, and MaskFeat (Bao et al., 2021; He et al., 2021; Xie et al., 2021; Wei et al., 2021) which predict targets lacking contextual information. Below, we detail how we parameterize the teacher which predicts the network representations that will serve as targets as well as how we construct the final target vectors to be predicted by the model in student-mode.
+
+Teacher Parameterization. The encoding of the un-masked training sample is parameterized by an exponentially moving average (EMA) of the model parameters  $(\theta; \text{Tarvainen & Valpola 2018; Grill et al. 2020; Caron et al. 2021})$  where the weights of the model in target-mode  $\Delta$  are:
+
+$$
+\Delta \gets \tau \Delta + (1 - \tau) \theta
+$$
+
+We use a schedule for  $\tau$  that linearly increases this parameter from  $\tau_0$  to the target value  $\tau_{e}$  over the first  $\tau_{n}$  updates after which the value is kept constant for the remainder of training. This strategy results in the teacher being updated more frequently at the beginning of training, when the model is random, and less frequently later in training, when good parameters have already been learned. We found it more efficient and slightly more accurate to share the parameters of the feature encoder and the positional encoder between the teacher and student networks.
+
+Targets. Training targets are constructed based on the output of the top  $K$  blocks of the teacher network for time-steps which are masked in student-mode. $^{5}$  The output of block  $l$  at time-step  $t$  is denoted as  $a_{t}^{l}$ . We apply a normalization to each block to obtain  $\hat{a}_{t}^{l}$  before averaging the top  $K$  blocks  $y_{t} = \frac{1}{K}\sum_{l = L - K + 1}^{L}\hat{a}_{t}^{l}$  for a network with  $L$  blocks in total to obtain the training target  $y_{t}$  for time-step  $t$ . This creates training targets that are to be regressed by the model when in student mode. In preliminary experiments we found that averaging performed as well as predicting each block separately with a dedicated projection while enjoying the advantage of being more efficient.
+
+Normalizing the targets helps prevent the model from collapsing into a constant representation for all time-steps and it also prevents layers with high norm to dominate the target features. For speech representations, we use instance normalization (Ulyanov et al., 2016) without any learned parameters over the current input sample since neighboring representations are highly correlated due to the small stride over the input data, while for NLP and vision we found parameter-less layer normalization (Ba et al., 2016) to be sufficient. Variance-Invariance-Covariance regularization (Bardes et al., 2021) also addresses this problem but
+
+we found the above strategy to perform well and it does not introduce additional hyper-parameters.
+
+# 3.4. Objective
+
+Given contextualized training targets  $y_{t}$ , we use a Smooth L1 loss to regress these targets:
+
+$$
+\mathcal {L} (y _ {t}, f _ {t} (x)) = \left\{ \begin{array}{l l} \frac {1}{2} (y _ {t} - f _ {t} (x)) ^ {2} / \beta & | y _ {t} - f _ {t} (x) | \leq \beta \\ (| y _ {t} - f _ {t} (x) | - \frac {1}{2} \beta) & \text {o t h e r w i s e} \end{array} \right.
+$$
+
+where  $\beta$  controls the transition from a squared loss to an  $L_{1}$  loss, depending on the size of the gap between the target  $y_{t}$  and the model prediction  $f_{t}(x)$  at time-step  $t$ . The advantage of this loss is that it is less sensitive to outliers, however, we need to tune the setting of  $\beta$ .
+
+# 4. Experimental Setup
+
+We experiment with two model sizes: data2vec Base and data2vec Large, containing either  $L = 12$  or  $L = 24$  Transformer blocks with  $H = 768$  or  $H = 1024$  hidden dimension (with  $4 \times H$  feed-forward inner-dimension). EMA updates are performed in fp32 for numerical stability (Manohar et al., 2021).
+
+# 4.1. Computer Vision
+
+We embed images of  $224 \times 224$  pixels as patches of  $16 \times 16$  pixels (Dosovitskiy et al., 2020). Each patch is linearly transformed and a sequence of 196 representations is input to a standard Transformer. We follow BEiT (Bao et al., 2021) by masking blocks of multiple adjacent patches where each block contains at least 16 patches with a random aspect ratio. Different to their work, we found it more accurate to mask  $60\%$  of the patches instead of  $40\%$ . We use randomly applied resized image crops, horizontal flipping, and color jittering (Bao et al., 2021). We use the same modified image both in teacher mode and student mode.
+
+ViT-B models are pre-trained for 800 epochs. As batch size we use 2,048 for ViT-B and 8,192 for ViT-L. We use Adam (Kingma & Ba, 2015) and a cosine schedule (Loshchilov & Hutter, 2016) with a single cycle where we warm up the learning rate for 40 epochs to 0.002 for ViT-B and for 80 epochs to 0.001 for ViT-L after which the learning rate is annealed following the cosine schedule. For ViT-B and ViT-L, we use  $\beta = 2$ ,  $K = 6$  and  $\tau = 0.9998$  as a constant value with no schedule which worked well. We use stochastic depth with rate 0.2 (Huang et al., 2016). For ViT-L, we train for 1,600 epochs in total, the first 800 epochs use  $\tau = 0.9998$ , we then reset the learning rate schedule and the teacher weights to the student and continue for another 800 epochs with  $\tau = 0.9999$ .
+
+For image classification we mean-pool the output of the
+
+last Transformer block and input it to a softmax-normalized classifier. We fine-tune ViT-B for 100 epochs and ViT-L for 50 epochs using Adam and a cosine schedule where we warm up the learning rate for 20 epochs to 0.004 for ViT-B and for 5 epochs to 0.004 for ViT-L after which the learning rate follows the cosine schedule. We build on the open source implementation of BEiT (Bao et al., 2021).
+
+# 4.2. Speech Processing
+
+Models are implemented in fairseq (Ott et al., 2019) and take as input  $16\mathrm{kHz}$  waveform which is processed by a feature encoder (Baevski et al., 2020b) containing seven temporal convolutions with 512 channels, strides  $(5,2,2,2,2,2,2)$  and kernel widths  $(10,3,3,3,3,2,2)$ . This results in an encoder output frequency of  $50\mathrm{Hz}$  with a stride of about  $20\mathrm{ms}$  between each sample, and a receptive field of 400 input samples or  $25\mathrm{ms}$  of audio. The raw waveform input to the encoder is normalized to zero mean and unit variance.
+
+The masking strategy for the Base model is also identical to Baevski et al. (2020b): we sample  $p = 0.065$  of all time-steps to be starting indices and mask the subsequent ten time-steps. This results in approximately  $49\%$  of all time-steps to be masked for a typical training sequence. During pre-training we linearly anneal  $\tau$  using  $\tau_0 = 0.999$ ,  $\tau_e = 0.9999$  and  $\tau_n = 30,000$ , average the top  $K = 8$  blocks as targets and found a simple L2 loss to work well.
+
+We optimize with Adam (Kingma & Ba, 2015), with a peak learning rate of  $5 \times 10^{-4}$  for data2vec Base. The Base model uses a tri-stage scheduler which linearly warms up the learning rate over the first  $3\%$  of updates, holds it for  $90\%$  of updates and then linearly decays it over the remaining  $7\%$ . We train data2vec Base for 400K updates with a batch size of 63 minutes of audio (61M frames). We follow the fine-tuning regime of wav2vec 2.0 (Baevski et al., 2020b) whose hyper-parameters depend on the labeled data setup.
+
+# 4.3. Natural Language Processing
+
+We build on the BERT re-implementation RoBERTa (Liu et al., 2019) available in fairseq (Ott et al., 2019). The input data is tokenized using a byte-pair encoding (Sennrich et al., 2016) of 50K types and the model learns an embedding for each type (Devlin et al., 2019; Liu et al., 2019) Once the data is embedded, we apply the BERT masking strategy to  $15\%$  of uniformly selected tokens:  $80\%$  are replaced by a learned mask token,  $10\%$  are left unchanged and  $10\%$  are replaced by randomly selected vocabulary token; we do not use the next-sentence prediction task. We also consider the wav2vec 2.0 strategy of masking spans of four tokens.
+
+For pre-training we use  $\tau_0 = 0.999$ ,  $\tau_e = 0.9999$  and  $\tau_n = 100,000$ ,  $K = 10$  and set  $\beta = 4$ . The model is optimized with Adam over 1M updates using a tri-stage
+
+Table 1. Computer vision: top-1 validation accuracy on ImageNet-1K with ViT-B and ViT-L models. data2vec ViT-B was trained for 800 epochs and ViT-L for 1,600 epochs. We distinguish between individual models and setups composed of multiple models (BEiT/PeCo train separate visual tokenizers and PeCo also distills two MoCo-v3 models).  
+
+<table><tr><td></td><td>ViT-B</td><td>ViT-L</td></tr><tr><td colspan="3">Multiple models</td></tr><tr><td>BEiT (Bao et al., 2021)</td><td>83.2</td><td>85.2</td></tr><tr><td>PeCo (Dong et al., 2022)</td><td>84.5</td><td>86.5</td></tr><tr><td colspan="3">Single models</td></tr><tr><td>MoCo v3 (Chen et al., 2021b)</td><td>83.2</td><td>84.1</td></tr><tr><td>DINO (Caron et al., 2021)</td><td>82.8</td><td>-</td></tr><tr><td>MAE (He et al., 2021)</td><td>83.6</td><td>85.9</td></tr><tr><td>SimMIM (Xie et al., 2021)</td><td>83.8</td><td>-</td></tr><tr><td>iBOT (Zhou et al., 2021)</td><td>83.8</td><td>-</td></tr><tr><td>MaskFeat (Wei et al., 2021)</td><td>84.0</td><td>85.7</td></tr><tr><td>data2vec</td><td>84.2</td><td>86.6</td></tr></table>
+
+learning rate schedule (5%, 80% and 15% of updates for warm-up, holding and linearly decaying, respectively). The peak learning rate is  $2 \times 10^{-4}$ . We train on 16 GPUs with a total batch size of 256 sequences and each sequence is up to 512 tokens. For downstream tasks, we fine-tune the pretrained model with four different learning rates ( $1 \times 10^{-5}$ ,  $2 \times 10^{-5}$ ,  $3 \times 10^{-5}$ ,  $4 \times 10^{-5}$ ) and choose the one which performs best across all considered NLP downstream tasks.
+
+# 5. Results
+
+# 5.1. Computer Vision
+
+To evaluate our approach for computer vision, we pretrain data2vec on the images of the ImageNet-1K training set (Deng et al., 2009) and fine-tune the resulting model for image classification using the labeled data of the same benchmark ( $\S 4.1$ ). Following standard practice, models are evaluated in terms of top-1 accuracy on the validation set. We distinguish between results based on a single self-supervised model, and results which train a separate visual tokenizer on additional data (Bao et al., 2021) or distill other self-supervised models (Dong et al., 2022).
+
+Table 1 shows that data2vec outperforms prior work with ViT-B and ViT-L in the single model setting and all prior work for ViT-L. Predicting contextualized latent representations in a masked prediction setup can perform very well compared to approaches which predict local targets such as the original input pixels (He et al., 2021; Xie et al., 2021), engineered image features (Wei et al., 2021) or visual tokens (Bao et al., 2021). It also outperforms prior self-distillation methods (Caron et al., 2021) which regressed
+
+Table 2. Speech processing: word error rate on the Librispeech test-other test set when fine-tuning pre-trained models on the Libri-light low-resource labeled data setups (Kahn et al., 2020) of  $10\mathrm{min}$ , 1 hour, 10 hours, the clean 100h subset of Librispeech and the full 960h of Librispeech. Models use the 960 hours of audio from Librispeech (LS-960) as unlabeled data. We indicate the language model used during decoding (LM). Results for all dev/test sets and other LMs can be found in the supplementary material (Table 6).  
+
+<table><tr><td rowspan="2"></td><td rowspan="2">Unlabeled data</td><td rowspan="2">LM</td><td colspan="5">Amount of labeled data</td></tr><tr><td>10m</td><td>1h</td><td>10h</td><td>100h</td><td>960h</td></tr><tr><td colspan="8">Base models</td></tr><tr><td>wav2vec 2.0 (Baevski et al., 2020b)</td><td>LS-960</td><td>4-gram</td><td>15.6</td><td>11.3</td><td>9.5</td><td>8.0</td><td>6.1</td></tr><tr><td>HuBERT (Hsu et al., 2021)</td><td>LS-960</td><td>4-gram</td><td>15.3</td><td>11.3</td><td>9.4</td><td>8.1</td><td>-</td></tr><tr><td>WavLM (Chen et al., 2021a)</td><td>LS-960</td><td>4-gram</td><td>-</td><td>10.8</td><td>9.2</td><td>7.7</td><td>-</td></tr><tr><td>data2vec</td><td>LS-960</td><td>4-gram</td><td>12.3</td><td>9.1</td><td>8.1</td><td>6.8</td><td>5.5</td></tr><tr><td colspan="8">Large models</td></tr><tr><td>wav2vec 2.0 (Baevski et al., 2020b)</td><td>LS-960</td><td>4-gram</td><td>10.3</td><td>7.1</td><td>5.8</td><td>4.6</td><td>3.6</td></tr><tr><td>HuBERT (Hsu et al., 2021)</td><td>LS-960</td><td>4-gram</td><td>10.1</td><td>6.8</td><td>5.5</td><td>4.5</td><td>3.7</td></tr><tr><td>WavLM (Chen et al., 2021a)</td><td>LS-960</td><td>4-gram</td><td>-</td><td>6.6</td><td>5.5</td><td>4.6</td><td>-</td></tr><tr><td>data2vec</td><td>LS-960</td><td>4-gram</td><td>8.4</td><td>6.3</td><td>5.3</td><td>4.6</td><td>3.7</td></tr></table>
+
+the final layer of the student network while inputting two different augmented versions of an image to the student and teacher networks.
+
+# 5.2. Speech and Audio Processing
+
+For speech processing, we pre-train data2vec on the 960 hours of speech audio data from Librispeech (LS-960). This dataset contains relatively clean speech audio from read audiobooks in English and is a standard benchmark in the speech community. To get a sense of performance in different resource settings, we fine-tune models for automatic speech recognition using different amounts of labeled data, ranging from just 10 minutes to 960 hours. We also compare to other work from the literature, including wav2vec 2.0 (Baevski et al., 2020b) and HuBERT (Hsu et al., 2021), two popular algorithms for speech representation learning relying on discrete units of speech.
+
+Table 2 shows improvements for most labeled data setups with the largest gains for 10 minutes of labeled data (20% relative WER improvement) for the Base models. For Large models, there are strong improvements for the smallest labeled data setups, and comparable performance for the resource-rich settings of 100 hours and 960 hours of labeled data where performance is generally saturating for many models (Zhang et al., 2020; Chung et al., 2021). Our results suggest that learning discrete units is not required when rich contextualized targets are used and that learning contextualized targets during pre-training improves performance.
+
+To further validate our approach for speech, we also trained a model on the AudioSet benchmark (Gemmeke et al., 2017). The pre-training setup is the same as Librispeech, but we set
+
+Table 3. Audio event classification: mean average precision (mAP) on the eval set of AudioSet when pre-training on all of AudioSet (AS) and/or Librispeech (LS) and fine-tuning on the 20K subset.  
+
+<table><tr><td></td><td>Pretrain data</td><td>mAP</td></tr><tr><td>SSAST (Gong et al., 2021)</td><td>AS, LS</td><td>31.0</td></tr><tr><td>MaskSpec (Chong et al., 2022)</td><td>AS</td><td>32.3</td></tr><tr><td>MAE-AST (Baade et al., 2022)</td><td>AS, LS</td><td>30.6</td></tr><tr><td>data2vec</td><td>AS</td><td>34.5</td></tr></table>
+
+$K = 12$  and train for  $200\mathrm{K}$  updates with a batch size of 94.5 minutes of audio. We apply DeepNorm (Wang et al., 2022) and layer normalization of the targets to stabilize training. Finetuning is done on the balanced subset over  $13\mathrm{k}$  updates with a batch size of 21.3 minutes. Similar to Srivastava et al. (2021), we use linear softmax pooling (Wang et al., 2018b) and mixup (Tokozume et al., 2017) with probability 0.7. We add a single linear projection layer into 527 audioset classes and set the projection learning rate to  $2e - 4$ . The pre-trained parameters are trained with a learning rate of  $3e - 5$  and we use masking during fine-tuning similar to Baevski et al. (2020b) with  $p = 0.45$  and mask length of 4. The results (Table 3) show that data2vec can outperform a comparable setup that uses the same pre-training and fine-tuning data.
+
+# 5.3. Natural Language Processing
+
+To get a sense of how data2vec performs for language, we adopt the same training setup as BERT (Devlin et al., 2019) by pre-training on the Books Corpus (Zhu et al., 2015) and English Wikipedia data over 1M updates and a batch size of 256 sequences. We evaluate on the General Language
+
+Table 4. Natural language processing: GLUE results on the development set for single-task fine-tuning of individual models. For MNLI we report accuracy on both the matched and unmatched dev sets, for MRPC and QQP, we report the unweighted average of accuracy and F1, for STS-B the unweighted average of Pearson and Spearman correlation, for CoLA we report Matthews correlation and for all other tasks we report accuracy. BERT Base results are from Wu et al. (2020) and our baseline is RoBERTa re-trained in a similar setup as BERT. We also report results with wav2vec 2.0 style masking of spans of four BPE tokens with no unmasked tokens or random targets.  
+
+<table><tr><td></td><td>MNLI</td><td>QNLI</td><td>RTE</td><td>MRPC</td><td>QQP</td><td>STS-B</td><td>CoLA</td><td>SST</td><td>Avg.</td></tr><tr><td>BERT (Devlin et al., 2019)</td><td>84.0/84.4</td><td>89.0</td><td>61.0</td><td>86.3</td><td>89.1</td><td>89.5</td><td>57.3</td><td>93.0</td><td>80.7</td></tr><tr><td>Baseline (Liu et al., 2019)</td><td>84.1/83.9</td><td>90.4</td><td>69.3</td><td>89.0</td><td>89.3</td><td>88.9</td><td>56.8</td><td>92.3</td><td>82.5</td></tr><tr><td>data2vec</td><td>83.2/83.0</td><td>90.9</td><td>67.0</td><td>90.2</td><td>89.1</td><td>87.2</td><td>62.2</td><td>91.8</td><td>82.7</td></tr><tr><td>+ wav2vec 2.0 masking</td><td>82.8/83.4</td><td>91.1</td><td>69.9</td><td>90.0</td><td>89.0</td><td>87.7</td><td>60.3</td><td>92.4</td><td>82.9</td></tr></table>
+
+Understanding Evaluation (GLUE) benchmark (Wang et al., 2018a) which includes tasks for natural language inference (MNLI, QNLI, RTE), sentence similarity (MRPC, QQP and STS-B), grammaticality (CoLA), and sentiment analysis (SST-2). We fine-tune data2vec separately on the labeled data provided by each task and report the average accuracy on the development sets over five fine-tuning runs. We compare to the published BERT results as well as to the results we obtain by retraining RoBERTa in the current setup (Baseline; Liu et al. 2019) which provides a more suitable baseline to data2vec since we build on their open source code.
+
+The results (Table 4) show that data2vec outperforms the RoBERTa baseline. When we mask spans of four BPE tokens with masking probability 0.35 (Baevski et al., 2020b), then results improve further. This strategy does not leave tokens unmasked or uses random targets as for BERT ( $\S 4.3$ ).
+
+To our knowledge this is the first successful pre-trained NLP model which does not use discrete units (words, subwords, characters or bytes) as the training target. Instead, the model predicts a contextualized latent representation emerging from self-attention over the entire unmasked text sequence. This enables a learning task where the model needs to predict targets with specific properties of the current text sequence rather than representations which are generic to every text sequence in which the particular discrete unit occurs. Moreover, the set of training targets is not fixed, i.e., not a closed vocabulary, and the model can choose to define
+
+$^{6}$ MNLI (Multi Genre Natural Language Inference; Williams et al. 2018), Stanford Question Answering Dataset (QNLI; Rajpurkar et al. 2016), Recognizing Textual Entailment (RTE; Dagan et al. 2006; Haim et al. 2006; Giampiccolo et al. 2007; Bentivogli et al. 2009), and we exclude Winograd NLI task from our results similar to Devlin et al. (2019), Microsoft Research Paragraph Corpus (MRPC; Dolan & Brockett 2005), Quora Question Pairs benchmark (QQP), and the Semantic Textual Similarity Benchmark (STS-B; Cer et al. 2018), Corpus of Linguistic Acceptability (CoLA; Warstadt et al. 2018), Stanford Sentiment Treebank (SST-2; Socher et al. 2013)
+
+<sup>7</sup>We used a cosine learning rate schedule for this result.
+
+new targets as it sees fit, akin to an open vocabulary setting.
+
+# 5.4. Ablations
+
+Layer-averaged Targets. One of the main differences of our method compared to BYOL is the use of targets which are based on averaging multiple layers from the teacher network (§3.3). This idea was partly inspired by the fact that the top layers of wav2vec 2.0 do not perform as well for downstream tasks as layers in the middle of the network (Baevski et al., 2021; Pasadena et al., 2021).
+
+In the next experiment, we measure performance for all three modalities when averaging  $K = 1, \dots, 12$  layers where  $K = 1$  corresponds to predicting only the top layer similar to BYOL. For faster experimental turn-around, we train Base models with  $L = 12$  layers in total. For speech, we pre-train for 200K updates on Librispeech, fine-tune on the 10 hour labeled split of Libri-light (Kahn et al., 2019) and report word error rate without a language model on dev-other. For NLP, we report the average GLUE score on the validation set (\$5.3) and for computer vision we pre-train models for 300 epochs and report the top-1 accuracy on ImageNet (\$5.1).
+
+Figure 2 shows that targets based on multiple layers improves over using only the top layer ( $K = 1$ ) for all modalities. Using all layers is generally a good choice and only slightly worse than a carefully tuned value of  $K$ . Neural networks build features over multiple layers and different types of features are extracted at different layers. Using features from multiple layers enriches the self-supervised task and improves accuracy.
+
+Target Contextualization. Teacher representations are based on self-attention over the entire input which results in contextualized targets. This distinguishes data2vec from other self-supervised approaches which construct a learning task by predicting or reconstructing local parts of the input ( $\S 2$ ). This poses the natural question of whether contextualized targets are required for data2vec to work well.
+
+![](images/a5d389f6a419703047c2fee1b365071f5f0d50299f9b2857300d552468266289.jpg)  
+(a) Speech
+
+![](images/38b47047ad870d79e615ff9f3781243b827443724210ec653d9e4b2fa48ec135.jpg)  
+(b) NLP  
+Figure 2. Predicting targets which are the average of multiple layers is more robust than predicting only the top most layer ( $K = 1$ ) for most modalities. We show the performance of predicting the average of  $K$  teacher layer representations (§3.3). The effect is very pronounced for speech and NLP while for vision there is still a slight advantage of predicting more than a single layer.
+
+![](images/40ed046f1d4b174d20528fdec0d0ad3258b83e1d48b271955bd03664c2a6f7fd.jpg)  
+(c) Vision
+
+![](images/161142234a73257bc28d51175ad548fac58ec6971f8cf4853f34ab1edd5d0deb.jpg)
+
+![](images/2abfdcb4e0667466ceff2520cb00a211e8a089779a4194341c4a9957e6e6fc7f.jpg)  
+(a) Speech  
+(b) Vision  
+Figure 3. Contextualized target representations improve accuracy. We mask all but a fraction of the data when constructing the target representations in the teacher for pre-training and report downstream performance for speech and vision. Time-steps for speech are more fine-grained (20ms patches) compared to vision (16x16 pixels per patch).
+
+Table 5. Effect of using different features from the teacher model as targets: we compare using the output of the self-attention module, the feed-forward module (FFN) as well as after the final residual connection (FFN + residual) and layer normalization (End of block). Results are not directly comparable to the main results since we use a reduced setup ( $\S 5.4$ ).  
+
+<table><tr><td>Layer</td><td>WER</td></tr><tr><td>self-attention</td><td>100.0</td></tr><tr><td>FFN</td><td>13.1</td></tr><tr><td>FFN + residual</td><td>14.8</td></tr><tr><td>End of block</td><td>14.5</td></tr></table>
+
+In order to get a better sense of this, we construct target representations which do not have access to the entire input sample but rather only a pre-defined fraction of it. Concretely, we restrict the self-attention mechanism of the teacher to only be able to access a portion of the input surrounding the current time-step. Once the model is trained, we fine-tune it so that it can access the full context size. Figure 3 shows that larger context sizes lead to better downstream performance. The best accuracy is achieved when the entire input sample is visible. This shows that richer target representations can indeed lead to better performance.
+
+Target Feature Type. Transformers blocks contain several layers which can each serve as targets. To get a sense of how different layers impact performance, we pre-train speech models on Librispeech using different layers as target features. Table 5 shows that the output of the feedforward network (FFN) block works best while the output of the self-attention block does not yield a usable model. We believe this is because the self-attention is before the residual connection and features are heavily biased towards other time-steps. This issue is alleviated by the use of FFN features since these also include the features before the self-attention.
+
+# 6. Discussion
+
+# Modality-specific Feature Extractors and Masking.
+
+Our primary goal is to design a single learning mechanism for different modalities. Despite the unified learning regime, we still use modality-specific feature extractors and masking strategies. This makes sense given the vastly different nature of the input data: for example, in speech we learn from a very high resolution input (16 kHz waveform) which contains hundreds of thousands of samples for typical utterances. To process this, we apply a multilayer convolutional neural network to obtain a 50 Hz feature sequence. For NLP, word input sequences have vastly lower resolution which can be directly embedded in distributional space via a lookup table. The type of data also impacts how we should mask the input to create a challenging learning task: removing individual words provides a sufficiently challenging task but for speech it is necessary to mask spans since adjacent audio samples are highly correlated with each other. Relatedly, there has been recent work on a Transformer architecture that can directly operate on the raw data of different modalities without modality-specific feature encoders (Jaegle et al., 2021b;a). Their work is focused on supervised learning for classification tasks and we believe that our work is complementary.
+
+Structured and Contextualized Targets. One of the main differences of data2vec to most other masked prediction work (Devlin et al., 2019; Baevski et al., 2020b; Ling et al., 2020; Bao et al., 2021; He et al., 2021; Wei et al., 2021) is that the features of the training targets are contextualized since the features are built with self-attention over the entire unmasked input in teacher mode. And while BYOL (Grill et al., 2020) and DINO (Caron et al., 2021) also use latent target representations based on the entire input, their focus is on learning transformation-invariant representations instead of structural information within a sample.
+
+One exception is HuBERT (Hsu et al., 2021) which builds a fixed set of discrete target units by clustering Transformer layer representations. In comparison, data2vec has no limitation on the number of target units. Instead of representing each instance of particular discrete target unit with the same set of features, data2vec can build target features that are specific to the current sequence.
+
+For NLP, we believe data2vec is the first work that does not rely on pre-defined target units. Most other work uses either words, sub-words (Radford et al., 2018; Devlin et al., 2019), characters (Tay et al., 2021) or even bytes (Xue et al., 2021). Aside, defining word boundaries is not straightforward for some Asian languages. Contextualized targets enable integrating features from the entire sequence into the training target which provides a richer self-supervised task. Further
+
+more, the representation of each instance of a particular unit (word/sub-word/character/byte) can differ for the masked prediction task. This enables to associate a different meaning to a particular depending on the context it occurs in. It also relieves the model from the need to learn a single set of features for a target unit that fits all instances of this unit.
+
+Representation Collapse. A common failure mode when learning targets is for similar representations for targets to be produced resulting in a trivial task (Jing et al., 2021). To deal with this, contrastive models such as wav2vec 2.0 (Baevski et al., 2020b) use the same target representation both as a positive and a negative example. BYOL (Grill et al., 2020) do not optimize the teacher parameters to minimize the loss and VicReg (Bardes et al., 2021) adds an explicit loss encouraging variance among different representations.
+
+We found that collapse is most likely to happen in the following scenarios: First, the learning rate is too large or the learning rate warmup is too short which can often be solved by tuning the respective hyperparameters. Second,  $\tau$  is too low which leads to student model collapse and is then propagated to the teacher. This can be addressed by tuning  $\tau_0$ ,  $\tau_e$  and  $\tau_n$ . Third, we found collapse to be more likely for modalities where adjacent targets are very correlated and where longer spans need to be masked, e.g., speech. We address this by promoting variance through normalizing target representations over the sequence or batch (Grill et al., 2020). For models where targets are less correlated, such as vision and NLP, momentum tracking is sufficient.
+
+# 7. Conclusion
+
+Recent work showed that uniform model architectures can be effective for multiple modalities (Jaegle et al., 2021b). Whereas we show that a single self-supervised learning regime can be effective for vision, speech and language. The key idea is to regress contextualized representations based on a partial input view. data2vec outperforms prior self-supervised work on ImageNet-1K for ViT-B and ViT-L single models, it improves over prior work on speech recognition for the low-resource Libri-light setups, and it outperforms RoBERTa on GLUE in the original BERT setup.
+
+A single learning method for multiple modalities will make it easier to learn across modalities and future work may investigate tasks such as audio-visual speech recognition or cross-modal retrieval. Our approach still uses modality-specific input encoders and we adopt modality-specific masking strategies which future work may unify.
+
+# Acknowledgements
+
+We thank Brenden Lake, Dhruv Batra, Marco Baroni and Laurens van der Maaten for helpful discussions.
+
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+
+# A. Extended speech processing results
+
+Table 6. Speech processing: word error rate on the Librispeech dev/test sets when training on the Libri-light low-resource labeled data setups of  $10\mathrm{min}$  1 hour, 10 hours and the clean 100h subset of Librispeech. Models use the audio of Librispeech (LS-960) as unlabeled data.  
+
+<table><tr><td rowspan="2">Model</td><td rowspan="2">Unlabeled data</td><td rowspan="2">LM</td><td colspan="2">dev</td><td colspan="2">test</td></tr><tr><td>clean</td><td>other</td><td>clean</td><td>other</td></tr><tr><td>10 min labeled</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>wav2vec 2.0 Base (Baevski et al., 2020b)</td><td>LS-960</td><td>4-gram</td><td>8.9</td><td>15.7</td><td>9.1</td><td>15.6</td></tr><tr><td>Hubert Base (Hsu et al., 2021)</td><td>LS-960</td><td>4-gram</td><td>9.1</td><td>15.0</td><td>9.7</td><td>15.3</td></tr><tr><td>data2vec Base</td><td>LS-960</td><td>4-gram</td><td>7.3</td><td>11.6</td><td>7.9</td><td>12.3</td></tr><tr><td>1h labeled</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>wav2vec 2.0 Base (Baevski et al., 2020b)</td><td>LS-960</td><td>4-gram</td><td>5.0</td><td>10.8</td><td>5.5</td><td>11.3</td></tr><tr><td>Hubert Base (Hsu et al., 2021)</td><td>LS-960</td><td>4-gram</td><td>5.6</td><td>10.9</td><td>6.1</td><td>11.3</td></tr><tr><td>data2vec Base</td><td>LS-960</td><td>4-gram</td><td>4.0</td><td>8.5</td><td>4.6</td><td>9.1</td></tr><tr><td>10h labeled</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>wav2vec 2.0 Base (Baevski et al., 2020b)</td><td>LS-960</td><td>4-gram</td><td>3.8</td><td>9.1</td><td>4.3</td><td>9.5</td></tr><tr><td>Hubert Base (Hsu et al., 2021)</td><td>LS-960</td><td>4-gram</td><td>3.9</td><td>9.0</td><td>4.3</td><td>9.4</td></tr><tr><td>data2vec Base</td><td>LS-960</td><td>4-gram</td><td>3.3</td><td>7.5</td><td>3.9</td><td>8.1</td></tr><tr><td>100h labeled</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Noisy student (Park et al., 2020)</td><td>LS-860</td><td>LSTM</td><td>3.9</td><td>8.8</td><td>4.2</td><td>8.6</td></tr><tr><td>IPL (Xu et al., 2020)</td><td>LL-60K</td><td>4-gram+Transf.</td><td>3.2</td><td>6.1</td><td>3.7</td><td>7.1</td></tr><tr><td>SlimIPL (Likhomanenko et al., 2021)</td><td>LS-860</td><td>4-gram+Transf.</td><td>2.2</td><td>4.6</td><td>2.7</td><td>5.2</td></tr><tr><td>wav2vec 2.0 Base (Baevski et al., 2020b)</td><td>LS-960</td><td>4-gram</td><td>2.7</td><td>7.9</td><td>3.4</td><td>8.0</td></tr><tr><td>Hubert Base (Hsu et al., 2021)</td><td>LS-960</td><td>4-gram</td><td>2.7</td><td>7.8</td><td>3.4</td><td>8.1</td></tr><tr><td>data2vec Base</td><td>LS-960</td><td>4-gram</td><td>2.2</td><td>6.4</td><td>2.8</td><td>6.8</td></tr></table>
+
+# B. Comparison of loss functions
+
+Table 7 shows that different choices of the loss function have a relatively small effect on final performance.  
+Table 7. Different pre-training losses on Librispeech dev-other (no language model).  
+
+<table><tr><td></td><td>WER</td></tr><tr><td>L2</td><td>17.1</td></tr><tr><td>L1</td><td>17.2</td></tr><tr><td>Smooth L1 (β = 0.08)</td><td>17.2</td></tr><tr><td>Smooth L1 (β = 0.25)</td><td>16.8</td></tr><tr><td>Smooth L1 (β = 0.5)</td><td>16.8</td></tr><tr><td>Smooth L1 (β = 1)</td><td>17.3</td></tr></table>
+
+# C. Speech masking parameter ablation
+
+Our method requires different masking parameters for each modality and this makes intuitive sense: masking  $15\%$  of inputs is effective for text but not for images since text tokens are highly semantic and it is sufficient to mask a smaller proportion of the input to construct a useful task. We rely largely on settings from the literature, except for images, where we found that a higher masking rate compared to BEiT works slightly better. When we tried tuning masking hyperparameters for other
+
+modalities than vision, we did not see significant improvements (e.g., see Table 8) for speech. The small improvements we saw are likely to disappear after adding a language model.
+
+Table 8. Ablation of speech masking parameters. Results are on Librispeech dev-other without a language model.  
+
+<table><tr><td></td><td>WER</td></tr><tr><td>baseline (mask prob = 0.65, mask len = 10)</td><td>17.1</td></tr><tr><td>mask prob = 0.8</td><td>17.2</td></tr><tr><td>mask prob = 0.5</td><td>17.3</td></tr><tr><td>mask len = 5</td><td>22.1</td></tr><tr><td>mask len = 15</td><td>18.9</td></tr></table>

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+# Moshe Eliasof<sup>1</sup> Eldad Haber<sup>2</sup> Eran Treister<sup>1</sup>
+
+# Abstract
+
+Graph Convolutional Networks (GCNs), similarly to Convolutional Neural Networks (CNNs), are typically based on two main operations - spatial and point-wise convolutions. In the context of GCNs, differently from CNNs, a pre-determined spatial operator based on the graph Laplacian is often chosen, allowing only the point-wise operations to be learnt. However, learning a meaningful spatial operator is critical for developing more expressive GCNs for improved performance. In this paper we propose pathGCN, a novel approach to learn the spatial operator from random paths on the graph. We analyze the convergence of our method and its difference from existing GCNs. Furthermore, we discuss several options of combining our learnt spatial operator with point-wise convolutions. Our extensive experiments on numerous datasets suggest that by properly learning both the spatial and point-wise convolutions, phenomena like over-smoothing can be inherently avoided, and new state-of-the-art performance is achieved.
+
+# 1. Introduction
+
+The study of Graph Convolutional Networks (GCNs) has gained large popularity in recent years (Bruna et al., 2013; Defferrard et al., 2016; Kipf & Welling, 2016; Bronstein et al., 2017; Monti et al., 2017) in a wide variety of fields and applications such as computer graphics and vision (Boscaini et al., 2016; Monti et al., 2017; Wang et al., 2018; Eliasof & Treister, 2020), Bioinformatics (Strokach et al., 2020; Jumper et al., 2021), node classification (Kipf & Welling, 2016; Chen et al., 2020; Chamberlain et al., 2021) and others. The common ingredient that most of the methods share
+
+$^{1}$ Department of Computer Science, Ben-Gurion University, Israel.  $^{2}$ Department of Earth, Ocean and Atmospheric Sciences, University of British Columbia, Canada.. Correspondence to: Moshe Eliasof <aliasof@post.bgu.ac.il>, Eran Treister <erant@cs.bgu.ac.il>.
+
+Proceedings of the  $39^{th}$  International Conference on Machine Learning, Baltimore, Maryland, USA, PMLR 162, 2022. Copyright 2022 by the author(s).
+
+![](images/c74afd7c9742ef846ddac02f0001bd714f2aac0a078579cc03cae3d35fcb74ab.jpg)  
+Figure 1. The spatial operator induced by a smoothing kernel on different graphs. The vertex with dashed outline is the path origin.
+
+is the use of a pre-determined spatial operator, often times based on the graph Laplacian. While this choice is intuitive and effective, it induces limitations on the behaviour of GCNs. First, it is limited in the aspect of the expressiveness of the networks. Unlike CNNs (Krizhevsky et al., 2012; He et al., 2016; Howard et al., 2017), where both the spatial filters (e.g.,  $3 \times 3$  depth-wise convolutions) and point-wise  $(1 \times 1)$  convolutions are learnt, here only the latter are left to be determined. Secondly, it is well known (Wu et al., 2019) that the Laplacian operator, when applied as in (Kipf & Welling, 2016) smoothes the input features, and therefore a recurrent application of it may lead to over-smoothing, resulting in typically shallow networks. This phenomenon is well documented and studied in the field of GCNs (Wu et al., 2019; Zhao & Akoglu, 2020; Chen et al., 2020; Chamberlain et al., 2021; Eliasof et al., 2021). In this work we propose pathGCN – a novel approach that overcomes the limitations above, based on aggregation from random paths defined over the graph vertices. We show that using this approach it is possible to define spatial operators similarly to the ones used in 2D convolutions on images. Such operators have variable aperture and coefficients that may increase the expressiveness of GCNs. In addition, since the coefficients of the operator are learnt, its eigenvalues can be rather different than those of the graph Laplacian. This implies that the learnt kernels can take different roles, from smoothing to edge-detecting (or sharpening) operators. An example of the effective spatial operators that are induced by the smoothing spatial kernel [0.8, 1.0, 0.6] is presented in Fig. 1, where we can see that the spatial operator is dependent both on the graph topology and the spatial kernel.
+
+We provide an analysis of our method to motivate our approach in Sec. 3, and present actual learnt kernels by our network in Fig. 4, which suggests that greater spatial flexibility is required to obtain better performance and to avoid over-smoothing, as reflected in our experiments in Sec. 4.
+
+Our contributions are as follows:
+
+- We introduce pathGCN - a novel approach for learning expressive spatial operators for GCNs from random paths. pathGCN supports several formulations, similarly to standard CNNs - ranging from a global to per layer and per channel learnt spatial operators.  
+- We provide an analysis of the behaviour of our pathGCN, and present a stochastic path training policy.  
+- Our experiments reveal the significance of the learnt spatial operator by obtaining and improving the state-of-the-art accuracy on various benchmarks, while also inherently preventing over-smoothing.
+
+# 2. Related work
+
+# 2.1. Graph convolutional networks
+
+Notations. Assume we are given an undirected graph defined as  $\mathcal{G} = (\mathcal{V},\mathcal{E})$  where  $\mathcal{V}$  is a set of  $n$  vertices (nodes) and  $\mathcal{E}$  is a set of  $m$  edges. Let us denote by  $\mathbf{f}_i\in \mathbb{R}^c$  the feature vector that resides at the  $i$ -th node of  $\mathcal{G}$  with  $c$  being the number of channels. Also, we denote the adjacency matrix  $\mathbf{A}$ , where  $\mathbf{A}_{ij} = 1$  if there exists an edge  $(i,j)\in \mathcal{E}$  and the diagonal degree matrix  $\mathbf{D}$  where  $\mathbf{D}_{ii}$  equals to the degree of the  $i$ -th node. The graph Laplacian is given by  $\mathbf{L} = \mathbf{D} - \mathbf{A}$ , and its symmetric normalized formulation reads  $\mathbf{L}^{sym} = \mathbf{I} - \mathbf{D}^{-\frac{1}{2}}\mathbf{A}\mathbf{D}^{-\frac{1}{2}}$  where  $\mathbf{I}$  is the identity operator. Let us also denote the adjacency and degree matrices after adding a self-loop to the nodes by  $\tilde{\mathbf{A}}$  and  $\tilde{\mathbf{D}}$  respectively, and accordingly define the normalized Laplacian of  $\mathcal{G}$  with added self-loops by  $\tilde{\mathbf{L}}^{sym} = \mathbf{I} - \tilde{\mathbf{D}}^{-\frac{1}{2}}\tilde{\mathbf{A}}\tilde{\mathbf{D}}^{-\frac{1}{2}}$ . It follows that the spatial operation from GCN (Kipf & Welling, 2016), induced by the graph Laplacian is given by  $\tilde{\mathbf{P}} = \mathbf{I} - \tilde{\mathbf{L}}^{sym}$ . We refer the readers to (Wu et al., 2019) for more information.
+
+The spatial operator in GCNs. GCNs typically involve two main ingredients: spatial and point-wise  $(1 \times 1)$  convolutions to mix the channels. The majority of GCNs employ a spatial operator based on the graph Laplacian, followed by a point-wise convolution. For example, GCN (Kipf & Welling, 2016) is given by:
+
+$$
+\mathbf {f} ^ {(l + 1)} = \sigma \left(\mathbf {S} ^ {(l)} \mathbf {f} ^ {(l)} \mathbf {W} ^ {(l)}\right) \tag {1}
+$$
+
+where  $\mathbf{S}^{(l)} = \tilde{\mathbf{P}}$ , and  $\mathbf{W}^{(l)}$  is a  $1 \times 1$  convolution operator. The combination of the operator  $\tilde{\mathbf{P}}$  and a learnable  $1 \times 1$  convolution continued in a series of works (Wu et al., 2019; Chen et al., 2020; Zhou et al., 2021). For instance, the spatial operation in GCNII (Chen et al., 2020) can be obtained by replacing  $\mathbf{S}^{(l)}$  with:
+
+$$
+\mathbf {S} ^ {(l)} \left(\mathbf {f} ^ {(l)}, \mathbf {f} ^ {(0)}\right) = (1 - \alpha^ {(l)}) \tilde {\mathbf {P}} \mathbf {f} ^ {(l)} + \alpha^ {(l)} \mathbf {f} ^ {(0)}, \tag {2}
+$$
+
+with  $\alpha^{(l)}\in [0,1]$  being a hyper-parameter and  $\mathbf{f}^{(0)}$  are the features of the first (embedding) layer, which is also similar to APPNP (Klicpera et al., 2019). Another recent example is EGNN (Zhou et al., 2021) which performs
+
+$$
+\mathbf {S} ^ {(l)} \left(\mathbf {f} ^ {(l)}, \mathbf {f} ^ {(0)}\right) = \left(1 - c _ {\min }\right) \tilde {\mathbf {P}} \mathbf {f} ^ {(l)} + \alpha \mathbf {f} ^ {(l)} + \beta \mathbf {f} ^ {(0)}, \tag {3}
+$$
+
+where  $c_{min} = \alpha +\beta$ , and  $\alpha, \beta$  are learnt scalars. While the methods above achieve impressive accuracy, they are still limited in their spatial expressiveness due to their dependence on  $\tilde{\mathbf{P}}$ , which may lead to sub-optimal performance.
+
+On the other hand, there are methods that allow the freedom of learning a rich spatial operator. For example, ChebNet (Defferrard et al., 2016) learns dense convolutions through polynomial parameterization of the graph Laplacian. While from a conceptual perspective, ChebNet should be able to learn diverse operators and prevent over-smoothing, it still exhibits a degradation in performance when adding more layers, as discussed in (Levie et al., 2018) and according to our experiments as presented in Fig. 3. In addition, GDC (Gasteiger et al., 2019) propose to impose constraints on the filters of ChebNet to obtain diffusion kernels. Another network to consider is MoNet (Monti et al., 2017), which learns local patch operators by a mixture of Gaussian. While the Gaussian parameterization can yield more expressive kernels and favourable results compared to GCN (Kipf & Welling, 2016), it is more computationally demanding and challenging to train as more layers are stacked.
+
+# 2.2. Random walk on graphs
+
+The concept of random walk is useful in many applications and domains, from graph node classification (Perozzi et al., 2014; Nikolentzos & Vazirgiannis, 2020) to RNA disease association (Lei & Bian, 2020) and mesh denoising (Sun et al., 2008). In the context of graphs in machine and deep learning, multiple methods utilized random walks for different purposes. DeepWalk (Perozzi et al., 2014) is a two-step algorithm for node embedding learning, based on random walks and the SkipGram (Mikolov et al., 2013) method. RWGNN (Nikolentzos & Vazirgiannis, 2020) learns a set of hidden graphs which are then compared with an input graph using a differentiable mutual random walk counting procedure. Our approach is different as we utilize the random walk to effectively sample paths, and employ them to learn a spatial operator by the means of a convolution kernel. Another difference is that RGWNN considers the graph classification problem, as it employs the random walk kernel which is a scalar binary function of two graphs, whose output discards the notion of the graph itself. Therefore, it is not straight-forward to use it for node classification tasks. The recent PAN (Ma et al., 2020) parameterizes the convolution kernel by a polynomial of the adjacency matrix, where the coefficients are the learnt parameters, motivated by path integral theory. However, as its formulation consists of
+
+learning non-negative weights, it is prone to over-smoothing and reduced expressiveness. Another related work that uses random walk strategy is GraphSAGE (Hamilton et al., 2017) where neighbourhood sampling of  $k$  hops is performed. This method is based on two steps in which an aggregation from two subsequent hops is performed, followed by a  $1 \times 1$  convolution. This is different than ours, as we first extract the complete path, and learn a convolution kernel based on the original information along this path, while the former iteratively aggregates from subsequent hops.
+
+# 3. Method
+
+# 3.1. Learning spatial operators
+
+In this section we motivate the need for learnt spatial operators. To do that, let us first consider a standard CNN where the data resides on a simple uniform mesh-grid. We note that CNNs and GCNs both represent data with geometrical features. However, while CNNs are networks that operate on a simple mesh-grid graph where pixels (nodes) are linked based on their location, and the local geometry of the graph is fixed, GCNs can be thought of as unstructured meshes where the local geometry varies.
+
+Given a feature tensor  $\mathbf{f} \in \mathbb{R}^{n \times c}$ , the convolution in CNNs denoted by  $\mathbf{Kf}$ , is a linear operation where each input channel and each output channel has its own spatial operator. Thus, each linear operator  $\mathbf{K}$  consists of  $c \times c$  different spatial convolutions represented by the tensor  $\mathbf{K}$ . Furthermore, the convolutions can have a variable aperture (i.e., kernel size) obtaining a larger field of view, and are typically learnt per-layer, yielding a highly expressive set of operators.
+
+In contrast, many popular and recent GCNs (Kipf & Welling, 2016; Wu et al., 2019; Chen et al., 2020; Chamberlain et al., 2021; Zhou et al., 2021) and others employ a pre-determined spatial operator, often times guided by the graph Laplacian which is determined solely by the topology of the graph  $\mathcal{G}$ , coupled with a  $1 \times 1$  convolution to mix the channels. Therefore, by comparing GCNs to CNNs it is notable that the former have significantly fewer degrees of freedom with respect to their spatial operation. Indeed, 2D CNNs with a spatial kernel of size  $k$  and  $c$  channels optimize  $c \times c \times k \times k$  parameters, while GCNs typically only optimize the  $1 \times 1$  convolution, yielding  $c \times c$  parameters. In addition to the possible expressiveness issue, the frequent of the graph Laplacian can lead to undesired phenomena such as oversmoothing (Wu et al., 2019; Zhao & Akoglu, 2020; Chen et al., 2020). Some attempts to overcome the expressiveness limitation consider using polynomials of the graph Laplacian, stabilizing them by constructing a Chebyshev basis (Defferrard et al., 2016). However, it imposes high computational cost due to the frequent computations of the Chebyshev polynomial and fully-connected convolution fil
+
+ters. Furthermore, it is difficult to train such a network due to its eigenvalues distribution, as demonstrated in (Levie et al., 2018). With respect to the over-smoothing phenomenon, various techniques were proposed (Zhao & Akoglu, 2020; Chen et al., 2020; Rong et al., 2020). While those methods indeed aid the over-smoothing issue, they do not inherently change the smoothing behaviour of GCNs, but rather ease the smoothing process.
+
+In what follows, we present a methodology that allows the construction of a graph convolution that is similar to the standard convolution on a regular mesh grid which allows greater expressiveness and inherently does not over-smooth.
+
+# 3.2. From fixed to variable spatial operator
+
+The non-constant topology of the graph is the major obstacle in generating a meaningful spatial convolution in GCNs. We now show that this can be addressed by using random walks.
+
+To this end, we consider a path on the graph, on which the weights that parameterize the spatial operator are learnt. Therefore, a transition function that dictates some traversal strategy on the graph is required in order to obtain a path like input to our network. Specifically, we adopt the graph random walk generator from node2vec (Grover & Leskovec, 2016), as implemented in PyTorch-Geometric (Fey & Lenssen, 2019), for its simplicity and efficient implementation. We note, however, that a different transition function could also be used or learnt.
+
+Assume first that we have a single channel feature tensor  $\mathbf{f} \in \mathbb{R}^{n \times 1}$ , and a path of length  $k$  is given. Let us denote the learnt spatial parameters by  $\mathbf{s} \in \mathbb{R}^k$ , and let  $y_j = (j_0, \dots, j_{k-1})$  be a tuple of node indices of a single random path of nodes of length  $k$ , starting from node  $j_0 = j$ . The convolution over a single path for the  $j$ -th node is defined by the following linear operator:
+
+$$
+\mathbf {K} _ {y _ {j}} (\mathbf {s}) \mathbf {f} = \sum_ {i = 0} ^ {k - 1} s _ {i} f _ {j _ {i}}. \tag {4}
+$$
+
+That is, the features of the nodes on the path  $y_{j}$  are weighted by the corresponding learnt parameter in  $\mathbf{s}$ , and summed to get the feature of the  $j$ -th node—the node where the path originates from.
+
+More generally, instead of considering a single path, let us sample  $p$  different paths, and accordingly define the paths convolution as the average over all sampled paths
+
+$$
+\mathbf {K} _ {\mathcal {Y} _ {j}} (\mathbf {s}) \mathbf {f} = \frac {1}{p} \sum_ {y _ {j} \in \mathcal {Y} _ {j}} \mathbf {K} _ {y _ {j}} (\mathbf {s}) \mathbf {f} \tag {5}
+$$
+
+where  $y_{j}\in \mathcal{V}_{j}$  is a path from a set of  $p$  random walks starting from the  $j$ -th node.
+
+![](images/704c92d593ea4126aa19b2e50b07c83fd6a28cac94ee300d3c14a929e9c05b7e.jpg)  
+Figure 2. The effective spatial operator from a walk of length 3 starting at the dashed node, ranging from 1 to 1000 random walks.
+
+![](images/d6e09aa41c2f5bb0c9339ce6dc2851ba7088406aa817ea27196821ecfcc844c1.jpg)
+
+![](images/f183f089601999a5b2356a5eaa7b8d95514ac686dd6a5cd6d40737de86d024b8.jpg)
+
+![](images/5cc67e1bdbbe389616a9616418b24b6e8f53b8809c77cda3309075c416787a5f.jpg)
+
+![](images/b9a4da272fc4e2a05f34613cc3515cea596909fa870e9083a773ee31eef35c0e.jpg)
+
+![](images/04c9490a2f20fa801a1c54964a9ae6704ad0a64eb85aad7f8875543bb500e0cb.jpg)
+
+![](images/43103508d57351b2784230fa2e4f89c488fbb7d2b669bc33bc5d5921c6b83b64.jpg)
+
+# 3.3. Constructing pathGCN
+
+So far we defined the path convolution, which operates in the spatial domain. In what follows, we omit s for brevity, and denote by  $\mathbf{K}_{\mathcal{V}}$  the pathConv module described in Eq. (5) on all the nodes in  $\mathcal{V}$  given their corresponding path realizations  $\mathcal{V}$ . To obtain a complete network, we need to add the channel mixing convolution and the non-linear activation  $\sigma$ , as follows
+
+$$
+\mathbf {f} ^ {(l + 1)} = \sigma \left(\mathbf {W} ^ {(l)} \mathbf {K} _ {\mathcal {Y}} ^ {(l)} \mathbf {f} ^ {(l)}\right), \tag {6}
+$$
+
+where in our implementation  $\sigma$  is ReLU and  $\mathbf{W}^{(l)}$  is a  $1\times 1$  (i.e., point-wise) convolution. The simplest utilization of pathConv applies a single spatial operator parameterized by  $\mathbf{s}\in \mathbb{R}^k$ , shared among all channels and all  $L$  layers. While this is more flexible than using  $\tilde{\mathbf{P}}$  as in Eq. (1), it can be further generalized. Scaling up, it is also possible to learn a different parameterization of the spatial operator per layer, that is,  $\mathbf{s}\in \mathbb{R}^{L\times k}$ . In practice and inspired by modern CNNs (Sandler et al., 2018; Ephrath et al., 2020), we found that learning a depth-wise spatial operator (i.e., a spatial operator per channel and layer such that  $\mathbf{s}\in \mathbb{R}^{L\times c\times k}$ ) followed by a  $1\times 1$  convolution leads to favorable performance both accuracy- and computationally-wise across all the considered data sets in this paper, as reflected in our experiments in Sec. 4. We refer to this architecture as pathGCN. We note that it is also possible to learn different spatial operators for all pairs of channels, per layer - similarly to a standard fully-connected CNN.
+
+Finally, learning the spatial convolution allows for a variety of operators, from non-smoothing (e.g., edge-detection filters) to smoothing operators. Thus, as we demonstrate in the experiments section, our network achieves state-of-the-art performance, and does not suffer from over-smoothing.
+
+# 3.4. Convergence analysis
+
+The process defined in Eq. (4)-(5) represents a simple stochastic process to build a spatial operation for a single channel that depends on two quantities. First, it depends on the algorithm used to sample the set of paths  $\mathcal{V}$ , and second, it depends on the learnt weights  $s$ . A natural question that
+
+arises is - does this process converge, and to what? To answer this question we need to make a mild assumption on the random walk process. Namely, we assume that the process is Markovian, that is, at each state the walk can visit any neighbouring node of the current node at an equal probability. In the following, we show that at the sampling limit (i.e.,  $p \to \infty$ ), there exists a stationary distribution of the paths, which induces a spatial operator sampled by our method, as exemplified in Fig. 2.
+
+Let us consider the simple case of a convolution on a general path of length 2, which can be written as
+
+$$
+\mathbf {K} _ {y _ {j}} (\mathbf {s}) \mathbf {f} = s _ {0} f _ {j _ {0}} + s _ {1} f _ {j _ {1}} = s _ {0} f _ {j} + s _ {1} f _ {j _ {1}}. \tag {7}
+$$
+
+The second transition follows from the fact that the path originates from the  $j$ -th node, i.e.,  $f_{j_0} = f_j$ . Note, that  $j_1$  can represent any of the immediate neighbours of the  $j$ -th node. Therefore, the expectation of (7) which corresponds to all the nodes  $\mathcal{V}$  is given by
+
+$$
+\mathbb {E} _ {y} \left(\mathbf {K} _ {y} (\mathbf {s}) \mathbf {f}\right) = s _ {0} \mathbf {f} + s _ {1} \left(\mathbf {A D} ^ {- 1}\right) \mathbf {f} \tag {8}
+$$
+
+where  $\mathbf{A}$  is the adjacency matrix and  $\mathbf{D}$  is a diagonal matrix with the degree of each node.
+
+More generally, let us consider a path of an arbitrary length  $k$ . In this case we have that the expectation term reads
+
+$$
+\mathbb {E} _ {y} \left(\mathbf {K} _ {y} (\mathbf {s}) \mathbf {f}\right) = \left(\sum_ {i = 0} ^ {k - 1} s _ {i} \left(\mathbf {A D} ^ {- 1}\right) ^ {i}\right) \mathbf {f} \tag {9}
+$$
+
+where  $(\mathbf{AD}^{-1})^0 = \mathbf{I}$ . We note that the transition matrix  $\mathbf{AD}^{-1}$  is column-stochastic, and therefore its eigenvalues are bounded between  $[-1, 1]$ . Thus, this polynomial formulation is stable and does not diverge for an appropriate choice of coefficients  $\mathbf{s}$ .
+
+Furthermore, Eq. (9) is a deterministic representation of the process given in Eq. (5). If the number of path-realizations,  $p$  is large then the results from both are similar. For short paths, a direct evaluation of Eq. (9) can be computationally advantageous compared to its stochastic implementation. However, for long paths, computing the powers of the adjacency matrix times a vector can be expensive. The average
+
+over paths is, in this case, an economical way to approximate the process, avoiding repeated matrix multiplications.
+
+Lastly, as we show in our experiments, the stochastic nature of the process has additional advantages. In particular, it allows for better trainability and generalization, compared to the deterministic form. Indeed, a significant increase in the value of  $p$  causes performance degradation, as reported in Fig. 5. For inference purposes, we may use both the stochastic or deterministic formulations from Eq. (5) and (9) respectively, yielding similar results for both.
+
+# 3.5. Computational cost and number of parameters
+
+We compare the cost of our pathGCN with the methods considered in Eq. (1)-(3). On the spatial side, our pathGCN involves  $n \times k \times p$  operations rather than  $n \times d$  for the considered methods, where  $d$  is the mean node degree of the graph. Therefore, if  $k \times p$  is larger than  $d$ , our method requires more computations. Nonetheless, our pathGCN has a wider aperture as it considers paths of  $k$  nodes, thus its field of view is larger. We also compare the number of trainable parameters, which further highlights the difference of our approach. Recall that our pathGCN learns the spatial operator in addition to a  $1 \times 1$  convolution, which is also present in other methods. That is, per pathGCN layer, the spatial weights that parameterize  $\mathbf{K}_y$  require  $c \times k$  parameters, while the  $1 \times 1$  convolution  $\mathbf{W}$  requires  $c \times c$  parameters, where typically  $k$  is significantly smaller than  $c$ . In our experiments, different values of  $k$  and  $p$  were examined, and we found that setting  $k$  between 3 to 7 and  $p$  between 5 to 10 achieves better or on par with state-of-the-art models while keeping the computational cost reasonable.
+
+We also consider the stochastic implementation of a random walk over an arbitrary graph. Sampling a single path of length  $k$  for all nodes in  $\mathcal{V}$  requires  $n \times k$  operations<sup>1</sup>. Therefore, if the number of paths  $p$  is smaller than the average node degree  $d$  of the graph, then the cost of random walk sampling is smaller than the cost of applying the adjacency matrix as in (9). Furthermore, in the deterministic case, a spatial operation on a path of length 2 requires  $n \times d$  computations. Extending it to a path of length  $k$  requires  $n \times d \times (k - 1)$  operations, realized by polynomials of degree  $k - 1$  of the adjacency matrix. We note that Eq. (9), which is the deterministic form of our method, can also be used for inference purposes.
+
+# 4. Experiments
+
+We demonstrate our pathGCN on node classification and protein-protein interaction (Hamilton et al., 2017), followed by an ablation study in order to gain a profound understand-
+
+Table 1. Node classification datasets statistics.  
+
+<table><tr><td>Dataset</td><td>Classes</td><td>Nodes</td><td>Edges</td><td>Features</td></tr><tr><td>Cora</td><td>7</td><td>2,708</td><td>5,429</td><td>1,433</td></tr><tr><td>CiteSeer</td><td>6</td><td>3,327</td><td>4,732</td><td>3,703</td></tr><tr><td>Pubmed</td><td>3</td><td>19,717</td><td>44,338</td><td>500</td></tr><tr><td>Chameleon</td><td>5</td><td>2,277</td><td>36,101</td><td>2,325</td></tr><tr><td>Cornell</td><td>5</td><td>183</td><td>295</td><td>1,703</td></tr><tr><td>Texas</td><td>5</td><td>183</td><td>309</td><td>1,703</td></tr><tr><td>Wisconsin</td><td>5</td><td>251</td><td>499</td><td>1,703</td></tr><tr><td>PPI</td><td>121</td><td>56,944</td><td>818,716</td><td>50</td></tr><tr><td>Wiki-CS</td><td>10</td><td>11,701</td><td>216,123</td><td>300</td></tr><tr><td>Actor</td><td>5</td><td>7,600</td><td>33,544</td><td>932</td></tr><tr><td>Ogbn-arxiv</td><td>40</td><td>169,343</td><td>1,166,243</td><td>128</td></tr></table>
+
+Table 2. Summary of semi-supervised node classification accuracy (%)  
+
+<table><tr><td>Method</td><td>Cora</td><td>Citeseer</td><td>Pubmed</td></tr><tr><td>ChebNet</td><td>81.2</td><td>69.8</td><td>74.4</td></tr><tr><td>GCN</td><td>81.1</td><td>70.8</td><td>79.0</td></tr><tr><td>GAT</td><td>83.1</td><td>70.8</td><td>78.5</td></tr><tr><td>APPNP</td><td>83.3</td><td>71.8</td><td>80.1</td></tr><tr><td>JKNET</td><td>81.1</td><td>69.8</td><td>78.1</td></tr><tr><td>GCNII</td><td>85.5</td><td>73.4</td><td>80.3</td></tr><tr><td>GRAND</td><td>84.7</td><td>73.6</td><td>81.0</td></tr><tr><td>PDE-GCN</td><td>84.3</td><td>75.6</td><td>80.6</td></tr><tr><td>EGNN</td><td>85.7</td><td>-</td><td>80.1</td></tr><tr><td>pathGCN (Ours)</td><td>85.8</td><td>75.8</td><td>82.7</td></tr></table>
+
+ing of our method. In all experiments, we employ a network that is comprised of an embedding layer ( $1 \times 1$  convolution), followed by a sequence of pathGCN layers, whose final output is fed to a  $1 \times 1$  convolution layer which acts as a classifier. A detailed description of the network architecture is given in Appendix A. We use the Adam (Kingma & Ba, 2014) optimizer in all experiments, and perform grid search over the hyper-parameters of our network. The selected hyper-parameters are reported in Appendix B. The objective function in all experiments is the cross-entropy loss, besides inductive learning on PPI where we use the binary cross-entropy loss. Our code is implemented using PyTorch (Paszke et al., 2019) and PyTorch-Geometric (Fey & Lenssen, 2019) and trained on an Nvidia Titan RTX GPU.
+
+We show that for all the considered tasks and datasets, whose statistics are provided in Tab. 1, our method is either better or on par with other state-of-the-art models.
+
+![](images/d4af74e0a34c82ac54d8580ab342c1e0f847cddddf4e8e6b647cf0320ff4ed3a.jpg)  
+Figure 3. ChebNet vs pathGCN on semi-supervised node classification.
+
+# 4.1. Semi-supervised node classification
+
+Here, we use three datasets - Cora, Citeseer and Pubmed (Sen et al., 2008). For all datasets we use the standard training/validation/testing split as in (Yang et al., 2016), with 20 nodes per class for training, 500 validation nodes and 1,000 testing nodes and follow the training scheme of (Chen et al., 2020). For comparison, we consider various models like ChebNet (Defferrard et al., 2016), GCN (Kipf & Welling, 2016), GAT(Veličković et al., 2018), Inception (Szegedy et al., 2017), APPNP (Klicpera et al., 2019), JKNet (Xuet al., 2018), DropEdge (Rong et al., 2020), GCNII(Chenet al., 2020), GRAND (Chamberlain et al., 2021), PDE-GCN (Eliasof et al., 2021) and EGNN (Zhou et al., 2021).
+
+As discussed in Sec. 3, our pathGCN is constructed such that wider spatial operators are obtained, allowing for improved expressiveness, namely, compared to methods that are based on the graph Laplacian or its proxies. As objectively portrayed by Tab. 2 and 3, our pathGCN is capable of obtaining higher accuracy on all three datasets. A bold improvement is obtained on Pubmed, where an accuracy of  $82.7\%$  is achieved, compared to previously state-of-the-art  $\mathrm{GRAND}_{\mathrm{nl - rw}}$  with  $81.0\%$ . Our approach also benefits from the inherent absence of over-smoothing, as the spatial operator is fully learnt. This is also validated by inspecting the learnt kernels of different layers of the network. Indeed, as shown in Fig. 4, some layers perform smoothing by averaging, while others act as edge-detection filters by considering the difference of neighbouring nodes along the path.
+
+# 4.2. Fully-supervised node classification
+
+To further validate our method, we employ a total of 10 datasets. First, we follow (Pei et al., 2020) and examine our pathGCN on Cora, Citeseer, Pubmed, Chameleon (Rozemberczki et al., 2021), Cornell, Texas and Wisconsin. We also use the same train/validation/test splits of  $60\%$ ,  $20\%$ ,  $20\%$ ,
+
+Table 3. Semi-supervised node classification accuracy  $(\%)$ . - indicates not available results.  
+
+<table><tr><td rowspan="2">Dataset</td><td rowspan="2">Method</td><td colspan="6">Layers</td></tr><tr><td>2</td><td>4</td><td>8</td><td>16</td><td>32</td><td>64</td></tr><tr><td rowspan="11">Cora</td><td>GCN</td><td>81.1</td><td>80.4</td><td>69.5</td><td>64.9</td><td>60.3</td><td>28.7</td></tr><tr><td>GCN (Drop)</td><td>82.8</td><td>82.0</td><td>75.8</td><td>75.7</td><td>62.5</td><td>49.5</td></tr><tr><td>JKNet</td><td>-</td><td>80.2</td><td>80.7</td><td>80.2</td><td>81.1</td><td>71.5</td></tr><tr><td>JKNet (Drop)</td><td>-</td><td>83.3</td><td>82.6</td><td>83.0</td><td>82.5</td><td>83.2</td></tr><tr><td>Incep</td><td>-</td><td>77.6</td><td>76.5</td><td>81.7</td><td>81.7</td><td>80.0</td></tr><tr><td>Incep (Drop)</td><td>-</td><td>82.9</td><td>82.5</td><td>83.1</td><td>83.1</td><td>83.5</td></tr><tr><td>GCNII</td><td>82.2</td><td>82.6</td><td>84.2</td><td>84.6</td><td>85.4</td><td>85.5</td></tr><tr><td>GCNII*</td><td>80.2</td><td>82.3</td><td>82.8</td><td>83.5</td><td>84.9</td><td>85.3</td></tr><tr><td>PDE-GCND</td><td>82.0</td><td>83.6</td><td>84.0</td><td>84.2</td><td>84.3</td><td>84.3</td></tr><tr><td>EGNN</td><td>83.2</td><td>-</td><td>-</td><td>85.4</td><td>-</td><td>85.7</td></tr><tr><td>pathGCN (Ours)</td><td>84.2</td><td>84.5</td><td>84.6</td><td>85.1</td><td>85.4</td><td>85.8</td></tr><tr><td rowspan="10">CiteSeer</td><td>GCN</td><td>70.8</td><td>67.6</td><td>30.2</td><td>18.3</td><td>25.0</td><td>20.0</td></tr><tr><td>GCN (Drop)</td><td>72.3</td><td>70.6</td><td>61.4</td><td>57.2</td><td>41.6</td><td>34.4</td></tr><tr><td>JKNet</td><td>-</td><td>68.7</td><td>67.7</td><td>69.8</td><td>68.2</td><td>63.4</td></tr><tr><td>JKNet (Drop)</td><td>-</td><td>72.6</td><td>71.8</td><td>72.6</td><td>70.8</td><td>72.2</td></tr><tr><td>Incep</td><td>-</td><td>69.3</td><td>68.4</td><td>70.2</td><td>68.0</td><td>67.5</td></tr><tr><td>Incep (Drop)</td><td>-</td><td>72.7</td><td>71.4</td><td>72.5</td><td>72.6</td><td>71.0</td></tr><tr><td>GCNII</td><td>68.2</td><td>68.8</td><td>70.6</td><td>72.9</td><td>73.4</td><td>73.4</td></tr><tr><td>GCNII*</td><td>66.1</td><td>66.7</td><td>70.6</td><td>72.0</td><td>73.2</td><td>73.1</td></tr><tr><td>PDE-GCND</td><td>74.6</td><td>75.0</td><td>75.2</td><td>75.5</td><td>75.6</td><td>75.5</td></tr><tr><td>pathGCN (Ours)</td><td>74.3</td><td>74.8</td><td>75.4</td><td>75.3</td><td>75.6</td><td>75.8</td></tr><tr><td rowspan="11">Pubmed</td><td>GCN</td><td>79.0</td><td>76.5</td><td>61.2</td><td>40.9</td><td>22.4</td><td>35.3</td></tr><tr><td>GCN (Drop)</td><td>79.6</td><td>79.4</td><td>78.1</td><td>78.5</td><td>77.0</td><td>61.5</td></tr><tr><td>JKNet</td><td>-</td><td>78.0</td><td>78.1</td><td>72.6</td><td>72.4</td><td>74.5</td></tr><tr><td>JKNet (Drop)</td><td>-</td><td>78.7</td><td>78.7</td><td>79.7</td><td>79.2</td><td>78.9</td></tr><tr><td>Incep</td><td>-</td><td>77.7</td><td>77.9</td><td>74.9</td><td>-</td><td>-</td></tr><tr><td>Incep (Drop)</td><td>-</td><td>79.5</td><td>78.6</td><td>79.0</td><td>-</td><td>-</td></tr><tr><td>GCNII</td><td>78.2</td><td>78.8</td><td>79.3</td><td>80.2</td><td>79.8</td><td>79.7</td></tr><tr><td>GCNII*</td><td>77.7</td><td>78.2</td><td>78.8</td><td>80.3</td><td>79.8</td><td>80.1</td></tr><tr><td>PDE-GCND</td><td>79.3</td><td>80.6</td><td>80.1</td><td>80.4</td><td>80.2</td><td>80.3</td></tr><tr><td>EGNN</td><td>79.2</td><td>-</td><td>-</td><td>80.0</td><td>-</td><td>80.1</td></tr><tr><td>pathGCN (Ours)</td><td>81.8</td><td>81.8</td><td>82.4</td><td>82.5</td><td>82.4</td><td>82.7</td></tr></table>
+
+respectively, and report the average performance over 10 random splits from (Pei et al., 2020). We fix the number of channels to 64 and perform grid search to determine the hyper-parameters. We compare our network with GCN, GAT, Geom-GCN (Pei et al., 2020), APPNP, JKNet, Inception, GCNII and PDE-GCN in Tab. 4. Our experiments read improvement across all data-sets compared to all the considered methods. For instance, we obtain  $90.02\%$  accuracy on Cora with our pathGCN, compared to  $88.49\%$  of GCNII* and  $88.60\%$  of PDE-GCN. In addition, we examine our pathGCN on larger datasets using the standard train/validation/test splits of Actor (Pei et al., 2020), Ogbn-axiv (Hu et al., 2020) and Wiki-CS (20 random splits) (Mernyei & Cangea, 2020) in Tab. 5 and 6 – where again we see accuracy improvement across all considered datasets.
+
+![](images/7246d47e626ffc29528d6e83962ef558c5ecab9c5171f8ceeb1069e290ece7bf.jpg)  
+Figure 4. A visualization of the learnt spatial operator  $\mathbf{K}_{\mathcal{V}}$  from a 8-layer pathGCN trained on Cora, applied on the dashed node in the graph (top), and its corresponding learnt weights (bottom).
+
+Table 4. Fully-supervised node classification accuracy  $(\%)$ . (L) denotes the number of layers.  
+
+<table><tr><td>Method</td><td>Cora</td><td>Cite.</td><td>Pubm.</td><td>Cham.</td><td>Corn.</td><td>Texas</td><td>Wisc.</td></tr><tr><td>GCN (Kipf &amp; Welling, 2016)</td><td>85.77</td><td>73.68</td><td>88.13</td><td>28.18</td><td>52.70</td><td>52.16</td><td>45.88</td></tr><tr><td>GAT (Veličković et al., 2018)</td><td>86.37</td><td>74.32</td><td>87.62</td><td>42.93</td><td>54.32</td><td>58.38</td><td>49.41</td></tr><tr><td>Geom-GCN-I (Pei et al., 2020)</td><td>85.19</td><td>77.99</td><td>90.05</td><td>60.31</td><td>56.76</td><td>57.58</td><td>58.24</td></tr><tr><td>Geom-GCN-P (Pei et al., 2020)</td><td>84.93</td><td>75.14</td><td>88.09</td><td>60.90</td><td>60.81</td><td>67.57</td><td>64.12</td></tr><tr><td>Geom-GCN-S (Pei et al., 2020)</td><td>85.27</td><td>74.71</td><td>84.75</td><td>59.96</td><td>55.68</td><td>59.73</td><td>56.67</td></tr><tr><td>APPNP (Klicpera et al., 2019)</td><td>87.87</td><td>76.53</td><td>89.40</td><td>54.30</td><td>73.51</td><td>65.41</td><td>69.02</td></tr><tr><td>JKNet (Xu et al., 2018)</td><td>85.25 (16)</td><td>75.85 (8)</td><td>88.94 (64)</td><td>60.07 (32)</td><td>57.30 (4)</td><td>56.49 (32)</td><td>48.82 (8)</td></tr><tr><td>JKNet (Drop) (Rong et al., 2020)</td><td>87.46 (16)</td><td>75.96 (8)</td><td>89.45 (64)</td><td>62.08 (32)</td><td>61.08 (4)</td><td>57.30 (32)</td><td>50.59 (8)</td></tr><tr><td>Incep (Drop) (Rong et al., 2020)</td><td>86.86 (8)</td><td>76.83 (8)</td><td>89.18 (4)</td><td>61.71 (8)</td><td>61.62 (16)</td><td>57.84 (8)</td><td>50.20 (8)</td></tr><tr><td>GCNII (Chen et al., 2020)</td><td>88.49 (64)</td><td>77.08 (64)</td><td>89.57 (64)</td><td>60.61 (8)</td><td>74.86 (16)</td><td>69.46 (32)</td><td>74.12 (16)</td></tr><tr><td>GCNII* (Chen et al., 2020)</td><td>88.01 (64)</td><td>77.13 (64)</td><td>90.30 (64)</td><td>62.48 (8)</td><td>76.49 (16)</td><td>77.84 (32)</td><td>81.57 (16)</td></tr><tr><td>PDE-GCN(M(Eliasof et al., 2021)</td><td>88.60 (16)</td><td>78.48 (32)</td><td>89.93 (16)</td><td>66.01 (16)</td><td>89.73 (64)</td><td>93.24 (32)</td><td>91.76 (16)</td></tr><tr><td>pathGCN (Ours)</td><td>90.02 (64)</td><td>78.95 (32)</td><td>90.42 (64)</td><td>66.79 (16)</td><td>91.35 (8)</td><td>95.14 (16)</td><td>93.53 (16)</td></tr></table>
+
+Table 5. Fully-supervised node classification accuracy  $(\%)$  
+
+<table><tr><td>Method</td><td>Actor</td><td>Ogbn-arxiv</td></tr><tr><td>GCN (Kipf &amp; Welling, 2016)</td><td>26.86</td><td>71.74</td></tr><tr><td>GAT (Veličković et al., 2018)</td><td>28.45</td><td>71.89</td></tr><tr><td>APPNP (Klicpera et al., 2019)</td><td>31.26</td><td>71.82</td></tr><tr><td>Geom-GCN-P (Pei et al., 2020)</td><td>31.63</td><td>-</td></tr><tr><td>JKNet (Xu et al., 2018)</td><td>29.81</td><td>72.19</td></tr><tr><td>SGC (Wu et al., 2019)</td><td>30.98</td><td>69.20</td></tr><tr><td>GCNII (Chen et al., 2020)</td><td>32.87</td><td>72.74</td></tr><tr><td>EGNN (Zhou et al., 2021)</td><td>-</td><td>72.70</td></tr><tr><td>GRAND (Chamberlain et al., 2021)</td><td>-</td><td>72.23</td></tr><tr><td>pathGCN (Ours)</td><td>37.54</td><td>72.83</td></tr></table>
+
+# 4.3. Inductive learning
+
+We employ the PPI dataset (Hamilton et al., 2017) for the inductive learning task. We use a 8 layer pathGCN, without weight-decay, dropout of 0.2 and a learning rate of 0.001. We compare our results with various methods like GraphSAGE, GAT, JKNet, GeniePath, Cluster-GCN, GCNII and others, and present the micro-averaged F1 score in in Tab.
+
+Table 6. Node classification on Wiki-CS.  
+
+<table><tr><td>Method</td><td>Accuracy (%)</td></tr><tr><td>GCN (Kipf &amp; Welling, 2016)</td><td>77.19</td></tr><tr><td>GAT (Veličković et al., 2018)</td><td>77.65</td></tr><tr><td>SuperGAT (Kim &amp; Oh, 2020)</td><td>77.90</td></tr><tr><td>APPNP (Klicpera et al., 2019)</td><td>79.84</td></tr><tr><td>pathGCN (Ours)</td><td>80.02</td></tr></table>
+
+7. We note that our pathGCN achieves a score of 99.61, superior to methods like GAT, JKNet, GeniePath, PDE-GCN, and slightly above GCNII* with a score of 99.58.
+
+# 4.4. Graph classification
+
+Our previous experiments considered various datasets and settings of the node classification task. To further demonstrate the efficacy of our pathGCN we experiment with graph classification on the popular TUDatasets (Morris et al., 2020). We follow the same experimental settings from (Xu et al., 2019) and report the 10 fold cross-validation performance on MUTAG, PTC, PROTEINS and NCI1 datasets. The hyper-parameters are determined by grid search, as
+
+Table 7. Inductive learning on PPI dataset. Results are reported in micro-averaged F1 score.  
+
+<table><tr><td>Method</td><td>Micro-averaged F1</td></tr><tr><td>GraphSAGE (Hamilton et al., 2017)</td><td>61.20</td></tr><tr><td>VR-GCN (Chen et al., 2018)</td><td>97.80</td></tr><tr><td>GaAN (Zhang et al., 2018a)</td><td>98.71</td></tr><tr><td>GAT (Veličković et al., 2018)</td><td>97.30</td></tr><tr><td>JKNet (Xu et al., 2018)</td><td>97.60</td></tr><tr><td>GeniePath (Liu et al., 2018)</td><td>98.50</td></tr><tr><td>Cluster-GCN (Chiang et al., 2019)</td><td>99.36</td></tr><tr><td>GCNII (Chen et al., 2020)</td><td>99.54</td></tr><tr><td>GCNII* (Chen et al., 2020)</td><td>99.58</td></tr><tr><td>PDE-GCNM (Eliasof et al., 2021)</td><td>99.18</td></tr><tr><td>pathGCN (ours)</td><td>99.61</td></tr></table>
+
+Table 8. TUDatasets graph classification accuracy  $(\%)$  
+
+<table><tr><td>Model</td><td>MUTAG</td><td>PTC</td><td>PROTEINS</td><td>NCI1</td></tr><tr><td>DGCNN</td><td>85.8±1.8</td><td>58.6±2.5</td><td>75.5±0.9</td><td>74.4±0.5</td></tr><tr><td>IGN</td><td>83.9±13.0</td><td>58.5±6.9</td><td>76.6±5.5</td><td>74.3±2.7</td></tr><tr><td>GIN</td><td>89.4±5.6</td><td>64.6±7.0</td><td>76.2±2.8</td><td>82.7±1.7</td></tr><tr><td>CIN</td><td>92.7±6.1</td><td>68.2±5.6</td><td>77.0±4.3</td><td>83.6±1.4</td></tr><tr><td>GSN</td><td>92.7±7.5</td><td>68.2±7.2</td><td>76.6±5.0</td><td>83.5±2.0</td></tr><tr><td>pathGCN (ours)</td><td>94.7±4.7</td><td>75.2±5.3</td><td>80.4±4.2</td><td>83.3±1.3</td></tr></table>
+
+in (Xu et al., 2019) and are reported in Appendix B. We compare our pathGCN with recent methods like DGCNN (Zhang et al., 2018b), IGN (Maron et al., 2018),GIN (Xu et al., 2019), CIN (Bodnar et al., 2021) and GSN (Bouritsas et al., 2022). The results are summarized in Tab. 8, where our pathGCN shows better or similar results compared to the considered methods, further highlighting the efficacy of our approach.
+
+# 4.5. Inference and runtimes
+
+In this section we compare the inference accuracy of our stochastic approach with its deterministic form. Later, we report the runtimes of our method.
+
+As discussed in Sec. 3.4, both the stochastic and deterministic forms of our pathGCN can be used during inference. In Tab. 9, we show that after training our stochastic pathGCN, it is possible to obtain similar results (with up to  $0.1\%$  accuracy difference) by its deterministic form as in Eq. (9). The stochastic results are averaged over 10 inference runs, and we also present their standard deviation.
+
+Next, we present the training and inferences times, as well as path sampling time of our pathGCN and compare it with
+
+Table 9. Deterministic vs stochastic pathGCN inference accuracy  $(\%)$  
+
+<table><tr><td>Inference</td><td>Cora</td><td>Cite.</td><td>Pub.</td><td>ogbn-arxiv</td><td>Wisc.</td><td>PPI</td></tr><tr><td>Determin.</td><td>85.8</td><td>75.8</td><td>82.7</td><td>72.84</td><td>93.51</td><td>99.61</td></tr><tr><td rowspan="2">Stochastic</td><td>85.8</td><td>75.8</td><td>82.7</td><td>72.83</td><td>93.53</td><td>99.61</td></tr><tr><td>±0.29</td><td>±0.34</td><td>±0.34</td><td>±0.13</td><td>±0.21</td><td>±0.02</td></tr></table>
+
+Table 10. Computation times (in ms) on Cora.  
+
+<table><tr><td>Model</td><td>Path samp.</td><td>Training</td><td>Inference</td><td>Acc (%)</td></tr><tr><td>GCN (2 layers)</td><td>-</td><td>4.07</td><td>1.97</td><td>81.1</td></tr><tr><td>GCNII (2 layers)</td><td>-</td><td>4.24</td><td>1.95</td><td>82.2</td></tr><tr><td>GCNII (8 layers)</td><td>-</td><td>13.05</td><td>6.87</td><td>84.2</td></tr><tr><td>GAT (2 layers)</td><td>-</td><td>5.27</td><td>1.96</td><td>83.1</td></tr><tr><td>pathGCNk=2,p=5 (2 layers)</td><td>0.378</td><td>3.48</td><td>1.67</td><td>81.3</td></tr><tr><td>pathGCNk=5,p=5 (2 layers)</td><td>0.384</td><td>5.35</td><td>2.17</td><td>84.2</td></tr></table>
+
+GCN, GCNII and GAT, in Tab. 10. For reference, we also report the node classification accuracy of the different methods. We see that the sampling time is relatively small, and that our pathGCN has a similar runtime to GCN, GCNII and GAT, while obtaining similar or better accuracy. For example, our pathGCN with  $k = 5$ ,  $p = 5$  obtains an accuracy of  $84.2\%$  on Cora and requires 5.35 milliseconds (ms) for a training iteration, while GCNII requires 8 layers and 13.05 ms to achieve the same accuracy.
+
+# 4.6. Ablation study
+
+In this section we consider the different possible variants of our pathGCN, and the influence of our its hyper-parameters the path length  $k$  and the number of paths  $p$ .
+
+As discussed in Sec. 3, our method can be formulated in various manners. That is, it is possible to learn a global spatial operator, which is most similar to GCN (Kipf & Welling, 2016), only there it is fixed and not learnt. We denote this variant by pathGCN $_G$ . In addition, we can learn a spatial operator per layer that is shared among the channels, denoted by pathGCN $_{PL}$ . The next step is our pathGCN from Sec. 3.3, which is more similar to CNNs, by utilizing a depth-wise spatial operator (i.e., per channel and layer). As depicted in Tab. 11, the global variant, pathGCN $_G$  yields the least attractive performance among the three considered variants. This is not surprising, as a global operator is learnt, which is less expressive than pathGCN $_{PL}$  and pathGCN. Still, it is interesting to see that no over-smoothing is evident. As discussed in (Wu et al., 2019), a recurrent application of a fixed operator leads to over-smoothing. However, the learnt spatial operator of pathGCN $_G$  is variable. Following that, the per layer variant, pathGCN $_{PL}$  prevents over-smoothing and also further improves accuracy. This is obtainable as the network has the freedom to learn a variety of kernels, which may function as smoothing kernels or edge detectors.
+
+![](images/5275f2b660d59887b3f73702244eac0cc343bac749fb7d83f4a5caff73ea1339.jpg)  
+Figure 5. Semi-supervised node classification accuracy  $(\%)$ , as a function of number of paths  $p$  and path length  $k$ .
+
+![](images/7e9dcc05fd08ce173e132cccb3bade500c974a769c3ff5a3ed52788491fb5232.jpg)
+
+Table 11. Variants of pathGCN on semi-supervised classification. Results are reported in accuracy  $(\%)$  
+
+<table><tr><td rowspan="2">Dataset</td><td rowspan="2">Method</td><td colspan="6">Layers</td></tr><tr><td>2</td><td>4</td><td>8</td><td>16</td><td>32</td><td>64</td></tr><tr><td rowspan="3">Cora</td><td>pathGCNG</td><td>83.0</td><td>81.9</td><td>81.5</td><td>81.2</td><td>81.4</td><td>82.0</td></tr><tr><td>pathGCNPL</td><td>82.9</td><td>83.3</td><td>83.5</td><td>83.8</td><td>84.1</td><td>84.7</td></tr><tr><td>pathGCN</td><td>84.2</td><td>84.5</td><td>84.6</td><td>85.1</td><td>85.4</td><td>85.8</td></tr><tr><td rowspan="3">CiteSeer</td><td>pathGCNG</td><td>73.1</td><td>71.9</td><td>72.0</td><td>71.9</td><td>72.6</td><td>71.7</td></tr><tr><td>pathGCNPL</td><td>73.4</td><td>73.6</td><td>74.0</td><td>74.3</td><td>74.5</td><td>75.0</td></tr><tr><td>pathGCN</td><td>74.3</td><td>74.8</td><td>75.4</td><td>75.3</td><td>75.6</td><td>75.8</td></tr><tr><td rowspan="3">Pubmed</td><td>pathGCNG</td><td>80.9</td><td>80.4</td><td>81.1</td><td>81.0</td><td>80.4</td><td>80.8</td></tr><tr><td>pathGCNPL</td><td>81.1</td><td>81.2</td><td>81.5</td><td>82.0</td><td>82.2</td><td>82.1</td></tr><tr><td>pathGCN</td><td>81.8</td><td>81.8</td><td>82.4</td><td>82.5</td><td>82.4</td><td>82.7</td></tr></table>
+
+Finally, we see that the additional degrees of freedom in the depth-wise operator in pathGCN lead to overall better performance, also as depicted in our other experiments.
+
+As for the hyper-parameters of our method, we report in Fig. 5 the influence of the number of paths  $p$  and path length  $k$  on the performance of our pathGCN on semi-supervised node classification using Cora, CiteSeer and Pubmed. In all the experiments we use a 2 layer pathGCN, fix one hyperparameter, and vary the other. Specifically, for the evaluation of the influence of number of paths  $p$ , we fix the path length to  $k = 5$ , and the results in Fig. 5 suggest that indeed the stochastic nature of our method is beneficial to obtain higher accuracy, since we see a consistent accuracy degradation trend as  $k$  is increased past 20. In addition, we present in Fig. 5, that by fixing  $p = 10$  and examining a variable path length  $k$  from 1 to 11, accuracy improves and stagnates at  $k = 7$ . We can see that when  $k = 1$ , our pathGCN behaves similarly to an MLP (Qi et al., 2017) (as it considers only the self node), and that in some cases, increasing the kernel size caused a slight performance degradation.
+
+# 5. Conclusion
+
+In this paper we propose a new approach for learning the spatial operators for GCNs. Our motivation stems from the
+
+need for deep GCNs that have expressive spatial kernels, similar to standard CNNs that do not over-smooth. Our approach leverages on paths defined on the graph, to enable the learning of such operators, further bridging the gap between GCNs and CNNs.
+
+Just as the Laplacian is not the sole spatial operator used on images in CNNs, it may also not necessarily be optimal in the case of graphs and GCNs. To this end we propose pathGCN which replaces the Laplacian based operator by a fully learnt kernel. Indeed, our experiments reveal that more expressive kernels can be learnt based on the data and task at hand, leading to consistently better accuracy on numerous applications and datasets and without over-smoothing.
+
+# Acknowledgements
+
+The research reported in this paper was supported by the Israel Innovation Authority through Avatar consortium. In addition, this work was supported in part by the Israeli Council for Higher Education (CHE) via the Data Science Research Center, Ben-Gurion University of the Negev, Israel. ME is supported by Kreitman High-tech scholarship.
+
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+
+# A. Architecture in details
+
+We elaborate on the architecture that was used in our experiments. As discussed in Sec. 4, our network is comprised of an embedding layer ( $1 \times 1$  convolution), a sequence of pathGCN layers, and a closing (projection) layer ( $1 \times 1$  convolution). Throughout this section,  $c_{in}$  and  $c_{out}$  denote the input and output channels, respectively, and  $c$  denotes the number of features in hidden layers (which is a hyperparameter, as given in Appendix B). We denote the number of pathGCN blocks by  $L$ , and the dropout probability by  $p_{drop}$ . Our architecture for node classification tasks is described in Tab. 12, which is similar to the architecture found in GCNII (Chen et al., 2020), only with our pathGCN instead. In Tab. 13 we present our network for graph classification tasks, which is based on the one in GIN (Xu et al., 2019).
+
+Table 12. pathGCN architecture for node classification.  
+
+<table><tr><td>Input size</td><td>Layer</td><td>Output size</td></tr><tr><td>n × cin</td><td>1 × 1 Dropout(pdrop)</td><td>n × cin</td></tr><tr><td>n × cin</td><td>1 × 1 Convolution</td><td>n × c</td></tr><tr><td>n × c</td><td>ReLU</td><td>n × c</td></tr><tr><td>n × c</td><td>L× pathGCN block</td><td>n × c</td></tr><tr><td>n × c</td><td>1 × 1 Dropout(pdrop)</td><td>n × c</td></tr><tr><td>n × c</td><td>1 × 1 Convolution</td><td>n × cout</td></tr></table>
+
+Table 13. pathGCN architecture for graph classification.  
+
+<table><tr><td>Input size</td><td>Layer</td><td>Output size</td></tr><tr><td>n × cin</td><td>1 × 1 Convolution</td><td>n × c</td></tr><tr><td>n × c</td><td>ReLU</td><td>n × c</td></tr><tr><td>n × c</td><td>L × [ pathGCN, BN</td><td>n × c</td></tr><tr><td></td><td>, 1 × 1 Convolution, ReLU ]</td><td></td></tr><tr><td>n × c</td><td>1 × 1 Add-pool</td><td>1 × c</td></tr><tr><td>1 × c</td><td>1 × 1 Convolution</td><td>1 × c</td></tr><tr><td>1 × c</td><td>1 × 1 Dropout(pdrop)</td><td>1 × c</td></tr><tr><td>1 × c</td><td>1 × 1 Convolution</td><td>1 × cout</td></tr></table>
+
+# B. Hyper-parameters details
+
+We provide the selected hyper-parameters in our experiments. We denote the learning rate of our pathGCN layers by  $LR_{GCN}$ , and the learning rate of the  $1 \times 1$  opening (embedding) and closing (classifier) layers by  $LR_{oc}$ . Also, the weight decay for the opening and closing layers is denoted by  $WD_{oc}$ , and for the pathGCN layers by  $WD_{GCN}$ .
+
+Table 14. Semi-supervised node classification hyper-parameters  
+
+<table><tr><td>Dataset</td><td>LRGCN</td><td>WDGCN</td><td>LRoc</td><td>WDoc</td><td>c</td><td>pdrop</td><td>k</td><td>p</td></tr><tr><td>Cora</td><td>1·10-3</td><td>2·10-5</td><td>1·10-2</td><td>1·10-5</td><td>64</td><td>0.6</td><td>5</td><td>5</td></tr><tr><td>Citeseer</td><td>1·10-3</td><td>1·10-5</td><td>7·10-3</td><td>5·10-5</td><td>256</td><td>0.7</td><td>5</td><td>5</td></tr><tr><td>Pubmed</td><td>5·10-3</td><td>0</td><td>1·10-2</td><td>1·10-5</td><td>256</td><td>0.5</td><td>7</td><td>10</td></tr></table>
+
+Table 15. Fully-supervised node classification hyper-parameters  
+
+<table><tr><td>Dataset</td><td>LRGCN</td><td>WDGCN</td><td>LRoc</td><td>WDoc</td><td>c</td><td>pdrop</td><td>k</td><td>p</td></tr><tr><td>Cora</td><td>1·10-4</td><td>1·10-4</td><td>7·10-2</td><td>1·10-4</td><td>64</td><td>0.5</td><td>5</td><td>10</td></tr><tr><td>CiteSeer</td><td>3·10-4</td><td>5·10-5</td><td>8·10-3</td><td>1·10-4</td><td>64</td><td>0.5</td><td>5</td><td>10</td></tr><tr><td>Pubmed</td><td>1·10-4</td><td>2·10-4</td><td>1·10-2</td><td>1·10-6</td><td>64</td><td>0.5</td><td>7</td><td>10</td></tr><tr><td>Chameleon</td><td>5·10-4</td><td>1·10-5</td><td>5·10-3</td><td>3·10-5</td><td>64</td><td>0.5</td><td>3</td><td>10</td></tr><tr><td>Cornell</td><td>4·10-4</td><td>1·10-5</td><td>5·10-2</td><td>5·10-4</td><td>64</td><td>0.5</td><td>5</td><td>10</td></tr><tr><td>Texas</td><td>3·10-4</td><td>5·10-4</td><td>4·10-2</td><td>1·10-4</td><td>64</td><td>0.5</td><td>7</td><td>10</td></tr><tr><td>Wisconsin</td><td>3·10-4</td><td>2·10-4</td><td>1·10-2</td><td>5·10-5</td><td>64</td><td>0.5</td><td>7</td><td>10</td></tr><tr><td>Actor</td><td>2·10-4</td><td>1·10-4</td><td>8·10-2</td><td>5·10-4</td><td>64</td><td>0.5</td><td>7</td><td>10</td></tr><tr><td>Wiki-CS</td><td>3·10-2</td><td>1·10-4</td><td>7·10-3</td><td>1·10-5</td><td>64</td><td>0.3</td><td>7</td><td>5</td></tr><tr><td>Ogbn-arxiv</td><td>1·10-3</td><td>0</td><td>1·10-3</td><td>0</td><td>256</td><td>0.1</td><td>5</td><td>10</td></tr></table>
+
+# B.1. Semi-supervised node classification
+
+The hyper-parameters for this experiment are summarized in Tab. 14.
+
+# B.2. Fully-supervised node classification
+
+The hyper-parameters for this experiment are summarized in Tab. 15.
+
+# B.3. Inductive learning on PPI
+
+For this experiment we used  $LR_{oc} = LR_{GCN} = 0.001$ , with dropout probability  $p_{drop} = 0.2$ ,  $k = 5$  and  $p = 10$ , and no weight decay was used.
+
+# B.4. Graph classification
+
+The hyper-parameters in this experiment were chosen according to the protocol described in (Xu et al., 2019). We present the chosen hyper-parameters in Tab. 16. Throughout all the experiments in this section, no weight decay was used.
+
+# B.5. Ablation study
+
+In this experiment we used the same hyper-parameters as reported in Tab. 14, for the results in Tab. 11. For the results in Fig. 5, we use the same learning rate and weight decay, but  $k$  and  $p$  are as described in the main paper.
+
+Table 16. Graph classification hyper-parameters. BS denotes batch size.  
+
+<table><tr><td>Dataset</td><td>LRGCN</td><td>LRoc</td><td>c</td><td>pdrop</td><td>BS</td><td>p</td><td>k</td></tr><tr><td>MUTAG</td><td>0.01</td><td>0.01</td><td>32</td><td>0</td><td>32</td><td>5</td><td>5</td></tr><tr><td>PTC</td><td>0.01</td><td>0.01</td><td>32</td><td>0</td><td>32</td><td>5</td><td>5</td></tr><tr><td>PROTEINS</td><td>0.01</td><td>0.01</td><td>32</td><td>0</td><td>128</td><td>5</td><td>10</td></tr><tr><td>NCI1</td><td>0.01</td><td>0.01</td><td>32</td><td>0.5</td><td>32</td><td>5</td><td>10</td></tr></table>

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+# Variational Nearest Neighbor Gaussian Process
+
+Luhuan Wu<sup>1</sup> Geoff Pleiss<sup>2</sup> John Cunningham<sup>1,2</sup>
+
+# Abstract
+
+Variational approximations to Gaussian processes (GPs) typically use a small set of inducing points to form a low-rank approximation to the covariance matrix. In this work, we instead exploit a sparse approximation of the precision matrix. We propose variational nearest neighbor Gaussian process (VNNGP), which introduces a prior that only retains correlations within  $K$  nearest- neighboring observations, thereby inducing sparse precision structure. Using the variational framework, VNNGP's objective can be factorized over both observations and inducing points, enabling stochastic optimization with a time complexity of  $O(K^3)$ . Hence, we can arbitrarily scale the inducing point size, even to the point of putting inducing points at every observed location. We compare VNNGP to other scalable GPs through various experiments, and demonstrate that VNNGP (1) can dramatically outperform low-rank methods, and (2) is less prone to overfitting than other nearest neighbor methods.
+
+# 1. Introduction
+
+Gaussian processes (GPs) provide rich priors over functions (Rasmussen & Williams, 2006). While the GP posterior can be computed in closed form for regression models, variational posterior approximations (Titsias, 2009; Hensman et al., 2013; van der Wilk et al., 2020) have become increasingly popular as they offer numerous advantages. First, variational methods can utilize stochastic minibatching (Hensman et al., 2013), enabling inference for large-scale datasets. Second, variational methods are compatible with non-conjugate likelihoods (Hensman et al., 2015) and simplify inference for interdomain observation models (Lázaro-Gredilla & Figueiras-Vidal, 2009; Álvarez et al., 2010; van der Wilk et al., 2020; Wu et al., 2021). Finally,
+
+$^{1}$ Department of Statistics, Columbia University  $^{2}$ Zuckerman Institute, Columbia University. Correspondence to: Luhuan Wu <lw2827@columbia.edu>.
+
+Proceedings of the  $39^{th}$  International Conference on Machine Learning, Baltimore, Maryland, USA, PMLR 162, 2022. Copyright 2022 by the author(s).
+
+black box variational methods (Ranganath et al., 2014) make it possible to use GP as components of larger models, such as Deep Gaussian Processes (Damianou & Lawrence, 2013).
+
+To apply variational methods to large-scale GP, it is often necessary to make additional approximations (Titsias, 2009; Hensman et al., 2017). This is because making GP inference with  $N$  observations incurs a  $\mathcal{O}(N^3)$  computation. A popular variational approach that aids this issue is the Stochastic Variational Gaussian Process method (SVGP) (Hensman et al., 2013), which essentially forms a low-rank approximation to the prior covariance. These approximations offer well understood predictive properties (Bauer et al., 2016), probable performance guarantees (Burt et al., 2019), and competitive performance on numerous datasets. However, there are also many instances when low-rank approximations are ill suited. For example, spatiotemporal datasets often have intrinsically low lengthscale, meaning that the data can vary rapidly with small changes in space and time. In these settings, low-rank approximations cannot capture fast variations, often resulting in low fidelity or "blurry" predictions (e.g. Datta et al., 2016a; Pleiss et al., 2020; Wu et al., 2021).
+
+In this paper, we are interested in making variational GP methods more amenable to spatiotemporal data and other domains with intrinsically low lengthscale, while retaining (and even improving) scalability. To do so, we replace SVGP's low-rank approximation of the prior covariance matrix with a sparse approximation of the prior precision matrix. Whereas a low-rank approximation assumes that observations are explained through a small number of global latent variables, a sparse precision approximation instead assumes that observations are conditionally independent given their nearest neighboring observations. Mathematically, the joint GP prior is approximated by the product of one-dimensional conditionals, each of which only depends on a small subset of preceding observations based on some predetermined ordering. This assumption is quite reasonable in the context of spatiotemporal modeling. For example, it is reasonable to assume that the weather in Los Angeles and Maine are conditionally independent, given neighboring cities in California.
+
+Sparse precision approximations are common in spatiotemporal modeling throughout the geostatistics literature (e.g.
+
+Vecchia, 1988; Stein et al., 2004; Datta et al., 2016a; Guinness, 2018; Finley et al., 2019; Katzfuss et al., 2021), where they often achieve higher predictive fidelity than low-rank approximations. However, most existing approaches target exact or MCMC inference and are not compatible with variational methods. As a result, it is not straightforward to use sparse precision methods with non-conjugate or interdomain likelihoods, or as part of larger probabilistic models without resorting to sampling. Recently, Tran et al. (2021) made a first key step in this direction. They introduce Sparse Within Sparse Gaussian Processes (SWSGP), which combines a similar nearest neighbor approximation with SVGP. Instead of exploiting the sparse precision structure, SWSGP introduces sparsity by constructing a hierarchical prior. While successful, we will show an important technical opportunity remains, which we offer here.
+
+In this work, we propose a novel sparse precision approximation that is compatible with stochastic variational inference. Similar to SVGP, our proposed method—the Variational Nearest Neighbor Gaussian Process (VNNGP)  ${}^{1}$ —formulates the variational objective and predictive distribution through a set of  $M$  inducing points. Unlike SVGP, our method applies 1) a sparse precision approximation to the inducing point distribution and 2) a sparse dependency approximation between observations and inducing points. These approximations enable the VNNGP objective to factorize over data and inducing points. Consequently, we can minibatch over data and inducing points when evaluating the variational objective, resulting in constant time computational complexity with respect to both  $N$  and  $M$ . This significant reduction makes it possible to scale  $M$  well beyond the limits of SVGP, even to the point of placing inducing points at every observed location. Moreover, unlike SWSGP and other sparse approximations (Vecchia, 1988; Katzfuss et al., 2021), the VNNGP prior and variational distribution both constitute valid GPs, enabling it to more faithfully retain the features of the true GP prior. We compare VNNGP to SVGP, SWSGP, and other scalable methods on numerous toy experiments and benchmark datasets. Our results demonstrate that VNNGP can offer higher fidelity predictions than SVGP and is less prone to overfitting than SWSGP. These advantages are most pronounced on spatiotemporal datasets (including those with non-conjugate observational models), but hold for high dimensional datasets as well.
+
+$^{1}$ VNNGP is implemented in the GPyTorch library. See the example in https://docs.gpytorch.ai/en/stable/ examples/04_Variational_and_Approximate_GPs/VNNGP.html
+
+# 2. Background
+
+# 2.1. Gaussian Processes
+
+Consider any finite dimensional realization  $\mathbf{f} = \{f_i\}_{i=1}^N$  of a Gaussian process at locations  $\mathbf{x}_1, \mathbf{x}_2, \dots, \mathbf{x}_N \in \mathbb{R}^D$ :
+
+$$
+f (\cdot) \sim \mathcal {G P} (0, k (\cdot , \cdot)) \tag {1}
+$$
+
+$$
+f _ {i} \equiv f (\mathbf {x} _ {i}) \quad \text {f o r} i = 1: N
+$$
+
+where  $k(\cdot, \cdot)$  is a kernel function that encodes the properties of the prior. We can always factorize the joint probability of a GP realization into a chain of conditional probabilities subject to some ordering, for example
+
+$$
+p (\mathbf {f}) = p \left(f _ {1}\right) \prod_ {i = 2} ^ {N} p \left(f _ {i} \mid \mathbf {f} _ {1: i - 1}\right), \tag {2}
+$$
+
+where  $p(f_1) = \mathcal{N}(f_1|0,k_{1,1})$
+
+$$
+p \left(f _ {i} \mid \mathbf {f} _ {1: i - 1}\right) = \mathcal {N} \left(f _ {i} \mid \mathbf {k} _ {1: i - 1, i} ^ {\top} \mathbf {K} _ {1: i - 1, 1: i - 1} ^ {- 1} \mathbf {f} _ {1: i - 1}, \right. \tag {3}
+$$
+
+$$
+k _ {i, i} - \mathbf {k} _ {1: i - 1, i} ^ {\top} \mathbf {K} _ {1: i - 1, 1: i - 1} ^ {- 1} \mathbf {k} _ {1: i - 1, i})
+$$
+
+and we define  $k_{i,i} \equiv k(\mathbf{x}_i, \mathbf{x}_i)$ ,  $\mathbf{k}_{1:i-1,i} \equiv k(\mathbf{x}_{1:i-1}, \mathbf{x}_i)$ , and  $\mathbf{K}_{1:i-1,1:i-1} \equiv k(\mathbf{x}_{1:i-1,i}, \mathbf{x}_{1:i-1})$ . The following notations will apply this convention accordingly.
+
+# 2.2. Sparse Precision Approximations
+
+GP approximations based on sparse precision matrices remain popular for years, largely in the geostatistics community (Vecchia, 1988; Datta et al., 2016a; Finley et al., 2019; Katzfuss et al., 2021). They consider the following approximation to Eq 2
+
+$$
+p (\mathbf {f}) \approx p \left(f _ {1}\right) \prod_ {i = 2} ^ {N} p \left(f _ {i} \mid \mathbf {f} _ {n (i)}\right), \tag {4}
+$$
+
+where  $p(f_1) = \mathcal{N}(f_1|0,k_{1,1})$
+
+$$
+p \left(f _ {i} \mid \mathbf {f} _ {n (i)}\right) = \mathcal {N} \left(f _ {i} \mid \mathbf {k} _ {n (i), i} ^ {\top} \mathbf {K} _ {n (i), n (i)} ^ {- 1} \mathbf {f} _ {n (i)}, \right. \tag {5}
+$$
+
+$$
+k _ {i, i} - \mathbf {k} _ {n (i), i} ^ {\top} \mathbf {K} _ {n (i), n (i)} ^ {- 1} \mathbf {k} _ {n (i), i})
+$$
+
+and  $n(i)$  denotes the indices of  $K$  nearest neighbors of  $\mathbf{x}_i$  in  $\{\mathbf{x}_k\}_{k=1}^{i-1}$ . Typically  $K$  is set to be much smaller than  $N$ , and when  $K = N$  Eq 4 recovers the original GP. This approximation results in a sparse Cholesky factor of the precision matrix (Datta, 2021b). As a consequence, the model complexity scales  $O(NK^3)$ . It has seen many successful applications in spatiotemporal problems, e.g. predicting PM pollutant levels arocss Europe (Datta et al., 2016b), predicting forest canopy height (Finley et al., 2017), estimating forest biomass (Taylor-Rodriguez et al., 2019), etc.
+
+# 2.3. Stochastic Variational Gaussian Processes
+
+Stochastic variational Gaussian process (SVGP) (Hensman et al., 2013) defines a small set of  $M$  inducing points  $\mathbf{u} = \{u_j\}_{j=1}^M$  which are GP latent variables at locations  $\{\mathbf{z}_j\}_{j=1}^M$ . It considers the joint latent generative process:
+
+$$
+p (\mathbf {u}, \mathbf {f}) = \mathcal {N} \left(\left( \begin{array}{l} \mathbf {u} \\ \mathbf {f} \end{array} \right) | 0, \left( \begin{array}{c c} \mathbf {K} _ {\mathbf {z}, \mathbf {z}} & \mathbf {K} _ {\mathbf {z}, \mathbf {x}} \\ \mathbf {K} _ {\mathbf {x}, \mathbf {z}} & \mathbf {K} _ {\mathbf {x}, \mathbf {x}} \end{array} \right)\right) \tag {6}
+$$
+
+and an independent observation model
+
+$$
+p (\mathbf {y} \mid \mathbf {f}) = \prod_ {i = 1} ^ {N} p \left(y _ {i} \mid f _ {i}\right) \tag {7}
+$$
+
+where  $p(y_{i}|f_{i})$  is an arbitrary likelihood function. SVGP is optimized by maximizing the evidence lower bound (ELBO) given below:
+
+$$
+\mathcal {L} _ {\mathrm {S V G P}} = \sum_ {i = 1} ^ {N} \underset {q \left(f _ {i}\right)} {\mathbb {E}} \left[ \log p \left(y _ {i} \mid f _ {i}\right) \right] - \operatorname {K L} \left[ q (\mathbf {u}) \| p (\mathbf {u}) \right], \tag {8}
+$$
+
+where  $q(\mathbf{u})$  is the variational posterior for  $\mathbf{u}$  and  $q(f_i) = \int p(f_i|\mathbf{u})q(\mathbf{u})\mathrm{d}\mathbf{u}$  is the variational posterior for  $f_i$ . The SVGP approximation relies on the low-rank matrix  $\mathbf{K}_{\mathbf{x},\mathbf{z}}\mathbf{K}_{\mathbf{z},\mathbf{z}}^{-1}\mathbf{K}_{\mathbf{z},\mathbf{x}}$  instead of the full-rank matrix  $\mathbf{K}_{\mathbf{x},\mathbf{x}}$  to reduce the size of any matrix inversion to  $M$ . Therefore, we refer to it as a low-rank approximation.
+
+SVGP has several advantages compared to exact GP and other scalable methods. First, as is shown in Eq 8, the ELBO is factorized over data points. Therefore, it is amenable to stochastic optimization and the computational complexity reduces to  $O(M^3)$ . Moreover, due to its variational inference nature, SVGP can be applied to problems with non-conjugate likelihoods, inter-domain observations or latent variable models based on GPs. However, since SVGP assumes  $M \ll N$ , it may struggle to obtain accurate predictions for large-scale data that are not inherently low-rank structured (Wu et al., 2021; Tran et al., 2021).
+
+# 2.4. Other Related Works
+
+Variational GP inference was first pursued by Csato et al. (2000) and Gibbs & MacKay (2000). More recent works connect variational inference with scalable methods. Titsias (2009) ties inducing point approximations and variational inference, and Hensman et al. (2013) extends this approach to be compatible with minibatched optimization. Recent extensions focus on inducing points in linearly transformed domains (Lázaro-Gredilla & Figueiras-Vidal, 2009; van der Wilk et al., 2020), inducing points with exploitable algebraic kernel structures (Wilson & Nickisch, 2015; Wu et al., 2021), and inducing points with separable structure (Cheng & Boots, 2017; Salimbeni et al., 2018; Shi et al., 2020). Beyond inducing point approximations, Hensman et al. (2017) and John & Hensman (2018) combine variational inference with finite basis approximations.
+
+Nearest neighbor approximations. In addition to the sparse precision approximations pursued by the geostatistics community (see introduction), the machine learning community has proposed nearest neighbor GP approximations (Kim et al., 2005; Nguyen-Tuong et al., 2008; Gramacy & Apley, 2015; Park & Apley, 2018; Jankowiak & Pleiss, 2021). Besides SWSGP which is introduced before, we review three methods that are most related to our method. Bui & Turner (2014) proposes a sparse prior by imposing a tree or chain structure over inducing points and calibrating the posterior approximation with a KL divergence minimization. This approach uses a message-passing inference algorithm and learns the hyperparameters by maximizing the marginal likelihood over inducing points. Stochastic gradient descent GP (sgGP) forms a (biased) stochastic GP objective that optimizes  $K$ -nearest- neighboring data points per iteration (Chen et al., 2020). This method is an approximation to exact GP inference, and therefore is limited to Gaussian likelihood. Liu & Liu (2019) use amortized variational inference for data points within a small neighborhood, where the variational covariance is constructed to have a sparse Cholesky factor. The latter two methods have computational complexity cubic in the number of nearest neighbors.
+
+# 3. Variational Nearest Neighbor GP
+
+In this section, we develop VNNGP, a highly scalable variational GP method. VNNGP considers the same observation model in Eq 7 and makes the following nearest neighbor approximations to the latent generative process
+
+$$
+p (\mathbf {u}) \approx \prod_ {j = 1} ^ {M} p \left(u _ {i} \mid \mathbf {u} _ {n (j)}\right), \quad p (\mathbf {f} \mid \mathbf {u}) \approx \prod_ {i = 1} ^ {N} p \left(f _ {i} \mid \mathbf {u} _ {n (i)}\right) \tag {9}
+$$
+
+where each conditional denotes a standard GP predictive distribution,  $n(j)$  denotes the inducing point indices corresponding to  $\mathbf{z}_j$ 's (at most)  $K$  nearest neighbors selected from  $\{\mathbf{z}_1, \dots, \mathbf{z}_{j-1}\}$  (with a special case of  $n(1) = \emptyset$ ), and we overload the notation  $n(i)$  to denote the inducing point indices of  $\mathbf{x}_i$ 's  $K$  nearest neighbors in all  $\{\mathbf{z}_j\}_{j=1}^M$ .
+
+Because of this construction, VNNGP forms a sparse approximation to the prior precision matrix  $\mathbf{K}_{\mathbf{z},\mathbf{z}}^{-1}$  (Datta, 2021b). Its Cholesky factor has at most  $K + 1$  non-zero elements per row, corresponding to the  $K$ -nearest-neighbor dependency structure. Figure 1 plots the Cholesky factor of  $\mathbf{K}_{\mathbf{z},\mathbf{z}}^{-1}$  for  $M = 20$  inducing points with varying number of nearest neighbors  $K$ . We further see that, as  $K$  increases, the Cholesky factor becomes denser and approaches the exact one. In this case,  $K = 10$  yields a quality approximation. Note that the sparsity pattern of the Cholesky factor depends on the ordering of inducing points and the nearest neighbor sets. In Figure 1 we use the coordinate ordering of 1-dimensional inducing points, thereby rendering a banded sparsity pattern.
+
+![](images/47d2d9f1ac9aea09dea6827b6ddf7b5b9825b9d08ba0a983134307950d35e5ea.jpg)  
+Figure 1. The Cholesky factor of prior precision matrix for  $M = 20$  inducing points (with even spacing of 1) by VNNGP with different number of nearest neighbors  $K$ . We use a squared exponential kernel with outputscale 1 and lengthscale 1. For each row of the Cholesky factor, at most  $K + 1$  elements are non-zero, and when  $K = 10$  the approximation is qualitatively close to the exact one.
+
+Ordering of inducing points. The nearest neighbor approximation in Eq 9 is subject to a particular ordering of inducing points. In all following experiments, we apply random ordering and we find that different orderings do not lead substantial differences on VNNGP's performance.
+
+Variational family. We consider a special choice of the variational distribution  $q(\mathbf{f}, \mathbf{u})$  that will greatly simplify the computation without sacrificing much prediction accuracy. Specifically, we use a mean-field variational approximation for  $\mathbf{u}$  and approximate the posterior for  $\mathbf{f}$  with the nearest neighbor prior:
+
+$$
+q (\mathbf {u}) = \prod_ {j = 1} ^ {M} q (u _ {j}) = \prod_ {j = 1} ^ {M} \mathcal {N} (u _ {j} | m _ {j}, s _ {j}), \tag {10}
+$$
+
+$$
+q (\mathbf {f}) = \prod_ {i = 1} ^ {N} q \left(f _ {i}\right) = \prod_ {i = 1} ^ {N} \int p \left(f _ {i} \mid \mathbf {u} _ {n (i)}\right) q \left(\mathbf {u} _ {n (i)}\right) \mathrm {d} \mathbf {u} _ {n (i)}. \tag {11}
+$$
+
+The choice of mean-field or sparse variational approximations are common for large-scale GP when full-rank approximations are impractical (Liu & Liu, 2019; Wu et al., 2021; Tran et al., 2021). Empirically, we found that VN-NGP with a mean-field variational distribution outperforms SVGP with a full-rank one for most cases. However, we also discuss the extension to a more expressive variational family in appendix B.
+
+Optimization objective is to maximize the VNNGP's ELBO as follows
+
+$$
+\mathcal {L} _ {\mathrm {V N N G P}} = \underbrace {\sum_ {i = 1} ^ {N} \underset {q (f _ {i})} {\mathbb {E}} [ \log p \left(y _ {i} \mid f _ {i}\right) ]} _ {\text {d a t a l i k e h o o d}} - \underbrace {\operatorname {K L} [ q (\mathbf {u}) \| p (\mathbf {u}) ]} _ {\text {K L d i v e r g e n c e}}, \tag {12}
+$$
+
+Note that the ELBO breaks into a data likelihood term and a KL divergence term. In the following, we first discuss the computation of these two terms. After that, we present the advantages of VNNGP. Finally, we compare VNNGP to other related works based on nearest neighbor approximations.
+
+# 3.1. Computing the Data Likelihood Term
+
+Similar to SVGP, the data likelihood term for VNNGP factorizes over data points. Unlike SVGP that considers all  $M$  inducing points in making the prediction for one data point, VNNGP makes use of the sparse prior in Eq 9 such that the prediction is based on only  $K$  nearest inducing points. As a result, VNNGP's data likelihood term is reduced to
+
+$$
+\sum_ {i = 1} ^ {N} \int p \left(f _ {i} \mid \mathbf {u} _ {n (i)}\right) q \left(\mathbf {u} _ {n (i)}\right) \log p \left(y _ {i} \mid f _ {i}\right) d f _ {i} d \mathbf {u} _ {n (i)} \tag {13}
+$$
+
+where only  $K$  nearest inducing points are involved in computing the expected log likelihood for each data point.
+
+# 3.2. Computing the KL Divergence
+
+We now show how we can write KL divergence as a summation over inducing points. Note that it is not possible for the standard SVGP: SVGP's KL requires accessing parameters for all inducing points at the same time.
+
+We first break the prior into a chain of conditionals using the nearest neighbor assumption in Eq 9 and utilize the mean-field assumption in Eq 10:
+
+$$
+\begin{array}{l} \operatorname {K L} \left[ q (\mathbf {u}) \| p (\mathbf {u}) \right] = \operatorname {K L} \left[ \prod_ {j = 1} ^ {M} q (u _ {j}) \| \prod_ {j = 1} ^ {M} p (u _ {j} | \mathbf {u} _ {n (j)}) \right] \\ = \underset {q (\mathbf {u})} {\mathbb {E}} \left[ \sum_ {j = 1} ^ {M} \log \frac {q \left(u _ {j}\right)}{p \left(u _ {j} \mid \mathbf {u} _ {n (j)}\right)} \right] \tag {14} \\ \end{array}
+$$
+
+exchanging the order of summation and expectation and integrating out irrelevant variables
+
+$$
+\begin{array}{l} = \sum_ {j = 1} ^ {M} \underset {q \left(\mathbf {u} _ {n (j)}\right)} {\mathbb {E}} \left[ \mathbb {E} _ {q \left(u _ {j}\right)} \left[ \log \frac {q \left(u _ {j}\right)}{p \left(u _ {j} \mid \mathbf {u} _ {n (j)}\right)} \right] \right] (15) \\ = \sum_ {j = 1} ^ {M} \mathbb {E} _ {q \left(\mathbf {u} _ {n (j)}\right)} \left[ \mathrm {K L} \left[ q \left(u _ {j}\right) \| p \left(u _ {j} \mid \mathbf {u} _ {n (j)}\right) \right] \right] (16) \\ \end{array}
+$$
+
+where
+
+$$
+\begin{array}{l} p \left(u _ {j} \mid \mathbf {u} _ {n (j)}\right) = \mathcal {N} \left(u _ {j} \mid \mathbf {k} _ {n (j), j} ^ {\top} \mathbf {K} _ {n (j), n (j)} ^ {- 1} \mathbf {u} _ {n (j)}, \right. (17) \\ k _ {j, j} - \mathbf {k} _ {n (j), j} ^ {\top} \mathbf {K} _ {n (j), n (j)} ^ {- 1} \mathbf {k} _ {n (j), j}). (18) \\ \end{array}
+$$
+
+# 3.3.Advantages
+
+Having factorized the data likelihood term and KL divergence over data points and inducing points, we are now ready to present VNNGP's key advantages as follows.
+
+Stochastic optimization. The summation structure of VN-NGP's ELBO, as detailed in Eq 13 and 16, makes it immediately available for stochastic optimization: in every training
+
+iteration, we randomly sample a mini-batch of training data indices  $\mathcal{I} = \{i_k\}_{k=1}^{N_b}$  and a mini-batch of inducing point indices  $\mathcal{J} = \{j_l\}_{l=1}^{M_b}$ ; we then optimize the unbiased estimate of the ELBO as follows
+
+$$
+\begin{array}{l} \mathcal {L} _ {\text {V N N G P}} \approx \frac {N}{N _ {b}} \sum_ {i \in \mathcal {I}} \underset {q \left(f _ {i}\right)} {\mathbb {E}} [ \log p \left(y _ {i} \mid f _ {i}\right) ] \tag {19} \\ - \frac {M}{M _ {b}} \sum_ {j \in \mathcal {J}} \underset {q (\mathbf {u} _ {n (j)})} {\mathbb {E}} \left[ \mathrm {K L} \left[ q (u _ {j}) \| p (u _ {j} | \mathbf {u} _ {n (j)}) \right] \right]. \\ \end{array}
+$$
+
+We include the exact computation of ELBO in Appendix A. Note that the sampling distributions for data points and inducing points can be arbitrary as long as both remain marginally a uniform distribution. We emphasize that this is a key advantage over SVGP: we are not able to perform stochastic optimization for inducing points with SVGP.
+
+Computational complexity. Once the nearest neighbor structure is computed, optimizing the objective in Eq 19 requires  $O((N_b + M_b)K^3)$  computations for inverting  $K \times K$  kernel matrices induced by nearest neighbor mini-batches. See Appendix A for mathematical details.
+
+The computational overhead comes from determining nearest neighbor structures for observations and inducing points. The brute-force complexity of finding any point's  $K$  nearest neighbors within  $M$  points is  $O(KM)$ . With the help of modern similarity search packages such as Johnson et al. (2017), we are able to dramatically speed up billion-scale similarity search with gpus. For example, for medium-sized datasets, e.g. Protein ( $N = 25.6\mathrm{K}$ ,  $D = 9$ ), it takes no more than 30 seconds to build up nearest neighbor structures with  $K = 256$  and  $M = N$  on an NVIDIA RTX2080 gpu; for the largest dataset we experiment with (Covtype:  $N = 372\mathrm{K}$  and  $D = 54$ ), it takes approximately 12 minutes with  $K = 256$  and  $M = N$ .
+
+More inducing points. Since the training complexity is free of  $N$  and  $M$ , we are able to place inducing points at every observed location and scale  $M$  to  $N$ . We enjoy several benefits by doing so. First, we greatly boost the model capacity, as opposed to a low-rank approximation where several observations "share" one inducing point. Moreover, we avoid optimizing inducing point locations as the standard SVGP training procedure requires. This is a huge advantage since it saves training cost and the optimization of inducing locations is in general non-convex and more difficult. Finally, we only need to compute the nearest neighbor structure once before training, or otherwise we need to recompute it frequently since inducing locations are being updated.
+
+# 3.4. Comparison to Related Methods
+
+The method most related to VNNGP is the Sparse Within Sparse Gaussian Process (SWSGP) method of Tran et al.
+
+(2021). SWSGP also builds upon SVGP and further imposes sparsity over  $M$  inducing points  $\mathbf{u}$  by defining a hierarchical prior. Essentially, in each training iteration, SWSGP randomly samples a minibatch of data point indices  $\mathcal{I} = \{i_k\}_{k=1}^{N_b}$  and optimizes the following ELBO
+
+$$
+\begin{array}{l} \mathcal {L} _ {\mathrm {S W S G P}} = \frac {N}{N _ {b}} \sum_ {i \in \mathcal {I}} \mathbb {E} _ {q (f _ {i})} [ \log p (y _ {i} | f _ {i}) ] \\ - \frac {1}{N _ {b}} \sum_ {i \in \mathcal {I}} \operatorname {K L} \left[ q \left(\mathbf {u} _ {n (i)}\right) \| p \left(\mathbf {u} _ {n (i)}\right) \right], \tag {20} \\ \end{array}
+$$
+
+where  $n(i)$  are indices of inducing point locations from  $\{\mathbf{z}_j\}_{j=1}^M$  that are top- $K$  nearest to  $\mathbf{x}_i$ . Similar to VNNGP, evaluating Eq. 20 requires  $O(N_b K^3)$  computation.
+
+There are two key differences between VNNGP and SWSGP. First, the two methods introduce sparsity through a different generative process. SWSGP applies a hierarchical prior; Consequently, the marginal prior for either  $p(\mathbf{f})$  or  $p(\mathbf{u})$  is no longer Gaussian. VNNGP, on the other hand, considers a sparse approximation to the prior precision matrix. The resulting process is still a GP but under an approximated kernel. While it is not immediately clear how these differing marginal distribution impact downstream performance, in this sense the VNNGP generative process more faithfully replicates the exact GP prior.
+
+The second difference is the training objective. As can be seen, the first term of Eq. 20 (data likelihood) is identical to that of VNNGP (Eq. 19). However, the KL divergence term is where two objectives differ. SWSGP optimizes a "local" KL divergence that only involves inducing points within a local neighborhood around current batch of training data. In fact, we prove (see appendix C) that SWSGP's KL divergence always underestimates the SVGP KL divergence:
+
+$$
+\operatorname {K L} \left[ q \left(\mathbf {u} _ {n (i)}\right) \| p \left(\mathbf {u} _ {n (i)}\right) \right]) \leq \operatorname {K L} \left[ q (\mathbf {u}) \| p (\mathbf {u}) \right], \tag {21}
+$$
+
+for any subset  $n(i)$ . Additionally, the amount of underestimation depends on size of  $n(i)$ .
+
+We hypothesize that SWSGP's KL, due to its "local" characteristic, may not sufficiently regularize the variational distribution—especially when  $K \ll M$ —which may make the model more prone to overfitting. Conversely, though VNNGP's KL does not equal the exact KL, it does (in expectation) consider the joint distribution over all inducing points. Optimizing this term will regularize the variational distribution towards the GP prior at all input locations. We investigate our hypotheses in the experiments section.
+
+Another method that also combines nearest neighbor approximations with variational inference is Amortized Inference GP (Liu & Liu, 2019). However, their method avoids computing certain terms of the KL divergence, and in doing so makes it impossible to simultaneously optimize kernel
+
+![](images/a1dfe56cafd0b777b7ea2a191991eff833482a640f6c8e23a6ee4dce643bbb0e.jpg)  
+Figure 2. Test set RMSE and NLL as a function of number of nearest neighbors  $K$  for VNNGP (green) on Elevators (left column) and UKHousing datasets (right column). Results for SVGP (blue dashed) and exact GP (purple dotted) are included as baselines.
+
+hyperparameters using the ELBO. By contrast, the VNNGP objective enables gradient-based kernel learning.
+
+# 4. Experiments
+
+We first demonstrate how sensitive our model is to the number of nearest neighbors  $K$ . Recognizing that our method is similar to SWSGP, our next task is to characterize the differences between two methods and investigate our hypotheses in Sec 3.4. Finally, we evaluate VNNGP and other methods on a wide range of real regression and classification datasets. Unless stated otherwise, for all experiments, VNNGP applies a random ordering of inducing points, VNNGP and SWSGP set inducing points at observed locations and use a mean-field variational approximation, and SVGP uses 1024 inducing points and a full-rank variational approximation.
+
+# 4.1. Sensitivity to number of nearest neighbors  $K$
+
+Our first goal is to see how the number of nearest neighbors  $K$  impact VNNGP's performance. We choose two datasets, Elevators with dimension  $D = 16$  (Asuncion & Newman, 2007), and UKHousing (https://landregistry.data.gov.uk/), a spatial dataset with  $D = 2$ , as we expect low-rank approximations to work better in high dimensional settings while nearest neighbor methods are more suitable for spatial problems that are sparse in nature. We fit VNNGP with  $K \in [32, 64, 128, 256]$ . Since these two datasets are modeled with Gaussian likelihood, we are able to fit exact GP (Wang et al., 2019) as a baseline. We also include SVGP as a low-rank variational alternative. We present their test set predictive performance, the root mean squared error (RMSE) and negative log likelihood (NLL), in Figure 2. From the figure, we see that as  $K$  increases, both test RMSE and NLL for VNNGP decrease, suggesting a better performance. By comparing the three methods, we note that exact GP does not always yield the best RMSE or NLL. This result could
+
+be due to model misspecification, as observed in previous literature (Wang et al., 2019; Potapczynski et al., 2021). Moreover, SVGP performs better on the high-dimensional Elevators dataset, whereas VNNGP with  $K = 32$  already outperforms SVGP and is less sensitive to  $K$  on the spatial UKHousing dataset.
+
+# 4.2. Comparison to SWSGP
+
+Building on the analysis in Section 3.4, we now aim to compare VNNGP to the related SWSGP method. Recall that the primary difference between these methods is the training objective (the KL divergence term in particular). We first empirically demonstrate how these two objectives differ, and then study its surprisingly significant impact on model fitting and selection.
+
+KL divergence comparison. We first investigate the difference between the VNNGP and SWSGP KL divergence terms. Specifically, we are interested in how their KL terms change as we vary  $K$ . We initialize VNNGP and SWSGP with the same configuration: the inducing points are placed on a 1-dimensional grid with even spacing of 1, the variational means  $\{m_i\}$  are obtained by sampling a GP under a squared exponential kernel with lengthscale 5 and outputscale 1, and the variational variances  $\{s_i\}$  are set to 1. For both methods, we compute their KL terms by varying the kernel lengthscale parameter and # of nearest neighbors  $K$ . A special note is that when  $K = M$ , KL by either method should be equal to the KL by SVGP in Eq. 8.
+
+We plot the SWSGP and VNNGP KL terms as a function of  $K$  under different kernel lengthscales in Figure 3. From this figure, we first confirm that VNNGP and SWSGP recover the SVGP KL divergence (dotted blue line) at  $K = M$ . Moreover, both methods tend to underestimate KL (recall from Section 3.4 that SWSGP provably underestimates this term). Notably, VNNGP converges rapidly to the true KL divergence, especially for small lengthscale settings. For example, when the lengthscale is 5, the VNNGP KL divergence is indistinguishable from the true KL divergence for all  $K \geq 5$ . We include results under different settings in appendix D.1 and observe similar patterns.
+
+Fitting models in high observational noise setting. We now study the impact of KL divergence on the inferred posterior distribution. Recall that the KL term of the ELBO regularizes the variational distribution towards the GP prior. If the regularization is not sufficient, the ELBO will be dominated by the data likelihood term, which could overfit the predictive distribution to the observed data.
+
+Hence, we conduct a simulation experiment in the presence of high observational noise: We draw  $N = 50$  observations from a GP under a squared exponential kernel with length-
+
+![](images/36ac73a7ff94029472543ee8e4736080373bf01de6e0e3eda9fd4f24d8502d57.jpg)  
+Figure 3. KL divergence computed by VNNGP (green) and SWSGP (orange) as a function of  $K$  for different prior kernel lengthscales. Both methods tend to underestimate the KL (SWSGP provably), and converges to the exact value computed by SVGP (blue dotted) as  $K \to M$ . Moreover, VNNGP's KL is closer to the exact one compared to SWSGP for the same  $K$  across all cases.
+
+![](images/2f8bfc01d33f1efc6c13f66719a500c0d021c8b8fcced4c93a9832d8b05f6d3d.jpg)
+
+![](images/c212f56a2f28775ae027857544f416fc693b995f479165fd56284d41960e33a2.jpg)
+
+![](images/9d1bd5429381c77288d972d43bcf3a444c1d6858f9eb57bb9282d20042ba2262.jpg)
+
+![](images/5cac50ca69c733f96caeafadcbcc0b4a01172aa9494e7ec525516c43e6b4329b.jpg)
+
+![](images/bac3e0d2e8897442539017a8bf9edadd33c20cb74e0c9d642601c61e0779cbb6.jpg)
+
+![](images/044f317266301aab5a78e61d3401fff0d89d6baa0015f1375d4279f4e0fc52d6.jpg)
+
+![](images/11c536dac247f26d8f0046bb5ddbc269d2402ae43fd6457db28d8130a05ceebc.jpg)  
+Figure 4. Posterior by Exact GP (purple), SVGP (blue), VNNGP (green) and SWSGP (orange), with  $95\%$  uncertainty interval. The grey line is to the true data generating process, and scattering black dots are highly noisy observations.
+
+![](images/73da0afea7aa3975272e66864fd78dba95b1af64d53cbad3ee64e13e91e41e0c.jpg)
+
+scale 5 and outputscale 5, and add iid noise from  $\mathcal{N}(0,5)$ . There are two dense clusters of observations around  $x = 0$  and  $x = 50$ , and scarcely scattered observations in between. We fit four models—exact GP, SVGP, VNNGP and SWSGP—to the observed data. We fix all hyperparameters corresponding to the true data generating process, and only optimize variational parameters. For the three variational methods, inducing points are set to the locations of observed data  $(M = N)$ . All variational models use mean-field variational approximations. Both VNNGP and SWSGP use  $K = 5$  nearest neighbors.
+
+In Fig 4, we display the true data generating process, noisy observations and posteriors of all four models. The SVGP and VNNGP posteriors are qualitatively close to that of exact GP, whereas SWSGP's posterior deviates from the underlying GP function in the data-sparse region. We suspect SWSGP's behavior to be due to its underestimating local KL divergence. As we discuss in Section 3.4, the "local" KL divergence of SWSGP will not regularize the variational distribution towards a globally smooth distribution. In particular, the SWSGP KL divergence concentrates on the data-dense regions, leading to a lack of regularization in the data-sparse region. By contrast, the VNNGP model is effectively regularized towards the GP prior throughout the input space.
+
+Model selection. Finally, we turn to model selection by maximizing the ELBO. Our goal is to see whether VNNGP or SWSGP is sensitive to different settings of hyperparameters, in particular, the likelihood noise.
+
+We first fit exact GP to the UKHousing dataset under a Matern 5/2 kernel and a Gaussian likelihood. We copy and fix the exact GP kernel hyperparameters to SVGP, VNNGP, and SWSGP models. However, we vary the value of likelihood noise from 0.001 to 1.0. We then optimize only the variational parameters of these three methods for 300 epochs. In Figure 5, we plot the negative training ELBO and test NLL as a function of likelihood noise for each method. We denote the noise parameters that maximize each method's training objective, as well as the noise parameter selected by the exact GP. In agreement with prior work (Bauer et al., 2016), we find that SVGP tends to overestimate the noise parameter. Both nearest neighbor methods tend to underestimate the likelihood noise. However, the ELBO of SWSGP monotonically decreases as likelihood noise decreases, whereas the VNNGP ELBO achieves an optimum as a noise of 0.1. If we use the ELBO as a model selection criterion (as is often the case for variational methods), the resulting SWSGP model will have very low likelihood noise which results in poor test NLL. Conversely, the VNNGP training ELBO is highly predictive of the test NLL, as both are roughly optimized as the same likelihood noise value. We note that SWSGP's behavior in this setting is likely due to having  $M = N$ . We repeat this experiment for SWSGP models that use  $M \ll N$  inducing points in Appendix D.2, and find that these models favor more reasonable likelihood noise values. These observations suggest that the ELBO is a good model selection criteria for VNNGP; however, ELBO optimization may result in extreme hyperparameters for SWSGP with large  $M$ .
+
+# 4.3. Predictive performance on real datasets
+
+We conduct an extensive evaluation of our method on real datasets. We consider a wide range of high dimensional and spatiotemporal datasets from the UCI repository (Asuncion & Newman, 2007). In addition we include three spatial datasdets, UKHousing as mentioned in Section 4.1, Precipi
+
+<table><tr><td rowspan="2">Dataset</td><td rowspan="2">N</td><td rowspan="2">D</td><td rowspan="2">ExactGP</td><td rowspan="2">sgGP</td><td colspan="4">NLL</td></tr><tr><td>SVGP</td><td>SWSGP-M</td><td>SWSGP-N</td><td>VNNGP</td></tr><tr><td>PoleTele</td><td>9.6K</td><td>26</td><td>-.509 ± .005</td><td>-.558 ± .016</td><td>-.708 ± .003</td><td>-.898 ± .029</td><td>-.962 ± .023</td><td>-1.160 ± .014</td></tr><tr><td>Elevators</td><td>10.6K</td><td>16</td><td>.464 ± .011</td><td>.432 ± .013</td><td>.417 ± .017</td><td>.691 ± .049</td><td>.767 ± .026</td><td>.463 ± .020</td></tr><tr><td>Bike</td><td>11.1K</td><td>17</td><td>-.744 ± .007</td><td>53.364 ± 27.690</td><td>-1.849 ± .016</td><td>-1.684 ± .024</td><td>-1.405 ± .096</td><td>-1.690 ± .057</td></tr><tr><td>Kin40K</td><td>25.6K</td><td>8</td><td>-.149 ± .001</td><td>.701 ± .139</td><td>-.399 ± .003</td><td>-.747 ± .016</td><td>-1.010 ± .005</td><td>-1.016 ± .004</td></tr><tr><td>Protein</td><td>25.6K</td><td>9</td><td>1.044 ± .003</td><td>.914 ± .004</td><td>.967 ± .005</td><td>.909 ± .006</td><td>1.505 ± .080</td><td>.671 ± .009</td></tr><tr><td>KEGG</td><td>31.2K</td><td>20</td><td>-.713 ± .004</td><td>-1.013 ± .035</td><td>-1.000 ± .011</td><td>-1.039 ± .010</td><td>14.925 ± .838</td><td>-1.039 ± .018</td></tr><tr><td>KEGGU</td><td>40.7K</td><td>26</td><td>-.471 ± .003</td><td>-.673 ± .005</td><td>-.680 ± .002</td><td>-.719 ± .004</td><td>41.675 ± .427</td><td>-.715 ± .004</td></tr><tr><td>Precipitation</td><td>64.8K</td><td>3</td><td>—</td><td>—</td><td>.818 ± .004</td><td>.396 ± .001</td><td>1.809 ± .224</td><td>.145 ± .002</td></tr><tr><td>UKHousing</td><td>116K</td><td>2</td><td>.703 ± .003</td><td>.598 ± .001</td><td>.879 ± .004</td><td>.455 ± .003</td><td>1.050 ± .013</td><td>.367 ± .003</td></tr><tr><td>3DRoad</td><td>278K</td><td>3</td><td>.993 ± .000</td><td>.712 ± .001</td><td>.325 ± .005</td><td>.584 ± .003</td><td>-1.476 ± .079</td><td>-.048 ± .406</td></tr><tr><td>Covtype</td><td>372K</td><td>54</td><td>—</td><td>—</td><td>.234 ± .001</td><td>.172 ± .001</td><td>.069 ± .000</td><td>.132 ± .000</td></tr></table>
+
+Table 1. Test set NLL (mean ± 1 standard error over 3 random seeds) on high-dimensional datasets (top 7 ones) and spatial datasets (bottom 4 ones). VNNGP achieves lowest NLL for most datasets. The result for RMSE is included in Table 2 in appendix.
+
+![](images/1d36696d377139ffa8406bc0c6c92b6be0a2b60aba2297656598a38b3585b04a.jpg)  
+Figure 5. Negative training ELBO (solid line) and test NLL (dotted line) as a function of the likelihood noise by SVGP (blue), VNNGP (green) and SWSGP (orange) on UKHousing dataset. All ELBO and NLL values are scaled to [0,1]. The x-axis is in log-scale. The noise value picked by maximizing ELBO for each method is indicated by the vertical dashed line with corresponding color. The vertical purple line indicates the value (0.28) learned by exact GP.
+
+tation (a monthly precipitation dataset with  $D = 3$ ) (Lyon, 2004; Lyon & Barnston, 2005) and Covtype (a tree cover dataset with  $D = 54$ ) $^{2}$  (Blackard & Dean, 1999). Our goal is primarily to compare VNNGP to other variational methods; however, we include the additional baselines of Exact GP—utilizing the methodology of Wang et al. (2019)—and sgGP (Chen et al., 2020)—which is a non-variational nearest neighbor benchmark. We do not compare to the Amortized Inference GP since it is not amenable to kernel learning. For VNNGP and SWSGP, we tune the number of nearest neighbors  $K$  in {32, 256}. Additionally, we include two variants of the SWSGP method: with SWSGP- $N$  we set all inducing points at observed locations, and with SWSGP- $M$  we allow using  $M < N$  inducing points – the latter is what Tran et al. (2021) originally experimented with. In particular for SWSGP- $M$ , we recompute nearest neighbors every epoch and we tune  $M$  in {1024, 2048, 4096, 8192}. For sgGP, the subsampled datasets are constructed by selecting a random point  $x$ ,  $y$  and its 15 nearest neighbors as in
+
+Chen et al. (2020). Each dataset is randomly split to  $64\%$  training,  $16\%$  validation and  $20\%$  testing sets. Precipitation dataset uses a student-t likelihood and Covtype dataset uses a bernoulli likelihood, otherwise a Gaussian likelihood is used. All kernels are Matern 5/2 with a separate lengthscale per dimension. We run each model with three random seeds. For each random seed, VNNGP applies different random ordering of inducing points. See appendix D.3.1 for more details.
+
+We report test NLL in Table 1, and the result for test RMSE is included in Table 2 of appendix. From the tables, we make three key comparisons. (i) Exact GP and sgGP cannot run on problems with non-Gaussian likelihoods (Precipitation and Covtype) while variational methods are still viable. (ii) VNNGP substantially outperforms other nearest neighbor methods. Specifically, we note that sgGP and SWSGP-  $N$  sometimes obtain a very high NLL, while SWSGP-  $M$  does not suffer from this issue. We suspect this result is attributed to that sgGP and SWSGP-  $N$  tend to learn a very small likelihood noise on certain datasets (we report the learned noise in Table 3 of appendix); Also recall from Section 4.2 that maximizing the ELBO of SWSGP-  $N$  leads to extremely small noise values. (iii) While there are a few datasets (Elevators and Bike) where SVGP excels, VNNGP outperforms SVGP especially on large-scale, spatial datasets. SVGP's performance is limited by using  $M = 1024 \ll N$  inducing points, which is detrimental on these problems, while VNNGP has the ability to scale  $M$  to  $N$  with a sparse approximation.
+
+Finally, we note that both VNNGP and SWSGP-  $N$  use much more parameters than SVGP. It is natural to ask whether this fact will impede their convergence speeds. In Figure 6, we plot the training loss versus number of training iterations for SVGP, VNNGP and SWSGP-  $N$  on Bike and UKHousing. We observe that all three methods reach convergence by the end of training. Moreover, VNNGP's convergence speed is comparable to SVGP. This result demonstrates that the larger number of model parameters does not hurt VNNGP's optimization process, making
+
+![](images/f833a1292c350f3b683da26c9c4a72939317a3371650fd48f46faa05614b58b1.jpg)  
+Figure 6. Training loss versus iterations on Bike and UKhousing. The training loss is averaged over every 100 iterations and all loss values are standardized. Despite VNNGP has more parameters, its convergence speed is comparable to SVGP.
+
+![](images/c8dd9a284f63d57083c56709ae2893c364addd903f0094dfee73614d0b92e218.jpg)
+
+it a practical model to use.
+
+# 5. Discussion
+
+In this work, we propose VNNGP, a scalable GP method that combines a sparse precision approximation with variational inference. This sparse approximation allows VNNGP to use orders of magnitude more inducing points than other variational methods. We perform an extensive empirical evaluation and show that VNNGP obtains strong performance compared to other baselines.
+
+We consider two extensions for future works. Similar to Liu & Liu (2019) and Jafrasteh et al. (2021), we can apply amortized learning to variational parameters. This could greatly reduce the number of parameters and potentially accelerate optimization. Another extension is to select nearest neighbors using metrics other than Euclidean distance. For example, Kang & Katzfuss (2021) suggest that the prior covariance function can be a metric to select the nearest neighbor set. This could enhance VNNGP's accuracy on problems with anisotropic or periodic covariance structure.
+
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+
+# A. Computing the ELBO for VNNGP
+
+In this section, we derive the exact computation of the ELBO for VNNGP and analyze the computational complexity.
+
+Following Eq 12, and using the derivations from Eq 13 and Eq 16, we have
+
+$$
+\begin{array}{l} \mathcal {L} _ {\mathrm {V N N G P}} = \underbrace {\sum_ {i = 1} ^ {N} \mathbb {E} _ {q (\mathbf {f} , \mathbf {u})} \left[ \log p \left(y _ {i} \mid f _ {i}\right) \right]} _ {\text {d a t a l i k e h o o d}} - \underbrace {\operatorname {K L} \left[ q (\mathbf {u}) \| p (\mathbf {u}) \right]} _ {\text {K L d i v e r g e n c e}}, (22) \\ = \underbrace {\sum_ {i = 1} ^ {N} \int q \left(f _ {i}\right) \log p \left(y _ {i} \mid f _ {i}\right) d f _ {i}} _ {\text {d a t a l i k e l i h o o d}} - \underbrace {\sum_ {j = 1} ^ {M} \mathbb {E} _ {q \left(\mathbf {u} _ {n (j)}\right)} \left[ \mathrm {K L} \left[ q \left(u _ {j}\right) \| p \left(u _ {j} \mid \mathbf {u} _ {n (j)}\right) \right] \right]} _ {\text {k l d i v e r g e n c e}}. (23) \\ \end{array}
+$$
+
+where
+
+$$
+\begin{array}{l} q \left(f _ {i}\right) = \mathbb {E} _ {q (\mathbf {u})} \left[ p \left(f _ {i} \mid \mathbf {u} _ {n (i)}\right) \right] (24) \\ = \mathcal {N} \left(f _ {i} \mid \mathbf {k} _ {n (i), i} ^ {\top} \mathbf {K} _ {n (i), n (i)} ^ {- 1} \mathbf {m} _ {n (i)}, k _ {i, i} - \mathbf {k} _ {n (i), i} ^ {\top} \mathbf {K} _ {n (i), n (i)} ^ {- 1} \mathbf {k} _ {n (i), i} + \mathbf {k} _ {n (i), i} ^ {\top} \mathbf {S} _ {n (i)} \mathbf {k} _ {n (i), i}\right), (25) \\ \end{array}
+$$
+
+and  $\mathbf{S}_{n(i)}$  is a diagonal matrix with diagonal elements being  $\mathbf{s}_{n(i)}$ .
+
+- The data likelihood term is tractable for Gaussian likelihood; otherwise, techniques such as MCMC sampling are required. The major complexity in evaluating each expected likelihood term comes from computing  $q(f_{i})$ , which essentially resides at inverting the  $K \times K$  matrix  $\mathbf{K}_{n(i),n(i)}$  which takes  $O(K^3)$  time.  
+- To compute the KL divergence term, we first compute each conditional KL term. Given fixed  $\mathbf{u}_{n(j)}$ ,
+
+$$
+\begin{array}{l} \operatorname {K L} \left[ q \left(u _ {j}\right) \| p \left(u _ {j} \mid \mathbf {u} _ {j}\right) \right] = \operatorname {K L} \left[ \mathcal {N} \left(u _ {j} \mid m _ {j}, s _ {j}\right) \| \mathcal {N} \left(u _ {j} \mid \mathbf {b} _ {j} ^ {\top} \mathbf {u} _ {n (j)}, f _ {j}\right) \right] (26) \\ = \frac {1}{2} \left[ \log f _ {j} - \log s _ {j} - 1 + f _ {j} ^ {- 1} s _ {j} + f _ {j} ^ {- 1} \left(m _ {j} - \mathbf {b} _ {j} ^ {\top} \mathbf {u} _ {n (j)}\right) ^ {2} \right], (27) \\ \end{array}
+$$
+
+where  $f_{j} = k_{j,j} - \mathbf{k}_{n(j),j}^{\top}\mathbf{K}_{n(j),n(j)}^{-1}\mathbf{k}_{n(j),j}$  and  $\mathbf{b}_j = \mathbf{K}_{n(j),n(j)}^{-1}\mathbf{k}_{n(j),j}$ . Integrating over  $q(\mathbf{u}_{n(j)})$  and summing over inducing points we obtain,
+
+$$
+\operatorname {k l} \text {d i v e r g e n c e} = \frac {1}{2} \sum_ {j = 1} ^ {M} \log f _ {j} - \log s _ {j} - 1 + f _ {j} ^ {- 1} \left[ s _ {j} + \left(\mathbf {b} _ {j} ^ {2}\right) ^ {\top} \mathbf {s} _ {n (j)} + \left(m _ {j} - \mathbf {b} _ {j} ^ {\top} \mathbf {m} _ {n (j)}\right) ^ {2} \right], \tag {28}
+$$
+
+where  $\mathbf{b}_j^2$  denotes the element-wise squared of vector  $\mathbf{b}_j$ . Computing each summation term in the KL divergence requires inverting an  $K\times K$  matrix, i.e.  $\mathbf{K}_{n(j),n(j)}$ , which is  $O(K^3)$  complexity.
+
+Therefore, the overall complexity of estimating the ELBO using a mini-batch of  $N_{b}$  data points and a mini-batch of  $M_{b}$  inducing points is  $O(N_{b}K^{3} + M_{b}K^{3})$ .
+
+# B. Extension to more expressive variational family
+
+We now consider more expressive variational approximations and discuss how to mini-batch the VNNGP's ELBO computation. We assume the following variational family:
+
+$$
+q (\mathbf {u}) = \mathcal {N} (\mathbf {u} | \mathbf {m}, \tilde {\mathbf {L}} \tilde {\mathbf {L}} ^ {\top}) \tag {29}
+$$
+
+$$
+q (\mathbf {f} | \mathbf {u}) = \prod_ {i = 1} ^ {N} p \left(f _ {i} \mid \mathbf {u} _ {n (i)}\right) \tag {30}
+$$
+
+where  $\tilde{\mathbf{L}}$  is the lower-triangular Cholesky factor for variational posterior covariance (potentially full-rank). The mean-field case discussed in appendix A is a special case where we make  $\tilde{L}$  diagonal.
+
+Similarly, the ELBO under this variational family decomposes into the data likelihood term and the KL divergence term as in Eq 23 with the modification that
+
+$$
+\begin{array}{l} q \left(f _ {i}\right) = \mathbb {E} _ {q (\mathbf {u})} \left[ p \left(f _ {i} \mid \mathbf {u} _ {n (i)}\right) \right] (31) \\ = \mathcal {N} \left(f _ {i} \mid \mathbf {k} _ {n (i), i} ^ {\top} \mathbf {K} _ {n (i), n (i)} ^ {- 1} \mathbf {m} _ {n (i)}, k _ {i, i} - \mathbf {k} _ {n (i), i} ^ {\top} \mathbf {K} _ {n (i), n (i)} ^ {- 1} \mathbf {k} _ {n (i), i} + \mathbf {k} _ {n (i), i} ^ {\top} \tilde {\mathbf {L}} _ {n (i)} \tilde {\mathbf {L}} _ {n (i)} ^ {\top} \mathbf {k} _ {n (i), i}\right). (32) \\ \end{array}
+$$
+
+- The likelihood computation is similar to the mean-field case in appendix A, except that we are now integrating over  $q(f_i)$  in Eq 32 instead of  $q(f_i)$  in Eq 25. The former involves a potentially full-rank  $K \times K$  variational covariance  $\tilde{\mathbf{L}}_{n(i)}\tilde{\mathbf{L}}_{n(i)}^{\top}$  while the latter involves a diagonal one.  
+- KL divergence.
+
+For mathematical convenience, we consider the Cholesky composition of prior precision for inducing points  $\mathbf{K}_{\mathbf{z},\mathbf{z}}^{-1} = \mathbf{L}^{\top}\mathbf{L}$  where  $\mathbf{L}$  is a lower triangular matrix. Each row of  $\mathbf{L}$  could be separately computed at  $O(K^{3})$  cost (see Datta (2021a) for details). The KL divergence can be computed as follows:
+
+$$
+\begin{array}{l} \operatorname {K L} [ q (\mathbf {u}) \| p (\mathbf {u}) ] = \operatorname {K L} \left[ \mathcal {N} (\mathbf {m}, \tilde {\mathbf {L}} \tilde {\mathbf {L}} ^ {\top}) \| \mathcal {N} (0, (\mathbf {L} ^ {\top} \mathbf {L}) ^ {- 1}) \right] (33) \\ = \frac {1}{2} \left[ - 2 \log | \mathbf {L} | - 2 \log | \tilde {\mathbf {L}} | - M + \operatorname {t r} \left(\mathbf {L} ^ {\top} \mathbf {L} \tilde {\mathbf {L}} \tilde {\mathbf {L}} ^ {\top}\right) + \mathbf {m} ^ {\top} \mathbf {L} ^ {\top} \mathbf {L} \mathbf {m} \right] (34) \\ \end{array}
+$$
+
+Here we take a special treatment of the trace term:
+
+$$
+\begin{array}{l} \operatorname {t r} \left(\mathbf {L} ^ {\top} \mathbf {L} \tilde {\mathbf {L}} \tilde {\mathbf {L}} ^ {\top}\right) = \operatorname {t r} \left(\mathbf {L} \tilde {\mathbf {L}} \left(\mathbf {L} \tilde {\mathbf {L}}\right) ^ {\top}\right) (\text {b y}) \\ = \sum_ {i = 1} ^ {M} \sum_ {j = 1} ^ {M} \left(\mathbf {L} \tilde {\mathbf {L}}\right) _ {i j} ^ {2} (\text {b y d e f i n i t i o n o f t r a c e}) (36) \\ = \sum_ {i = 1} ^ {M} \sum_ {j = 1} ^ {M} \sum_ {k = 1} ^ {M} L _ {i j} ^ {2} \tilde {L} _ {j k} ^ {2} (\text {e x p a n d i n g t h e (L} \tilde {\mathbf {L}} _ {i j}) \text {t e r m s}) (37) \\ = \sum_ {i = 1} ^ {M} \sum_ {j = 1} ^ {M} \sum_ {k \in n (i) \cup \{i \}} \mathbf {L} _ {i k} ^ {2} \tilde {\mathbf {L}} _ {k j} ^ {2} (\text {u s i n g r o w - s p a r s i t y o f L}) (38) \\ \end{array}
+$$
+
+Now we re-write the KL by decomposing each term:
+
+$$
+\begin{array}{l} \operatorname {K L} [ q (\mathbf {u}) \parallel p (\mathbf {u}) ] = \frac {1}{2} \left[ - 2 \sum_ {i = 1} ^ {M} \log L _ {i i} - 2 \sum_ {i = 1} ^ {M} \tilde {L} _ {i i} - M + \sum_ {i = 1} ^ {M} \sum_ {j = 1} ^ {M} \sum_ {k \in n (i) \cup \{i \}} \mathbf {L} _ {i k} ^ {2} \tilde {\mathbf {L}} _ {k j} ^ {2} + \sum_ {i = 1} ^ {M} \left(\sum_ {k \in n (i) \cup \{i \}} L _ {i k} m _ {k}\right) ^ {2} \right] (39) \\ = \frac {1}{2} \sum_ {i = 1} ^ {M} \left[ - 2 \log L _ {i i} - 2 \tilde {L} _ {i i} - 1 + \sum_ {k \in n (i) \cup \{i \}} \mathbf {L} _ {i k} ^ {2} \sum_ {j = 1} ^ {M} \tilde {\mathbf {L}} _ {k j} ^ {2} + \left(\sum_ {k \in n (i) \cup \{i \}} L _ {i k} m _ {k}\right) ^ {2} \right], (40) \\ \end{array}
+$$
+
+which again factorizes over inducing points  $i = 1:M$ . However, one should note that computing each KL term now scales  $O(MK^3)$  due to the complexity in trace term  $\sum_{k\in n(i)\cup \{i\}}\sum_{j = 1}^{M}\mathbf{L}_{ik}^{2}\tilde{\mathbf{L}}_{kj}^{2}$ . One can alleviate this issue by (i) obtaining a stochastic estimate for the trace term (i.e. sub-sample  $j$ ) (ii) introduce sparsity structure in  $\tilde{\mathbf{L}}$  / variational posterior.
+
+In summary, we explore the possibility of extending VNNGP to more expressive variational family. We show that under the full-rank variational family, the VNNGP's ELBO still factorizes over data points and inducing points. Computing each term now takes more computations and but techniques can be applied to improve the scalability.
+
+# C. KL divergence for SWSGP
+
+Here we re-state and prove the two claims about KL divergence for SWSGP in Section 3.4.
+
+Consider a fixed prior  $p(\mathbf{u})$  and a fixed variational posterior  $q(\mathbf{u})$  over  $M$  inducing points  $\mathbf{u}$ . Then we claim that
+
+1. SWSGP's KL divergence is always underestimating, if not equal to, SVGP's KL divergence. That is, for any subset  $S \in \{1, 2, \dots, M\}$ .
+
+$$
+\operatorname {K L} [ q (\mathbf {u} _ {\mathcal {S}}) \| p (\mathbf {u} _ {\mathcal {S}}) ] \leq \operatorname {K L} [ q (\mathbf {u}) \| p (\mathbf {u}) ], \tag {41}
+$$
+
+2. The amount of underestimation depends on the size of the nearest neighbor set. More precisely, for any two subsets  $S_1$  and  $S_2$  such that  $S_1 \subset S_2 \subset \{1, \dots, M\}$ ,
+
+$$
+\operatorname {K L} \left[ q \left(\mathbf {u} _ {\mathcal {S} _ {1}}\right) \| p \left(\mathbf {u} _ {\mathcal {S} _ {1}}\right) \right] \leq \operatorname {K L} \left[ q \left(\mathbf {u} _ {\mathcal {S} _ {1}}\right) \| p \left(\mathbf {u} _ {\mathcal {S} _ {2}}\right) \right]. \tag {42}
+$$
+
+The proof for the above claims will immediately follow once we prove the following proposition.
+
+Proposition C.1. Let  $p(\mathbf{u})$  and  $q(\mathbf{u})$  be two distributions for an  $M$ -dimensional random variable  $\mathbf{u}$ , and let  $\mathbf{u}_1$  be any sub-vector of  $\mathbf{u}$ , then
+
+$$
+\operatorname {K L} \left[ q \left(\mathbf {u} _ {1}\right) \| p \left(\mathbf {u} _ {1}\right) \right] \leq \operatorname {K L} \left[ q (\mathbf {u}) \| p (\mathbf {u}) \right]. \tag {43}
+$$
+
+Proof. The proof relies on the chain rule of probability distribution, and the fact that the KL divergence is non-negative.
+
+Denote  $\mathbf{u}_2$  as the remaining sub-vector by taking  $\mathbf{u}_1$  out of  $\mathbf{u}$ . It follows that
+
+$$
+\begin{array}{l} \operatorname {K L} [ q (\mathbf {u}) \| p (\mathbf {u}) ] (44) \\ = \int q (\mathbf {u}) \log \frac {q (\mathbf {u})}{p (\mathbf {u})} d \mathbf {u} (45) \\ = \iint q \left(\mathbf {u} _ {1}\right) q \left(\mathbf {u} _ {2} \mid \mathbf {u} _ {1}\right) \log \frac {q \left(\mathbf {u} _ {1}\right) q \left(\mathbf {u} _ {2} \mid \mathbf {u} _ {1}\right)}{p \left(\mathbf {u} _ {1}\right) p \left(\mathbf {u} _ {2} \mid \mathbf {u} _ {1}\right)} d \mathbf {u} _ {1} d \mathbf {u} _ {2} (46) \\ = I + I I, (47) \\ \end{array}
+$$
+
+where
+
+$$
+I \equiv \int q \left(\mathbf {u} _ {1}\right) \log \frac {q \left(\mathbf {u} _ {1}\right)}{p \left(\mathbf {u} _ {1}\right)} \left(\int q \left(\mathbf {u} _ {2} \mid \mathbf {u} _ {1}\right) d \mathbf {u} _ {2}\right) d \mathbf {u} _ {1} = \mathrm {K L} \left[ q \left(\mathbf {u} _ {1}\right) \| p \left(\mathbf {u} _ {1}\right) \right], \tag {48}
+$$
+
+$$
+I I \equiv \int q \left(\mathbf {u} _ {1}\right) \left(\int q \left(\mathbf {u} _ {2} \mid \mathbf {u} _ {1}\right) \log \frac {q \left(\mathbf {u} _ {2} \mid \mathbf {u} _ {1}\right)}{p \left(\mathbf {u} _ {2} \mid \mathbf {u} _ {1}\right)}\right) d \mathbf {u} _ {1} = \int q \left(\mathbf {u} _ {1}\right) \mathrm {K L} \left[ q \left(\mathbf {u} _ {2} \mid \mathbf {u} _ {1}\right) \| p \left(\mathbf {u} _ {2} \mid \mathbf {u} _ {1}\right) \right] d \mathbf {u} _ {1}. \tag {49}
+$$
+
+Note that for each  $\mathbf{u}_1$  fixed, KL  $[q(\mathbf{u}_2|\mathbf{u}_1)\parallel p(\mathbf{u}_2|\mathbf{u}_1)]$  defines a (conditional) KL divergence and is non-negative. Therefore, KL  $[q(\mathbf{u})\parallel p(\mathbf{u})] = I + II\geq I$ , i.e. Eq 43 holds.
+
+# D. Experiment details and additional results
+
+# D.1. KL divergence comparison
+
+In Figure 7, we vary the setting in Section 4.2: different orderings of inducing points, different kernel functions, and real data v.s. synthetic data. From this figure, we observe similar patterns as from Figure 3.
+
+# D.2. Model selection
+
+We include additional results for the model selection experiment in Section 4.2. We refer the models in Section 4.2 by VNNGP-  $N$  and SWSGP-  $N$  that set inducing points on observed locations. Here, we consider the alternative versions, VNNGP-  $M$  and SWSGP-  $M$ , which use  $M \ll N$  inducing points for  $M \in \{1024, 8192\}$ , and the inducing point locations are learned during optimization. Correspondingly, the nearest neighbors are updated every training epoch for VNNGP and SWSGP. The other configurations are the same as Section 4.2 (especially, SVGP uses 1024 inducing points for all times). All inducing point locations are initialized by k-means clustering of observed locations.
+
+Comparing Figure 8 to Figure 5, we can see that (1) VNNGP-  $M$  and VNNGP-  $N$  have consistent behaviors in model selection. (2) While maximizing the ELBO of SwSGP-  $N$  tends to favor a model with small likelihood noise, this is not the case for SwSGP-  $M$ . As the noise parameter becomes extremely small, the training ELBO of SwSGP-  $M$  decreases. As a result, optimizing SwSGP-  $M$  leads to a model that has a "medium" likelihood noise and a low test NLL. These observations suggest that maximizing the ELBO for SwSGP-  $N$  (and potentially for SwSGP-  $M$  with  $M$  close to  $N$ ) can lead to undesired model selection outcome, while this does not hold for SwSGP-  $M$  with a medium size of inducing points.
+
+![](images/f0a66fc5be1477730b39d60660466e769767dc3ce542a9d553e51c4d26f6615d.jpg)
+
+![](images/add643e8edcdf02eeca7310a870df9c030ac0522a076c0d597c632607d86d9c9.jpg)
+
+![](images/315e25b89debd9d891a8545875365adfb022daddfe931f34a485246cdc031e92.jpg)
+
+![](images/a16ac438d9c7ab3d05c819d2c93b69b30970dc5b93360600d1073809ae3f14c7.jpg)
+
+![](images/3dcd5aa7e6a1eea7ed4d4d2f656765559da389003379952cf4ba9ff9c592668c.jpg)
+
+![](images/102c0766ddc4e41a415a53df0f2acbdb59480a2841b548e0e03bd6fd30f436d1.jpg)
+
+![](images/d632d873425dd0878346e5684b6fa0e9c440c5c8449c65b7904eb71faabd5539.jpg)
+
+![](images/485c5bd2432344e457ecd5ea3c8c53b2107b6428a8462b5407dedf1fb4ccf237.jpg)
+
+![](images/d3f327f0d0bd70e2897beefc5dbf04d5520004704720d95fb6623d60d1b28c5e.jpg)
+
+![](images/adfa471b3e2a1844aa6c95069df79a85aaf47db76791bbac79c1439c1998076a.jpg)
+
+![](images/c4cd1be6d4a1e7dd13e9daae2fe678f0665b795c419a0eaf4e39b6e4a37516be.jpg)
+
+![](images/98961659767c20e14ee7c1287269edcbe81af9e4098a731f53e12aa6a1160966.jpg)
+
+![](images/06cc1a8febb7ae9051a5e3748b492261412343194182f898ff7b09ae8551424f.jpg)
+
+![](images/fcca8ca97a791a2587d71782e1e8fd10f9dd6d0c2dfc0490603602acd45167be.jpg)
+
+![](images/4a1a402b84ad647d7a2a27dc10404b26423c6bebd85a9018726d8ce2236303b1.jpg)
+
+![](images/f68b0577c0fa7d132f3ae0c69ebc9484241c5cc7d41915c864cff28e5aab2ec8.jpg)  
+Figure 7. KL comparison under different settings than the setting of Figure 3. Row 1: A different random ordering of inducing points. Row 2: Coordinate ordering of inducing points. Row 3: Matern (2.5) kernel. Row 4: Real dataset (3DRoad).
+
+![](images/61f743b8a5a743ec79af88a37bc25fb4a455db9fc5180a4b91a3c98366060e89.jpg)
+
+![](images/71fb1b4d400e0f9ba4ed4e58338bffda6d5c36a79313ed081028b377b8050a10.jpg)  
+Figure 8. Negative ELBO in the last training iteration (solid line) and test set NLL (dotted line) as a function of the likelihood noise parameter by SVGP (blue), VNNGP (green) and SWSGP (orange) on UKHousing dataset. All ELBO and NLL values are scaled to [0, 1]. The x-axis (Noise) is in log-scale. The best noise parameter picked by maximizing ELBO for each method is indicated by the vertical dashed line with corresponding color. The vertical purple line indicates the noise parameter  $(= 0.28)$  learned by exact GP. Left: both VNNGP and SWSGP use  $M = 1024$  inducing points. Right: both VNNGP and SWSGP use  $M = 8192$  inducing point. For all cases, SVGP use 1024 inducing points.
+
+# D.3. Predictive performance on real dataset
+
+Here we provide training details and additional results for Section 4.3.
+
+# D.3.1. TRAINING DETAILS
+
+For all methods, we use an Adam optimizer and a MultiStepLR scheduler dropping the learning rate by a factor of 10 at the  $75\%$  and  $90\%$  of the optimization iterations; all kernels are Matern 5/2 kernel with separate lengthscale per dimension; the kernel lengthscales, outputscale and likelihood noise parameter (if any) are all initialized as 0.6931, except that exact GP is initialized with lengthscale  $= 0.01$  on 3DRoad dataset. All hyperparameters are picked by the best validation NLL.
+
+All methods are trained with  $\{300, 500\}$  iterations and learning rate of  $\{0.005, 0.01\}$  for datasets of size below  $50K$ , and with  $\{100, 300\}$  iterations and learning rate of  $\{0.005, 0.001\}$  for above  $50K$ .
+
+For sgGP, we train using minibatches of 16 data points. As suggested by Chen et al. (2020), the minibatches are constructed by sampling one training data point and selecting its 15 nearest neighbors. To accelerate optimization, we accumulate the gradients of 1,024 minibatches before performing an optimization step (these 1,024 minibatch updates can be performed in parallel, enabling GPU acceleration).
+
+For SVGP, we use  $M = 1024$  inducing points and a full-rank variational approximation. The inducing point locations are initialized by k-means clustering of observed locations
+
+VNNGP, SWSGP-  $M$  and SWSGP-  $N$  use a mean-field variational approximation. SWSGP-  $M$  chooses  $M \in \{1024, 2048, 4096, 8192\}$ , and the inducing point locations are initialized by k-means clustering of observed locations. All three methods tune  $K$  in  $\{32, 256\}$ .
+
+# D.3.2. PREDICTIVE RMSE TABLE
+
+The test set RMSE results for reported in Table 2. From the table, we observe that VNNGP obtains comparable RMSE values to other state-of-the-arts and excels at some datasets.
+
+<table><tr><td rowspan="2">Dataset</td><td rowspan="2">N</td><td rowspan="2">D</td><td rowspan="2">ExactGP</td><td rowspan="2">sgGP</td><td colspan="4">RMSE</td></tr><tr><td>SVGP</td><td>SWSGP-M</td><td>SWSGP-N</td><td>VNNGP</td></tr><tr><td>PoleTele</td><td>9.6K</td><td>26</td><td>.095 ± .000</td><td>.092 ± .002</td><td>.113 ± .000</td><td>.101 ± .002</td><td>.098 ± .003</td><td>.091 ± .002</td></tr><tr><td>Elevators</td><td>10.6K</td><td>16</td><td>.360 ± .006</td><td>.358 ± .006</td><td>.366 ± .006</td><td>.399 ± .008</td><td>.456 ± .001</td><td>.373 ± .004</td></tr><tr><td>Bike</td><td>11.1K</td><td>17</td><td>.054 ± .003</td><td>.400 ± .153</td><td>.028 ± .001</td><td>.037 ± .002</td><td>.042 ± .006</td><td>.043 ± .003</td></tr><tr><td>Kin40K</td><td>25.6K</td><td>8</td><td>.099 ± .001</td><td>.458 ± .076</td><td>.145 ± .001</td><td>.118 ± .002</td><td>.097 ± .001</td><td>.096 ± .001</td></tr><tr><td>Protein</td><td>25.6K</td><td>9</td><td>.533 ± .002</td><td>.534 ± .006</td><td>.633 ± .003</td><td>.600 ± .004</td><td>.592 ± .005</td><td>.565 ± .003</td></tr><tr><td>KEGG</td><td>31.2K</td><td>20</td><td>.085 ± .002</td><td>.087 ± .003</td><td>.089 ± .001</td><td>.086 ± .001</td><td>.085 ± .002</td><td>.085 ± .001</td></tr><tr><td>KEGGU</td><td>40.7K</td><td>26</td><td>.117 ± .000</td><td>.122 ± .001</td><td>.122 ± .000</td><td>.118 ± .001</td><td>.116 ± .001</td><td>.118 ± .000</td></tr><tr><td>Precipitation</td><td>64.8K</td><td>3</td><td>—</td><td>—</td><td>.635 ± .004</td><td>.485 ± .001</td><td>.506 ± .001</td><td>.432 ± .002</td></tr><tr><td>UKHousing</td><td>116K</td><td>2</td><td>.402 ± .001</td><td>.359 ± .001</td><td>.582 ± .003</td><td>.382 ± .001</td><td>.498 ± .000</td><td>.346 ± .002</td></tr><tr><td>3DRoad</td><td>278K</td><td>3</td><td>.165 ± .000</td><td>.187 ± .001</td><td>.329 ± .001</td><td>.432 ± .001</td><td>.097 ± .017</td><td>.298 ± .064</td></tr><tr><td>Covtype</td><td>372K</td><td>54</td><td>—</td><td>—</td><td>.671 ± .001</td><td>.657 ± .001</td><td>.670 ± .001</td><td>.671 ± .001</td></tr></table>
+
+# D.3.3. LEARNED LIKELIHOOD NOISE TABLE
+
+We report the learned value of likelihood noise for all models in Table 3. Note that Covtype dataset uses Bernoulli likelihood which does not have noise parameter, therefore it is not included.
+
+We observe in Table 1 that sgGP and SWSGP-  $N$  can sometimes obtain a very high NLL values. We suspect that is due to they learn a particularly small likelihood noise on corresponding datasets. For example, sgGP's NLL is approximately 53 on Bike, and its learned noise is around 0; SWSGP-  $N$  's NLL is approximately 41 on KEGGU, and its learned noise is around 0.
+
+Table 2. Test set RMSE (mean ± 1 standard error over 3 random seeds) on high-dimensional datasets (top 8 ones) and spatial datasets (bottom 4 ones).  
+
+<table><tr><td rowspan="2">Dataset</td><td rowspan="2">N</td><td rowspan="2">D</td><td rowspan="2">ExactGP</td><td rowspan="2">sgGP</td><td colspan="4">noise</td></tr><tr><td>SVGP</td><td>SWSGP-M</td><td>SWSGP-N</td><td>VNNGP</td></tr><tr><td>PoleTele</td><td>9.6K</td><td>26</td><td>.020 ± .000</td><td>.002 ± .000</td><td>.015 ± .000</td><td>.005 ± .000</td><td>.000 ± .000</td><td>.002 ± .000</td></tr><tr><td>Elevators</td><td>10.6K</td><td>16</td><td>.166 ± .001</td><td>.102 ± .001</td><td>.132 ± .001</td><td>.062 ± .000</td><td>.000 ± .000</td><td>.067 ± .000</td></tr><tr><td>Bike</td><td>11.1K</td><td>17</td><td>.016 ± .000</td><td>.000 ± .000</td><td>.002 ± .000</td><td>.002 ± .000</td><td>.000 ± .000</td><td>.001 ± .000</td></tr><tr><td>Kin40K</td><td>25.6K</td><td>8</td><td>.020 ± .000</td><td>.000 ± .000</td><td>.031 ± .000</td><td>.009 ± .000</td><td>.000 ± .000</td><td>.001 ± .000</td></tr><tr><td>Protein</td><td>25.6K</td><td>9</td><td>.182 ± .003</td><td>.022 ± .002</td><td>.417 ± .001</td><td>.339 ± .001</td><td>.000 ± .000</td><td>.018 ± .000</td></tr><tr><td>KEGG</td><td>31.2K</td><td>20</td><td>.019 ± .000</td><td>.006 ± .000</td><td>.008 ± .000</td><td>.007 ± .000</td><td>.000 ± .000</td><td>.005 ± .000</td></tr><tr><td>KEGGU</td><td>40.7K</td><td>26</td><td>.023 ± .000</td><td>.013 ± .000</td><td>.015 ± .000</td><td>.014 ± .000</td><td>.000 ± .000</td><td>.013 ± .000</td></tr><tr><td>Precipitation</td><td>64.8K</td><td>3</td><td>—</td><td>—</td><td>.182 ± .001</td><td>.030 ± .000</td><td>.000 ± .000</td><td>.005 ± .000</td></tr><tr><td>UKHousing</td><td>116K</td><td>2</td><td>.280 ± .001</td><td>.112 ± .000</td><td>.345 ± .002</td><td>.144 ± .000</td><td>.015 ± .000</td><td>.070 ± .003</td></tr><tr><td>3DRoad</td><td>278K</td><td>3</td><td>.584 ± .000</td><td>.001 ± .000</td><td>.126 ± .000</td><td>.203 ± .001</td><td>.000 ± .000</td><td>.001 ± .000</td></tr><tr><td>Covtype</td><td>372K</td><td>54</td><td>—</td><td>—</td><td>.000 ± .000</td><td>.000 ± .000</td><td>.000 ± .000</td><td>.000 ± .000</td></tr></table>
+
+Table 3. The learned likelihood noise (mean ± 1 standard error over 3 random seeds) on high-dimensional datasets (top 8 ones) and spatial datasets (bottom 3 ones).

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+Jangho Kim\*12 Jun-Tae Lee\*1 Simyung Chang 1 Nojun Kwak 2
+
+# Abstract
+
+With the development of deep learning (DL) technologies, the demand for DL-based services on personal devices, such as mobile phones, also increases rapidly. In this paper, we propose a novel personalization method, Variational On-the-Fly Personalization. Compared to the conventional personalization methods that require additional fine-tuning with personal data, the proposed method only requires forwarding a handful of personal data on-the-fly. Assuming even a single personal data can convey the characteristics of a target person, we develop the variational hyper-personalizer to capture the weight distribution of layers that fits the target person. In the testing phase, the hyper-personalizer estimates the model's weights on-the-fly based on personality by forwarding only a small amount of (even a single) personal enrollment data. Hence, the proposed method can perform the personalization without any training software platform and additional cost in the edge device. In experiments, we show our approach can effectively generate reliable personalized models via forwarding (not back-propagating) a handful of samples.
+
+# 1. Introduction
+
+In recent years, most of the deep learning researches have paid attention to developing universal models with sophisticated architectures using a large-scale dataset covering an entire target domain. However, in edge devices, such as mobile phones and IoT sensors, deep models are required to process (learn or infer) a personal domain where data
+
+*Equal contribution 1Qualcomm AI Research, an initiative of Qualcomm Technologies, Inc. Jangho Kim completed the research in part during an internship at Qualcomm Technologies, Inc. 2Seoul National University, South Korea. Correspondence to: Jangho Kim <kjh91@snu.ac.kr>, Jun-Tae Lee <juntlee@qti.Qualcomm.com>, Simyung Chang <simychan@qti.Qualcomm.com>, Nojun Kwak <nojunk@snu.ac.kr>.
+
+Proceedings of the  $39^{th}$  International Conference on Machine Learning, Baltimore, Maryland, USA, PMLR 162, 2022. Copyright 2022 by the author(s).
+
+![](images/2abe5abea2584c6f12a896aa465e7c40af5d55d62a2d74ed407bc479fd34f1e3.jpg)  
+Figure 1: Our personalization scenario on edge devices with practically crucial constraints (few-shot target, unsupervised, source-free, and training-free).
+
+are generated in a specific environment (Wang et al., 2019b; Chen & Ran, 2019). For example, data could be always collected by a specific user, then the device may never confront other personal domains. In this case, using a universal model is inefficient, and also its performance may degrade in some personal domains. Hence, to successfully deploy deep learning algorithms in edge devices, it is significant to specialize a model to the data distribution of the personal domain, i.e., personality, in question.
+
+Despite the importance of personalization, there has been little progress due to the following practical constraints of edge devices, which are depicted in Fig. 1. First, it is hard to access the source data (generic data) used for training (source-free). Second, only a few user-specific target data (personal data) can be available (few-shot). Third, for usability, personal data are preferable to be unlabeled since a service requiring user's annotation drastically reduces usability (unsupervised). Also, on-device training with user's data suffers from some non-algorithmic constraints (training-free). For example, collecting personal data to train the models may cause privacy concerns. And model training requires an on-device training platform and much more hardware resources (memory and computational units) than inference.
+
+Various existing works have tried to qualify a model to specified domains. Nevertheless, they do not completely meet the aforementioned constraints, and are not directly applicable to our personalization task. Fallah et al. (2020) localized multiple models to different personal domains, federating to
+
+produce a universal one. However, in the localized learning, a plenty of annotated personal data are required. Although Motiian et al. (2017) and Luo et al. (2017) adapted a model to a target domain with a few data, they need source domain data as well as the labels of the few target data. While unsupervised domain adaptation was devised by Long et al. (2016), enough target and source domain data are required. Recent source-free domain adaptation techniques (Yang et al., 2021; Kundu et al., 2020) adapted a model without any source domain data and label of target domain data, but they still use abundant target domain data. Commonly, those existing methods are not training-free. Therefore, it is still challenging to flexibly personalize a model to a given personal domain using only a small-sized personal data.
+
+To attain the personalization satisfying the constraints shown in Fig. 1, we introduce a novel on-the-fly personalization paradigm. Specifically, given a personal data, we compute the weights of layers specialized to its personality on-the-fly via forwarding only a few personal data. To this end, we propose a variational on-the-fly personalization (VoP) method. The key of our method lies in a small detachable module, the variational hyper-personalizer which is trained to produce an approximated posterior distribution of weights of a layer based on the personality. Assuming the data in a personal domain share the same personality, we set the target prior distribution as the averaged distribution, called prototype, of the personal data in the variational inference of the hyper-personalizer. In the testing phase, by forwarding a small amount of testing data with the same personality, called enrollment data, the hyper-personalizer estimates the posterior distribution of weights in the layer according to the personality represented by the enrollment data. Then, a personalized layer is generated from the estimated posterior distribution. Since the hyper-personalizer is detached from the model after generating the personalized weights of the layer, we can obtain a personalized model without increasing computational cost.
+
+To show the effectiveness of the proposed variational on-the-fly personalization, we first apply it to two tasks strongly relevant to personalization: keyword spotting and open-set speaker verification. Then, we extend our VoP to few-shot classification task, as well. For each task, we show that our VoP successfully increases the performance of a baseline method on the considered public benchmark dataset.
+
+The contribution of our work is summarized as follows:
+
+- To the best of our knowledge, we propose the first on-the-fly personalization method to specialize a model to a given personality.  
+- We formulate the personalization via forwarding as a variational inference problem, and solve it by considering the prototype distribution as the target posterior distribution of the personal layer's weights.
+
+- By forwarding a few personal data, the variational hyperpersonalizer captures the weight distribution of layers, and produces the personalized layer weights.  
+- We analyze the effectiveness of the proposed method on various tasks such as keyword spotting, open-set speech verification, and few-shot classification tasks.
+
+# 2. Related work
+
+Neural networks for edge devices Many on-device deep learning services have been introduced, then personalization of deep learning algorithms gets important.
+
+Keyword spotting (KWS) recognizes a specific keyword to wake up a device. As KWS models are frequently deployed and used in edge devices, diverse methods have been developed for efficient KWS in the devices, such as lightening model sizes (Tang & Lin, 2017; Zhang et al., 2017; Tang & Lin, 2018), or reducing computation (Choi et al., 2019; Li et al., 2020). Although KSW deals with an enrolled user's voice rather than anonymous ones, research on personalization has not been actively studied.
+
+Speaker verification (SV) also requires personalization for accurate verification for a target user. To obtain robust target speaker's embedding, metric learning-based methods are widely addressed (Wang et al., 2019a; Chung et al., 2020; Ko et al., 2020). However, while they aim to make a better embedder in the training phase, our VoP can personalize the model by forwarding a handful of personal data on-the-fly.
+
+Recently, to address the degeneration issue of federated learning for non-IID local data, personalized federated learning was devised. In this line of research, several works decentralized the model-agnostic meta-learning (Jiang et al., 2019; Fallah et al., 2020; Deng et al., 2020). Other approaches mixed the global and personal (local) models (Deng et al., 2020; Hanzely & Richtárik, 2020). While these methods are akin to our VoP in that the global and local layers are used separately, unlike ours, they inevitably require training models on personal data.
+
+Existing personalization methods are mostly tailored to specific tasks, and also require task-specific training for target personality. Whereas, as a versatile algorithm for diverse tasks, the proposed VoP uses no extra training on devices. This enables a lightweight personalized neural network platform for edge devices that may not support training.
+
+Few-shot embedding adaptation To overcome the need for massive training data and generalize the model with a few limited examples, lots of approaches have been attempted in many research fields. In the field of few-shot domain adaptation (Tzeng et al., 2015; Motiian et al., 2017; Luo et al., 2017), a model trained on a source domain is adopted to a heterogeneous target domain with a few labeled target data. However, those methods highly rely on prior information from a large-scale source data.
+
+In few-shot classification (Snell et al., 2017; Iakovleva et al., 2020; Ye et al., 2020; Gordon et al., 2018; Das et al., 2022), a model meta-learns how to build embedding with a few examples in source classes with sufficient labeled data, and then the model is applied to tasks sampled from unseen target classes, where each task consists of a few labeled support set and a query set. Recently, Gordon et al. (2018) exploited a Bayesian framework to relieve the model uncertainty by insufficient support examples, where the weight of the classification layer is generated for each task in the testing phase. To further tailor the Bayesian framework to this task, Iakovleva et al. (2020) regularized the classifier generator based on the relationship between support and query sets. While these Bayesian approaches have relevance with our VoP, they only focus on the weight generation of the final linear layer based on query set for the few-shot classification task. Contrarily, we supposed that the inherent personality can be conveyed by even a single personal sample for generic use in various personalization tasks with scarce personal data. Also, we designed VoP to personalize both convolutional and linear layers in various depths.
+
+# 3. Method
+
+In this section, we first introduce the framework of the proposed VoP. Note that VoP is generally applicable to personalize deep networks, via only forwarding a small amount of samples on-the-fly. Next, we describe the variational inference process of the hyper-personalizer which produces the personalized weight for a corresponding layer. Hypermodules (Ha et al., 2017; Ballé et al., 2018; Minnen et al., 2018) have been widely adopted in diverse tasks, but our variational hyper-personalizer is specially designed for personalization. Lastly, we analyze the convergence of VoP.
+
+# 3.1. Overall Framework
+
+Fig. 2 depicts the framework of the proposed VoP. Our network consists of an encoding module  $f_{\psi}$  and a detachable hyper-personalizer  $q_{\theta}$ . Then, the network is learned to solve a classification problem considering different personalities of input samples. To this end, we suppose a dataset  $D = \{(s_i, y_i, r_i)\}_{i=1}^N$  with multiple personalities. A sample  $s_i$  is annotated by a class label  $y_i$  and a personality label  $r_i$  where  $y_i \in [C]$ ,  $r_i \in [R]$  ([n] is a positive integer set  $\{1, \dots, n\}^1$ .
+
+In the training stage, the encoding module, which is shared across the personalities of inputs, first extracts features of input samples. Then, the encoded features are forwarded to the hyper-personalizer. For each personality, the hyper-personalizer performs the variational inference where the true posterior of the sample-specific weights of the corresponding layer is approximated. Then, the sample-specific
+
+# Algorithm 1 Variational On-the-Fly Personalization (VoP)
+
+1:  $P \gets$  The number of iterations  
+2: Given  $D = \{(s_i, y_i, r_i)\}_{i=1}^N$  where  $y_i \in [C], r_i \in [R]$  
+3:  $\psi, \theta \gets$  Initialization  
+4: [Training]  
+5: for  $\operatorname{Iter} = 1, \dots, P$  do  
+6:  $D^{M} = \{(s_{i},y_{i},r_{i})\}_{i = 1}^{M}\gets$  Minibatch of  $M$  samples (drawn from  $D$  ), where  $y_{i}\in \mathcal{C}^{M}\subset [C],r_{i}\in \mathcal{R}^{M}\subset [R]$  
+7:  $\{x_{i}\}_{i = 1}^{M}\gets$  Extract features from  $\{s_i\}_{i = 1}^M$  , using the encoding module  $f_{\psi}$  
+8:  $\{\mu_i,\Sigma_i\}_{i = 1}^M\gets$  Outputs from  $q_{\theta}$  
+9:  $\{\bar{\mu}_k\}_{k = 1}^{|R^M|},\{\bar{\Sigma}_k\}_{k = 1}^{|R^M|}\gets$  Obtain proto-means and protovariances using (9)  
+10:  $\{\omega_{i}\}_{i = 1}^{M}\gets$  Sample sample-specific weights using  $p(\epsilon)$  based on  $\{\mu_i,\Sigma_i\}_{i = 1}^M$  
+11: Update  $\psi, \theta$  by minimizing  $\widehat{\mathcal{L}}_{VoP}$  in (12)  
+12: end for  
+13: [Testing] to generate the  $k$ -th personality model  
+14:  $D^{E} = \{(s_{i},y_{i},r_{i} = k)\}_{i = 1}^{E}$ : Few enrollment samples for the  $k$ -th personality  
+15: Obtain  $\{\bar{\mu}_k\}$  using  $f_{\psi}, q_{\theta}$  and  $D^{E}$  
+16:  $\{\bar{\omega}_k\} \gets$  Set  $\bar{\mu}_k$  as the personal weights, which maximize the likelihood for the personal samples  
+17: Abolish  $q_{\theta}$
+
+layer is generated from the approximated posterior. Finally, the sample-specific layer yields the personalized feature (or prediction if it is the last layer).
+
+Note that, in the testing phase, the hyper-personalizer is used only once for a few enrollment samples, in order to generate the weights of the personalized layer for the personality corresponding to the enrollment samples on-the-fly without any back-propagation.
+
+# 3.2. Variational On-the-Fly Personalization
+
+The goal of the hyper-personalizer is to find a proper weight distribution shared across the samples with the same personality. For this purpose, for a personality  $k$ , we estimate the sample-specific true posterior from the encoded sample feature  $x_{i}^{k}$  ( $= f_{\psi}(s_{i})$  where  $r_{i} = k$ ), because it is difficult to directly find the true posterior from raw input samples. However, in general, the true posterior of the weight  $\omega$ ,  $p(\omega | x_{i}^{k}, y_{i}^{k})$ , is intractable (Gal, 2016).
+
+Accordingly, in order to approximate the true posterior  $p(\omega |x_i^k,y_i^k)$  , we design a variational distribution  $q_{\theta}(\omega |x_i^k)$  parameterized by  $\theta$  . Then, we minimize the Kullback-Leibler (KL) divergence between the two distributions:
+
+$$
+\begin{array}{l} K L \left(q _ {\theta} \left(\omega \mid x _ {i} ^ {k}\right) | | p \left(\omega \mid x _ {i} ^ {k}, y _ {i} ^ {k}\right)\right) = \int q _ {\theta} \left(\omega \mid x _ {i} ^ {k}\right) \ln \frac {q _ {\theta} \left(\omega \mid x _ {i} ^ {k}\right)}{p \left(\omega \mid x _ {i} ^ {k} , y _ {i} ^ {k}\right)} d \omega \\ = \int q _ {\theta} \left(\omega \mid x _ {i} ^ {k}\right) \ln \left(\frac {q _ {\theta} \left(\omega \mid x _ {i} ^ {k}\right)}{p \left(\omega \mid x _ {i} ^ {k}\right)} \frac {p \left(y _ {i} ^ {k} \mid x _ {i} ^ {k}\right)}{p \left(y _ {i} ^ {k} \mid x _ {i} ^ {k} , \omega\right)}\right) d \omega \\ = K L \left(q _ {\theta} \left(\omega \left| x _ {i} ^ {k}\right) \right| | p (\omega \left| x _ {i} ^ {k}\right)\right) \\ - \mathbb {E} _ {\omega \sim q _ {\theta} (\omega | x _ {i} ^ {k})} [ \ln p \left(y _ {i} ^ {k} \mid x _ {i} ^ {k}, \omega\right) ] + \ln p \left(y _ {i} ^ {k} \mid x _ {i} ^ {k}\right). \tag {1} \\ \end{array}
+$$
+
+![](images/ba6b7f4737ffeb6d5bc0e2e91c6315b97df3b8af6d6b6b5f5476b0c57a18aa69.jpg)  
+Figure 2: The overall process of the proposed VoP. Colors in input samples represent personalities for each input sample. In the training phase, VoP trains the encoding module and hyper-personalizer to estimate sample-specific weights via black dashed and blue bold arrows. At testing phase, for each personality, VoP generates personal weights by forwarding a few enrollment samples via black and red dashed arrows, once. After generating the personal weights (model), VoP abolishes the hyper-personalizer, and then only utilizes the encoding module and the personal weights. More details are in Sec. 3.
+
+In (1), the minimization of the KL divergence is equivalent to maximizing the evidence lower bound (ELBO):
+
+$$
+\begin{array}{l} \mathbb {E} _ {\omega \sim q _ {\theta} \left(\omega \mid x _ {i} ^ {k}\right)} \left[ \ln p \left(y _ {i} ^ {k} \mid x _ {i} ^ {k}, \omega\right) \right] - K L \left(q _ {\theta} \left(\omega \mid x _ {i} ^ {k}\right) \mid \mid p \left(\omega \mid x _ {i} ^ {k}\right)\right) \\ \leq \ln p \left(y _ {i} ^ {k} \mid x _ {i} ^ {k}\right). \tag {2} \\ \end{array}
+$$
+
+This approximation process is known as variational inference (VI) (Hoffman et al., 2013). In other words, we approximate the true posterior distribution by minimizing the following objective function (negative ELBO):
+
+$$
+\begin{array}{l} \mathcal {L} _ {V I} (\theta , (x _ {i}, y _ {i}, k)) = - \mathbb {E} _ {\omega \sim q _ {\theta} \left(\omega \mid x _ {i} ^ {k}\right)} [ \ln p \left(y _ {i} ^ {k} \mid x _ {i} ^ {k}, \omega\right) ] \tag {3} \\ + K L \left(q _ {\theta} \left(\omega \mid x _ {i} ^ {k}\right) | | p (\omega \mid x _ {i} ^ {k})\right). \\ \end{array}
+$$
+
+In the minimization of the objective function, the first term induces  $q_{\theta}(\omega |x_i^k)$  to predict the correct output and the second KL term encourages the variational distribution to resemble the sample-specific prior  $p(\omega |x_i^k)$ .
+
+Note that the first RHS term requires estimation by sampling  $\omega$  from the variational distribution  $q_{\theta}(\omega |x_i^k)$ . For the sampling, we assume the true posterior  $p(\omega |x_i^k,y_i^k)$  and the sample-specific prior  $p(\omega |x_i^k)$  to be uncorrelated multivariate Gaussian. Then, the hyper-personalizer calculates parameters for the variational distribution  $q_{\theta}(\omega |x_i^k)$  following the multivariate Gaussian,  $q_{\theta}(\omega |x_i^k) = \mathcal{N}(\omega |\mu_i,\Sigma_i)$ , with  $\Sigma_{i} = \mathrm{diag}(\sigma_{i}^{2})$ . Here,  $\mu_{i}$  and  $\sigma_{i}^{2}$  are outputs of the hyper-personalizer which is a multi-layer perceptron (MLP) with a single hidden layer ( $\mu_{i},\sigma_{i}^{2}\in \mathbb{R}^{Z}$  where  $Z$  is the dimension of  $\omega$ ).
+
+After calculating  $\mu_{i},\sigma_{i}^{2}$ , we apply the reparameterization trick for a pathwise derivative estimator (Kingma & Welling, 2013; Gal, 2016). To reparameterize a random variable  $\omega$  following  $q_{\theta}(\omega |x_i^k)$ , we exploit a noise variable  $\epsilon$  following  $p(\epsilon)$ . Then,  $\omega$  can be represented by a non-random differen
+
+tiable function  $g(\epsilon ,x_i^k;\theta)$
+
+$$
+\omega = g \left(\epsilon , x _ {i} ^ {k}; \theta\right) = \mu_ {i} + \sigma_ {i} \odot \epsilon \quad \text {w i t h} \quad \epsilon \sim \mathcal {N} (0, I), \tag {4}
+$$
+
+where  $\odot$  denotes elementwise multiplication. From this, we can rewrite (3) w.r.t.  $p(\epsilon)$  (for convenience,  $KL(q_{\theta}(\omega |x_i^k))||p(\omega |x_i^k))$  is referred to as  $KL$ ):
+
+$$
+\begin{array}{l} \mathcal {L} _ {V I} (\theta , \left(x _ {i}, y _ {i}, k\right)) = - \int q _ {\theta} \left(\omega \mid x _ {i} ^ {k}\right) \ln p \left(y _ {i} ^ {k} \mid x _ {i} ^ {k}, \omega\right) d \omega + K L \\ = - \int p (\epsilon) \ln p \left(y _ {i} ^ {k} \mid x _ {i} ^ {k}, g \left(\epsilon , x _ {i} ^ {k}; \theta\right)\right) d \epsilon + K L \tag {5} \\ \end{array}
+$$
+
+whose first term can be approximated with Monte Carlo (MC) estimator as follows:
+
+$$
+\begin{array}{l} - \int p (\epsilon) \ln p \left(y _ {i} ^ {k} \mid x _ {i} ^ {k}, g \left(\epsilon , x _ {i} ^ {k}; \theta\right)\right) d \epsilon \\ \simeq - \frac {1}{L} \sum_ {l = 1} ^ {L} \ln p \left(y _ {i} ^ {k} \mid x _ {i} ^ {k}, g \left(\epsilon_ {l}, x _ {i} ^ {k}; \theta\right)\right), \tag {6} \\ \end{array}
+$$
+
+where  $\epsilon_{l}$  is independently sampled from  $p(\epsilon)$ . We can train  $\theta$  with the MC estimator:
+
+$$
+\begin{array}{l} \mathcal {L} _ {M C} (\theta , (x _ {i}, y _ {i}, k)) = - \frac {1}{L} \sum_ {l = 1} ^ {L} \ln p \left(y _ {i} ^ {k} \mid x _ {i} ^ {k}, g \left(\epsilon_ {l}, x _ {i} ^ {k}; \theta\right)\right) \\ + K L \left(q _ {\theta} \left(\omega \left| x _ {i} ^ {k}\right) \right| | p \left(\omega \left| x _ {i} ^ {k}\right)\right)\right). \tag {7} \\ \end{array}
+$$
+
+To compute the KL divergence in (7), we assume that the sample-specific prior of  $x_{i}$  follows the distribution of its personality  $r_i = k$ :
+
+$$
+p \left(\omega \mid x _ {i} ^ {k}\right) \simeq p (\omega | k). \tag {8}
+$$
+
+Then, we define the prototype distribution for the personality by the mean and variance of a proto-set  $S_{k}$ . This proto-set, which is pre-defined for each  $k$ , contains the mean and variance for each of a few samples belonging to the  $k$ th
+
+personality. Hence, the proto-mean  $\bar{\mu}_k$  and proto-variance  $\bar{\Sigma}_k$  are obtained by
+
+$$
+\bar {\mu} _ {k} = \frac {1}{| \mathcal {S} _ {k} |} \sum_ {\mu_ {j} \in \mathcal {S} _ {k}} \mu_ {j}, \quad \bar {\Sigma} _ {k} = \frac {1}{| \mathcal {S} _ {k} |} \sum_ {\Sigma_ {j} \in \mathcal {S} _ {k}} \Sigma_ {j} \tag {9}
+$$
+
+where  $\mu_{j}$  and  $\Sigma_{j}$  are the means and variances from the hyper-personalizer for the  $j$ th sample in  $\mathcal{S}_k$ , respectively.
+
+For brevity, we assume that  $\Sigma$  and  $\bar{\Sigma}$  are diagonal, i.e.,  $\Sigma = \mathrm{diag}(\sigma^2)$  and  $\bar{\Sigma} = \mathrm{diag}(\bar{\sigma}^2)$ . Hence, the KL divergence term of (7) can be calculated analytically (See Appendix A):
+
+$$
+\begin{array}{l} K L (q _ {\theta} (\omega | x _ {i} ^ {k}) | | p (\omega | x _ {i} ^ {k})) \simeq \frac {1}{2} \sum_ {z = 1} ^ {Z} (\ln \bar {\sigma} _ {(k, z)} ^ {2} - \ln \sigma_ {(i, z)} ^ {2} - 1 \\ + \frac {\sigma_ {(i , z)} ^ {2}}{\bar {\sigma} _ {(k , z)} ^ {2}} + \frac {\left(\mu_ {(i , z)} - \bar {\mu} _ {(k , z)}\right) ^ {2}}{\bar {\sigma} _ {(k , z)} ^ {2}}) \tag {10} \\ \end{array}
+$$
+
+where we use  $p(\omega |x_i^k)\simeq p(\omega |k) = \mathcal{N}(\omega |\bar{\mu}_k,\mathrm{diag}(\bar{\sigma}_k^2))$  and  $z\in [Z]$  is the index of multivariate Gaussian dimension.
+
+Based on the sample-specific MC estimator in (7), the overall loss function for all  $N$  samples is defined as follows:
+
+$$
+\begin{array}{l} \mathcal {L} _ {V o P} = \frac {1}{N} \sum_ {i = 1} ^ {N} \left(- \frac {1}{L} \sum_ {l = 1} ^ {L} \ln p \left(y _ {i} ^ {r _ {i}} \mid x _ {i} ^ {r _ {i}}, g \left(\epsilon_ {l}, x _ {i} ^ {r _ {i}}; \theta\right)\right) \right. \tag {11} \\ + \alpha K L \left(q _ {\theta} \left(\omega \left| x _ {i} ^ {r _ {i}}\right) \right| | p (\omega \left| x _ {i} ^ {r _ {i}}\right)\right)). \\ \end{array}
+$$
+
+Here,  $\alpha$  is a scale hyper-parameter to balance between the negative log likelihood and the KL divergence. We can construct an unbiased stochastic estimator  $\hat{\mathcal{L}}_{VoP}$  to (11) by data sub-sampling, i.e., minibatch:
+
+$$
+\begin{array}{l} \widehat {\mathcal {L}} _ {V o P} = \frac {1}{M} \sum_ {i = 1} ^ {M} \left(- \frac {1}{L} \sum_ {l = 1} ^ {L} \ln p \left(y _ {i} ^ {r _ {i}} \mid x _ {i} ^ {r _ {i}}, g \left(\epsilon_ {l}, x _ {i} ^ {r _ {i}}; \theta\right)\right) \right. \tag {12} \\ + \alpha K L \left(q _ {\theta} \left(\omega \left| x _ {i} ^ {r _ {i}}\right) \right| | p \left(\omega \left| x _ {i} ^ {r _ {i}}\right)\right)\right). \\ \end{array}
+$$
+
+where  $M$  is the minibatch size. Here, we set  $L$  to 1 following (Kingma & Welling, 2013). VoP can train both the encoding module  $f_{\psi}$  and the hyper-personalizer  $q_{\theta}$  with this VoP loss function (12). At test time, for each personality, the hyper-personalizer inferences only a few enrollment samples  $D^{E}$  to generate personalized weights  $(\bar{\omega})$  on the fly. During this forwarding, we generate personalized weights based on the proto-mean, which can maximize the likelihood of personal samples. Note that, after obtaining the personalized weights  $(\bar{\omega})$  for every personality, the hyper-personalizer is discarded. Thus, only the encoding module and the personalized layers are left. The overall process is described in Algorithm 1.
+
+# 3.3. Analysis on VoP convergence
+
+One may wonder if the sample mean and variance diverge as the iteration goes on because the proto-mean and protovariance in (9) are not fixed during training. Fig. 3 left
+
+![](images/c84e53fe54f0ab0c2029c9d51f776d0af2ee8a94beef67da6ed4dba1f99016e0.jpg)  
+Figure 3: Convergence of VoP: Computation graph of VoP (left) and loss curve on the KWS task (right).
+
+![](images/26de7d61e67ba65db554752788dca6147f4a41722bcced8bf9a92fe16771a9ae.jpg)
+
+is the computation graph of our VoP in training where  $p_i \triangleq [\mu_i, \sigma_i^2]^T$ . The weight-loss relation of a network with nonlinearity is generally nonconvex, and the convergence proof of SGD is not straightforward and is an active research area. However, when the loss function is convex w.r.t. an intermediate variable (e.g. softmax output), we expect a better convergence behavior.
+
+Theorem 3.1. In VoP,  $\theta \rightarrow KL$  and  $w \rightarrow L$  are nonconvex, but  $p_i \rightarrow KL$  (see dashed box in Fig. 3 left) is convex when  $\bar{\sigma}^2 \leq 2\sigma_i^2$ :
+
+Proof. From (10),  $KL(\mu_i)$  is convex because of the last quadratic term and  $\mu_{i} - \bar{\mu} = \mu_{i} - \frac{1}{N}\sum_{j}\mu_{j} = \frac{N - 1}{N}\mu_{i} - X$  where  $X$  denotes other terms independent of  $\mu_{i}$ . Likewise, if we take  $\frac{\partial^2KL}{(\partial\sigma_i^2)^2}$ , it is nonnegative when  $\bar{\sigma}^2\leq 2\sigma_i^2$ . Thus,  $KL(\sigma_i^2)$  is convex in this condition. If  $KL$  is convex w.r.t  $p_i = [\mu_i,\sigma_i^{2}]^T$ , it can be quadratically approximated near the optimal solution  $p_i = \bar{p}$  such that  $KL\simeq \frac{1}{2}\sum_{i}(p_{i} - \bar{p})^{T}\Lambda (p_{i} - \bar{p})$ , where  $\Lambda$  is a positive definite matrix.
+
+Let  $p_i' = \Lambda^{\frac{1}{2}}p_i$  and taking the gradient descent step on  $p_i'$ , it becomes  $p_i'^+ = p_i' - \eta \nabla p_i' = (1 - \beta)p_i' + \beta \bar{p}'$  where  $\beta = \eta \frac{N - 1}{N}$ . Computing  $p_i'^+ - \bar{p}'$ , it becomes  $p_i'^+ - \bar{p}' = (1 - \beta)(p_i' - \bar{p}')$ , which means that  $p_i'$  (thus  $p_i$ ) moves toward the mean  $\bar{p}'(\bar{p})$  as iterations go on. By aggregating all  $p_i$ 's, we can show that  $\bar{p}^+ = \frac{1}{N}\sum_{i}p_{i}^{+} = \bar{p}$  and it is fixed under gradient descent. Therefore, we can prove the convergence of  $p_i$  when only  $KL$  term is used.
+
+In reality, as  $w$  is updated to minimize  $L$ ,  $\bar{p}$  is not fixed and moves according to  $w$  during training. However, as  $KL$  enforces contraction towards  $\bar{p}$ ,  $p_i$  does not diverge and we can interpret  $KL$  as a regularizer (See Fig. 4). We also show the converging loss curve for the KWS task in Fig. 3 right.
+
+# 4. Experiments
+
+To verify the proposed VoP, we apply it to three tasks: keyword spotting, speaker verification and few-shot classification. For each task, we apply VoP employing baseline networks where VoP and the baseline model share the same architecture except the hyper-personalizer consisting of MLPs. Note that, in testing, as we discard the hyper-personalizer after generating personalized layers, the VoP-learned baselines have the same capacity as the original ones. We provide further experiments and analysis in Appendix C, D, E.
+
+![](images/c3ed4faf7882f1edc60ced5d3846386c4e5431c6be917d603a2dd34994d044e0.jpg)  
+(a) Means from two models with two PIDs
+
+![](images/b650c5951f0ee05ccc898c5301f3293be5931f7f6dab84688a7d38a74f2d540b.jpg)  
+Figure 4: Visualizations by PCA on outputs of the hyper-personalizers of VoP and vanilla models under two identical personal identifications (PIDs), i.e., personality labels. (a) shows the means  $(\mu_{i}$ 's) from hyper-personalizers of the two models in the same PCA space. To see the personalization in closer views, (b) and (c) display the means  $(\mu_{i}$ 's) and the proto-means  $(\bar{\mu})$  from the hyper-personalizers of the vanilla and VoP models, respectively.
+
+![](images/73f0d50db46c8d80a40eeae9db702872d70f89a31418444d84acc76a4c396a59.jpg)  
+(b) Means from the vanilla with two PIDs  
+(c) Means from VoP with two PIDs
+
+Table 1: Relative variances of the vanilla and VoP models depending on PCA axes for each PIDs from Figure 4a.  
+
+<table><tr><td>Method</td><td>PID1 - PC1</td><td>PID1 - PC2</td><td>PID2 - PC1</td><td>PID2 - PC2</td></tr><tr><td>Vanilla</td><td>148.20</td><td>933.84</td><td>90.01</td><td>1140.58</td></tr><tr><td>VoP</td><td>1.00</td><td>1.53</td><td>1.13</td><td>2.05</td></tr></table>
+
+# 4.1. Keyword Spotting
+
+We apply the proposed VoP on Keyword spotting (KWS) which is a classification task to detect pre-defined keywords. Due to the diversity of the texture of voices, it is crucial to personalize the KWS algorithm to individual speakers. Therefore, in this task, personality refers to an individual speaker. We employ the 'cnn-one-stride1' (Tang et al., 2018) as the baseline network. Using our VoP, we personalize the last two fully connected layers (remaining layers are the encoding module). In training, following the most basic setting (the optimizer and total training epochs) of (Tang & Lin, 2017), we use a larger minibatch size (512) to compute the proto-means and the proto-variances of different personalities, together. Also, for the stable learning of VoP, we experimentally set both learning rate and  $\alpha$  in (12) as 0.0005, and use no learning rate decay. In testing, we forward the enrollment set of 5 samples per each personality.
+
+Dataset We use Qualcomm Keyword Speech (Kim et al., 2019) dataset where wav files are recorded with  $16\mathrm{kHz}$  with mono channel in 16 bits. It consists of 4,270 utterances spoken by 42-50 individuals which are annotated by six classes: four English keyword classes ('Hi Galaxy,' 'Hey Snapdragon,' 'Hi Lumina,' and 'Hey Android') and two non-keyword classes ('silence' and 'unknown.') We address both closed-set and open-set tasks. In the closed-set setting, including 42 speakers, the dataset is divided to  $80\%$  training and  $20\%$  testing sets. In the open-set setting, the dataset consists of 37 speakers for training and five ones for testing. Note that the five speakers cannot be seen on training.
+
+Analysis on Efficacy of VoP As aforementioned, our VoP loss consists of the negative log likelihood and the KL divergence. While the negative log likelihood is a cross-entropy loss for the classification task (task loss), the KL divergence helps to learn the personality. Therefore, to explore the effectiveness of the proposed VoP in personalization, we compare two models: VoP model trained with our VoP loss, and vanilla model with only task-specific cross-entropy loss. Note that both models have the same architecture including the encoding modules and hyper-personalizer. We train those two models on Qualcomm Keyword Speech dataset.
+
+For intuitive comparison, we visualize the means of the hyper-personalizer, using the principal component analysis (PCA). Firstly, we compare the vanilla and VoP models in terms of the means computed by their hyper-personalizer for two different PIDs (PID1 and PID2). To compare the VoP and vanilla models according to the dispersion of means, we re-scale each mean by  $L1$  normalization and compute a covariance matrix from the sample-wise concatenated normalized means. In Fig. 4a, we show the means of several testing samples from the two models for each PID, regardless of the class labels. The blue and red circles represent the means from VoP model for PIDs 1 and 2, respectively. Similarly, the green and black circles are those for the vanilla models. We see that the means from our VoP model are more concentrated compared to the means from the vanilla model. In Table 1, for the PIDs, we also provide the variances of the normalized means on each PCA axes (PC1 and PC2). We set the variance of VoP on PC1 as 1 and report the relative variances for clear understanding. Compared to the vanilla model, all the variances of the VoP model are even and very small. Thus, the KL divergence in (10) pushes each output of hyper-personalizer to mimic the distribution of the sample-specific prior following the distribution of the corresponding personality. This sample-wise regulation can give representativeness of personality to each sample, which is very important in the on-the-fly personalization.
+
+Table 2: Keyword spotting accuracy(%) onQualcommKeyword Speech dataset.  
+
+<table><tr><td>Method</td><td>Closed-set</td><td>Open-set</td></tr><tr><td>Baseline</td><td>87.46 ± 1.68</td><td>74.45 ± 0.77</td></tr><tr><td>Baseline w/ Dropout</td><td>81.77 ± 1.75</td><td>77.35 ± 1.90</td></tr><tr><td>Baseline w/ samovar (2fc)</td><td>17.25 ± 1.42</td><td>24.35 ± 1.84</td></tr><tr><td>Baseline w/ samovar (1fc)</td><td>87.47 ± 1.27</td><td>81.43 ± 1.02</td></tr><tr><td>VoP</td><td>92.80 ± 1.40</td><td>83.60 ± 0.84</td></tr></table>
+
+Table 3: Ablation studies on Qualcomm Keyword Speech dataset under closed-set setting.  
+
+<table><tr><td>Method</td><td>No. hidden units</td><td>α</td><td>Accuracy(%)</td></tr><tr><td>VoP with one hyper-personalizer</td><td>32</td><td>5e-4</td><td>42.35 ± 1.34</td></tr><tr><td>VoP</td><td>32</td><td>5e-4</td><td>92.80 ± 1.40</td></tr><tr><td>VoP (α = 5e-5)</td><td>32</td><td>5e-5</td><td>88.10 ± 0.84</td></tr><tr><td>VoP (α = 0; Vanilla)</td><td>32</td><td>0</td><td>68.40 ± 2.12</td></tr><tr><td>VoP with 64 hidden units</td><td>64</td><td>5e-4</td><td>85.71 ± 0.70</td></tr></table>
+
+Finally, we verify the effect of the VoP loss on the personalization of the hyper-perosnitzer for multiple PIDs. For this purpose, we visualize the means, outputs of hyper-perosnizers, of two different PIDs, for the VoP and vanilla models in Figs. 4b and 4c, respectively. The light red and light blue circles represent means from the hyper-perosnizer respective to PID regardless of four classes. The red and blue crosses denote the proto-mean of PID1 and PID2, respectively. In vanilla case depicted in Fig. 4b, outputs are scattered sporadically and there is little correlation among the means within the same PID. Interestingly, in Fig. 4c, which is the case of the VoP model, means are well clustered near the proto-mean according to each PID. Note that the KL divergence of the VoP loss enforce the mean of a sample to follow the distribution of the corresponding personality. Namely, the VoP loss has no explicit design, such as contrastive loss, to separate the samples with different PIDs. Nevertheless, the KL divergence successfully groups the samples depending on their personalities.
+
+From this analysis, the KL divergence loss considers estimating the reliable variational distribution to a sample-specific prior. By minimizing cross-entropy and KL divergence together by the VoP loss, each personalized weight distribution can be unique during optimized to both the target task and the personality. Also, sample-wise generated model can achieve representativeness corresponding to personality with sample-wise KL regularization.
+
+Result We compare our VoP with multiple variants of the baseline model. As in (Tang et al., 2018), we first include the dropout regularization on the baseline (Baseline w/ Dropout). Also, we applied samovar (Iakovleva et al., 2020), a Bayesian weight generation technique, to the baseline (Baseline w/ samovar). Since samovar generates layer's weights based on the set-to-set relationship between support and query sets for few-shot classification task, we apply
+
+Table 4: Adequacy for no. the enrollment samples.  
+
+<table><tr><td>No. samples</td><td>5</td><td>4</td><td>3</td><td>2</td><td>1</td></tr><tr><td>Accuracy(%)</td><td>92.8±1.4</td><td>92.3±1.8</td><td>93.3±1.0</td><td>92.7±1.6</td><td>92.1±1.7</td></tr></table>
+
+samovar splitting a minibatch into two sets (considering  $30\%$  for each personality as their support set) for weight generation. For the proposed VoP, we personalize two last fully-connected (fc) layers.
+
+In Table 2, we report the accuracy scores. In the open-set setting, the dropout regularization is slightly effective. But, it results in a degraded performance on the closed-set setting. Whereas, the proposed VoP significantly improves the baseline in both settings by  $4.99\%$  and  $9.15\%$ . As samovar is designed for the last classification layer, we first apply it on the last fc layer of the baseline. Also, for fair comparison with our VoP, we also apply samovar on the last two fc layers. In the open-set setting, samovar decently personalizes the last fc layer, but is less effective on the closed-set setting. Also, it fails to personalize more than the final classification layer, yielding severe performance drop in both settings. Note that, our VoP outperforms all the variants of the baseline at least  $4.98\%$  and  $2.23\%$  on closed- and open-set settings, respectively.
+
+This is because hyper-personalizer of VoP is well-designed for giving the representativeness to each generated model from the single sample. Namely, the hyper-personalizer is beneficial for such few enrollment samples. Hence, our VoP is effective to the KWS task which needs personalization.
+
+Ablation studies To analyze the efficacy of VoP, we provide ablation studies for several important factors: the number of the hyper-personalizer, the number of hidden units per hyper-personalizer, and  $\alpha$ . In Table 3, we report the accuracy scores of five variants where the best performed one is bold-faced. Notice that we personalize the last fully-connected layer when using one hyper-personalizer, and the last two ones for two hyper-personalizers. When using less number of the hyper-personalizer, the accuracy score is degraded by  $50.10\%$  due to weak personalization. Next, as qualitatively studied in Sec. 4.1, on using the vanilla VoP ( $\alpha = 0$ ), the personalization is unreliable compared to  $\alpha = 5\mathrm{e}-4$ . Similar tendency is shown for  $\alpha = 5\mathrm{e}-5$ . Furthermore, when doubling the number of hidden units, VoP suffers from performance degradation, as well. From these ablation studies, we see that those factors are important for successful personalization via forwarding in VoP. We also show the relationship between the test performance and the size of enrollment sample in Table 4. Until reducing the number of enrollment samples from 5 to 2, the performance is maintained. And, at using just one sample for enrollment, there is only  $0.6\%$  performance degradation. This means that VoP gives representativeness to the independent sample by regularizing every generated model from a single sample with the KL loss on training.
+
+Table 5: Open set speaker verification equal error rates (lower is better) on VoxCeleb1 test set.  
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">Objective</td><td colspan="2">Equal error rates(%)</td></tr><tr><td>Fast ResNet-34</td><td>VGG-M-40</td></tr><tr><td rowspan="2">Baseline VoP</td><td rowspan="2">Angular Prototypical</td><td>5.46±0.02</td><td>8.13±0.24</td></tr><tr><td>5.24±0.08</td><td>7.91±0.14</td></tr></table>
+
+# 4.2. Speaker Verification
+
+Speaker verification (SV) is a task to dichotomize whether an utterance belongs to a specific speaker based on the enrolled utterances. To apply the proposed VoP to this task, we employ the Fast ResNet-34 (Chung et al., 2020) and the VGG-M-40 (Nagrani et al., 2017) as baseline networks. Here, considering each speaker as a personality, we personalize the last fully connected layer of each baseline. Note that, in the benchmark VoxCeleb1 (Nagrani et al., 2017) dataset, all testing speakers are not pre-defined in the training set. Hence, both baselines are designed to solve metric learning problem in their literature. To apply VoP, we take the angular prototypical loss, commonly used in the baselines, as our task loss. Then, we use these two samples to calculate each proto-mean and proto-variance. For each 200-sized minibatch, we randomly select one person for generating personal weights from proto-mean. As VoP computes the metric loss in the same space, it helps to generate metric learning space for a given proto-set.  $\alpha$  is set as 1e-7.
+
+Dataset VoxCeleb1 contains 153,516 utterances for 1,251 celebrities, which are extracted from YouTube videos. This data consists of  $55\%$  of the male speakers and  $45\%$  of the female speakers, which is gender balanced. The speakers cover a different accents, ethnicities and ages.
+
+Result Following the typical evaluation protocol (Chung et al., 2020), we sample 10 four-second temporal crops at regular interval from each test sample, and compute similarities with all possible pairs  $(10 \times 10)$ . Table 5 shows the equal error rates on the VoxCeleb1 test set. VoP generates personalized weights with the 10 crops from a target speaker. We see that the proposed VoP show better performance than both baselines due to personalized metric feature space.
+
+# 4.3. Few-Shot Classification
+
+We also apply our VoP on few-shot classification task on the miniImageNet (Vinyals et al., 2016). We employ the Prototypical Net (Snell et al., 2017) (Proto-Net) as the baseline, where the embedding architecture is composed of four convolutional blocks. Each block is comprised of  $64 \times 3 \times 3$  convolutions, batch normalization layer, a ReLU nonlinearity and a  $2 \times 2$  max-pooling layer. When applied to the images of miniImageNet, this architecture results in a 1,600-dimensional output space. In this task, we consider the personality of each episode. Namely, the hyper-personalizer is trained to produce the weight distribution of each episode
+
+Table 6: Few-shot classification accuracy on miniImageNet. The baseline is re-implemented following the literature.  
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">Backbone</td><td colspan="2">5-way accuracy(%)</td></tr><tr><td>1-shot</td><td>5-shot</td></tr><tr><td rowspan="2">Baseline VoP</td><td rowspan="2">ConvNet-64</td><td>49.42 ± 0.73</td><td>67.39 ± 0.62</td></tr><tr><td>51.43 ± 0.54</td><td>68.44 ± 0.64</td></tr></table>
+
+from the support set. Specifically, we personalize the convolution filter of the last block. Since the target categories keep changed in each episode, the baseline doesn't include the classification layer. Hence, we only add the KL divergence loss on their basic loss where  $\alpha$  is empirically set to 5e-4. In testing, the personalized embedding architecture can produce embedded features specialized to the personality of the given episode via forwarding the support set. Except for the loss function, we follow the experimental setting of the baseline e.g. learning rate, optimizer, episode configuration.
+
+Dataset The miniImagenet dataset consists of 60,000 color images of size  $84 \times 84$  with 100 classes where each class includes 600 images. We followed the split introduced by (Vinyals et al., 2016): 64, 16, and 20 classes for training, validation and testing, respectively. The 16 validation classes is used for monitoring generalization performance.
+
+Result Following the standard setting adopted by the ProtoNet, we conducted 5 way 1-shot and 5-shot classification tasks. We train using 30-way episodes for 1-shot classification and 20-way episodes for 5-shot classification. We match train shot to test shot and each class contains 15 query points per episode. Table C shows the accuracy scores for 1-shot 5-way and 5-shot 5-way where each is averaged over 600 test episodes and are reported with  $95\%$  confidence intervals. Even though using the same architecture and computational cost with the baseline, the proposed VoP yields higher accuracy scores for both tasks by  $2.01\%$  and  $1.05\%$ , each. Accordingly, we see that the proposed method is effective to personalize the embedding space by forwarding merely a few training examples.
+
+# 5. Conclusions
+
+We proposed Variational On-the-Fly Personalization (VoP), a novel personalization method that can produce a personalized network via forwarding a small amount of personal data on-the-fly. The proposed VoP can effectively estimate the weight distribution suitable for an individual without additional training using a large amount of personal data. Through extensive experiments on the three important tasks including the open-set problem, we showed that VoP successfully generates an accurately personalized model without increasing the computational cost. Also, as additional training is not required, it can be easily applied to various scenarios on edge devices e.g. mobile and IoT devices.
+
+# Acknowledgements
+
+Nojun Kwak was supported by the National Research Foundation of Korea (NRF) grant (2021R1A2C3006659) and IITP grant [NO.2021-0-01343, Artificial Intelligence Graduate School Program (Seoul National University)], both funded by the Korea government (MSIT). The research that is the subject of this paper, and the paper itself, was solely funded by Qualcomm Technologies, Inc.
+
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+
+# A. Proof for  $KL(q_{\theta}(\omega |x_i^k)||p(\omega |x_i^k))$
+
+In eq. (10), we analytically calculated the KL divergence with two assumptions:
+
+- The sample-specific prior follows the distribution of the personality, i.e.  $p(\omega | x_i^k) \simeq p(\omega | k)$ .  
+- The variational distribution and the personality distribution are an uncorrelated Gaussian distribution.
+
+Based on the assumptions, we prove eq. (10):
+
+$$
+\begin{array}{l} K L \left(q _ {\theta} \left(\omega \left| x _ {i} ^ {k}\right) \right| | p \left(\omega \left| x _ {i} ^ {k}\right)\right)\right) \simeq K L \left(\mathcal {N} \left(\omega \mid \mu_ {i}, \Sigma_ {i}\right) \right\lvert   \left| \mathcal {N} \left(\omega \mid \bar {\mu} _ {k}, \bar {\Sigma} _ {k}\right)\right) \\ = \mathbb {E} _ {\omega \sim q _ {\theta}} \left[ \ln q _ {\theta} \left(\omega | x _ {i} ^ {k}\right) - \ln p (\omega | x _ {i} ^ {k}) \right] \\ = \mathbb {E} _ {\omega \sim q _ {\theta}} \left[ \frac {1}{2} \ln \frac {| \bar {\Sigma} _ {k} |}{| \Sigma_ {i} |} - \frac {1}{2} (\omega - \mu_ {i}) ^ {T} \Sigma_ {i} ^ {- 1} (\omega - \mu_ {i}) + \frac {1}{2} (\omega - \bar {\mu} _ {k}) ^ {T} \bar {\Sigma} _ {k} ^ {- 1} (\omega - \bar {\mu} _ {k})) \right] \\ = \frac {1}{2} \left(\ln \frac {\left| \bar {\Sigma} _ {k} \right|}{\left| \Sigma_ {i} \right|} - \mathbb {E} _ {\omega \sim q _ {\theta}} \left[ \operatorname {t r} \left[ (\omega - \mu_ {i}) (\omega - \mu_ {i}) ^ {T} \Sigma_ {i} ^ {- 1} \right] \right] + \mathbb {E} _ {\omega \sim q _ {\theta}} \left[ (\omega - \bar {\mu} _ {k}) ^ {T} \bar {\Sigma} _ {k} ^ {- 1} (\omega - \bar {\mu} _ {k}) \right]\right) \\ = \frac {1}{2} (\ln \frac {| \bar {\Sigma} _ {k} |}{| \Sigma_ {i} |} - \mathrm {t r} [ \Sigma_ {i} \Sigma_ {i} ^ {- 1} ] + \mathbb {E} _ {\omega \sim q _ {\theta}} [ (\omega - \bar {\mu} _ {k}) ^ {T} \bar {\Sigma} _ {k} ^ {- 1} (\omega - \bar {\mu} _ {k}) ]) \\ = \frac {1}{2} \left(\ln \frac {\left| \bar {\Sigma} _ {k} \right|}{\left| \Sigma_ {i} \right|} - Z + \operatorname {t r} \left\{\bar {\Sigma} _ {k} ^ {- 1} \Sigma_ {i} \right\} + \left(\mu_ {i} - \bar {\mu} _ {k}\right) ^ {T} \bar {\Sigma} _ {k} ^ {- 1} \left(\mu_ {i} - \bar {\mu} _ {k}\right)\right) \\ = \frac {1}{2} (\ln \prod_ {z = 1} ^ {Z} \bar {\sigma} _ {(k, z)} ^ {2} - \ln \prod_ {z = 1} ^ {Z} \sigma_ {(i, z)} ^ {2} - Z + \sum_ {z = 1} ^ {Z} \frac {\sigma_ {(i , z)} ^ {2}}{\bar {\sigma} _ {(k , z)} ^ {2}} + \sum_ {z = 1} ^ {Z} \frac {(\mu_ {(i , z)} - \bar {\mu} _ {(k , z)}) ^ {2}}{\bar {\sigma} _ {(k , z)} ^ {2}}) \\ = \frac {1}{2} \sum_ {z = 1} ^ {Z} (\ln \bar {\sigma} _ {(k, z)} ^ {2} - \ln \sigma_ {(i, z)} ^ {2} - 1 + \frac {\sigma_ {(i , z)} ^ {2}}{\bar {\sigma} _ {(k , z)} ^ {2}} + \frac {(\mu_ {(i , z)} - \bar {\mu} _ {(k , z)}) ^ {2}}{\bar {\sigma} _ {(k , z)} ^ {2}}) \\ \end{array}
+$$
+
+where  $z \in [Z]$  is the index of multivariate Gaussian dimension.
+
+# B. Architecture of Hyper-Personalizer
+
+![](images/822a05072706b625ed82199ed3aad87b1e01888303c4739bd80f5f0e26f43c9b.jpg)  
+Figure 5: The overall architecture of the hyper-personalizer.  
+Fig. 5 depicts the detailed architecture of the hyper-personalizer in the proposed VoP. Similar to the encoder of VAE (AntixK, 2018), our hyper-personalizer is designed by a multi-layer perceptron with a hidden layer which is shared to two output layers.
+
+For  $i$ th sample, taking the spatially average-pooled encoded feature as an input, the hyper-personalizer produces two outputs:  $\mu_i$  and  $\sigma_i^2$  (both are in  $\mathbb{R}^Z$ ). The sizes of  $\mu_i$  and  $\sigma_i^2$  are equal to that of a target personalized layer (reparameterized sample-specific weight). Hence, the hyper-personalizer computes the mean and variance for each element of the sample-specific weight.
+
+# C. Comparison with Other Bayesian Approaches in Few-Shot Classification
+
+Table 7: Comparison of the proposed VoP and another Bayesian approach, samovar (Iakovleva et al., 2020), in few-shot classification.  
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">Features</td><td colspan="2">5-way Acc.(%)</td><td colspan="2">ΔAcc. to baseline</td></tr><tr><td>1-shot</td><td>5-shot</td><td>1-shot</td><td>5-shot</td></tr><tr><td rowspan="2">Baseline (versa, (Gordon et al., 2018)) samovar</td><td rowspan="2">Conv-5</td><td>53.4 ± 1.9</td><td>67.4 ± 0.9</td><td>-</td><td>-</td></tr><tr><td>52.4 ± 0.3</td><td>69.8 ± 0.2</td><td>-1.0</td><td>1.6</td></tr><tr><td rowspan="2">Baseline (Proto-Net, (Snell et al., 2017)) VoP (Ours)</td><td rowspan="2">Conv-4</td><td>49.4 ± 0.7</td><td>67.4 ± 0.6</td><td>-</td><td>-</td></tr><tr><td>51.4 ± 0.5</td><td>68.4 ± 0.6</td><td>2.0</td><td>1.0</td></tr></table>
+
+Table 8: Comparison of the proposed VoP and existing few-shot classification methods for different backbone features.  
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">Features</td><td colspan="2">5-way Acc. (%)</td></tr><tr><td>1-shot</td><td>5-shot</td></tr><tr><td>Matching Nets (Vinyals et al., 2016)</td><td></td><td>46.6</td><td>60.0</td></tr><tr><td>Meta LSTM (Ravi &amp; Larochelle, 2016)</td><td></td><td>43.4 ± 0.8</td><td>60.6 ± 0.7</td></tr><tr><td>MAML (Finn et al., 2017)</td><td>Conv-4</td><td>48.7 ± 1.8</td><td>63.1 ± 0.9</td></tr><tr><td>RelationNets (Sung et al., 2018)</td><td></td><td>50.4 ± 0.8</td><td>65.3 ± 0.7</td></tr><tr><td>Proto-Net (Snell et al., 2017)</td><td></td><td>49.4 ± 0.7</td><td>67.4 ± 0.6</td></tr><tr><td>VoP (Ours)</td><td></td><td>51.4 ± 0.5</td><td>68.4 ± 0.6</td></tr><tr><td>versa (Gordon et al., 2018)</td><td></td><td>53.4 ± 1.9</td><td>67.4 ± 0.9</td></tr><tr><td>samovar (Iakovleva et al., 2020)</td><td>Conv-5</td><td>52.4 ± 0.3</td><td>69.8 ± 0.2</td></tr><tr><td>VoP (Ours)</td><td></td><td>52.8 ± 0.2</td><td>70.3 ± 0.3</td></tr></table>
+
+In Sec. 4.3, we extended the proposed VoP to the few-shot classification. In this field, versa, a Bayesian approach, has tried to generate the weight of the last linear classification layer, for given task (support and query sets). While our VoP is based on the sample-specific posterior approximation for personalization, it starts from the formulation of the posterior approximation of the entire support set. Namely, versa is consciously designed to this few-shot classification task. On the top of the ConvNet backbone, versa generates weight and bias vectors for each class in class independent manner. To this end, it manipulates the size of conv-4 block feature with another  $1 \times 1$  convolution layer, so that their embedding backbone, called as Conv-5, is larger than ours. Although versa uses more computation and is fitted to this few-shot classification task, the proposed VoP, simple extension to this task, outperforms it by  $1.0\%$  on the 5-shot setting in Table 7. Also, as shown in Table 8, when using the same backbone (Conv-5), our VoP achieves higher accuracy than versa by a large margin  $2.9\%$  in the 5-shot setting.
+
+Similar to versa, samovar also generates the task-specific weight of the classification layer. Based on the versa's formulation, samovar exploits the relationship between the support set and the union of support and query sets. In specific, the posterior for the union set is approximated, using the prior given by the support set. Hence, samovar is more deliberately formulated for this few-shot classification task.
+
+Furthermore, samovar has several advantages in the experimental setting. First, samovar employs versa as the baseline which is better than our baseline (Proto-Net). versa inevitably requires another  $1 \times 1$  convolution layer and generates the linear classification layer on top of the layer. Whereas, we personalize the Conv-4 block (the last layer of ConvNet backbone), itself. Hence, we doesn't need more computation. Second, samovar exploits task conditioning and auxiliary co-training schemes to further improve the performance while VoP doesn't.
+
+Due to the different assumption in formulation, we can't direct apply VoP on the versa baseline. Also, the source code of samovar is not released, and it was impossible to reproduce samovar. Accordingly, it is difficult to compare our VoP and samovar on fair experimental setting of the baseline and w/o additional schemes. Therefore, keeping the difference in backbones, we first compare VoP and samovar in terms of the improvement to the corresponding baselines ( $\Delta$ Acc.) in Table 7. In the 5-shot setting, samovar yields better  $\Delta$ Acc. than ours by  $0.6\%$ , since it is based on the set-to-set relationship. Our VoP method is not designed to this few-shot classification task. Nevertheless, VoP is successfully extended to task in both 1- and 5-shot settings (better than the proto-net baseline by  $2.0\%$  and  $1.0\%$ , respectively). Especially, the proposed VoP achieves a larger  $\Delta$ Acc. than samovar in challenging 1-shot setting ( $2.0\%$  vs  $-1.0\%$  in  $\Delta$ Acc.). Next, in Table 8 we apply the same backbone of samovar (Conv-5) to our VoP. Here, VoP outperforms samovar for both settings.
+
+Table 9: Comparison of VoP and BatchNorm Calibration (BC) in KWS and few-shot classification for accuracy  $(\%)$ .  
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">Keyword spotting</td><td colspan="2">Few-shot classification</td></tr><tr><td>1-shot</td><td>5-shot</td></tr><tr><td>Baseline</td><td>87.46 ± 1.68</td><td>49.42 ± 0.73</td><td>67.39 ± 0.62</td></tr><tr><td>BatchNorm Calibration</td><td>90.70 ± 1.03</td><td>49.62 ± 0.38</td><td>67.19 ± 0.44</td></tr><tr><td>VoP</td><td>92.80 ± 1.40</td><td>51.43 ± 0.54</td><td>68.44 ± 0.64</td></tr></table>
+
+Table 10: Comparison of VoP and the simple fine-tuning of the global model with five enrollment samples using different learning rates (lr) in KWS according to accuracy (\%).  
+
+<table><tr><td>Method</td><td>Baseline</td><td>Baseline w/ Dropout</td></tr><tr><td>Pre-trained</td><td>87.46 ± 1.68</td><td>81.77 ± 1.75</td></tr><tr><td>Fine-tuning (lr=1e-3)</td><td>58.55 ± 0.92</td><td>59.90 ± 0.57</td></tr><tr><td>Fine-tuning (lr=1e-4)</td><td>72.95 ± 0.77</td><td>77.10 ± 0.98</td></tr><tr><td>Fine-tuning (lr=1e-5)</td><td>87.87 ± 0.09</td><td>78.95 ± 0.21</td></tr><tr><td>Fine-tuning (lr=1e-6)</td><td>87.80 ± 0.28</td><td>77.70 ± 0.84</td></tr><tr><td>VoP</td><td>92.80 ± 1.40</td><td>-</td></tr></table>
+
+# D. Comparison with Other Naive Techniques
+
+On the KWS tasks with the closed-set setting, we compare the proposed VoP with two naive techniques, BatchNorm Calibration (BC) (Jin et al., 2020) and fine-tuning the global model with few enrollment samples (5), which are straightforwardly applicable to the personalization task. Since BC (Jin et al., 2020) inevitably requires the batch normalization layer, we apply it to a network with batch normalization layers. In specific, we pass the target data through the network, and then re-calculate the batch normalization parameters for their personalization. As demonstrated in Table 9, BC improves the baseline by  $3.3\%$ , but is worse than our VoP. This result means that the personalization of only the parameters of the batch normalization layers has a limitation since most of parameters are still fixed (not personalized). Contrarily, our VoP can effectively regularize the entire parameters of the target layers to represent discriminative characteristics among personalities.
+
+Next, we compare the proposed VoP with the simple fine-tuning of the pre-trained global models varying the learning rates. In Table 10, the pre-trained (Baseline) is Baseline of Tables 2 and 9, and the pre-trained (Baseline w/ Dropout) is Baseline w/ Dropout of Table 2. We fine-tune each model with five enrolment samples for 10 times on different learning rates. In the case of Baseline w/ Dropout, the fine-tuning degrades the pre-trained global model for all the learning rates. Only when the dropout scheme is removed (Baseline), fine-tuning with such small learning rates can increase the performance of the pre-trained model, but the gaps are meagre  $(0.41\%$  and  $0.34\%)$ . On the other hand, the proposed VoP notably outperforms all the variants of the fine-tuning technique, and achieves significant performance improvement of the pre-trained baseline.
+
+# E. Computational Overhead
+
+The overhead of the hyper-personalizer (HP) itself is large as in the conventional hyper-network approaches. But, after training, in VoP, this overhead happens during enrollment process only. As we abolish the HP after enrollment, computation cost of VoP is same with the baseline in testing. We measure the training overhead by (Mac / no. params.) in Table 11.
+
+Table 11: Mac and parameters  
+
+<table><tr><td>Module</td><td>Baseline</td><td>1stHP</td><td>2nd HP</td></tr><tr><td>Mac</td><td>10M</td><td>88M</td><td>0.1M</td></tr><tr><td>Memory</td><td>0.95M</td><td>0.1M</td><td>0.05M</td></tr></table>
+
+# F. Privacy Issue
+
+In recent years, deep learning-based applications and services have focused on data privacy and security issues. Our proposed Variational On-the-Fly Personalization (VoP) also handles the private data. However, VoP is free from the above issues because VoP generates the personal weights at the edge device with few forwardings. In other words, at the server, the global model and hyper-personalizer are trained with the given training dataset. After training, the personalization process is on-the-fly conducted on the client's edge device without training, and the server does not access this process. We design the personalization process with a small amount of the enrollment samples at the personal device which can protect privacy.
+
+# G. Implementation Details
+
+For all experiments, we implement VoP using the Pytorch libraries with a single 1080ti GPU. For the keyword spotting and speaker verification tasks, we used the released official implementations (Castorini, 2017; ClovaAI, 2020) following a MIT License. In few-shot classification, we re-implemented the baseline (Snell et al., 2017) on the Pytorch libraries. All experiments are conducted three times with a GeForce GTX 1080 Ti GPU and we report the mean and standard deviation.

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+# Variational Sparse Coding with Learned Thresholding
+
+Kion Fallah<sup>1</sup> Christopher J. Rozell<sup>1</sup>
+
+# Abstract
+
+Sparse coding strategies have been lauded for their parsimonious representations of data that leverage low dimensional structure. However, inference of these codes typically relies on an optimization procedure with poor computational scaling in high-dimensional problems. For example, sparse inference in the representations learned in the high-dimensional intermediary layers of deep neural networks (DNNs) requires an iterative minimization to be performed at each training step. As such, recent, quick methods in variational inference have been proposed to infer sparse codes by learning a distribution over the codes with a DNN. In this work, we propose a new approach to variational sparse coding that allows us to learn sparse distributions by thresholding samples, avoiding the use of problematic relaxations. We first evaluate and analyze our method by training a linear generator, showing that it has superior performance, statistical efficiency, and gradient estimation compared to other sparse distributions. We then compare to a standard variational autoencoder using a DNN generator on the Fashion MNIST and CelebA datasets.
+
+# 1. Introduction
+
+Variational inference has become a ubiquitous tool in unsupervised learning of a distribution of latent features. These variational distributions can offer approximations in cases where inference over a true distribution is computationally expensive. Once inference is performed, latent features can be used for a variety of machine learning tasks, such as summarizing a dataset or training a generative model. The structure and statistical properties of latent features depend on the practitioner's choice of a prior distribution. Sparse distributions, in which only a few features are non-zero for
+
+![](images/9ff25e74d56835dea1eddcc9db0404e6a390d6e767840b814f0cf685393518d9.jpg)
+
+![](images/5626b4ad1863d7dde68e20a0ed4e2bfd760e271a46018f4ac6dac2f834bc78d1.jpg)  
+Figure 1. Comparison of standard black-box variational inference and our proposed approach for variational sparse coding. Orange arrows depict a forward pass, blue arrows depict automatic differentiation. Top: Sampling from standard variational inference approaches result in all features non-zero. Bottom: Our approach incorporates sparsity via a shifted soft-threshold function. We utilize a straight-through estimator, skipping the gradient of the shifted soft-threshold, for exact sparsity without numerical instability during training.
+
+each input data sample, have been favored for encouraging statistically efficient representations, especially when input data has low-dimensional structure (Olshausen & Field, 2004; Elad, 2010).
+
+To perform inference in sparse models with low computational cost, recent black-box variational inference (BBVI) (Ranganath et al., 2013) methods have been proposed to learn distribution parameters with DNNs using automatic differentiation (Kingma & Welling, 2014; Rezende et al., 2014). Unfortunately, these approaches either do not explicitly learn sparse features (Barello et al., 2018) or rely on relaxations that can lead to poor gradient estimation during training (Tonolini et al., 2020).
+
+Motivated by the success of soft-thresholding in iterative optimization procedures (Daubechies et al., 2003; Donoho & Johnstone, 1994), we propose a BBVI method to learn sparse distributions by thresholding samples drawn from a Laplacian or Gaussian distribution. We analytically show that thresholded samples from the former are identically distributed to a Spike-and-Slab distribution, giving practitioners control over the extent of sparsity by adjusting a
+
+threshold hyper-parameter. In cases where the degree of sparsity is not known beforehand, we also propose a technique to learn a distribution on the threshold parameter. To train our inference network, we apply a straight-through estimator (Bengio et al., 2013; Oord et al., 2018), leading to favorable training stability and gradient estimation. Finally, we propose a new sampling procedure that encourages feature reuse, encouraging the generator to learn more diverse features. We showcase the performance of our method compared to other inference strategies by training and analyzing a linear generator on whitened image patches (Olshausen & Field, 1996) and a DNN generator on the Fashion MNIST (Xiao et al., 2017) and CelebA (Liu et al., 2015) datasets<sup>1</sup>.
+
+# 2. Related Work
+
+# 2.1. MAP Estimate/Regressive Inference
+
+Sparsity models have a long history in statistical modeling with methods such as LASSO regularization (Tibshirani, 1996). These models often infer sparse latent features via a maximum a posteriori (MAP) estimate, requiring an iterative optimization procedure to be solved. The seminal work of (Olshausen & Field, 1996) proposes sparse codes as a means of unsupervised learning of a linear generator. Once trained, columns of the generator qualitatively resemble the receptive fields of mammalian cortical cells. Although methods exist to solve this optimization in discrete (Beck & Teboulle, 2009; Yang et al., 2012) and continuous (Rozell et al., 2008) time, their computational cost often scales poorly with dimensionality, making them prohibitive to use in modern deep learning settings.
+
+An alternative approach for inference has been to use DNNs to regress sparse codes for given input data. One method "unrolls" iterations of the ISTA algorithm (Gregor & LeCun, 2010). These methods are limited for unsupervised learning since they require ground truth codes as supervision during training.
+
+# 2.2. Variational Inference
+
+Early variational inference approaches applied exponential families for sparse coding, using iterative procedures to fit a variational posterior distribution (Girolami, 2001; Seeger, 2008). Later works explored variational methods for Spike-and-Slab models (Goodfellow et al., 2012; Sheikh & Lücke, 2016), using approximations to analytic solutions for faster inference. None of these approaches effectively scale inference to high-dimensional DNN representations.
+
+The proposal of BBVI, where DNNs are used to estimate parameters of a variational posterior distribution, has led
+
+to a leap in the computational efficiency of variational inference (Kingma & Welling, 2014; Rezende et al., 2014). As such, many recent methods have been proposed to train DNN inference networks with various prior distributions. These include Laplacian (Barello et al., 2018), Spike-and-Slab (Tonolini et al., 2020), and Beta-Bernoulli (Singh et al., 2017) distributions. Other work has incorporated sparsity through hierarchical posterior distributions (Salimans, 2016), evolutionary variational algorithms (Drefs et al., 2022), or group-sparsity in connections to generator networks (Ainsworth et al., 2018; Moran et al., 2021). We refer the reader to (Zhang et al., 2018) for a review of variational inference.
+
+# 2.3. Estimating the Variational Bound
+
+Various sampling approaches have been proposed to estimate the variational bound. (Cremer et al., 2017) proposes a sampling procedure based on a tighter bound on the data likelihood introduced in (Burda et al., 2016). Counterintuitively, using this tighter bound in training leads to a reduced signal-to-noise ratio in the inference network gradient (Rainforth et al., 2018). In (Grover et al., 2014), the authors apply rejection sampling using a computed acceptance probability for each sample. A later work (Bauer & Mnih, 2019) learns this acceptance probability with an additional DNN. Other work uses an alternative gradient estimator in BBVI which ignores certain terms during automatic differentiation to reduce variance (Roeder et al., 2017), with a bias-free estimator proposed by (Tucker et al., 2018).
+
+# 3. Methods
+
+# 3.1. Black-box Variational Inference
+
+The capability of DNNs as universal function approximators has been recently applied to learn complex distributions. Given a dataset of training samples  $[\mathbf{x}^1, \ldots, \mathbf{x}^N]^T \in \mathbb{R}^{N \times D}$  from target density  $p(\mathbf{x})$ , BBVI can be applied to train an inference network  $q_{\phi}(\mathbf{z} \mid \mathbf{x})$  to learn a distribution over latent features  $\mathbf{z}^k \in \mathbb{R}^d$ . This distribution can be applied in various machine learning tasks, such as training a generator  $p_{\theta}(\mathbf{x} \mid \mathbf{z})$ . To do this, one may employ the variational lower bound (ELBO) to maximize the marginal likelihood over the training samples (Jordan et al., 1998):
+
+$$
+\begin{array}{l} \log p _ {\boldsymbol {\theta}} (\mathbf {x}) = \log \mathbb {E} _ {p (\mathbf {z})} [ p _ {\boldsymbol {\theta}} (\mathbf {x} \mid \mathbf {z}) ] \\ = \log \mathbb {E} _ {p (\mathbf {z})} \left[ \frac {q _ {\phi} (\mathbf {z} \mid \mathbf {x})}{q _ {\phi} (\mathbf {z} \mid \mathbf {x})} p _ {\boldsymbol {\theta}} (\mathbf {x} \mid \mathbf {z}) \right] \\ \geq \mathbb {E} _ {q _ {\phi}} \left[ \log p _ {\boldsymbol {\theta}} (\mathbf {x} \mid \mathbf {z}) \right] - D _ {K L} \left(q _ {\phi} (\mathbf {z} \mid \mathbf {x}) \mid \mid p (\mathbf {z})\right) \\ = \mathcal {L} (\boldsymbol {\theta}, \phi ; \mathbf {x}). \tag {1} \\ \end{array}
+$$
+
+![](images/1da083969c06ee10ba339ce49af9f32a785a5514b6eaa331baad790c4d5f7549.jpg)  
+Figure 2. Visual depiction of the shifted soft-threshold applied to a Laplacian distribution. Probability mass  $\mu \pm \lambda$  is collapsed to the origin. The red points in the left are shifted to meet at  $\mu$  on the right.
+
+prior  $p(\mathbf{z})$  and a distribution family for the posterior  $q_{\phi}(\mathbf{z} \mid \mathbf{x})$ . In this work, we follow the common assumption of independence for both,  $p(\mathbf{z}) = \prod_{i=1}^{d} p(z_i), q_{\phi}(\mathbf{z} \mid \mathbf{x}) = \prod_{i=1}^{d} q_{\phi}(z_i \mid \mathbf{x})$ . For each prior distribution we consider, we will use the same distribution family for our variational posterior.
+
+Parameters  $(\phi, \theta)$  can be trained via a stochastic version of the variational expectation maximization algorithm (Neal & Hinton, 1998; Barello et al., 2018). Using batched training samples  $\mathbf{x}^k$ , this algorithm iterates in two steps from initializations  $(\phi^{(0)}, \pmb{\theta}^{(0)})$ :
+
+1. Expectation:  $\phi^{(t)} = \arg \max_{\phi}\mathcal{L}\left(\pmb{\theta}^{(t - 1)},\phi ;\mathbf{x}^k\right)$  
+2. Maximization:  $\pmb{\theta}^{(t)} = \arg \max_{\pmb{\theta}}\mathcal{L}\left(\pmb {\theta},\phi^{(t)};\mathbf{x}^k\right)$
+
+In the expectation step a differentiable transform  $g_{\phi}(\mathbf{x}^k,\epsilon^{j,k})$  of auxiliary samples  $\epsilon^{j,k}\sim p(\epsilon)$  is used to estimate the expectation over the inference network distribution. Coined the "reparameterization trick," this technique has been shown to reduce the variance of the gradient estimates through the inference network (Kingma & Welling, 2014; Rezende et al., 2014). In practice, the two steps of the EM algorithm are often approximated concurrently with a gradient step using a single sample  $J = 1$ .
+
+In the next sections, we will describe 1) a method for reparameterization that results in sparse latent features and 2) a new sampling procedure in the expectation step that improves performance for distributions with sparse priors.
+
+# 3.2. Reparameterization for Thresholded Samples
+
+A favorable choice of sparse prior distribution is the Spike-and-Slab  $p(z_{i}) = \gamma p(s_{i}) + (1 - \gamma)\delta (z_{i})$  (Mitchell & Beauchamp, 1988), which allows sparse random variables to be sampled by first sampling a Bernoulli random variable with probability  $\gamma$ . If the random variable is non-zero, then a sample is drawn from the slab distribution  $p(s_{i})$ . A Spike-and-Slab is favorable to setting  $p(z_{i})$  equal to Nor
+
+mal, Laplacian, or Cauchy distributions because these latter distributions will never result in samples with features exactly equal to zero. This allows one to benefit from precise sparsity for downstream computations (e.g., decoding, image generation). Unfortunately, current BBVI approaches depend on continuous approximations (controlled by temperature parameter  $\tau$ ) to each Bernoulli random variable  $\tau$  (Tonolini et al., 2020; Jang et al., 2017; Maddison et al., 2017). Tuning  $\tau$  requires a trade-off between variance and bias when estimating gradients, potentially leading to poor performance.
+
+Rather than parameterize  $p(z_{i})$  with discrete random variables, we propose an approach that applies shifted soft-thresholding of samples drawn from a Laplacian distribution. Let  $p(s) = \mathrm{Laplace}(\mu ,b)$  be a Laplacian pdf with shift  $\mu$  and scale  $b$ . We can draw a sample  $s\sim p(s)$  and apply the shifted soft-threshold function:
+
+$$
+\begin{array}{l} \mathcal {T} (s; \lambda , \mu) = \operatorname {s i g n} (s - \mu) \max  \left(| s - \mu | - \lambda , 0\right) \tag {2} \\ + \mathbb {I} \left[ | s - \mu | > \lambda \right] \mu . \\ \end{array}
+$$
+
+For ease of notation, we will use the shorthand  $\mathcal{T}_{\lambda}(s)$  to refer to  $\mathcal{T}(s;\lambda,\mu)$ . Intuitively, this function is a soft-threshold around  $\mu$ , where points mapped to  $\mu$  are set exactly to zero. This is visualized in Figure 2. When  $\mu = 0$ , this function is precisely the soft-threshold function (Donoho & Johnstone, 1994). Next, we will show how this shifted soft-threshold can be used to reparameterize to a Spike-and-Slab without needing discrete random variables.
+
+Proposition 3.1 Samples from a Laplace distribution  $s \sim p(s)$  passed through a shifted soft-threshold  $z = \mathcal{T}_{\lambda}(s)$  are equivalently distributed as a Spike-and-Slab distribution  $z \stackrel{i.i.d.}{\sim} p(z)$ .
+
+Note that under this rule for  $z$ ,  $p(z \mid |s - \mu| \leq \lambda) = \delta(z)$ , where the spike probability  $(1 - \gamma)$  can be found by computing  $p(|s - \mu| \leq \lambda)$ :
+
+$$
+\begin{array}{l} p (| s - \mu | \leq \lambda) = 2 \int_ {0} ^ {\lambda} \frac {1}{2 b} \exp (- s b ^ {- 1}) d s \\ = 1 - \exp (- \lambda b ^ {- 1}) = (1 - \gamma). \tag {3} \\ \end{array}
+$$
+
+We show in Appendix A that  $p(z \mid |s - \mu| > \lambda)$  is distributed as  $p(s)$ . Combining these facts and marginalizing over  $p(s)$  yields:
+
+$$
+z \sim \frac {\gamma}{2 b} \exp \left(\frac {- | z - \mu |}{b}\right) + (1 - \gamma) \delta (z).
+$$
+
+Using this technique, we can sample sparse random variables  $\mathbf{z}^k$  for a corresponding input data  $\mathbf{x}^k$  from an inference network trained via the EM algorithm (1). For a given input data sample, we encode the parameters of a base distribution that is either a Gaussian or Laplacian via  $q_{\phi}(\mathbf{s}\mid \mathbf{x}^{k})$ .
+
+These can be used to both compute the KL divergence and to reparameterize a sample to be passed through the shifted soft-threshold. The threshold parameter can be either set as a fixed hyperparameter  $\lambda = \lambda_0$  (equating to a fixed prior on the spike probability) or learned via variational inference with a factorial gamma prior  $p(\lambda) = \prod_{i=1}^{d} \Gamma(\alpha_0, \frac{\alpha_0}{\lambda_0})$  (Jankowiak & Obermeyer, 2018; Garrigues & Olshausen, 2010). In the case where we perform inference on  $\lambda$ , we assume independence with the base distribution  $p(\mathbf{s})$ .
+
+One challenge during training is dealing with the non-differentiable  $T_{\lambda}$  in the first term of (1). We have found that the subgradient of this function leads to shrinkage of the base distribution parameters, eventually leading to numerical instability or posterior collapse (Lucas et al., 2019). For further discussion on this issue, refer to Appendix C. As a result, we employ a straight-through estimator (Bengio et al., 2013; Jang et al., 2017; Oord et al., 2018), skipping the shifted soft-threshold when we differentiate the loss by passing the gradient from the generator directly to the base distribution parameters. This is depicted in Figure 1.
+
+This results in our final training objective with a reweighted KL term (Higgins et al., 2017). Following the notation of (Oord et al., 2018), we denote by  $\mathrm{sg}[\cdot]$  the stopgradient operator (identity in the forward pass with partial derivatives equal to zero).
+
+$$
+\widetilde {\mathbf {z}} ^ {k} = \mathbf {s} ^ {k} + \mathcal {T} _ {\boldsymbol {\lambda} ^ {k}} \left(\operatorname {s g} \left[ \mathbf {s} ^ {k} \right]\right) - \operatorname {s g} \left[ \mathbf {s} ^ {k} \right] \tag {4}
+$$
+
+$$
+\begin{array}{l} \mathcal {L} (\boldsymbol {\theta}, \phi ; \mathbf {x} ^ {k}) = \mathbb {E} _ {q _ {\phi}} \left[ \log p _ {\boldsymbol {\theta}} \left(\mathbf {x} ^ {k} \mid \widetilde {\mathbf {z}} ^ {k}\right) \right] \\ - \beta_ {1} D _ {K L} \left(q _ {\phi} (\mathbf {s} \mid \mathbf {x} ^ {k}) \mid \mid p (\mathbf {s})\right) \tag {5} \\ - \beta_ {2} D _ {K L} \left(q _ {\phi} (\boldsymbol {\lambda} \mid \mathbf {x} ^ {k}) | | p (\boldsymbol {\lambda})\right) \\ \end{array}
+$$
+
+Note that in the case where we fix  $\lambda^k = \lambda_0$ , the second term in (4) has no gradient and the third term in (5) is omitted. More details on reparameterization for the sampling procedure  $(\mathbf{s}^k,\pmb{\lambda}^k)\sim q_{\phi}(\mathbf{s},\pmb {\lambda}\mid \mathbf{x}^k)$  and on computing the loss terms in (5) are included in Appendix B.
+
+# 3.3. Max ELBO Sampling
+
+A favorable property of MAP estimates is that priority of which latent features are non-zero is given to those that best represent features in input data. These developed features lead to the highest decrease in loss, a property exploited in greedy MAP inference procedures (Tropp & Gilbert, 2007). Unfortunately, BBVI does not inherently have this property, with the factorial prior in the KL divergence term in (5) encouraging all features to be equally likely for each input data sample. To address this, we propose a sampling procedure that biases towards reuse of developed features.
+
+Before introducing our approach, we note the standard approach in BBVI for using multiple samples to approximate
+
+# Algorithm 1 Training with Thresholded Samples
+
+Input: Training batch  $\mathbf{x}^k$ , threshold hyper-parameter  $\lambda_0$ , whether to use a GammaPrior (along with gamma hyperprior  $\alpha_0$ ), network initializations  $(\phi^0, \theta^0)$ , number of samples  $J$ , and number of iterations  $T$ .
+
+for  $t = 1$  to  $T$  do
+
+$$
+\mathbf {f o r} j = 1 \mathbf {t o} J \mathbf {d o}
+$$
+
+$$
+\epsilon^ {j, k} \sim p (\epsilon)
+$$
+
+if GammaPrior then
+
+$$
+\left(\mathbf {s} ^ {j, k}, \boldsymbol {\lambda} ^ {j, k}\right) \gets g _ {\phi} \left(\mathbf {x} ^ {k}, \boldsymbol {\epsilon} ^ {j, k}\right)
+$$
+
+else
+
+$$
+\mathbf {s} ^ {j, k} \leftarrow g _ {\phi} \left(\mathbf {x} ^ {k}, \epsilon^ {j, k}\right)
+$$
+
+$$
+\boldsymbol {\lambda} ^ {j, k} \leftarrow \boldsymbol {\lambda} _ {0}
+$$
+
+end if
+
+$$
+\widetilde {\mathbf {z}} ^ {j, k} \leftarrow \mathbf {s} ^ {j, k} + \mathcal {T} _ {\lambda^ {j, k}} \left(\operatorname {s g} \left[ \mathbf {s} ^ {j, k} \right]\right) - \operatorname {s g} \left[ \mathbf {s} ^ {j, k} \right]
+$$
+
+$$
+\mathcal {L} ^ {j, k} \leftarrow \log p _ {\boldsymbol {\theta}} \left(\mathbf {x} ^ {k} \mid \widetilde {\mathbf {z}} ^ {j, k}\right) - \beta D _ {K L}
+$$
+
+end for
+
+$$
+\widehat {\mathcal {L}} ^ {k} = \arg \max  _ {j} \mathcal {L} ^ {j, k}
+$$
+
+$$
+\left(\boldsymbol {\phi} ^ {t}, \boldsymbol {\theta} ^ {t}\right) = \arg \max  \widehat {\mathcal {L}} ^ {k}
+$$
+
+end for
+
+the expectation with respect to  $q_{\phi}$  in (5). Given a sampling budget  $J$ , one may draw  $J$  i.i.d. samples  $\mathbf{z}^{j,k} \sim q_{\phi}(\mathbf{z} \mid \mathbf{x}^k)$  from the posterior and compute the ELBO for each sample.
+
+$$
+\mathcal {L} ^ {j, k} = \log p _ {\boldsymbol {\theta}} \left(\mathbf {x} ^ {j, k} \mid \mathbf {z} ^ {j, k}\right) - \beta D _ {K L} \left(q _ {\phi} \left(\mathbf {z} ^ {j, k} \mid \mathbf {x} ^ {k}\right) \mid \mid p (\mathbf {z})\right) \tag {6}
+$$
+
+Then, to estimate the training objective, one may average over the loss from each sample:
+
+$$
+\widehat {\mathcal {L}} _ {a v g} ^ {k} = \frac {1}{J} \sum_ {j = 1} ^ {J} \mathcal {L} ^ {j, k}. \tag {7}
+$$
+
+In the case where we have a sparse prior, each of the  $J$  samples taken here will have different support and thus encourage development of different latent features.
+
+Rather than give each sample equal probability, we introduce a new sampling strategy motivated by the approximation to expectations utilized in (Olshausen & Field, 1996; Connor et al., 2020). In these works, it is observed that the selected prior distribution concentrates most of its probability mass around the maximum value. This leads to an approximation of the expectation over the prior latent variable with a delta at the max value  $\widehat{\mathbf{z}}$ , motivating a MAP estimate:
+
+$$
+\begin{array}{l} \log \mathbb {E} _ {p (\mathbf {z})} [ p _ {\boldsymbol {\theta}} (\mathbf {x} \mid \mathbf {z}) ] \approx \log \int_ {z} p _ {\boldsymbol {\theta}} (\mathbf {x} \mid \mathbf {z}) p (\mathbf {z}) \delta (\widehat {\mathbf {z}}) d \mathbf {z} \\ = \max  _ {\mathbf {z}} \log p _ {\boldsymbol {\theta}} (\mathbf {x} \mid \mathbf {z}) + \log p (\mathbf {z}). \tag {8} \\ \end{array}
+$$
+
+Although our training setting differs in that we take our expectation over the variational posterior  $\mathbb{E}_{q_{\phi}}$ , in many cases
+
+we explicitly regularize this distribution with a KL divergence penalty against a sparse prior. To this same end, rather than approximate our expectation  $\mathbb{E}_{q_{\phi}}$  with the average over several samples, we approximate it with a single sample with the highest likelihood:
+
+$$
+\widehat {\mathcal {L}} _ {m a x} ^ {k} = \max  _ {j} \mathcal {L} ^ {j, k}. \tag {9}
+$$
+
+Although this provides a biased estimate of the loss, we will show in later sections that this heuristic provides an increase in performance and latent feature reuse during training. We combine this method with our sampling approach to outline our full training procedure in Algorithm 1.
+
+# 4. Experiments
+
+# 4.1. Linear Generator
+
+# 4.1.1. SPARSE CODING PERFORMANCE
+
+In our first experiment, we test different inference methods on sparse coding of the whitened image patches used in (Olshausen & Field, 1996). We train on 80,000 16x16 training patches with a dictionary of 256 elements (i.e., a linear generator  $\theta = \mathbf{A} \in \mathbb{R}^{256 \times 256}$ ). We start in this simplified setting since it allows us to compare against baseline sparse coding strategies (that use MAP estimates) and so we can analyze the dictionary we learn for each method. As a baseline, we infer coefficients  $\widehat{\mathbf{z}}^k$  using the FISTA algorithm (Beck & Teboulle, 2009) to optimize objective (10). This objective includes a data fidelity term, sparsity-encouraging  $\ell_1$  penalty on latent features, and a Frobenius norm regularizer on the dictionary to prevent unbounded growth:
+
+$$
+\min  _ {\mathbf {A}} \left\| \mathbf {x} ^ {k} - \mathbf {A} \widehat {\mathbf {z}} ^ {k} \right\| _ {2} ^ {2} + \lambda \| \widehat {\mathbf {z}} ^ {k} \| _ {1} + \kappa \| \mathbf {A} \| _ {F} ^ {2}. \tag {10}
+$$
+
+In all of our experiments, we set the sparsity hyperparameter such that roughly  $10\%$  features are non-zero for each batch of data. For FISTA, this occurs when  $\lambda = 20$ . We use this objective to measure validation performance of our variational methods.
+
+To train our inference networks, we use an ELBO objective with a Frobenius norm regularizer on the dictionary:
+
+$$
+\min  _ {\mathbf {A}, \phi} - \mathcal {L} (\mathbf {A}, \phi ; \mathbf {x} ^ {k}) + \kappa \| \mathbf {A} \| _ {F} ^ {2} \tag {11}
+$$
+
+$$
+\log p _ {\mathbf {A}} \left(\mathbf {x} ^ {k} \mid \mathbf {z} ^ {k}\right) = - \left\| \mathbf {x} ^ {k} - \mathbf {A} \mathbf {z} ^ {k} \right\| _ {2} ^ {2}, \tag {12}
+$$
+
+where  $\mathbf{z}^k$  is either found via the inference procedure outlined in section 3.2 or by previous methods that used Gaussian (Kingma & Welling, 2014), Laplacian (Barello et al., 2018), or Spike-and-Slab (Tonolini et al., 2020) prior distributions. For the Spike-and-Slab prior, we approximate each Bernoulli random variable with a straight-through estimator of a Gumbel-Softmax sample (Jang et al., 2017). For
+
+each method, we apply the sampling procedure outlined in Section 3.3 with  $J = 1$  and  $J = 20$ . Details on training hyper-parameters are provided in Appendix D.
+
+![](images/df4983bac4af16983470c2d736ce67d18c372505b15a3e8aeb87fac95d35f7a2.jpg)
+
+![](images/e4b5fcd81ecad85fd0780c96915b3ac22eea11caa8f331fdf8c02150733adaf5.jpg)
+
+![](images/908f0c8075878923f7ee39b90912d6aa543964bf8b8167017b9a99e94fb3c71b.jpg)
+
+![](images/caa6fc33d9efecdf333e8bbb32e68331dec105b94b465db51c6b21f7b7e35dc1.jpg)
+
+![](images/2653e3447ca7d78f833343e58b0310248bda474474e182e6b779dc9422fcac37.jpg)
+
+![](images/8d7667d034f8e54bea553bb9cfa6b92cceb4f3a04fedf4b6846668262fdb29a3.jpg)
+
+![](images/9e71fe40cb1132096245ea140e7ebb4215ecb16c2ce8d2bcee54f80d3229308e.jpg)
+
+![](images/1d1904870ef0e6b41d4765eb417b5e9ec05cdc4987b99d51234b19d892484c9a.jpg)
+
+![](images/793f796c506850efa17bfa37cd7c04b1cfdde306c93f1a6d2026f01e347b7949.jpg)  
+(a) FISTA
+
+![](images/15f00289b2c6e36ca5b7ad76b4f9ff57f95ffe07f33267d0af8214f4746abd4d.jpg)
+
+![](images/78a9caf3128501256e8417b6ba0e178bf3a7c95ca24928899bacb4e36f2f3540.jpg)
+
+![](images/8868ea3d6d753ec63bf0580a2e6228df98dac3ead8536b265288f62bccd46f19.jpg)  
+(b) Gaussian
+
+![](images/700a98293153931541977f1e0aca999c9fa7ed922bff303761d504bdfe9ab7b7.jpg)
+
+![](images/99ae4b4acb6e15b2aee9f074b572fb3b74065afe3ac4d7c884feeb03fd94bcf5.jpg)
+
+![](images/423c4627d2e17caf3a1e07f31e653ac57ca50e1a63951e7d1d8bb26a5b36d742.jpg)  
+(c) Laplacian
+
+![](images/d1de338bfc589484f4def2e9ebde6d312f4b6fca6c2e3274c972aa37affd34ef.jpg)
+
+![](images/e86fc2f6596c85eb5ae9b27d50e7e0593ba4c9cdfe7eebfeea9c5dc700918e2c.jpg)
+
+![](images/450145d29d031bc384d3081562d9fa25723a4ad8a8da7b35e6b9b65d11191537.jpg)
+
+![](images/73c03e4167a5501120d912264930f48583f37a76806020830754942447ab07ab.jpg)
+
+![](images/7f05a7c015d0edf2045a8a3f2942d98497b3ee4e5cc22851f8e92e5229161f04.jpg)
+
+![](images/de10557dc75bb9f3bc6beba794cec9bdf5887b311db00c1a75002d986339f687.jpg)
+
+![](images/4a9809ee297eb8f31fee87f41e4034f0d4e7f7a180b69944338ceed61ee8e6a0.jpg)
+
+![](images/fd6fd31d191ae61d5286792dc93374e0669238658a2bce50da98f4f53829d544.jpg)
+
+![](images/75eaaa7e9142ef4079a4906ae8cee4e7a6f513a0294a871744e4cde9fe5eee1e.jpg)
+
+![](images/db0744dae591cbc1f65662c4931b6c227984292c61ec01d328427db239dfa117.jpg)
+
+![](images/ea4b57a73c642a918d473608187feaef1a7d560cb11222e3b71d0058faa27d35.jpg)
+
+![](images/b6c5bcbc7b1ea40557ed4a3ac7f6fb87aaabbba19b804af1a21e92736b9bcbb8.jpg)
+
+![](images/cc4d1681393a2b9f24b1fdda3d18e1a67633d65eae080ee0c4ecb70e5dee88be.jpg)
+
+![](images/4de335b5f2518c4b81d97f95d27594e17cd3fce67676f2546cd52360029e0505.jpg)
+
+![](images/01a35a7db295689736d60eacf68824ae55684c925c4ae01d4ecac6eca719794e.jpg)
+
+![](images/e9a71cb5bd078b6ab11ad14ccdb036e4aa3fcbdba5cf2630e43013a424b5a90e.jpg)
+
+![](images/e2bf08c33c2c3db9a212af1c98e9e126d3cfccab314abcd21a8c67d00a36ca4d.jpg)
+
+![](images/6e2602289fa111513e9de5b0a9fe0803c3f7a3de77bfaf04ca552a34539f9734.jpg)
+
+![](images/57a7e316cc430f3737e3e8188bc9cea43063131186a8e9d58f3d78eb82d32250.jpg)
+
+![](images/a3ac47e94199fcb517ba4480951ee87baef41c0c983b54570e1162423a5c55ba.jpg)
+
+![](images/aa0599a7b6d92db94d4b5bb5e68463b06310e5a3dc3b0c5b1c0d52ed5da6f64b.jpg)  
+(d) Spike-and-Slab
+
+![](images/7883f953a01f62e2e44f3db1fdc5d760cdb952d466d6f0e50dc12706e788bb7c.jpg)
+
+![](images/8be91d796f5ab974fc0448f0eb4084027ccc24bcaffc5b2c1be0d09897f3de40.jpg)
+
+![](images/da3e861c7842c78750261d32be91252eb1fef5475c5b065937f1db4a2d607fe8.jpg)  
+(e) Thresholded Gaussian
+
+![](images/177cfe408ddfdf259dfbd48b757e4f2b0e8fcae196b62ed60c1f3976321a28c3.jpg)
+
+![](images/7cf47000c36e07fb3c9bbe9bc2f460e7ec525cc39f6784ef64502e5c91d64e59.jpg)
+
+![](images/62ee56786f7135ec6a56085e9d1eed45b66c420840a8d0d383be279c3fbcd34f.jpg)  
+(f) Thresholded Laplacian
+
+![](images/92a6e955a6b3fdc994ce6113d3777b285e09b20fa75f8c1a8c80e092e538b127.jpg)
+
+![](images/db3eb756845bf5ef5357e58ca17bc8e47c74824ede12de52e2b5b6736847848b.jpg)
+
+Figure 3. Entries of the learned linear generator (dictionary) for different approaches for inferring sparse codes. FISTA uses an iterative MAP estimate that optimizes objective (10) directly. Others use a variational approach with different prior distributions using  $J = 20$  samples. The Spike-and-Slab, Thresholded Gaussian, and Thresholded Laplacian methods learn dictionaries with most entries visually resembling the dictionary learned by FISTA.
+
+Figure 3 plots the columns of the learned dictionary for each inference method for  $J = 20$ . As expected, FISTA learns dictionary entries that qualitatively resemble Gabor wavelets (Olshausen & Field, 1996). The Laplacian prior (Barello et al., 2018) learns a few dictionary entries that resemble wavelets, with most entries resembling noise. Although the entries learned by all sparse priors resemble wavelets, only the Thresholded Laplacian is capable of learning this structure with  $J = 1$ . Full learned dictionaries for all methods under both sampling budgets are visualized in Appendix F.
+
+Table 1 compares average quantitative performance over three training runs for each inference method. The Laplacian prior with a fixed threshold performs the best among variational methods on objective (10). We note that methods that learn the threshold parameters tend to increase the number of active latent features, leading to an increase in L1 penalty in the validation loss. Although the Spike-and-Slab prior minimizes the importance weighted (IWAE) loss from (Burda et al., 2016) with  $K = 200$ , it performs worse on the sparse coding objective. We note that the IWAE loss measures the tightness of the ELBO bound (reconstruction + KL loss), whereas the thresholding in our method has a trade-off between deviation of the KL loss and posterior expressivity.
+
+Table 1. Performance of different inference methods with a linear generator on whitened image patches. Validation loss computed on objective (10). Multi-information measures the statistical efficiency of inferred codes. The importance weighted autoencoder (IWAE) loss (Burda et al., 2016) is a tight bound on the ELBO, but does not necessarily depict success in the sparse coding task. A table with standard deviations for  $J = 20$  is included in Table 6 in Appendix F  
+
+<table><tr><td rowspan="2">Method/Prior Distribution</td><td colspan="2">Validation Loss</td><td colspan="2">Multi-Information</td><td colspan="2">IWAE Loss</td></tr><tr><td>J=1</td><td>J=20</td><td>J=1</td><td>J=20</td><td>J=1</td><td>J=20</td></tr><tr><td>FISTA (baseline)</td><td>1.01E+02</td><td>-</td><td>8.75E+01</td><td>-</td><td>-</td><td>-</td></tr><tr><td>Gaussian</td><td>1.35E+03</td><td>1.35E+03</td><td>7.34E+02</td><td>7.36E+02</td><td>2.57E-01</td><td>2.16E-01</td></tr><tr><td>Laplacian</td><td>5.96E+02</td><td>5.79E+02</td><td>5.36E+02</td><td>5.34E+02</td><td>2.20E-01</td><td>2.22E-01</td></tr><tr><td>Spike-and-slab</td><td>2.52E+02</td><td>2.39E+02</td><td>1.94E+02</td><td>1.96E+02</td><td>2.41E-01</td><td>2.12E-01</td></tr><tr><td>Thresholded Gaussian</td><td>2.32E+02</td><td>2.30E+02</td><td>1.67E+02</td><td>1.76E+02</td><td>1.52E+00</td><td>1.33E+00</td></tr><tr><td>Thresholded Gaussian+Gamma</td><td>2.85E+02</td><td>2.70E+02</td><td>2.35E+02</td><td>2.29E+02</td><td>1.17E+00</td><td>1.13E+00</td></tr><tr><td>Thresholded Laplacian</td><td>1.98E+02</td><td>1.94E+02</td><td>1.80E+02</td><td>1.91E+02</td><td>9.30E-01</td><td>1.13E+00</td></tr><tr><td>Thresholded Laplacian+Gamma</td><td>2.23E+02</td><td>2.11E+02</td><td>2.38E+02</td><td>2.33E+02</td><td>9.81E-01</td><td>1.12E+00</td></tr></table>
+
+To measure the statistical efficiency in the inferred codes, we follow previous work in estimating the multi-information,  $\sum_{i=1}^{d} h(z_i) - h(\mathbf{z})$ , of the inferred codes (Fallah et al., 2020; Eichhorn et al., 2009; Bethge, 2006). This quantity is minimized when the features have high joint entropy while being independent of one another, denoting a reduction in redundancy. Among variational methods, the Thresholded Gaussian minimizes this quantity.
+
+To further understand the benefit of our proposed method, we investigate the quality of the gradient estimates through both the dictionary and inference network at the end of training. To measure this, we use the signal-to-noise ratio (SNR) metric proposed in (Rainforth et al., 2018):
+
+$$
+\mathrm {S N R} (\theta) = \left| \mathbb {E} \left[ \nabla_ {\theta} \mathcal {L} \right] / \sigma \left[ \nabla_ {\theta} \mathcal {L} \right] \right|.
+$$
+
+Where  $\theta$  is a single parameter from our dictionary or inference network and  $\mathcal{L}$  is our training loss. In our experiments, we average the SNR over all parameters in our inference network or dictionary.
+
+The benefit of measuring the gradient quality through a relative metric like SNR is that it weights the variance by the importance of each parameter. For example, dictionary entries that are rarely activated for a given input data will have a low expected gradient, but may have high variance. We evaluate the SNR over 1000 sparse code samples for each data point in our validation set. Figure 4 compares the SNR for both the dictionary and inference network for various methods. It can be seen that the Spike-and-Slab suffers from low SNR in the case of  $J = 1$ , with a significant increase for  $J = 20$ . We speculate that this metric explains the lack of clear wavelets in the dictionary learned by the Spike-and-Slab for  $J = 1$ .
+
+# 4.1.2. SAMPLING PERFORMANCE
+
+To test whether our sampling procedure leads to more feature reuse, we measure the consistency in the features used over several forward passes for fixed input data. Each forward pass leads to a sparse code with a different support set, indicating use of different features. Hence, we can compute the Jaccard index between the support set of several samples  $\mathbf{z}^{1,k},\ldots ,\mathbf{z}^{J,k}\sim q_{\phi}(\mathbf{z}\mid \mathbf{x}^k)$  for fixed input to measure consistency (Jaccard, 1912). Intuitively, the Jaccard index measures similarity between discrete sets, providing a value in [0, 1] based on how similar the support is over different forward passes. Let  $S(\mathbf{z}^{j,k})$  denote the support of  $\mathbf{z}^{j,k}$  (i.e.
+
+![](images/ce120a91be3f29d0c1553fece25abcac72e031c6b99ef9d64b588b97a0962093.jpg)  
+(a) Inference Network Gradient SNR
+
+![](images/cebf51128004f0709bd8e2f6982ae7cd61154ab5928870d06f811ea5f074ed0a.jpg)  
+(b) Generator Gradient SNR  
+Figure 4. Comparison of the SNR of the (a) inference network and (b) linear generator gradients at the end of training. Thresholded methods perform the best, with all sparse priors experiencing higher SNR with  $J = 20$  samples under max ELBO sampling.
+
+![](images/e1a0fa183daedf3224bfc902e6717b31b1d310a0a09d378a50b52a9886abc853.jpg)  
+(a)  $\mathrm{J} = {20}$
+
+![](images/07bbcdabe86f80e8e56c8cdc186f502ea156b98068a61604e87540084052fbb2.jpg)  
+(b)  $\mathrm{J} = {100}$  
+Figure 6. Selected entries of an estimated dictionary for variational autoencoders trained with a (a) Gaussian prior and (b) Thresholded Gaussian+Gamma prior with a 512 dimensional latent space<sup>3</sup>. For each method, the first row denotes dictionary entries with the highest magnitude and the bottom row denotes random entries from the middle of the dictionary.
+
+$i\in S(\mathbf{z}^{j,k})$  if  $z_{i}^{j,k}\neq 0)$  , the Jaccard index is computed as:
+
+$$
+C \left(\mathbf {z} ^ {m, k}, \mathbf {z} ^ {n, k}\right) = \frac {\left| S \left(\mathbf {z} ^ {m , k}\right) \cap S \left(\mathbf {z} ^ {n , k}\right) \right|}{\left| S \left(\mathbf {z} ^ {m , k}\right) \cup S \left(\mathbf {z} ^ {n , k}\right) \right|}. \tag {13}
+$$
+
+For any given input data  $\mathbf{x}^k$  we can draw  $J$  samples from our inference network and compute the average pairwise Jaccard index as  $\frac{1}{J(J - 1)}\sum_{m}^{J}\sum_{n\neq m}^{J}C(\mathbf{z}^{m,k},\mathbf{z}^{n,k})$ . To normalize for the fact that each training methods leads to a different number of non-zero dictionary entries (e.g., an inference network that only uses a subset of the available support is more likely to be consistent), we measure consistency using the support of features with  $\| \mathbf{a}_i\| _2^2 >1\mathrm{E} - 01$ .
+
+We use this metric to compare the consistency of inference networks trained under the average ELBO scheme from (7) and the proposed max ELBO scheme in (9) using  $J = [20,100]$  samples. We observe that both methods result in  $10\%$  of features non-zero on average.
+
+We plot the consistency on the validation set averaged over three training runs as a histogram in Figure 5. It can be seen that the max ELBO sampling procedure leads to higher consistency (indicating reuse of developed features) compared to average sampling. Qualitatively, the dictionary entries learned in the average sampling scheme has several repeated entries, indicating poor reuse in comparison to the qualitatively diverse entries learned by max sampling. We visualize the full dictionary for both in Figure F.2 in Appendix F.
+
+![](images/191e4ed3d2ab5ae24b01309e856fcedf7f9b5f17fd5310f8ea725254b2145b90.jpg)  
+(a) Gaussian
+
+![](images/c5f03ef2536605688ed0046eb8fbb5d58842160d2f34a0f8f295d54e4fe273b1.jpg)  
+Figure 5. Histogram of the average pairwise Jaccard index over 20 forward passes on two trained inference networks. The pairwise Jaccard index measures the consistency of the support set of inferred codes. Max ELBO sampling leads to a higher average Jaccard index, indicating reuse of developed latent features.  
+(b) Thresholded Gaussian+Gamma
+
+# 4.2. DNN Generator
+
+We next investigate the efficacy of variational sparse coding in unsupervised learning on the Fashion MNIST (FMNIST) (Xiao et al., 2017) and CelebA datasets (Liu et al., 2015). We follow a similar methodology as the previous section, except that we apply a DNN generator in place of the linear dictionary (more details in Appendix E). Following the principle of overcomplete representations in sparse coding, where the robustness of latent features increases with the dimensionality (Olshausen, 2013), we compare performance as we sweep dimensionality.
+
+Figure 7 shows the quantitative performance of different prior methods trained with either increasing latent dimensionality or percentage of latent features non-zero. For both datasets, as the dimensionality increases with a fixed degree of sparsity, it can be seen that the Gaussian prior (corresponding to a variational autoencoder (Kingma & Welling, 2014; Higgins et al., 2017)) has a significant increase in redundancy, measured via the multi-information, with a slight increase in the validation Fréchet inception distance (FID) (Heusel et al., 2017) and MSE. Alternatively, all the sparse priors show superior scaling with dimensionality, with the Thresholded Gaussian and Thresholded Gaussian+Gamma depicting superior performance to the Spike-and-Slab model from Tonolini et al. (2020). Additionally, all thresholded methods scale better with MSE as the degree of sparsity (corresponding to the  $\%$  of features non-zero) increases.
+
+We observe the same qualitative generative interpretability
+
+![](images/c44e4740b55723201f23bb7c15080de23551a3e8de9e413e0d6ac716f7793352.jpg)  
+(a) FMNIST MI
+
+![](images/60e0ecc2f779cd72c7fa77a245a97f570d1b23d5b6ac4104ec13ff0f941cd33d.jpg)  
+(b) FMNIST FID
+
+![](images/f7458bf3b4096978ddd7b3a64b2f9a09412669ca75ceb16b23046b43c80e3b9c.jpg)  
+(c) FMNIST MSE
+
+![](images/e4d61700749c9d5f82a227296e875bc0f87232e4ef9aff94423e4d9f2aedf000.jpg)  
+(d) FMNIST MSE per % Sparsity
+
+![](images/9057a63a5b3afc7e5efa1f377d3c46218fcd7c96c4359ed7fe5ae6ee4d6ca424.jpg)  
+(e) CelebA MI
+
+![](images/8337500d48da8fa58e1f0092747363eb2bdaee764807cea0757a5ebd399bf92d.jpg)  
+(f) CelebA FID
+
+![](images/6c86565582c76bf6f1962cb73c0f8cdbae9735e9e77846574e78c5fd7d09b9c7.jpg)  
+(g) CelebA MSE
+
+![](images/b9f2491376b357b63217b72ead8d80ac3d804158a7c2545ff026256af094d555.jpg)  
+(h) CelebA MSE per % Sparsity
+
+when sweeping each individual latent feature as in previous work that investigates Gaussian priors (Higgins et al., 2017) and sparse priors (Tonolini et al., 2020). In these works, it is observed that each latent feature corresponds to a specific semantic change in the generated image (such as changes in skin color, background color, hairstyle, etc.). Unfortunately, in the case where one trains a DNN generator, they are unable to observe the generator features the same way one could with the columns of the linear dictionary.
+
+To this end, we estimate a linear dictionary for each fully trained CelebA inference network to visualize what is encoded by each latent feature. Given a collection of data samples  $\mathbf{X} = \left[\mathbf{x}^{1},\dots ,\mathbf{x}^{n}\right]^{T}$  and corresponding inferred codes  $\mathbf{Z} = \left[\mathbf{z}^{1},\dots ,\mathbf{z}^{n}\right]^{T}$  from an inference network with fixed weights fully trained with a DNN generator  $\mathbf{z}^k\sim q_\phi (\mathbf{z}\mid \mathbf{x}^k)$ , one can estimate a linear dictionary as (Isely et al., 2010; Fallah et al., 2020):
+
+$$
+\mathbf {X} = \mathbf {A Z}
+$$
+
+$$
+\widehat {\mathbf {A}} = \left(\mathbf {X} \mathbf {Z} ^ {T}\right) \left(\mathbf {Z} \mathbf {Z} ^ {T}\right) ^ {- 1}.
+$$
+
+We use this technique with each trained inference network to analyze which features in the training dataset are represented by each latent feature. We plot selected entries from dictionaries estimated for the Gaussian and Thresholded Gaussian+Gamma in Figure 6. Interestingly, the first eight entries from the Gaussian prior resemble the qualitative features observed in the  $\beta$ -VAE (Higgins et al., 2017). Poor statistical efficiency in the Gaussian prior indicates
+
+![](images/f1111a8d95b6d7cfb71692b87172adaaa89a9b6a61cd9ab1e56267760659c398.jpg)  
+Figure 7. Comparison of variational autoencoder with different prior distributions as the latent feature dimensionality increases on the Fashion MNIST (a-d) and CelebA (e-h) datasets. (a, e) Multi-information scales at a higher rate with the Gaussian prior, indicating inferior statistical efficiency at higher latent dimensions. (b, f) The Fréchet Inception Distance of images reconstructed by learned DNN generator. (c, g) Mean squared error on validation data for a fixed degree of sparsity. (d, h) Mean squared error for a fixed dimensionality with an increase in the degree of sparsity (indicating the % of non-zero latent features set by the prior).  
+Figure 8. Estimated dictionary entries for latent features with the highest absolute point-biserial correlation with the CelebA binary attribute labels. From top to bottom, the rows denote the Gaussian prior, Thresholded Gaussian prior, and Thresholded Gaussian+Gamma prior.
+
+that certain latent features are not encoding useful information from training data, leading to corresponding estimated dictionary entries that resemble noise (visualized in the second row). Meanwhile, all the entries for the sparse prior more clearly resemble features in the training dataset. While the qualitative results here are similar to previous research comparing learned features from principal component analysis (PCA) and independent component analysis (ICA) on face datasets (Bartlett et al., 2002), a detailed quantitative comparison with those results is beyond the scope of the present paper. We visualize more estimated dictionary plots in Appendix G.
+
+Finally, we use the binary attribute labels included with the CelebA dataset to measure correlation between latent fea
+
+tures and attributes in the input data. Given vectors of 40 binary attributes for each data sample  $\mathbf{y}^1, \ldots, \mathbf{y}^N$  and corresponding latent features  $\mathbf{z}^1, \ldots, \mathbf{z}^N$  from a trained inference network, we can compute the point-biserial correlation coefficient between each attribute  $m$  and each dimension of latent feature  $n$  as:
+
+$$
+\rho_ {p b} ^ {m, n} = \frac {M _ {1} - M _ {0}}{s _ {N}} \frac {\sqrt {n _ {1} n _ {0}}}{\sqrt {N}}.
+$$
+
+Where  $M_1$  and  $M_0$  are the sample mean of latent features  $z_n^k$  corresponding to when attribute label  $y_m^k$  equals one or zero, respectively.  $s_N$  is the sample standard deviation over all features, and  $n_1$  and  $n_0$  respectively denote the total count of attribute labels that equal 1 and 0.
+
+We plot the estimated dictionary entry corresponding to the feature with the highest absolute correlation for selected attributes in Figure 8. Noting that features can be positive or negative for a given input data, it can be seen that the sparse priors lead to dictionary entries with features that visually resemble the semantic meaning of each attribute label.
+
+# 5. Discussion
+
+In this work, we have presented a new method for variational sparse coding by thresholding samples from an inference network. This simple approach avoids discrete random variables in its parameterization, and we show that it leads to superior performance and gradient estimation. Compared to a standard Gaussian prior, it provides more statistical efficiency in its use of latent features. One area of future work may consider alternative discrete random variable estimators that employ control variates (Tucker et al., 2017). Another interesting direction is the substitution of the KL divergence with distance metrics that perform better when the prior lies on a manifold (as is the case with sparse signals) (Patrini et al., 2020).
+
+# 6. Acknowledgements
+
+The authors would like to thank Adam Charles, Francesco Tonolini, and Linxing Preston Jiang for their correspondence regarding their experience with variational sparse coding. This work was partially supported by NSF CAREER award CCF-1350954, ONR grant number N00014-15-1-2619 and AFRL grant number FA8750-19-C-0200.
+
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+Tonolini, F., Jensen, B. S., and Murray-Smith, R. Variational Sparse Coding. In Proceedings of The 35th Uncertainty in Artificial Intelligence Conference, pp. 690-700. PMLR, August 2020. URL https://proceedings.mlrpress/v115/tonolini20a.html. ISSN: 2640-3498.  
+Tropp, J. A. and Gilbert, A. C. Signal Recovery From Random Measurements Via Orthogonal Matching Pursuit. IEEE Transactions on Information Theory, 53(12):4655-4666, December 2007. ISSN 1557-9654. doi: 10.1109/TIT.2007.909108. Conference Name: IEEE Transactions on Information Theory.  
+Tucker, G., Mnih, A., Maddison, C. J., Lawson, D., and Sohl-Dickstein, J. REBAR: Low-variance, unbiased gradient estimates for discrete latent variable models. arXiv:1703.07370 [cs, stat], November 2017. URL http://arxiv.org/abs/1703.07370. arXiv: 1703.07370.  
+Tucker, G., Lawson, D., Gu, S., and Maddison, C. J. Doubly Reparameterized Gradient Estimators for Monte Carlo Objectives. arXiv:1810.04152 [cs, stat], November 2018. URL http://arxiv.org/abs/1810.04152.arXiv:1810.04152.  
+Xiao, H., Rasul, K., and Vollgraf, R. Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms, 2017. URL https://arxiv.org/abs/1708.07747.
+
+Yang, A. Y., Zhou, Z., Ganesh, A., Sastry, S. S., and Ma, Y. Fast L1-Minimization Algorithms For Robust Face Recognition. arXiv:1007.3753 [cs], August 2012. URL http://arxiv.org/abs/1007.3753.arXiv:1007.3753.  
+Zhang, C., Butepage, J., Kjellstrom, H., and Mandt, S. Advances in Variational Inference. arXiv:1711.05597 [cs, stat], October 2018. URL http://arxiv.org/abs/1711.05597. arXiv:1711.05597.
+
+# A. Thresholded Laplacian Derivation
+
+To determine the distribution of our thresholded samples  $p(z)$ , we marginalize the conditional distribution with respect to  $p(s)$ . The shifted soft-threshold can be viewed as a shift of the pdf of  $p(s)$  towards the mean value  $\mu$ , as depicted in Figure 2.
+
+We thus break up the marginalization over  $p(s)$  into two disjoint regions: the region that gets collapsed into the origin  $p(|s - \mu| \leq \lambda)$  and the regions that get shifted towards the mean  $p(|s - \mu| \geq \lambda)$ . We can then write the expectation:
+
+$$
+\begin{array}{l} p (z) = \mathbb {E} _ {p (s)} [ p (z \mid s) ] \\ = p (z \mid | s - \mu | \leq \lambda) p (| s - \mu | \leq \lambda) \\ + p (z \mid | s - \mu_ {0} | \geq \lambda) p (| s - \mu | \geq \lambda) \\ = (1 - \gamma) \delta (z) + \gamma p (z | | s - \mu | \geq \lambda) \\ \end{array}
+$$
+
+Where  $(1 - \gamma) = 1 - \exp (-\lambda b^{-1})$  and  $\gamma = \exp (-\lambda b^{-1})$  as shown in the main text.
+
+To derive  $p(z \mid |s - \mu| \geq \lambda)$ , we write the pdf of  $p(s)$  with the shift applied from soft-thresholding:
+
+$$
+\begin{array}{l} \left\{ \begin{array}{l l} \frac {1}{2 b} \exp \left(- \frac {| s - \mu + \lambda |}{b}\right) & s - \mu \leq 0 \\ \frac {1}{2 b} \exp \left(- \frac {| s - \mu - \lambda |}{b}\right) & s - \mu > 0 \end{array} \right. \\ = \frac {1}{2 b} \exp \left(- \frac {| s - \mu + \operatorname {s i g n} (s - \mu) \lambda |}{b}\right) \\ = \frac {1}{2 b} \exp \left(- \frac {| s - \mu |}{b}\right) \exp \left(- \lambda b ^ {- 1}\right) \\ \end{array}
+$$
+
+Where the third line follows from the equality condition of the triangle inequality (both terms have the same sign) and from the fact that  $\lambda$  is always positive.
+
+This is precisely equal to  $p(z \mid |s - \mu| \geq \lambda)$  multiplied by a normalizing constant  $\frac{1}{Z}$  to become a valid distribution (accounting for the probability mass that was moved to zero). This normalizing constant is exactly one minus the mass moved to zero  $Z = \gamma = \exp(-\lambda b^{-1})$ . This leads to our final conditional probability:
+
+$$
+p (z \mid | s - \mu_ {0} | \geq \lambda) = \frac {1}{2 b} \exp \left(- \frac {| s - \mu |}{b}\right) \tag {14}
+$$
+
+# B. Reparameterization Details for Thresholded Samples
+
+To reparameterize thresholded samples, we first draw samples  $s_i^k$  for each feature space dimension from a base distribution that is either Gaussian or Laplacian. For an input data  $\mathbf{x}^k$ , our encoder network outputs a shift and log-scale for each dimension of our latent feature for both distributions. The KL divergence loss is always computed on the base distribution, summing over each feature dimension.
+
+For a Gaussian distribution, we apply the standard reparameterization technique from (Kingma & Welling, 2014) using  $(\mu_i^k, 2\log \sigma_i^k)$  output from a DNN:
+
+$$
+\begin{array}{l} \epsilon_ {i} ^ {j, k} \sim \mathcal {N} (0, 1) \\ z _ {i} ^ {j, k} = g _ {\phi} \left(\mathbf {x} ^ {k}, \epsilon_ {i} ^ {j, k}\right) _ {i} = \sigma_ {i} ^ {k} \epsilon_ {i} ^ {j, k} + \mu_ {i} ^ {k} \\ \end{array}
+$$
+
+with a KL divergence depending on scale prior  $\sigma_0^2$  (Kingma & Welling, 2014):
+
+$$
+D _ {K L} ^ {\mathcal {N}} = \left(\left(\mu_ {i} ^ {k}\right) ^ {2} + \left(\sigma_ {i} ^ {k}\right) ^ {2}\right) \left(2 \sigma_ {0}\right) ^ {- 1} + \log \frac {\sigma_ {0}}{\sigma_ {i} ^ {k}} - \frac {1}{2}
+$$
+
+For a Laplacian distribution, we apply the reparameterization rule from (Connor et al., 2020) using  $(\mu_i^k,\log b_i^k)$  output from a DNN:
+
+$$
+\epsilon_ {i} ^ {j, k} \sim \mathrm {U n i f} \left(- \frac {1}{2}, \frac {1}{2}\right)
+$$
+
+$$
+z _ {i} ^ {j, k} = g _ {\phi} \left(\mathbf {x} ^ {k}, \epsilon_ {i} ^ {j, k}\right) _ {i} = - b \operatorname {s i g n} \left(\epsilon_ {i} ^ {j, k}\right) \log \left(1 - 2 \epsilon_ {i} ^ {j, k}\right)
+$$
+
+Applying the KL divergence loss derived in (Meyer, 2021) with a scale prior  $b_{0}$ :
+
+$$
+D _ {K L} ^ {L} = \frac {| \mu_ {i} ^ {k} |}{b _ {0}} + \frac {b _ {i} ^ {k} \exp \left(- | \mu_ {i} ^ {k} | (b _ {i} ^ {k}) ^ {- 1}\right)}{b _ {0}} + \log \frac {b _ {0}}{b _ {i} ^ {k}} - 1
+$$
+
+To prevent numerical instability during training, we apply clamping to the log term in  $g_{\phi}(\mathbf{x}^k,\epsilon_i^{j,k})_i$  to have a minimum value of 1E-06. Additionally, we multiply a warm-up variable  $\omega$  to the inferred scale variable  $b_{i}^{k}$ . At the beginning of training we set this variable  $\omega^{(0)} = 0.1$  and increment it at each training iteration by 2E-04 until it reaches a maximum value of 1.0.
+
+When applying thresholding to either distribution, we either apply a fixed threshold set as a hyper-parameter  $\lambda_{i} = \lambda_{0}$  or infer a threshold for each feature dimension. To perform reparameterization, we follow the pathwise derivative methods proposed in (Jankowiak & Obermeyer, 2018), parameterized by a neural network that outputs  $(\log \alpha_{i}^{k}, \log \beta_{i}^{k})$ . For numerical stability, we clamp these values to be between  $[1\mathrm{E} - 06, 1\mathrm{E} + 06]$ . We use both the reparameterization and KL divergence implementations included in PyTorch 1.10 (Paszke et al., 2019). In all cases, we set the prior  $\alpha_{0} = 3$ ,  $\beta_{0} = \frac{3}{\lambda_{0}}$ . We find that on average this encourages the inferred threshold values to equal  $\lambda_{0}$ .
+
+Our reparameterization procedure differs from the Spike-and-Slab from (Tonolini et al., 2020) in a few ways. First and foremost, we avoid the need of applying continuous approximations to Bernoulli random variables by applying shifted soft-thresholding. This also results in our model not needing to estimate a spike probability for each latent
+
+variable, but rather just parameters for the base distribution. Finally, rather than use the KL divergence for Spike-and-Slabs derived in (Tonolini et al., 2020), we only penalize the base distribution with a KL divergence.
+
+# C. Straight-Through Estimator Details
+
+The shifted soft-threshold is non-differentiable, with a subgradient of 0 when a latent component  $z_{i} = 0$ . This is problematic since it serves as a block of gradient flowing from the generator to the inference network whenever a latent feature is not used (a common occurrence when sparsity is imposed on latent features). In these circumstances, although the gradient is blocked from the generator, a gradient is still computed for the KL divergence term. This empirically causes numerical instability (gradients go to NaN) or the inference network to primarily minimize the KL divergence term, leading to posterior collapse (Lucas et al., 2019). Note that this same problem would occur with a reparameterization procedure that simply sets  $z_{i} = 0$  without applying  $\mathcal{T}_{\lambda}$  in the cases where  $\mathcal{T}_{\lambda}$  would result in a zero-valued latent component.
+
+Posterior collapse measures the extent to which the inference network is indistinguishable from the prior (Lucas et al., 2019):
+
+$$
+\mathbb {P} _ {\mathbf {x} \sim p (\mathbf {x})} \left[ D _ {K L} \left(q _ {\phi} \left(z _ {i}, \mathbf {x}\right) \| p \left(z _ {i}\right)\right) \leq \epsilon \right] \geq 1 - \delta . \tag {15}
+$$
+
+To measure another tangible problem of all latent features values going to zero, we also introduce the metric of feature collapse to measure whether inferred latent features  $\widehat{\mathbf{z}}$  are zero for most images  $\mathbf{x}$ :
+
+$$
+\mathbb {P} _ {\mathbf {x} \sim p (\mathbf {x})} \left[ | \widehat {z} _ {i} | \leq \epsilon \right] \geq 1 - \delta . \tag {16}
+$$
+
+We measure these two quantities in Table 2 using subgradient and straight-through estimators. It can be observed that using a single sample, the subgradient leads to numerical instability in all three runs. Increasing the number of samples avoids this instability, at the cost of complete posterior and feature collapse in all three runs. Meanwhile, the straight-through estimator avoids this issue.
+
+# D. Linear Generator Training Details
+
+For each method, we train using 80,000 image patches and validate using 16,000 image patches of dimension  $16 \times 16$  with a dictionary of 256 entries. We train for 300 epochs using a batch size of 100. Our initial learning rate for the dictionary is  $5\mathrm{E} - 01$  and we apply an exponential decay by a factor of 0.99 each epoch. Our inference network is trained with an initial learning rate of  $1\mathrm{E} - 02$ , using an SGD+Nesterov optimizer with a CycleScheduler. For FISTA, we set  $\kappa = 1\mathrm{E} - 03$ . For the variational methods,
+
+we set  $\kappa = 1\mathrm{E} - 04$ , and the scale parameter of our prior  $p(\mathbf{z})$  equal to 0.1 when applicable. Unless specified, we set the KL weight as  $\beta = 1\mathrm{E} - 02$  for each method. We run each method for three trials, using the random seeds  $\{0,1337,747\}$ .
+
+For FISTA, we multiply  $\lambda$  by a warmup parameter  $\omega$  that starts at  $\omega^{(0)} = 0.1$  and is incremented by  $1\mathrm{E} - 04$  each training iteration, capping at a maximum value of  $\omega^{(t)} = 1.0$ . We find that this prevents all dictionary entries from going to zero when training with a Frobenius norm regularizer.
+
+For the Spike-and-Slab, we use the same warmup strategy proposed in (Tonolini et al., 2020). We set  $\omega^{(0)} = 0.0$ , incrementing by  $2\mathrm{E} - 04$  each iteration after iteration 1500. This parameter is also capped out at  $\omega^{(t)} = 1.0$ . Additionally, we apply a warmup to the temperature parameter  $\tau$  for the Gumbel-Softmax used in the Spike-and-Slab. This starts at  $\tau^{(0)} = 1.0$  and is decreased by a multiplicative factor of 0.9995 until it reaches a minimum value of  $\tau^{(t)} = 5\mathrm{E} - 01$ . We set the prior for the spike probability as 0.1.
+
+For the Thresholded Gaussian and Laplacian methods, we use the analytic results to set  $\lambda_0$  respectively as 0.52 and 0.25. We find that these values lead to  $10\%$  of the latent features non-zero for each input data on average. When using a Gamma prior on the threshold parameter, we reduce the corresponding KL weight by a multiplicative factor of 1E-01 (resulting in  $\beta_{1} = 1\mathrm{E} - 02$  and  $\beta_{2} = 1\mathrm{E} - 03$ ). We set  $\alpha_0 = 3$  for each method that uses a Gamma prior.
+
+Each inference network uses an MLP backbone with the same architecture, with a separate projection head for each distribution parameter. The architecture for the backbone is outlined in Table 3. For each distribution parameter, an additional  $256 \times 256$  linear layer is added on top of the output of this network.
+
+# E. DNN Generator Training Details
+
+For CelebA, we use 150,000 training samples and 19,000 validation samples. All samples are center cropped to  $140 \times 140$  pixels and then resized to  $64 \times 64$ . We also apply a random horizontal flip to training samples only. We train for 300 epochs using a batch size of 512 across two Nvidia RTX 3080s. We use an initial learning rate of 3E-04 using the Adam optimizer with  $\beta = (0.5, 0.999)$ , weight decay equal to 1E-05, and a sample budget of  $J = 10$ . Additionally, we use the automatic mixed precision (AMP) and DistributedDataParallel implementations included in PyTorch 1.10 (Paszke et al., 2019). We set  $\beta = 1\mathrm{E} - 03$  for the KL divergence term in all methods. In the case where we have a Gamma distribution on the threshold parameter, we multiply the KL term by a factor of 1E-01 (leading to  $\beta_{1} = 1\mathrm{E} - 03$  and  $\beta_{2} = 1\mathrm{E} - 04$ ). The network architecture for the encoder and the decoder are included in Table 4. All
+
+Table 2. Posterior collapse and feature collapse for different sparsity estimators. Thresholding methods use a straight-through (ST) estimator or a subgradient (SG) estimator. NaN denotes a run that failed due to gradients equal to NaN.  
+
+<table><tr><td rowspan="2">Method/Prior Distribution</td><td colspan="2">% Posterior Collapse (ε = 1E-02, δ = 5E-02)</td><td colspan="2">% Feature Collapse (ε = 1E-02, δ = 5E-02)</td></tr><tr><td>J=1</td><td>J=20</td><td>J=1</td><td>J=20</td></tr><tr><td>Thresholded Laplacian (SG)</td><td>NaN</td><td>100.00%</td><td>NaN</td><td>100.00%</td></tr><tr><td>Thresholded Laplacian (ST)</td><td>4.17%</td><td>4.56%</td><td>0.00%</td><td>0.00%</td></tr></table>
+
+Table 3. Inference Network Backbone for Linear Generator  
+
+<table><tr><td>Input ∈ R16×16</td></tr><tr><td>Linear: 256 × 512</td></tr><tr><td>ReLU</td></tr><tr><td>Linear: 512 × 1024</td></tr><tr><td>ReLU</td></tr><tr><td>Linear: 1024 × 512</td></tr><tr><td>ReLU</td></tr><tr><td>Linear: 512 × 256</td></tr><tr><td>ReLU</td></tr><tr><td>Output ∈ R256</td></tr></table>
+
+other methodology is kept consistent with the linear generator. We run each method for three trials, using the random seeds  $\{0,1337,747\}$ .
+
+For FMNIST, we train with all 60,000 training samples and evaluate on all 10,000 validation samples. Hyperparameters are kept the same as CelebA experiments, except for a KL divergence weight  $\beta = 1\mathrm{E} - 02$  and sampling budget  $J = 20$ . A smaller network architecture is used for FMNIST, depicted in Table 5.
+
+# F. Additional Results for Linear Generator
+
+F.1. Full Dictionary for Different Priors  
+F.2. Full Dictionary for Different Sampling Methods  
+G. Additional Results for DNN Generator  
+G.1. Recovered Dictionary at Different Dimensionality
+
+Table 4. Network Architecture for CelebA Experiments  
+
+<table><tr><td>Encoder Network</td><td>Decoder Network</td></tr><tr><td>Input ∈ R64×64×3</td><td>Input ∈ Rd</td></tr><tr><td>conv: chan: 64, Kern: 4, stride: 2, pad: 1</td><td>Linear: d × 1024 Units</td></tr><tr><td>BatchNorm: feat: 64</td><td>ReLU</td></tr><tr><td>ReLU</td><td>convTranpose: chan: 128, Kern: 4, stride: 2, pad: 1</td></tr><tr><td>conv: chan: 64, Kern: 4, stride: 2, pad: 1</td><td>BatchNorm: feat: 128</td></tr><tr><td>BatchNorm: feat: 64</td><td>ReLU</td></tr><tr><td>ReLU</td><td>convTranpose: chann: 128, Kern: 4, stride: 2, pad: 1</td></tr><tr><td>conv: chan: 128, Kern: 4, stride: 2, pad: 1</td><td>BatchNorm: feat: 128</td></tr><tr><td>BatchNorm: feat: 128</td><td>ReLU</td></tr><tr><td>ReLU</td><td>convTranpose: chan: 64, kernel: 4, stride: 2, pad: 1</td></tr><tr><td>conv: chan: 256, Kern: 4, stride: 2, pad: 1</td><td>BatchNorm: feat: 64</td></tr><tr><td>BatchNorm: feat: 256</td><td>ReLU</td></tr><tr><td>ReLU</td><td>convTranpose: chan: 64, kernel: 4, stride: 2, pad: 1</td></tr><tr><td>conv: chan: 256, Kern: 4, stride: 2, pad: 0</td><td>BatchNorm: feat: 64</td></tr><tr><td>BatchNorm: feat: 256</td><td>ReLU</td></tr><tr><td>ReLU</td><td>convTranpose: chan: 3, kernel: 4, stride: 2, pad: 1</td></tr><tr><td rowspan="2">Output ∈ R256</td><td>Sigmoid</td></tr><tr><td>Output ∈ R64×64×3</td></tr></table>
+
+Table 5. Network Architecture for FMNIST Experiments  
+
+<table><tr><td>Encoder Network</td><td>Decoder Network</td></tr><tr><td>Input ∈ R28×28×1</td><td>Input ∈ Rd</td></tr><tr><td>conv: chan: 64, Kern: 4, stride: 2, pad: 1</td><td>Linear: d × 128 Units</td></tr><tr><td>BatchNorm: feat: 64</td><td>ReLU</td></tr><tr><td>ReLU</td><td>convTranpose: chan: 128, Kern: 4, stride: 2, pad: 0</td></tr><tr><td>conv: chan: 64, Kern: 4, stride: 2, pad: 1</td><td>BatchNorm: feat: 128</td></tr><tr><td>BatchNorm: feat: 64</td><td>ReLU</td></tr><tr><td>ReLU</td><td>convTranpose: chann: 64, Kern: 4, stride: 1, pad: 0</td></tr><tr><td>conv: chan: 128, Kern: 4, stride: 1, pad: 0</td><td>BatchNorm: feat: 64</td></tr><tr><td>BatchNorm: feat: 128</td><td>ReLU</td></tr><tr><td>ReLU</td><td>convTranpose: chan: 64, kernel: 4, stride: 2, pad: 1</td></tr><tr><td>conv: chan: 64, Kern: 4, stride: 2, pad: 1</td><td>BatchNorm: feat: 64</td></tr><tr><td>BatchNorm: feat: 128</td><td>ReLU</td></tr><tr><td>ReLU</td><td>convTranpose: chan: 1, kernel: 4, stride: 2, pad: 1</td></tr><tr><td rowspan="2">Output ∈ R128</td><td>Sigmoid</td></tr><tr><td>Output ∈ R28×28×1</td></tr></table>
+
+Table 6. A copy of Table 1 from the main text with  $J = {20}$  samples with standard deviations included  
+
+<table><tr><td>Method/Prior Distribution</td><td>Validation Loss</td><td>Multi-Information</td><td>IWAE Loss</td></tr><tr><td>FISTA (baseline)</td><td>1.01E+02 ± 7.00E-01</td><td>8.75E+01 ± 5.43E-01</td><td>-</td></tr><tr><td>Gaussian</td><td>1.35E+03 ± 2.08E+00</td><td>7.36E+02 ± 3.06E-01</td><td>2.16E-01 ± 2.64E-04</td></tr><tr><td>Laplacian</td><td>5.79E+02 ± 1.51E+00</td><td>5.34E+02 ± 5.29E-01</td><td>2.22E-01 ± 7.97E-04</td></tr><tr><td>Spike-and-slab</td><td>2.39E+02 ± 4.27E+00</td><td>1.96E+02 ± 2.57E+00</td><td>2.12E-01 ± 2.64E-02</td></tr><tr><td>Thresholded Gaussian</td><td>2.30E+02 ± 1.87E+00</td><td>1.76E+02 ± 2.08E-01</td><td>1.33E+00 ± 3.80E-03</td></tr><tr><td>Thresholded Gaussian+Gamma</td><td>2.70E+02 ± 1.87E+00</td><td>2.29E+02 ± 3.79E-01</td><td>1.13E+00 ± 2.47E-03</td></tr><tr><td>Thresholded Laplacian</td><td>1.94E+02 ± 1.66E+00</td><td>1.91E+02 ± 1.53E-01</td><td>1.13E+00 ± 1.29E-02</td></tr><tr><td>Thresholded Laplacian+Gamma</td><td>2.11E+02 ± 1.47E+00</td><td>2.33E+02 ± 5.77E-01</td><td>1.12E+00 ± 8.09E-03</td></tr></table>
+
+![](images/fa945d1f0bda04724442178dbd74b30df0efa4bc74f754d7575967409c054d7a.jpg)  
+(a) Inference Network SNR
+
+![](images/d81b4cb6c345177e4bda8b3dce3b83f4e1710cf0d01784df7979b8636ccdbccf.jpg)  
+(b) Generator SNR
+
+![](images/4b3c0d68dd9b2e813a95c2825e79c8b807a3dc37bb0505c058fde2fa935d0105.jpg)  
+Figure 9. Comparison of SNR for (a) inference network and (b) linear generator for all methods using  $J = 1$  and  $J = 20$  with max ELBO sampling.  
+(a)  
+Figure 10. Full dictionary sorted by magnitude in descending order for FISTA with  $\lambda = 20$  and a Frobenius norm penalty of  $1\mathrm{E} - 03$ .
+
+![](images/55e52d447f2d389166193a65ac370fff4cf774b123754f19bdde4ff62ecaad47.jpg)  
+(a)  $J = 1$
+
+![](images/6fd72e8a8e651598288bb582a5256231290c8d9dde9cf8a66efc0dd87d2b860e.jpg)  
+(b)  $J = 20$
+
+![](images/3ddf86f675f1904640eeb1d9d010abb54cecbac7e461eb2ea0cf5732318b0c05.jpg)  
+Figure 11. Full dictionary sorted by magnitude in descending order for Gaussian prior with  $J = 1$  and  $J = 20$  under max ELBO sampling.  
+(a)  $J = 1$  
+Figure 12. Full dictionary sorted by magnitude in descending order for Laplacian prior with  $J = 1$  and  $J = 20$  under max ELBO sampling.
+
+![](images/8f3346ac40e00415b3597a972945d4015f9a52f89d595b70cca56388e045ef20.jpg)  
+(b)  $J = 20$
+
+![](images/aec8b2c96b11ba116e07bc2f33e3350c9c8be2e4ff7de8a868864edf28b2ab96.jpg)  
+(a)  $J = 1$
+
+![](images/4ac9fb82de171863790fff915c8bb01d3e7efa8bc42d6f2fbfc2610ec17c9526.jpg)  
+(b)  $J = 20$
+
+![](images/a0119be45c629991c9e2c9c99040651595db4ebe0522456b9eb03dd1653d1327.jpg)  
+Figure 13. Full dictionary sorted by magnitude in descending order for Spike-and-Slab prior with  $J = 1$  and  $J = 20$  under max ELBO sampling.  
+(a)  $J = 1$  
+Figure 14. Full dictionary sorted by magnitude in descending order for Thresholded Gaussian prior with  $J = 1$  and  $J = 20$  under max ELBO sampling.
+
+![](images/58f1f1a9b0b6609421fe4bf5d75ac02e98f779a97a8d2cd28a168a4376d943ed.jpg)  
+(b)  $J = 20$
+
+![](images/16efc86a946225d3c098cd06482fb00cbae9fedb94efa5e2cbbd62d87e7c45eb.jpg)  
+(a)  $J = 1$
+
+![](images/ec457dc45322b8518e70688f5efc77b97bce9b997634ee6cb7f7197e7d13d498.jpg)  
+(b)  $J = 20$
+
+![](images/68cab076288d03e2a8888d5ab83a905d4526a2b2d101009abf5173527659e7fc.jpg)  
+Figure 15. Full dictionary sorted by magnitude in descending order for Thresholded Gaussian + Gamma threshold prior prior with  $J = 1$  and  $J = 20$  under max ELBO sampling.  
+(a)  $J = 1$  
+Figure 16. Full dictionary sorted by magnitude in descending order for Thresholded Laplacian prior with  $J = 1$  and  $J = 20$  under max ELBO sampling.
+
+![](images/2289c4fa27c023947c322791fecebbf77bb76494568cfe8bec0924d6ac7d5049.jpg)  
+(b)  $J = 20$
+
+![](images/562b00f5838d308412149ef5a6070c7e0f4f75d93279b4a86852ed3d6c3942f4.jpg)  
+(a)  $J = 1$
+
+![](images/9f6e85be3415c3403410179498e8793959c4595e68cbb9f87f820600c3305b0a.jpg)  
+(b)  $J = 20$  
+Figure 17. Full dictionary sorted by magnitude in descending order for Thresholded Laplacian + Gamma threshold prior prior with  $J = 1$  and  $J = 20$  under max ELBO sampling.
+
+![](images/76c48696a6f3c24e0e9abcf49a39ace08f1db049ac85c31ccca237449e9f4095.jpg)  
+(a) Avg
+
+![](images/884acc3b88be1d23fb4905b5a5c795331ade8c259e239ceb7703d68d7977250d.jpg)  
+(b) IWAE
+
+![](images/07013ddaf78bd59a8ef03266d3e9ff7ff00e7232b55d4b21e4efe10c154473d1.jpg)  
+(c) Max  
+Figure 18. Full dictionary sorted by magnitude in descending order for different sampling methods using  $J = 100$  samples using the Thresholded Laplacian prior. Average sampling and IWAE sampling learn repetitive features in their dictionary since they do not encourage feature reuse during training. On the other hand, max ELBO sampling learns a larger dictionary of diverse features.
+
+![](images/9037b47876876d45fffc6e499a68d6aae2ded05add5fb21fa552226d9823cedf.jpg)  
+(a) 64 Dimensions
+
+![](images/fb49d29638610120b9c4d5600725dfe00931e167c818694e010446fe41b1db7c.jpg)  
+(b) 4096 Dimensions  
+Figure 19. Image reconstruction on CelebA using DNN generator for (a) 64 dimensions and (b) 4096 dimensions. From top to bottom, the rows denote the ground truth images, a Gaussian prior, a Spike-and-Slab prior, a Thresholded Gaussian prior, and a Thresholded Gaussian+Gamma prior.
+
+![](images/3a018a83a732fd52539767df4e2443696dea1bf97ae4b1058548fee4242759a8.jpg)  
+(a) Gaussian  
+Figure 20. 64 dimensional latent space. Sampled entries from an estimated dictionary using an inference network trained with a DNN generator on CelebA. Entries are shown in descending order of Frobenius norm magnitude, with each row taking linearly spaced entries from the full dictionary. The sparse prior is set so to encourage  $10\%$  of latent features to be non-zero.
+
+![](images/0f117af46ed1427f0c09dd90ac6d86726362cdbb27dc91fe3bb5455bb18eb69a.jpg)  
+(b) Thresholded Gaussian+Gamma
+
+![](images/ac321cd30906662b1893915663a0076e4db1ee3aff7d1965bb960e137f854555.jpg)  
+(a) Gaussian
+
+![](images/cf3b7aa205926d513c22bb0ae29d7b8e8d6ad0f3c316e55a2a877ef7b7511ad0.jpg)  
+(b) Thresholded Gaussian+Gamma
+
+![](images/4956562e5237acbb95ed36d397f7062084fbd9f4356562d0cf1bfc9b8d8c9f50.jpg)  
+Figure 21. 512 dimensional latent space. Sampled entries from an estimated dictionary using an inference network trained with a DNN generator on CelebA. Entries are shown in descending order of Frobenius norm magnitude, with each row taking linearly spaced entries from the full dictionary. The sparse prior is set so to encourage  $10\%$  of latent features to be non-zero.  
+(a) Gaussian  
+Figure 22. 2048 dimensional latent space. Sampled entries from an estimated dictionary using an inference network trained with a DNN generator on CelebA. Entries are shown in descending order of Frobenius norm magnitude, with each row taking linearly spaced entries from the full dictionary. The sparse prior is set so to encourage  $10\%$  of latent features to be non-zero.
+
+![](images/0bdfdc34f8357a7340cc6a6871ab8f50dd5b0a843fb69ba3019e045aaa1e36e0.jpg)  
+(b) Thresholded Gaussian+Gamma
+
+![](images/984a5b1e3eda498d3660f2d8b094c26310197a70c91255a3d97f1e16e3fece88.jpg)  
+(a) Gaussian  
+Figure 23. 4096 dimensional latent space. Sampled entries from an estimated dictionary using an inference network trained with a DNN generator on CelebA. Entries are shown in descending order of Frobenius norm magnitude, with each row taking linearly spaced entries from the full dictionary. The sparse prior is set so to encourage  $10\%$  of latent features to be non-zero.
+
+![](images/67881e4d2f43dcf8824920be42572201074fa1ec976c265c879803177fc47b4d.jpg)  
+(b) Thresholded Gaussian+Gamma

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+# Versatile Dueling Bandits: Best-of-both World Analyses for Online Learning from Relative Preferences
+
+Aadirupa Saha<sup>1</sup> Pierre Gaillard<sup>2</sup>
+
+# Abstract
+
+We study the problem of  $K$ -armed dueling bandit for both stochastic and adversarial environments, where the goal of the learner is to aggregate information through relative preferences of pair of decision points queried in an online sequential manner. We first propose a novel reduction from any (general) dueling bandits to multi-armed bandits which allows us to improve many existing results in dueling bandits. In particular, we give the first best-of both world result for the dueling bandits regret minimization problem—a unified framework that is guaranteed to perform optimally for both stochastic and adversarial preferences simultaneously. Moreover, our algorithm is also the first to achieve an optimal  $O\left(\sum_{i=1}^{K} \frac{\log T}{\Delta_i}\right)$  regret bound against the Condorcet-winner benchmark, which scales optimally both in terms of the arm-size  $K$  and the instance-specific suboptimality gaps  $\{\Delta_i\}_{i=1}^K$ . This resolves the long standing problem of designing an instance-wise gap-dependent order optimal regret algorithm for dueling bandits (with matching lower bounds up to small constant factors). We further justify the robustness of our proposed algorithm by proving its optimal regret rate under adversariably corrupted preferences—this outperforms the existing state-of-the-art corrupted dueling results by a large margin. In summary, we believe our reduction idea will find a broader scope in solving a diverse class of dueling bandits setting, which are otherwise studied separately from multi-armed bandits with often more complex solutions and worse guarantees. The efficacy of our proposed algorithms are empirically corroborated against state-of-the-art dueling bandit methods.
+
+# 1. Introduction
+
+Studies have shown that it is often easier, faster and less expensive to collect feedback on a relative scale rather than asking ratings on an absolute scale. E.g., to understand the liking for a given pair of items, say (A,B), it is easier for the users to answer preference-based queries like: "Do you prefer Item A over B?", rather than their absolute counterparts: "How much do you score items A and B in a scale of [0-10]?" From a system designer's point of view, exploiting such user preference information could greatly aid in improving systems performances, especially when data can be collected on a relative scale and online fashion; such as recommendation systems, crowd-sourcing platforms, training bots, multi-player games, search-engine optimization, online retail, just to name a few. In many real world problems, especially where human preferences are elicited in an online fashion, e.g., design of surveys and expert reviews, assortment selection, search engine optimization, recommender systems, ranking in multiplayer games, etc, or even more general reinforcement learning problems where rewards shaping is often a challenging problem (e.g. if multi-objective rewards etc.), and instead, a preference feedback is much easier to elicit.
+
+Due to the widespread applicability and ease of data collection with relative feedback, learning from preferences has gained much popularity in the machine learning community and widely studied as the problem of *Dueling-Bandits* (DB) over last decade (Yue et al., 2012; Ailon et al., 2014; Zoghi et al., 2014a;b; 2015a), which is an online learning framework that generalizes the standard multiarmed bandit (MAB) (Auer et al., 2002a) setting for identifying a set of 'good' arms from a fixed decision-space (set of items) by querying preference feedback of actively chosen item-pairs.
+
+tive of the learner is to minimize the regret with respect to a (or set of) 'best' arm(s) in hindsight.
+
+Related Works Over the years, the problem of Dueling Bandits has been studied with various objectives and generalizations. This includes analyzing the learning rate under various preference structures, such as total ordering, transitivity, stochastic triangle inequality (Falahatgar et al., 2017; Yue & Joachims, 2011), utility based preference structure (Ailon et al., 2014; Szorenyi et al., 2015; Saha & Gopalan, 2018a; Chen et al., 2018), or under any general pairwise preference matrices (Dudík et al., 2015; Jamieson et al., 2015; Komiyama et al., 2016). Consequently, depending on the underlying preference structure, the learner's performance has been evaluated w.r.t. different benchmarks including among other promising generalizations. The problem of stochasticueling bandits has been studied for both PAC (Falahatgar et al., 2018; Szorenyi et al., 2015; Sui et al., 2018) as well as regret minimization setting (Zoghi et al., 2014a; Yue & Joachims, 2009; Chen & Frazier, 2017; Zoghi et al., 2015a) under several notions of benchmarks including best arm identification (Saha & Gopalan, 2019b; Busa-Fekete et al., 2014; Falahatgar et al., 2017), top-set detection (Busa-Fekete et al., 2013; Saha & Gopalan, 2019a), ranking (Ren et al., 2018; Saha & Gopalan, 2018b), amongst many. Some recent works have also looked into the problem for adversarial sequence of preference matrices (Saha & Gupta, 2022; Gajane et al., 2015; Saha et al., 2021), robustness to corruptions (Agarwal et al., 2021), or extendingueling bandits to potentially infinite arm sets (Saha, 2021; Kumagai, 2017) and contextual scenarios (Dudík et al., 2015; Saha & Krishnamurthy, 2021). Another interesting line of work alongueling bandits is to study the implications for its subsetwise generalization (Ren et al., 2018; Sui et al., 2018; Brost et al., 2016; Chen et al., 2018), also studied as battling bandits (Saha & Gopalan, 2018a; 2019a; Bengs et al., 2021).
+
+Despite widespread surge of interest along this line of research and multiple attempts there are some fundamental long standing open questions in dueling bandits which are (surprisingly!) yet unresolved.
+
+Unresolved Question #1 One of the longest and most widely studied objective in stochasticueling bandit is to measure regret w.r.t. the Condorcet winner (CW) arm: Given a preference matrix  $\mathbf{P}$ , an arm  $k^{(cw)} \in [K]$  is termed as the CW of  $\mathbf{P}$  if  $P(k^{(cw)}, k) > 0.5 \forall k \in [K] \setminus \{k^{(cw)}\}$  (Zoghi et al., 2014a). Assuming  $P$  contains a CW, there have been several attempts to design an optimal regretueling bandit algorithm against the CW arm  $k^{(cw)}$  (see Eq. (2) for details) (Zoghi et al., 2014a; Wu & Liu, 2016; Komiyama et al., 2015; Bengs et al., 2021). Without loss of generality, assuming  $k^{(cw)} = 1$  and by denoting  $\Delta_i = P(1, i) - 0.5$  to be the suboptimality gap of item  $i$  w.r.t. the CW, it is well known that theueling bandit regret lower bound (w.r.t. the
+
+CW arm) is  $\Omega\left(\sum_{i=2}^{K} \frac{\log T}{\Delta_i}\right)$  (Yue et al., 2012; Komiyama et al., 2015). However, despite several attempts, it is still unknown how to design an order optimal dueling bandit algorithm for the CW regret. Existing upper-bounds suffer all suboptimal  $\Delta_{\min}^{-2}$  and/or  $K^2$  dependencies.
+
+Notably, under more restricted structures, e.g. total ordering (Yue & Joachims, 2011), or utility based preferences (Szorenyi et al., 2015; Saha & Gopalan, 2020), or even special preference structures where the suboptimality gaps of all items  $(\Delta_{i}, i \in [K] \setminus \{1\})$  are equal, the problem is easier to solve and tight regret guarantees are available with matching upper and lower bound analysis. However, for the case of any general preference matrix with CW, none of the existing attempts were able to close this regret analysis gap successfully (Zoghi et al., 2014a; Wu & Liu, 2016; Chen & Frazier, 2017; Komiyama et al., 2015; Saha & Gaillard, 2021). Subsequently, the natural questions to ask are:
+
+(1). Is the lower bound tight? (2). How to close the gap between the upper and lower bound for CW regret?
+
+Unresolved Question #2 Till date, all the proposed algorithms of dueling bandits need to know underlying preference structure ahead of time in order to yield optimal regret bounds. In fact, different algorithms have been proposed based on the nature/structures of the underlying preference matrices, e.g. (Yue & Joachims, 2011) for preferences with total orderings in terms of (relaxed) stochastic transitivity and stochastic triangle inequality; (Ailon et al., 2014; Gajane et al., 2015) for linear-utility based preferences, (Szorenyi et al., 2015) for BTL models, (Zoghi et al., 2014a; Komiyama et al., 2015; Wu & Liu, 2016) for stochastic preferences in presence of CW, (Saha et al., 2021; Gajane et al., 2015) for adversarial sequence of preferences, etc. However, it might not always be realistic to assume complete knowledge of the properties underlying preference matrices. Thus the daunting question to ask in this regard is
+
+Is it possible to design an order optimal 'best-of-both-world' algorithm for dueling bandits?
+
+That is, an algorithm that can adapt itself to the underlying structures of the preference environments and give optimal regret for both stochastic and adversarial settings? There has been a series of work on this line for the MAB framework (e.g., Bubeck & Slivkins, 2012; Auer & Chiang, 2016; Zimmert & Seldin, 2021), but unfortunately there has not been any existing 'best-of-both-world' attempt for generalueling bandits.
+
+Unresolved Question #3 In any real world situation, the true feedback are often corrupted with some form of system noise. Undoubtedly, the binary  $0/1$  bitueling preferences are extremely prone to such noises when the learner might get to observe a flipped bit (adversarily corrupted) instead
+
+of the true dueling feedback.
+
+Can we design an efficient dueling bandit algorithm which is robust to adversarial corruptions and provably optimal?
+
+Our Contributions In this paper, we answer all of the above three questions affirmatively. The list of our specific contributions can be summarized as follows:
+
+1. A novel insight on the reduction from DB to MAB.  
+Ailon et al. (2014) proposed a reduction from utility-based dueling bandits (which is a special class of DB) to standard MAB. We show that the latter can easily be extended to any general dueling bandit problems (including CW), with significant consequences on the state of the art results in dueling bandits theory, as discussed in the points below. We believe that the reduction will find wider application in solving a diverse class of dueling bandit settings, using analyses of their MAB counterparts, which are otherwise studied separately from MAB with often more complex solutions and worse guarantees (Sec. 4).  
+2. First Best-of-Both World regret for DB. Applying the above reduction to a Best-of-Both world algorithm from MAB, we provide an algorithm that simultaneously guarantees a pseudo-regret bound  $O(\sqrt{KT})$  in the adversarial setting and  $O(K\log (T) / \Delta_{\mathrm{min}})$  in the stochastic one. This leads to the first best-of-both world results forueling bandits (Sec. 5).  
+3. Optimal stochastic gap-dependent Regret. Our algorithm also provides the first optimal Condorcet regret, which suffers neither from a suboptimal dependence on  $\Delta_{\mathrm{min}}^{-2}$  nor from a quadratic dependence on the number of arms (Sec. 6.1).  
+4. Robustness to adversarial corruptions. Our algorithm is robust to adversarial corruptions and significantly improves existing results in DB (Agarwal et al., 2021) (Sec. 6.2).  
+5. Another easy algorithm for stochastic DB. We also propose a new elimination based algorithm with  $O\left(\sum_{i=2}^{K}\frac{K \log T}{\Delta_i}\right)$  Condorcet regret which is regret optimal up to a factor of  $K$  (Sec. 3).  
+6. Experimental evaluations. Finally we corroborate our theoretical results with extensive empirical evaluations (Sec. 7).
+
+# 2. Problem Formulation
+
+Notations. Decision space (or item/arm set)  $[K] := \{1, 2, \ldots, K\}$ . For any matrix  $\mathbf{M} \in \mathbb{R}^{K \times K}$ , we define  $m_{ij} := M(i,j)$ ,  $\forall i,j \in [K]$ .  $\mathbf{1}(\cdot)$  denotes the indicator random variable which takes value 1 if the predicate is true and 0 otherwise and  $\lesssim$  a rough inequality which holds up to universal constants. For any two items  $x,y \in [K]$ , we use the symbol  $x \succ y$  to denote  $x$  is preferred over  $y$ . By convention, we set  $\frac{0}{0} := 0.5$ . For any closed and convex set
+
+$S$ , we denote by  $\mathcal{I}_S(x) \coloneqq 0$ , if  $x \in S$ , and  $\mathcal{I}_S(x) \coloneqq \infty$  otherwise; for any  $x \in \mathbb{R}_+$  (notation used in Alg. 3).
+
+Consider any function  $f:\mathbb{R}^K\mapsto \mathbb{R}$ .  $\nabla f(\mathbf{x})$  denotes the gradient of function  $f$  at point  $\mathbf{x}\in \mathbb{R}^{K}$ . The convex conjugate (a.k.a. Fenchel conjugate) of  $f$  is defined by:
+
+$$
+f ^ {*} (\mathbf {y}) := \sup  _ {\mathbf {x} \in \mathbb {R} ^ {k}} \left(\langle \mathbf {x}, \mathbf {y} \rangle - f (\mathbf {x})\right).
+$$
+
+Thus we have for any any closed and convex set  $S$ ,  $(f + \mathcal{I}_S)^*(\mathbf{y}) = \max_{\mathbf{x} \in S} \left( \langle \mathbf{x}, \mathbf{y} \rangle - f(\mathbf{x}) \right)$ . Further, as claimed in (Zimmert & Seldin, 2019), differentiable and convex  $f$  with invertible gradient  $(\nabla f)^{-1}$ , it holds that
+
+$$
+\nabla (f + \mathcal {I} _ {S}) ^ {*} (\mathbf {y}) = \arg \max  _ {\mathbf {x} \in S} \left(\langle \mathbf {x}, \mathbf {y} \rangle - f (\mathbf {x})\right).
+$$
+
+Setup. We assume a decision space of  $K$  arms denoted by  $\mathcal{A} \coloneqq [K]$ . At each round  $t$ , the task of the learner is to select a pair of actions  $(k_{+1,t}, k_{-1,t}) \in [K] \times [K]$ , upon which a preference feedback  $o_t \sim \mathrm{Ber}(P_t(k_{+1,t}, k_{-1,t}))$  is revealed to the learner according to the underlying preference matrix  $\mathbf{P}_t \in [0,1]^{K \times K}$  (chosen adversarially), such that the probability of  $k_{+1,t}$  being preferred over  $k_{-1,t}$  is given by  $Pr(o_t = 1) \coloneqq Pr(k_{+1,t} \succ k_{-1,t}) = P_t(k_{+1,t}, k_{-1,t})$ , and hence  $Pr(o_t = 0) \coloneqq Pr(k_{-1,t} \succ k_{+1,t}) = 1 - P_t(k_{+1,t}, k_{-1,t})$ .
+
+Objective. Assuming the learner selects the duel  $(k_{+1,t}, k_{-1,t})$  at round  $t$ , one can measure its performance w.r.t. a single fixed arm  $k^* \in [K]^1$  in hindsight by calculating the regret w.r.t.  $k^* \in [K]$
+
+$$
+R _ {T} \left(k ^ {*}\right) := \sum_ {t = 1} ^ {T} \frac {1}{2} \left(P _ {t} \left(k ^ {*}, k _ {+ 1, t}\right) + P _ {t} \left(k ^ {*}, k _ {- 1, t}\right) - 1\right). \tag {1}
+$$
+
+For the stochastic setting where  $P_{t}$ s are fixed across all time steps  $t \in [T]$ , we denote  $P_{t} = P \forall t \in [T]$ . Further assuming there exists a Condorcet winner for  $P$ , i.e. fixed arm  $k^{(\mathrm{cw})} \in [K]$  such that  $P(k^{(\mathrm{cw})}, k) > 0.5 \forall k \in [K] \setminus \{k^{(\mathrm{cw})}\}$ , the above notion of regret boils down to the regret with respect to the Condorcet winner for the choice of  $k^* = k^{(\mathrm{cw})}$ , as widely studied in many dueling bandit literature (Zoghi et al., 2014a; Wu & Liu, 2016; Komiyama et al., 2015; Bengs et al., 2021), defined as:
+
+$$
+R _ {T} ^ {\left(\mathrm {c w}\right)} := \sum_ {t = 1} ^ {T} \frac {1}{2} \left(\Delta \left(k ^ {\left(\mathrm {c w}\right)}, k _ {+ 1, t}\right) + \Delta \left(k ^ {\left(\mathrm {c w}\right)}, k _ {- 1, t}\right)\right), \tag {2}
+$$
+
+where  $\Delta (i,j)\coloneqq P(i,j) - 1 / 2$  being the suboptimality gap of item  $i$  and  $j$  in terms of their relative preferences. Without
+
+loss of generality we will assume  $k^{(cw)} = 1$  throughout the rest of this paper (whenever relevant).
+
+# 3. Warm-Up: Near-Optimal Algorithm
+
+In this section, we propose an efficient UCB based algorithm for stochastic dueling bandit, which has a nearly optimal gap-dependent Condorcet regret of  $R_T^{(cw)} = O\left(\sum_{i=2}^K i \Delta(k^{(cw)}, i)^{-1} \log T\right)$  (assuming  $k^{(cw)} = 1$  is the Condorcet Winner). Note that existing dueling bandit algorithms, that satisfy a non-asymptotic Condorcet regret bound, suffer an additional constant of order  $\Delta_{\min}^{-2}$  (Bengs et al., 2021), which implies a worst-case regret of order  $O(T^{2/3})$  when  $\Delta_{\min} \to 0$ . The simple elimination algorithm below solves this drawback and depends only on  $\Delta_{\min}^{-1}$  but at the cost of a suboptimal quadratic dependence in the number of arms  $K$ . Despite our efforts, we could not avoid this suboptimal factor by following the classical stochastic dual bandit analysis. In the following sections, we will show how to easily reach the optimal dependence in both  $\Delta_{\min}$  and  $K$  connecting the dueling bandit (DB) problem to standard MAB.
+
+Main Ideas: Algorithm RR-DB The high-level idea of Algorithm 1 is to sequentially compare arms in a round-robin fashion and eliminate arms when they are significantly suboptimal compared to any other arm. Typically, after  $t$  rounds, a suboptimal arm  $k$  has been compared at least  $t / K$  times with the Condorcet winner. Denoting by  $\Delta_k$  its suboptimality gap, the arm is eliminated after at most  $t_k$  rounds, where  $(t_k / K)^{-1/2} \approx \Delta_k$ . At that time the arm has been played  $t_k / K$  times, yielding a regret of order  $(t_k / K) \times \Delta_k \approx (K / \Delta_k^2) \times \Delta_k = K / \Delta_k$ . Summing over the arms yields a final regret of order  $O(K^2 / \Delta_{\min})$ .
+
+Algorithm 1 RR-DB (Near Optimal DB)  
+1: input: Arm set:  $[K]$ , Confidence parameter  $\delta \in (0,1)$   
+2: init: Active arms:  $\mathcal{A}_1 \coloneqq [K]$ ,  $n_{ij}(t) \gets 0$ ,  $\forall i,j \in [K]$   
+3: for  $t = 1,2,\ldots,T$  do  
+4: Play  $(k_{+1,t}, k_{-1,t}) \in \mathrm{argmin}_{i,j \in \mathcal{A}_t} \{n_{ij}(t - 1)\}$   
+5: Observe  $o_t(k_{+1,t}, k_{-1,t}) = 1 - o_t(k_{+1,t}, k_{-1,t})$   
+6: for  $i,j \in \mathcal{A}_t$  do  
+7: Define  $\mathbb{1}_t(i,j) \coloneqq \mathbb{1}\{\{i,j\} = \{k_{-1,t}, k_{+1,t}\}\}$  and  
+[ n_{ij}(t) \coloneqq \sum_{s=1}^{t} \mathbb{1}_t(i,j) ]  
+[ \widehat{p}_{ij}(t) \coloneqq \frac{1}{n_{ij}(t)} \sum_{s=1}^{t} o_t(i,j) \mathbb{1}_t(i,j) ]  
+[ u_{ij}(t) \coloneqq \widehat{p}_{ij}(t) + \sqrt{\frac{\log(Kt / \delta)}{n_{ij}(t)}} ]  
+where we assume  $x/0 = +\infty$ .
+
+8: end for  
+9:  $\mathcal{A}_{t + 1} := \mathcal{A}_t \setminus \{i \in \mathcal{A}_t : \exists j \in \mathcal{A}_t \text{ s.t. } u_{ij}(t) < \frac{1}{2}\}$  
+10: end for
+
+Without loss of generality assume the Condorcet winner  $k^{(cw)} = 1$ , and denote  $\Delta_i = \Delta(1, i)$ ,  $\forall i \in [K] \setminus \{1\}$ .
+
+Theorem 1 (Regret analysis of RR-DB (Alg. 1) for Stochastic Preferences against Condorcet Winner). Assume any instance of stochastic dueling bandit problem with Condorcet Winner  $k^{(cw)} = 1$ . Let  $\delta \in (0,1/2)$ , for any  $T \geq 1$ , the regret of Algorithm 1 is upper-bounded with probability at least  $1 - \delta$  as
+
+$$
+R _ {T} ^ {(c w)} \leq \frac {K ^ {2}}{2} + 4 \sum_ {i = 2} ^ {K} \frac {(i - 1) \log (K T / \delta)}{\Delta_ {i}}.
+$$
+
+Further, when  $T \geq K^2$ , in the worst case (over the problem instance, i.e.  $\Delta_2, \ldots, \Delta_K$ ), the regret of Algorithm 1 can be upper bounded as:
+
+$$
+R _ {T} ^ {(c w)} \leq 2 K \sqrt {T \log (K T / \delta)}.
+$$
+
+Remark 1. In particular, our regret analysis shows that, except a logarithmic factor, the regret bound of  $RR - DB$  (Alg. 1) is off only by a multiplicative factor of  $K$ , as follows from the known  $\Omega \left( \sum_{k=1}^{K} \frac{\log T}{\Delta_k} \right)$  Condorcet winner regret lower bound (Yue et al., 2012; Komiyama et al., 2015).
+
+The proof is postponed to Appendix A.
+
+# 4. Key Idea: Reducing DB to MAB
+
+We now present a simple reduction from a multi-armed bandit algorithm to aueling bandit one. The latter was already proposed by Saha & Gupta (2022) to show worst-case guarantees and by Ailon et al. (2014) for utility basedueling bandits only. We recall it here since it is central to our analysis and we believe that it is of important interest for theueling bandit community that usually uses significantly different algorithms and analysis than the ones from standard multi-armed bandits.
+
+The main idea is to apply a multi-armed bandit algorithm independently to two players  $i \in \{-1, +1\}$  respectively with losses defined for any  $k \in [K]$  and  $t \in [T]$ , by
+
+$$
+\ell_ {i, t} (k) := o _ {t} \left(k _ {- i, t}, k\right),
+$$
+
+where  $o_{t}(k,k^{\prime}) = 1 - o_{t}(k^{\prime},k)$  for  $1 < k^{\prime} \leq K$  follows a Bernoulli with parameter  $P_{t}(k,k^{\prime})$  (and we assume  $o_{t}(k,k) = 1 / 2$ ).
+
+We show below that any MAB regret upper-bound satisfied by  $\mathcal{M}_i$  can be transformed into a DB regret upper-bound.
+
+Theorem 2 (DB Regret analysis of Alg. 2 in terms of the regret of the underlying MAB instances  $(\mathcal{M}_{-1},\mathcal{M}_{+1})$ ). Define for  $i\in \{-1,1\}$  and  $k\in [K]$  by
+
+$$
+R _ {i, T} (k) := \sum_ {t = 1} ^ {T} \ell_ {i, t} \left(k _ {i, t}\right) - \ell_ {i, t} (k)
+$$
+
+Algorithm 2 Reduction from DB to MAB  
+1: input: Arm set:  $[K]$ , two instances  $\mathcal{M}_i$  of an algorithm for MAB,  $i\in \{-1, + 1\}$    
+2: for  $t = 1,2,\ldots ,T$  do   
+3: for  $i\in \{+1, - 1\}$  do   
+4: receive  $p_{i,t}$  as suggested by  $\mathcal{M}_i$    
+5: sample  $k_{i,t}$  from the distribution  $p_{i,t}$    
+6: end for   
+7: Play duel  $(k_{+1,t},k_{-1,t})$    
+8: Observe preference feedback  $o_t(k_{+1,t},k_{-1,t})$  and set  $o_t(k_{-1,t},k_{+1,t}) = 1 - o_t(k_{+1,t},k_{-1,t})$    
+9: Feed  $\mathcal{M}_i$  with loss  $\ell_{i,t}(k)\coloneqq o_t(k_{-i,t},k_i,t)$  for  $i\in$ $\{-1, + 1\}$    
+10: end for
+
+the regret achieved by algorithm  $\mathcal{M}_i$ . Then, the expected regret (1) of Algorithm 2 for dueling bandits can be decomposed as
+
+$$
+\mathbf {E} \left[ R _ {T} (k) \right] = \frac {1}{2} \mathbf {E} \left[ R _ {- 1, T} (k) + R _ {+ 1, T} (k) \right].
+$$
+
+Proof. The proof follows from
+
+$$
+\begin{array}{l} \mathbf {E} \left[ \ell_ {- 1, t} (k) + \ell_ {+ 1, t} (k) \right] = \mathbf {E} \left[ o _ {t} \left(k _ {+ 1, t}, k\right) + o _ {t} \left(k _ {- 1, t}, k\right) \right] \\ = \mathbf {E} \left[ P _ {t} \left(k _ {+ 1, t}, k\right) + P _ {t} \left(k _ {- 1, t}, k\right) \right] \\ = 2 - \mathbf {E} \left[ P _ {t} \left(k, k _ {+ 1, t}\right) + P _ {t} \left(k, k _ {- 1, t}\right) \right] \\ \end{array}
+$$
+
+and
+
+$$
+\begin{array}{l} \mathbf {E} \left[ \ell_ {- 1, t} (k _ {+ 1, t}) + \ell_ {+ 1, t} (k _ {- 1, t}) \right] \\ = \mathbf {E} \left[ o _ {t} \left(k _ {+ 1, t}, k _ {- 1, t}\right) + o _ {t} \left(k _ {+ 1, t}, k _ {- 1, t}\right) \right] = 1. \\ \end{array}
+$$
+
+We conclude by summing over  $t = 1, \dots, T$  both equations and by substituting them into the definition of the regret  $R_{T}(k)$  in (1).
+
+Note that such a reduction can also be used to bound  $R_{T}(k)$  directly rather than its expectation. Saha & Gupta (2022) indeed use this reduction to show a  $O(\sqrt{KT})$  high-probability regret upper-bound for adversarial dueling bandit. They also obtain non-stationary regret bounds.
+
+The main message of this paper is that this reduction can be used to transpose many results from standard multi-armed bandit to general dueling bandits. For instance, applying a subroutine  $\mathcal{M}_i$  which is robust to delays (Thune et al., 2019; Zimmert & Seldin, 2020, e.g.,), one directly obtains a dueling bandit with the same robustness guarantees.
+
+As we said, this reduction is not new. However, to date, it has only been considered in two specific contexts: the adversarial setting with worst-case regret bounds of order  $O(\sqrt{KT})$  and the utility-based setting. Since the losses
+
+$\ell_{i,t} = o_t(k_{-i,t},k)$  are not i.i.d. but depend on an adaptive adversary which chooses  $k_{-i,t}$ , one cannot use stochastic bandit algorithms. And theueling bandit community usually needs to resort to more sophisticated algorithms and arguments to obtain logarithmic regret bounds for Condorcet stochasticueling bandits. The only stochasticueling bandit for which such a reduction was considered (Ailon et al., 2014; Zimmert & Seldin, 2021, e.g.,) was the utility based-dueling bandits, which is overly restrictive in practice. That is, when the preference matrix of the form  $P_{t}(k,k^{\prime}):(1 + u_{t}(k) + u_{t}(k^{\prime})) / 2$  for some sequence of utility vectors  $(u_{t})_{t\geq 1}$ . Utility basedueling bandit are known to be easily reduced to two independent multi-armed bandit problems (Ailon et al., 2014).
+
+Our main contribution is to show that this reduction can in fact be easily extended to the much weaker Condorcet winner hypothesis. To do this, as we show in the next sections, we simply apply the reduction with a best-of-both-worlds multi-armed bandit algorithm. As we will see, this recovers and improves the best existing upper bounds on the Condorcet regret forueling bandits.
+
+# 5. Best-of-Both Dueling: Optimal Algorithm for Stochastic and Adversarial DB
+
+This section contains our main result, which extends the best-of-both-worlds result from Zimmert & Seldin (2021) to dueling bandits. Moreover, it allows us to improve the best existing upper bound on the regret for stochastic and corrupted dueling bandits. The main idea is to apply the "reduction idea" of Algorithm 2 with the multi-armed bandit algorithm (Online Mirror Descent with Tsallis regularizer) of Zimmert & Seldin (2021).
+
+Online Mirror Descent (or famously abbreviated as OMD) is essentially a follow the regularized leader (FTRL) type algorithm, for any general choice of regularizer (Srebro et al., 2011; Lattimore & Szepesvári, 2020). The famous exponential weight algorithm in online learning (Cesa-Bianchi & Lugosi, 2006; Arora et al., 2012) (or EXP3 algorithm in MAB (Auer et al., 2002b)) is a special instance of OMD algorithm for (-ve) entropy based regularizer.
+
+It is important to note that, our reduction analysis of Algorithm 3 (see proof of Theorem 3) indicates that we could have used any best-of-both algorithm of MAB as the underlying blackbox algorithm, instead of OMD with Tsallis regularizer (Zimmert & Seldin, 2021). For example, the EXP3 based approach of Bubeck & Slivkins (2012) could have been a potential MAB blackbox algorithm as well, but they do not give optimal regret guarantees due to the use of 'entropic-regularizers' which is not ideal for obtaining the optimal instance (gap)-dependent stochastic regret for MAB. Hence Zimmert & Seldin (2021) motivated the use
+
+of OMD based Tsallis-Inf algorithm for the best-of both MAB algorithm and our analysis of Theorem 3 shows that the same works for the best-of both DB analysis as well.
+
+Of course, as for the classical adversarial multi-armed bandits, the losses  $\ell_{i,t}(k) = o_t(k_{-i,t},k)$  cannot be observed for all  $k\in [K]$ . Therefore, they are estimated in the algorithm with the importance weight estimators
+
+$$
+\widehat {\ell} _ {i, t} (k) = \left\{ \begin{array}{l l} \ell_ {i, t} (k) / p _ {i, t} (k) & \text {i f} k = k _ {i, t} \\ 0 & \text {o t h e r w i s e} \end{array} . \right. \tag {3}
+$$
+
+The resulted algorithm is described in Algorithm 3.
+
+Algorithm 3 Versatile-DB (Best-of-Both DB)
+
+1: input: Arm set:  $[K]$ , Regularizers:  $(\Psi_{t})_{t = 1,2,\ldots}$  
+2: init:  $\widehat{L}_{i,0} \gets \mathbf{0}_K$  for  $i \in \{+1, -1\}$  
+3: for  $t = 1,2,\ldots$  do  
+4: choose  $p_{i,t} = \nabla (\Psi_t + \mathcal{I}_\Delta)^*(-\widehat{L}_{i,t - 1})$  (here  $\Delta$  denotes the  $K$  -simplex. Refer to Sec. 2 for other notations)  
+5: For  $i\in \{+1, - 1\}$  , sample  $k_{i,t}$  from the distribution  $(p_{i,t}(1),\dots ,p_{i,t}(K))$  
+6: Observe preference feedback  $o_{t}(k_{+1,t}, k_{-1,t})$  
+7: Compute  $\widehat{\ell}_{i,t}(k)$  for  $i\in \{+1, - 1\}$  and  $k\in [K]$  using (3)  
+8: update  $\widehat{L}_{i,t} = \widehat{L}_{i,t - 1} + \widehat{\ell}_{i,t}$  
+9: end for
+
+Theorem 3 (Regret analysis of Versatile-DB (Alg. 3) for Adversarial and Stochastic Preferences). For any sequence of preference matrices  $\mathbf{P}_t$ , the pseudo-regret of Algorithm 3 with  $\Psi_t(w) = \sqrt{t} \sum_{k=1}^{K} \left( \frac{\sqrt{w_k} - w_k}{2} \right) / 8$  satisfies for any  $T \geq 1$
+
+$$
+\overline {{R}} _ {T} := \max  _ {k \in [ K ]} \mathbf {E} \left[ R _ {T} (k) \right] \leq 4 \sqrt {K T} + 1.
+$$
+
+Furthermore, if there exists a gap vector  $\Delta \in [0,1]^K$  with a unique zero coordinate  $k^* \in [K]$  and  $C \geq 0$  such that
+
+$$
+\bar {R} _ {T} \geq \frac {1}{2} \mathbf {E} \left[ \sum_ {t = 1} ^ {T} \sum_ {k \neq k ^ {*}} \left(p _ {+ 1, t} (k) + p _ {- 1, t} (k)\right) \Delta_ {k} \right] - C, \tag {4}
+$$
+
+the pseudo regret also satisfies
+
+$$
+\begin{array}{l} \bar {R} _ {T} \leq \sum_ {k \neq k ^ {*}} \frac {4 \log T + 1 2}{\Delta_ {k}} + 4 \log T + \frac {1}{\Delta_ {\min }} \\ + \frac {3}{2} \sqrt {K} + 8 + C. \\ \end{array}
+$$
+
+where  $\Delta_{\mathrm{min}} = \min_{k\neq k^{*}}\Delta_{k}$
+
+The proof is postponed to Appendix B. Note that the theorem largely follows from (and is itself highly similar to)
+
+the best-of-both worlds regret-bound of Zimmert & Seldin (2021, Theorem 1) for MAB. The proof is just a clever combination of their MAB analysis with our black box reduction (Theorem 2). But we believe that the simplicity of our approach is its strength that can benefit the community of dueling bandits. As we will see, several state-of-the-art results of dueling bandits can be simultaneously improved as a direct consequence of this theorem, including the order optimal regret bounds for stochastic DB against condorcet winner or DB with adversarial corruption, as discussed in Sec. 6.1 and 6.2 respectively.
+
+Note that for simplicity, we restricted ourselves to importance weighted estimators (3). By using more sophisticated variance reduced estimators, as in Zimmert & Seldin (2021), the multiplicative constants can be reduced. Furthermore, similar to Zimmert & Seldin (2021), the result holds only for the pseudo-regret and not for the true regret. Auer & Chiang (2016) have indeed proven that no optimal adversarial and stochastic high probability regret bounds can be obtained simultaneously for standard stochastic bandits. The learner must pay suboptimal logarithmic factors. The result can be extended toueling bandits.
+
+It is worth to emphasize that this is the first best-of-both worlds regret bound for general dueling bandits (the stochastic bound follows from the choice  $C = 0$ , see Sec 6.1). Zimmert & Seldin (2021, Cor. 10) obtain a similar result for the very same algorithm but for utility based dueling bandits only.
+
+Remark 2. Note that a single sub-routine of OMD to optimize the weights is actually enough to get the same regret guarantee. To do so, one samples both  $k_{-1,t}$  and  $k_{+1,t}$  independently from the same distribution  $\mathbf{p}_t = \nabla (\psi_t + \mathcal{I}_{\Delta})^* (-\widehat{L}_{t - 1})$ . Here,  $\widehat{L}_t = \sum_{s = 1}^t\widehat{\ell}_s\in \mathbb{R}_+^K$  and the importance weight estimator are defined for all  $k\in [K]$  by
+
+$$
+\widehat {\ell} _ {t} (k) = \frac {1}{2} \left(\widehat {\ell} _ {- 1, t} (k) + \widehat {\ell} _ {+ 1, t} (k)\right).
+$$
+
+Noting that  $\mathbf{E}[\widehat{\ell}_t(k)] = \mathbf{E}\big[P_t(k_{-1,t},k) + P_t(k_{+1,t},k)\big] / 2$  and  $\mathbf{E}\big[\sum_{k = 1}^{K}p_t(k)\widehat{\ell}_t(k)\big] = \mathbf{E}\big[P_t(k_{-1,t},k_{+1,t}) + P_t(k_{+1,t},k_{-1,t})\big] / 2 = 1 / 2,$  the proof follows similarly to the other one. Though the regret upper-bound is exactly the same, we believe that this version might lead to better performance because the two players share information.
+
+# 6. Improvements Over Existing Dueling Bandit Results
+
+Also our approach and analysis is rather simple, it allows to outperform the best existing regret-upper bounds for stochastic dueling bandits with or without corruption. We believe that the dueling bandit community will benefit from this reduction and that it may be applied to a wider scope such
+
+as to deal with non-stationarity, delays, or non-standard feedbacks (graphs) for which many results already exist in standard multi-armed bandits.
+
+# 6.1. Stochastic dueling bandits with Condorcet winner
+
+In stochastic dueling bandits, the preference matrices  $\mathbf{P}_t$  are fixed over time  $\mathbf{P}_t = \mathbf{P}$  for all  $t\geq 1$ . Under the Condorcet winner assumption there exists  $k^{(\mathrm{cw})}\in [K]$  such that  $P(k^{(\mathrm{cw})},k) > \frac{1}{2}$  for all  $k\neq k^{(\mathrm{cw})}$ . Then, the suboptimality gaps of all actions  $k\in [K]$  are defined as  $\Delta_k\coloneqq P(k^{(\mathrm{cw})},k) - \frac{1}{2}$ . Remarking that in this case the self-bounding assumption (4) is satisfied with  $C = 0$ , since
+
+$$
+\begin{array}{l} \overline {{R}} _ {T} = \frac {1}{2} \sum_ {t = 1} ^ {T} \mathbf {E} \left[ P _ {t} \left(k ^ {\left(\mathrm {c w}\right)}, k _ {+ 1, t}\right) + P _ {t} \left(k ^ {\left(\mathrm {c w}\right)}, k _ {- 1, t}\right) - 1 \right] \\ = \frac {1}{2} \sum_ {t = 1} ^ {T} \mathbf {E} \left[ \Delta_ {k _ {+ 1, t}} + \Delta_ {k _ {- 1, t}} \right] \\ = \frac {1}{2} \sum_ {t = 1} ^ {T} \mathbf {E} \left[ \sum_ {k \neq k ^ {(c w)}} (p _ {- 1, t} + p _ {+ 1, t}) \Delta_ {k} \right], \\ \end{array}
+$$
+
+we get the following corollary from Theorem 3.
+
+Corollary 1 (Regret analysis of Versatile-DB (Alg. 3) for Stochastic Preferences against Condorcet Winner). For stochastic dueling bandits with Condorcet winner, the pseudo-regret of Algorithm 3 with well-chosen parameters satisfies
+
+$$
+\bar {R} _ {T} \leq \sum_ {k \neq k ^ {(c w)}} \frac {4 \log T + 1 2}{\Delta_ {k}} + 4 \log T + \frac {1}{\Delta_ {\min }} + \frac {3}{2} \sqrt {K} + 8.
+$$
+
+Note that the above bound is the first pseudo-regret upperbound for stochastic dueling bandit that does not suffer from a  $\Delta_{\mathrm{min}}^{-2}$  dependence under the Condorcet winner assumption, without a quadratic dependence on the number of arms,  $K$  which is a concern when it comes to dealing with largescale problems. For instance, RUCB (Zoghi et al., 2014a) satisfies a regret bound of order  $O(K\log (T)\Delta_{\mathrm{min}}^{-2} + K^2)$ , MergeRUCB (Zoghi et al., 2015b) has linear dependence on  $K$  but suffers  $O(K\log (T)\Delta_{\mathrm{min}}^{-2})$ . Finally, RMED from (Komiyama et al., 2015) is asymptotically optimal when  $T\to \infty$  but also suffers from large constant terms ( $K^2$  and  $\Delta_{\mathrm{min}}^{-2}$ ) and is only valid for  $K\to \infty$ . We refer the reader to (Bengs et al., 2021) for existing results on stochastic dueling bandits.
+
+# 6.2. Corrupted regime
+
+Here, we consider the stochastic dueling bandit problem in the presence of adversarial corruptions. The robustness to adversarial corruption has known recent progress in the MAB setting (Gupta et al., 2019; Lykouris et al., 2018;
+
+Zimmert & Seldin, 2021) and has recently been extended to the DB framework (Agarwal et al., 2021). The preference matrices are fixed  $\mathbf{P}_t = \mathbf{P}$  for all  $t \geq 1$  and we assume the existence of a Condorcet winner  $k^{(cw)}$ . Furthermore, an adversary may corrupt the outcomes of some duels by replacing the results of the duels  $o_t(k,k')$  with corrupted ones  $\tilde{o}_t(k,k')$ . At the end of each round, the player only observes  $\tilde{o}_t(k_{+1,t},k_{-1,t})$ . The objective of the player is to minimize the pseudo-regret  $\bar{R}_T$  under a bounded total amount of corruption
+
+$$
+C := \sum_ {t = 1} ^ {T} \sum_ {k \neq k ^ {\left(\mathrm {c w}\right)}} \left| o _ {t} \left(k ^ {\left(\mathrm {c w}\right)}, k\right) - \tilde {o} _ {t} \left(k ^ {\left(\mathrm {c w}\right)}, k\right) \right|.
+$$
+
+We show here that similarly to what happens for standard Multi-armed bandits in Zimmert & Seldin (2021), this corrupted setting is a special case of the self-bounding assumption (4). Indeed, defining  $\tilde{\mathbf{P}}_t$  the corrupted preference matrices by  $\tilde{P}_t(k,k') = \mathbf{E}\big[\bar{o}_t(k,k')\big]$  and the corrupted pseudo-regret
+
+$$
+\widetilde {R} _ {T} (k) = \frac {1}{2} \mathbf {E} \left[ \sum_ {t = 1} ^ {T} \tilde {P} _ {t} (k, k _ {+ 1, t}) + P _ {t} (k, k _ {- 1, t}) - 1 \right],
+$$
+
+we have
+
+$$
+\begin{array}{l} \widetilde {R} _ {T} (k ^ {(c w)}) \\ = \frac {1}{2} \mathbf {E} \left[ \sum_ {t = 1} ^ {T} \tilde {o} _ {t} \left(k ^ {\left(\mathrm {c w}\right)}, k _ {+ 1, t}\right) + \tilde {o} _ {t} \left(k ^ {\left(\mathrm {c w}\right)}, k _ {- 1, t}\right) - 1 \right] \\ \geq \frac {1}{2} \mathbf {E} \left[ \sum_ {t = 1} ^ {T} o _ {t} \left(k ^ {\left(\mathrm {c w}\right)}, k _ {+ 1, t}\right) + o _ {t} \left(k ^ {\left(\mathrm {c w}\right)}, k _ {- 1, t}\right) - 1 \right] - C \\ = \bar {R} _ {T} - C \\ = \frac {1}{2} \sum_ {t = 1} ^ {T} \mathbf {E} \left[ \sum_ {k \neq k ^ {(c w)}} (p _ {- 1, t} + p _ {+ 1, t}) \Delta_ {k} \right] - C. \\ \end{array}
+$$
+
+Therefore, the corrupted regime satisfies the self-bounding assumption (4). Applying Theorem 3 on the corrupted regime and using that we also have  $\overline{R}_T(k^{(cw)}) \leq \widetilde{R}_T(k^{(cw)}) + C$ , we get the following corollary.
+
+Corollary 2 ([Regret analysis of Versatile-DB (Alg. 3) for Stochastic Preferences against Condorcet Winner with a maximum adversarial Corruption  $C$ ). For stochastic dueling bandits with Condorcet winner and adversarial corruptions (bounded by  $C$ ), the pseudo-regret  $\overline{R}_T$  Algorithm 3 is upper-bounded as
+
+$$
+\begin{array}{l} \bar {R} _ {T} \leq \sum_ {k \neq k ^ {(c w)}} \frac {4 \log T + 1 2}{\Delta_ {k}} + 4 \log T \\ + \frac {1}{\Delta_ {\operatorname* {m i n}}} + \frac {3}{2} \sqrt {K} + 8 + 2 C. \\ \end{array}
+$$
+
+Although Corollary 2 easily follows from Theorem 3 which itself easily follows from Theorem 1 of Zimmert & Seldin (2021), the latter result significantly improves upon the recent results onueling bandit with corruptions obtained by Agarwal et al. (2021). Indeed, the latter provide for the same setting and a significantly more sophisticated procedure a high-probability regret bound of order
+
+$$
+O \left(\frac {K ^ {2} C}{\Delta_ {\operatorname* {m i n}}} + \sum_ {k \neq k ^ {(c w)}} \frac {K ^ {2}}{\Delta_ {k} ^ {2}} \log \left(\frac {K}{\Delta_ {k}}\right) + \sum_ {k \neq k ^ {(c w)}} \frac {\log T}{\Delta_ {k}}\right).
+$$
+
+As often in theueling bandit literature, it suffers from both a quadratic dependence on the number of actions and  $\Delta_{\mathrm{min}}^{-1}$ . Furthermore, our regret bound is sublinear in  $T$  as soon as the corruption level is  $o(T)$  while Agarwal et al. (2021) can only afford  $o(\Delta_{\mathrm{min}}T / K^2)$ .
+
+Moreover, Zimmert & Seldin (2021) also provide an upper-bound for stochastic bandits with adversarial corruption. The latter is of order  $O\left(\sum_{k \neq k^{(cw)}} \frac{\log T}{\Delta_k} + \sqrt{C \sum_{k \neq k^{(cw)}} \frac{\log T}{\Delta_k}}\right)$ , which seems to outperform our bound when  $C$  is large. The difference is since they upper-bound the corrupted regret  $\widetilde{R}_T(k^{(cw)})$  and not  $\bar{R}_T$ .
+
+# 7. Experiments
+
+Algorithms. We compared the performances of the following algorithms: 1. VDB: Our proposed Versatile-DB (Alg. 3) algorithm. 2. RUCB: The algorithm proposed in Zoghi et al. (2014a) for K-armed stochastic dueling bandits for CW regret. (We set the algorithm parameter  $\alpha = 0.6$ ). 3. RMED: Another algorithm for CW regret as proposed in Komiyama et al. (2015). (We set the algorithm parameter  $f(K) = 0.3K^{1.01}$  as suggested in their experimental evaluation). 4. DTS: The double thompson sampling algorithm of Wu & Liu (2016). (Here again we set the similar algorithm parameter  $\alpha = 0.5$  as used in the experiments of Wu & Liu (2016)). 5. REX3: As introduced in Gajane et al. (2015). Note that their suggested optimal tuning parameters, i.e. the uniform exploration rate  $\gamma$  as well as the learning rate  $\eta$  requires the knowledge of problem dependent parameters  $\tau$  (see Thm. 1 of Gajane et al. (2015)) which are unknown to the learner. We used  $T$  in place of  $\tau$  henceforth.
+
+Performance Measures. We report the average cumulative regret (Eqn. (1)) of the algorithms averaged over 20 runs.
+
+# 7.1. Stochastic Preferences
+
+We compared their regret performance across the following stochastic environments:
+
+Constructing Preference Matrices (P). We use four different utility parameter  $\pmb{\theta} = (\theta_{1},\dots,\theta_{K})$  based preference
+
+models where the underlying preference model is defined as  $P(i,j) \coloneqq \frac{\theta_i}{\theta_i + \theta_j} \forall i, j \in [K]$ . The model is famously studied as BTL model or more generally Plackett-Luce model (Saha & Gopalan, 2019b; Chen et al., 2018; Negahban et al., 2017). Note this ensures  $P$  to have total-ordering (Yue & Joachims, 2011; Falahatgar et al., 2017).
+
+In particular we consider the following choices of  $\theta$ : 1. Trivial 2. Easy 3. Medium, and 4. Hard with their respective  $\theta$  parameters are given by: 1. Trivial:  $\theta(1) = 1$ ,  $\theta(2: K) = 0.5$ . 2. Easy:  $\theta(1: \lfloor K/2 \rfloor) = 1$ ,  $\theta(\lfloor K/2 \rfloor + 1: K) = 0.5$ . 3. Medium:  $\theta(1: \lfloor K/3 \rfloor) = 1$ ,  $\theta(\lfloor K/3 \rfloor + 1: \lfloor 2K/3 \rfloor) = 0.7$ ,  $\theta(\lfloor 2K/3 \rfloor + 1: K) = 0.4$ . 4. Hard:  $\theta(i) = 1 - (i-1)/K$ ,  $\forall i \in [K]$ . Note for each instance the best-arm is arm-1 and the optimal ordering on the arms is  $(1 > 2 > \ldots, K)$ . For the purpose of our experiments we set  $K = 10$ . We also evaluated the algorithms on two general  $10 \times 10$  preference matrices  $Car$  and  $Hurdy$  as also used in Niranjan & Rajkumar (2017); Saha & Gopalan (2018a).
+
+![](images/1f38e1edb5f73b4951280624d6d97035ba94f83f333373174f8a1724877fb03f.jpg)
+
+![](images/47974b3c17c9b1966f318bdf01d98416223db9dfc50882a611f9b021383d5222.jpg)
+
+![](images/56ed49ae22c287ea8db48cece7648cee596e57d6274852feb4a0e0ebcc7d9fe2.jpg)
+
+![](images/b30994f05d6a6be55996fe01e09b86218c96339abe1284aac95e86b8f876b40d.jpg)
+
+![](images/7a60378ddba866b9068d94a8b72b6d143668314b9f79d76ce1a70d5b526e0219.jpg)  
+Figure 1. Averaged cumulative regret over time
+
+![](images/9a40265dfb77b7a6dd88343ee73a1c226d8e352ccafa40f8790263272fb6cfb0.jpg)
+
+Regret vs Time. Fig. 1 shows the relative performances of different algorithms with time. As follows from the plots, in general VDB (Versatile-DB) outperforms the rest in all instances, with DTS being closely competitive in some cases. In terms of the problem hardness, as their names suggest too, the Trivial and Easy instances are easiest to learn as the best-vs-worst item preferences are well separated in these cases and the diversity of the item preferences across different groups are least. Consequently the algorithms yield slightly more regret on instance-Medium due to higher preference diversity, and the hardest instance being Hard where the algorithms require maximum time to converge, though VDB reaching the convergence fastest still.
+
+# 7.2. Corrupted Preferences
+
+We also evaluated the performances of algorithms in presence of corruption (Sec. 6.2). In particular, Fig. 2 and 3 respectively shows the relative performances of the algorithms
+
+![](images/7b329af286adac60ead11a5b8455e09c94f6322daaaf4c3f703bf6c97ec21638.jpg)  
+Figure 2. Averaged cumulative regret (20% corrupted feedback)
+
+![](images/05e033af7ef7cedc18e4238d5ba87a5fd9c192e3bce56017e827c84cd3fe111b.jpg)
+
+![](images/5662c791781bfb2dc4d0fb869e8820aff28451212e77b9e5fb47c5701c412eda.jpg)  
+Figure 3. Averaged cumulative regret (40% corrupted feedback)
+
+![](images/32992866ca96034fab4b701ced24f8b1d5ddc93681b46f630268d14be1f0c48a.jpg)
+
+with  $20\%$  and  $40\%$  corrupted feedback (at each round, we flip the winner feedback with that certain probability) respectively on Medium and Hard Plackett-Luce instances. As expected, the performances of all the algorithms decay significantly with increasing degree of feedback-corruption, however as before, VDB consistently performed best over all the baselines and tend to converge the fastest among all.
+
+# 7.3. Adversarial Preferences
+
+In Fig. 4, we also studied the case of adversarial preferences setup (as discussed in Thm. 3 or Sec. 5). The following two plots show the superior performance of VDB on two sequences of adversarial preferences for  $K = 10$  (1). Switching Best-Arm [Switch(500)]: Here we tweaked  $P_{t}$ s such that the best-arm changes after every 500 rounds, and Random-Walk [RW(0.01)]: Here we generated the sequences of preferences  $P_{t}(i,j)$  for all pairs of arms  $(i,j)$  as random walks with increment parameter  $\pm 0.01$ . Any values that fall outside [0, 1] are truncated back to [0, 1]. Both these settings were considered in (Neu & Valko, 2014; Saha et al., 2020; 2021) for generating adversarial sequences (of rewards or preferences).
+
+![](images/d47f6cd33ee3e14f4a32f415657f65f36e30881c4ff1acfb7bc3758432c17af2.jpg)  
+Figure 4. Avg. Regret for Adversarial preferences  $(K = 10)$
+
+![](images/ee12397bbe92546133e74276dd53db4cb0bfaa9a699a1739ee08191a47976cf8.jpg)
+
+# 8. Discussions
+
+We studied the problem of Versatile Dueling Bandits, which gives the first 'best-of-both' world result for the problem of Dueling Bandits. The crux of our analyses relies on a novel idea of decomposing the Dueling bandit regret into multiarmed bandit (MAB) regret by interpreting theueling preference feedback as a certain realization of adversarial reward sequence. An important byproduct of our best-of-both Dueling analysis is, this gives the first order
+
+optimal gap-dependent regret bound for  $K$ -armed stochastic dueling bandits, closing the decade-long open problem of tightness of 'Condorcet dueling bandit regret'. Further we also analyze the robustness of our algorithm under corrupted preference feedback setting, which provably improves over the state-of-the-art corrupted dueling bandits algorithms.
+
+Future Works. Proving the first best-of-both world result forueling bandits using our novel reduction idea is just a first step towards exploring the possibility of understanding how far this idea can be extended to apply existing multiarmed bandits results toueling bandits frameworks, instead of putting individual and isolated efforts in developingueling bandit algorithms, taking inspirations from existing MAB generalizations. Some such extensions could be to analyze dynamicueling bandit regret under nonstationary preferences (Wei & Luo, 2021; Chen et al., 2019; Besbes et al., 2015), item non-availability (Neu & Valko, 2014; Kanade et al., 2009), delayed feedback (Vernade et al., 2018; Pike-Burke et al., 2018; Thune et al., 2019), budget constraints (Immorlica et al., 2019; Zhou & Tomlin, 2018; Ding et al., 2013), or even the more general reinforcement learning (RL) scenarios (Auer et al., 2009; Talebi & Maillard, 2018; Ng et al., 2006), for which there are already well established theory of works with MAB framework. Also under what settings ofueling Bandits, its corresponding MAB counterpart based reductions are bound to fail?
+
+Finally it is worth mentioning that, an ambitious (and broad) objective along these line of thoughts is to understand the connection between different learning scenarios toueling bandits, e.g. feedback graphs (Alon et al., 2015; 2017), partial monitoring problems (Lattimore & Szepesvári, 2019; Mannor et al., 2014; Lin et al., 2014), markov games (Xie et al., 2020; Bai et al., 2020; Bai & Jin, 2020), etc. The obvious motivation being to understand how far we can reengineer the existing results from related learning literature to solve the online preference bandits problems.
+
+# Acknowledgement
+
+We thank the anonymous reviewers for their insightful suggestions to improve the paper.
+
+# References
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+
+# Supplementary: Versatile Dueling Bandits: Best-of-both World Analyses for Online Learning from Relative Preferences
+
+# A. Regret analysis of Algorithm 1
+
+Theorem 1 (Regret analysis of RR-DB (Alg. 1) for Stochastic Preferences against Condorcet Winner). Assume any instance of stochastic dueling bandit problem with Condorcet Winner  $k^{(cw)} = 1$ . Let  $\delta \in (0,1/2)$ , for any  $T \geq 1$ , the regret of Algorithm 1 is upper-bounded with probability at least  $1 - \delta$  as
+
+$$
+R _ {T} ^ {(c w)} \leq \frac {K ^ {2}}{2} + 4 \sum_ {i = 2} ^ {K} \frac {(i - 1) \log (K T / \delta)}{\Delta_ {i}}.
+$$
+
+Further, when  $T \geq K^2$ , in the worst case (over the problem instance, i.e.  $\Delta_2, \ldots, \Delta_K$ ), the regret of Algorithm 1 can be upper bounded as:
+
+$$
+R _ {T} ^ {(c w)} \leq 2 K \sqrt {T \log (K T / \delta)}.
+$$
+
+Proof of Theorem 1. Let us denote by
+
+$$
+u _ {i j} (t) := \widehat {p} _ {i j} (t) + c _ {i j} (t)
+$$
+
+for any pair  $(i,j)$  and time  $t$ , where
+
+$$
+c _ {i j} (t) := \sqrt {\frac {\log (K t / \delta)}{n _ {i j} (t)}},
+$$
+
+and assume  $\Delta_2 \leq \Delta_3 \leq \dots \leq \Delta_K$  without loss of generality. We will also assume the confidence bounds of Lem. 4 hold good for all  $t \in [T]$  and all pairs  $(i,j)$ , which is shown to hold good with probability at least  $1 - \delta$ . In particular, this implies that the best arm cannot be eliminated, i.e.,  $1 \in \mathcal{A}_t$  for all  $t \geq 1$ .
+
+We start by noting that if the worst arm  $K$  (since  $\Delta_K = \max_{i=2}^K \Delta_i$ , arm-K gets maximally beaten by the CW) is played at time  $t$ , it means  $u_{K1}(t) \geq \frac{1}{2}$ . However we also have,
+
+$$
+\begin{array}{l} u _ {K 1} (t) = \widehat {p} _ {K 1} (t) + c _ {K 1} (t) \\ \leq p _ {K 1} + 2 c _ {K 1} (t) = 1 / 2 - \Delta_ {K} + 2 c _ {K 1} (t), \\ \end{array}
+$$
+
+where the inequality holds due to Lem. 4 and the last equality holds by noting  $p_{K1} = 1 - p_{1K} = 1 - (1/2 + \Delta_K) = 1/2 - \Delta_K$ . So  $u_{K1}(t) > 1/2$  can only hold good if  $c_{K1}(t) > \Delta_K / 2$  which implies,
+
+$$
+n _ {K 1} (t) \leq \frac {4 \log (K t / \delta)}{\Delta_ {K} ^ {2}}. \tag {5}
+$$
+
+But by the our algorithm design since all the pairs are drawn in a round robin fashion, at any round  $t \in [T]$ , for any two distinct pairs  $(i,j)$  and  $(i',j')$  that are in  $\mathcal{A}_t$  note that
+
+$$
+\left| n _ {i j} (t) - n _ {i ^ {\prime} j ^ {\prime}} (t) \right| \leq 1. \tag {6}
+$$
+
+Thus the total regret incurred by Alg. 1 at rounds where  $K \in \{k_{+1,t}, k_{-1,t}\}$ , can be upper bounded as:
+
+$$
+\begin{array}{l} \sum_ {t = 1} ^ {T} \sum_ {k <   K} \mathbf {1} (\{k _ {+ 1, t}, k _ {- 1, t} \} = \{k, K \}) \frac {\Delta_ {K} + \Delta_ {k}}{2} \\ \leq \sum_ {k = 1} ^ {K - 1} n _ {k K} (T) \Delta_ {K} \leq (K - 1) (1 + n _ {K 1} (T)) \Delta_ {K} \\ \end{array}
+$$
+
+$$
+\begin{array}{l} \leq (K - 1) \left(1 + \frac {4 \log (K t / \delta)}{\Delta_ {K} ^ {2}}\right) \Delta_ {K} \\ = (K - 1) \left(\Delta_ {k} + \frac {4 \log (K t / \delta)}{\Delta_ {K}}\right). \\ \end{array}
+$$
+
+Similarly, note for any  $i \in \{2, 3, \dots, K - 1\}$ , we can upper bound the regret of rounds where  $i$  was played in the duel as:
+
+$$
+\begin{array}{l} \sum_ {t = 1} ^ {T} \sum_ {k <   i} \mathbf {1} \left(\left\{k _ {+ 1, t}, k _ {- 1, t} \right\} = \left\{k, i \right\}\right) \frac {\Delta_ {i} + \Delta_ {k}}{2} \\ \leq \sum_ {k = 1} ^ {i - 1} n _ {k i} (T) \Delta_ {i} \leq (i - 1) (1 + n _ {1 i} (T)) \Delta_ {i} \\ \leq (i - 1) \left(\Delta_ {i} + \frac {4 \log (K t / \delta)}{\Delta_ {i}}\right). \\ \end{array}
+$$
+
+Thus we can bound the total regret of Algorithm 1 as:
+
+$$
+\begin{array}{l} \sum_ {t = 1} ^ {T} \sum_ {i = 2} ^ {K} \sum_ {k = 1} ^ {i - 1} \mathbf {1} (\{k _ {+ 1, t}, k _ {- 1, t} \} = \{k, i \}) \frac {\Delta_ {i} + \Delta_ {k}}{2} \\ \leq \sum_ {i = 2} ^ {K} (i - 1) \left(\Delta_ {i} + \frac {4 \log (K t / \delta)}{\Delta_ {i}}\right) \\ \stackrel {(\Delta_ {i} \leq 1 / 2)} {\leq} \frac {K ^ {2}}{4} + 4 \sum_ {i = 2} ^ {K} (i - 1) \frac {\log (K t / \delta)}{\Delta_ {i}} \\ \end{array}
+$$
+
+which concludes the first half of the proof. Further, to show the second part of the claim (analyzing worst-case gap-independent regret bound of Algorithm 1), note that Eqn. (5) equivalently implies for any  $i \in [K] \setminus \{1\}$  and for any  $t \in [T]$ :
+
+$$
+\Delta_ {i} \leq \sqrt {\frac {4 \log (K t / \delta)}{n _ {i 1} (t)}}.
+$$
+
+Hence we can alternatively upper bound the regret as:
+
+$$
+\begin{array}{l} R _ {T} = \sum_ {i = 2} ^ {K} \sum_ {k = 1} ^ {i - 1} n _ {i k} (T) \frac {\Delta_ {i} + \Delta_ {k}}{2} \leq \sum_ {i = 2} ^ {K} \sum_ {k = 1} ^ {i - 1} n _ {i k} (T) \Delta_ {i} \\ \leq \sum_ {i = 2} ^ {K} \sum_ {k = 1} ^ {i - 1} n _ {i k} (T) \sqrt {\frac {4 \log (K T / \delta)}{n _ {i 1} (T)}} \\ \stackrel {(a)} {\leq} 2 \sum_ {i = 2} ^ {K} \sum_ {k = 1} ^ {i - 1} \sqrt {n _ {i k} (T) \log (K T / \delta)} \\ \stackrel {(b)} {\leq} 2 \sqrt {K ^ {2} \sum_ {i = 2} ^ {K} \sum_ {k = 1} ^ {i - 1} n _ {i k} (T) \log (K T / \delta)} \\ \leq 2 K \sqrt {T \log (K T / \delta)}, \\ \end{array}
+$$
+
+where  $(a)$  follows from the observation of Eqn. (6) which implies  $n_{i1}(T) \geq n_{ik}(T)$  when  $T \geq K^2$  and  $(b)$  from Jensen's inequality and  $\sum_{i=2}^{K} (i - 1) \leq K^2 / 2$ .
+
+Lemma 4. For any  $\delta \in (0,1 / 2)$ . Then, with probability at least  $1 - \delta$ , for any pair  $i,j \in [K]$  and any time  $t \in [T]$
+
+$$
+\widehat {p} _ {i j} (t) - c _ {i j} (t) \leq p _ {i j} \leq \widehat {p} _ {i j} (t) + c _ {i j} (t), \quad \forall t \in [ T ],
+$$
+
+where  $c_{ij}(t)\coloneqq \sqrt{\frac{\log(Kt / \delta)}{n_{ij}(t)}}$
+
+Proof. Let us denote by  $u_{ij}(t) \coloneqq \widehat{p}_{ij}(t) + c_{ij}(t)$  and  $\ell_{ij}(t) \coloneqq \widehat{p}_{ij}(t) - c_{ij}(t)$ . Note the inequality holds trivially at round  $t$ , for any pair  $(i,j)$  for which  $n_{ij}(t) = 0$  since in these cases  $\ell_{ij}(t) \leq 0$  and  $u_{ij}(t) \geq 1$ .
+
+Now consider any pair  $(i,j)$  and round  $t\in [T]$  such that  $n_{ij}(t) > 0$ . Note in this case by Hoeffding's Inequality:
+
+$$
+P r \left(\left| p _ {i j} - \widehat {p} _ {i j} (t) \right| > \sqrt {\frac {\ln (K t / \delta)}{n _ {i j} (t)}}\right) \leq 2 e ^ {- 2 n _ {i j} (t) \frac {\ln (K t / \delta)}{n _ {i j} (t)}} = \frac {2 \delta^ {2}}{K ^ {2} t ^ {2}} \leq \frac {\delta}{K ^ {2} t ^ {2}}.
+$$
+
+Taking union bound over all  $\binom{K}{2}$  pairs and time  $t\in [T]$  we get:
+
+$$
+\begin{array}{l} P r \left(\exists i, j \in [ K ], t \in [ T ] \text {s . t .} | p _ {i j} - \widehat {p} _ {i j} (t) | > \sqrt {\frac {\ln (K t / \delta)}{n _ {i j} (t)}}\right) \\ \leq \sum_ {t = 1} ^ {T} \sum_ {i = 2} ^ {K} \sum_ {j = 1} ^ {i} \frac {\delta}{K ^ {2} t ^ {2}} \leq \sum_ {t = 1} ^ {\infty} \frac {\delta}{2 t ^ {2}} \leq \frac {\delta \pi^ {2}}{1 2} \leq \delta , \\ \end{array}
+$$
+
+where in the second last inequality we used  $\sum_{t=1}^{\infty} \frac{1}{t^2} < \frac{\pi^2}{6}$ . This concludes the claim.
+
+![](images/e9fc0d39ce80d0a1db5d02eeb05262b46bd79849223e1e70d5bff3df44889c89.jpg)
+
+# B. Regret Analysis of Alg. 3
+
+Theorem 3 (Regret analysis of Versatile-DB (Alg. 3) for Adversarial and Stochastic Preferences). For any sequence of preference matrices  $\mathbf{P}_t$ , the pseudo-regret of Algorithm 3 with  $\Psi_t(w) = \sqrt{t}\sum_{k=1}^{K}(\sqrt{w_k} - w_k/2)/8$  satisfies for any  $T \geq 1$
+
+$$
+\overline {{R}} _ {T} := \max _ {k \in [ K ]} \mathbf {E} \bigl [ R _ {T} (k) \bigr ] \leq 4 \sqrt {K T} + 1.
+$$
+
+Furthermore, if there exists a gap vector  $\Delta \in [0,1]^K$  with a unique zero coordinate  $k^* \in [K]$  and  $C \geq 0$  such that
+
+$$
+\bar {R} _ {T} \geq \frac {1}{2} \mathbf {E} \left[ \sum_ {t = 1} ^ {T} \sum_ {k \neq k ^ {*}} \left(p _ {+ 1, t} (k) + p _ {- 1, t} (k)\right) \Delta_ {k} \right] - C, \tag {4}
+$$
+
+the pseudo regret also satisfies
+
+$$
+\begin{array}{l} \bar {R} _ {T} \leq \sum_ {k \neq k ^ {*}} \frac {4 \log T + 1 2}{\Delta_ {k}} + 4 \log T + \frac {1}{\Delta_ {\min }} \\ + \frac {3}{2} \sqrt {K} + 8 + C. \\ \end{array}
+$$
+
+where  $\Delta_{\mathrm{min}} = \min_{k\neq k^{*}}\Delta_{k}$
+
+Proof of Theorem 3. The analysis follows from carefully combining our reduction (Theorem 2) with Theorem 1 of (Zimmert & Seldin, 2021) for MAB to both of the players. Indeed, for each player  $i \in \{-1, 1\}$ , Algorithm 3 chooses  $k_{i,t}$  by following the decisions of Tsallis-INF (Zimmert & Seldin, 2021, Alg. 1) with  $\alpha = 1/2$ , symmetric regularization, learning rate  $\eta_t = 4/\sqrt{t}$  and losses  $\ell_{i,t}$  estimated in (3) with standard importance sampling (IW).
+
+Adversarial regime A direct application of Theorem 1 of (Zimmert & Seldin, 2021), upper-bounds the pseudo-regret for each player  $i \in \{-1, 1\}$  as
+
+$$
+\max  _ {k \in [ K ]} \mathbf {E} \left[ R _ {i, T} (k) \right] \leq 4 \sqrt {K T} + 1.
+$$
+
+Combining the about bounds with the reduction from MAB to DB (Theorem 2) yields the adversarial pseudo-regret upper-bound
+
+$$
+\mathbf {E} \big [ R _ {T} (k) \big ] = \frac {1}{2} \mathbf {E} \big [ R _ {- 1, T} (k) + R _ {+ 1, T} (k) \big ] \leq 4 \sqrt {K T} + 1.
+$$
+
+Adversarial regime with a self-bounding constraint Our self-bounding constraint is slightly different from that of (Zimmert & Seldin, 2021), since it involves both players simultaneously. This is necessary so that our gap vector  $\Delta$  can recover the standard suboptimality gaps used in stochastic dueling bandits. Thus, we cannot directly combine their result with our black-box reduction in this case. However, the proof largely follows their analysis, except that the upper-bounds on the regret of both players must be combined in the middle of their analysis, just before they apply their self-bounding constraint assumption. Thus, we give here only the modification to the proof of their Theorem 1.
+
+Following their proof of Thm. 1 until their pseudo-regret bound at the top of p. 23, we get for each player  $i \in \mathcal{I} := \{-1, +1\}$ :
+
+$$
+\mathbf {E} \left[ R _ {i, T} (k) \right] \leq \sum_ {k \neq k ^ {*}} \left(\sum_ {t = 1} ^ {T} \frac {\sqrt {\mathbf {E} [ p _ {i , t} (k) ]}}{\sqrt {t}} + \sum_ {t = T _ {0} + 1} ^ {T} \frac {\mathbf {E} [ p _ {i , t} (k) ]}{4 \sqrt {t}}\right) + M,
+$$
+
+where  $M := \sqrt{T_0} + \frac{3}{4}\sqrt{K} + 15 + 14K\log(T)$  and  $T_0 := \lceil \Delta_{\min}^{-2}/4 \rceil$ . Together with Theorem 2 and taking the max over  $k$ ,  $\overline{R}_T$  is thus upper-bounded by
+
+$$
+\frac {1}{2} \sum_ {i \in \mathcal {I}} \sum_ {k \neq k ^ {*}} \left(\sum_ {t = 1} ^ {T} \frac {\sqrt {\mathbf {E} [ p _ {i , t} (k) ]}}{\sqrt {t}} + \sum_ {t = T _ {0} + 1} ^ {T} \frac {\mathbf {E} [ p _ {i , t} (k) ]}{4 \sqrt {t}}\right) + M.
+$$
+
+Now, applying the self-bounding property (4) we get for any  $\lambda \in [0,1]$
+
+$$
+\overline {{R}} _ {T} \leq \overline {{R}} _ {T} + \lambda \Bigg (\overline {{R}} _ {T} - \frac {1}{2} \mathbf {E} \bigg [ \sum_ {t = 1} ^ {T} \sum_ {k \neq k ^ {*}} \left(p _ {+ 1, t} (k) + p _ {- 1, t} (k)\right) \Delta_ {k} \bigg ] + C \Bigg)
+$$
+
+Thus, combined with the previous bound using  $1 + \lambda \leq 2$
+
+$$
+\begin{array}{l} \overline {{R}} _ {T} \leq \frac {1}{2} \sum_ {i \in \mathcal {I}} \sum_ {k \neq k ^ {*}} \left(\sum_ {t = 1} ^ {T} \left(\frac {2 \sqrt {\mathbf {E} [ p _ {i , t} (k) ]}}{\sqrt {t}} - \lambda \Delta_ {k} \mathbf {E} [ p _ {i, t} ]\right) + \sum_ {t = T _ {0} + 1} ^ {T} \frac {\mathbf {E} [ p _ {i , t} (k) ]}{2 \sqrt {t}}\right) + 2 M + \lambda C. \\ \leq \sum_ {k \neq k ^ {*}} \left(\sum_ {t = 1} ^ {T _ {0}} \max  _ {z \geq 0} \left\{\frac {2 \sqrt {z}}{\sqrt {t}} - \lambda \Delta_ {k} z \right\} \sum_ {t = T _ {0} + 1} ^ {T} \max  _ {z \geq 0} \left\{\frac {2 \sqrt {z} + \frac {1}{2} z}{\sqrt {t}} - \lambda \Delta_ {i} z \right\}\right) + 2 M + \lambda C \\ \end{array}
+$$
+
+Now, we are back with the same upper-bound (Zimmert & Seldin, 2021) have in the middle of their page 23. Following their analysis by solving the optimization problems, summing over  $t$ , and optimizing  $\lambda$  concludes.

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